MSDS6373-Time Series-Air Quality
Sachin Chavan
03/21/2020
1 Abstract
Air pollution is one of the major concern in today’s world. It can have very serious cost, penalites and consequences for the health of human beings and also ruthlessly distresses natural bio-network and ecosystems.Out of five major air pollutants (round-level ozone, particle pollution also known as particulate matter, carbon monoxide, sulfur dioxide, and nitrogen dioxide) WHO has identified SPM (Suspended Particulated Matter) as most sinister in terms of its effects on health. You may experience health issues such as Irregular heartbeat,Respiratory symptoms like coughing, wheezing or difficulty breathing etc. Therefore measuring these pollutants on daily basis and forecast is vital for any government. Daily measuring and forecasts can help Governments in designing policies to control air pollution and issue a guidance to citizens of possible health impacts and precautions to be taken if air quality becomes bad or predicted to be bad on next day.
2 Intro to Dataset
With this little background, for academic purposes we have decided to use Air Quality dataset from UCI machine learning respository for time series term project. This data set includes hourly air pollutants data from 12 nationally-controlled air-quality monitoring sites. The air-quality data are from the Beijing Municipal Environmental Monitoring Center. The meteorological data in each air-quality site are matched with the nearest weather station from the China Meteorological Administration. The time period is from March 1st, 2013 to February 28th, 2017.
This hourly data set considers 6 main air pollutants and 6 relevant meteorological variables at multiple sites in Beijing.
| SrNo | Air Pollutants | Description |
|---|---|---|
| 1 | PM2.5 | PM2.5 concentration (ug/\(m^{3}\)) |
| 2 | PM10 | PM10 concentration (ug/\(m^{3}\)) |
| 3 | SO2 | Sulphur Dioxide concentration (ug/\(m^{3}\)) |
| 4 | NO2 | Nitrogen Dioxide concentration (ug/\(m^{3}\)) |
| 5 | CO | Carbon Monoxide concentration (ug/\(m^{3}\)) |
| 6 | O3 | Ozone concentration (ug/\(m^{3}\)) |
| 7 | TEMP | temperature (degree Celsius) |
| 8 | PRES | pressure (hPa) |
| 9 | DEWP | dew point temperature (degree Celsius) |
| 10 | RAIN | precipitation (mm) |
| 11 | wd | wind direction |
| 12 | WSPM | wind speed (m/s) |
Goal
Goal of this project is to develop a model to forecast on all major air pollutants PM2.5,PM10,SO2,NO2,CO and O3 at least for next 10 days and effect of these pollutants on temperature and rain (Multivariate Analysis)
This dataset contains observations from 12 Chinese cities. For initial analysis (EDA) we are using observations from the city called Dongsi and it will be extended to two more cities for the purpose of the project and will try fit best possible model based on analysis.
Following are city names :
| SrNo | City |
|---|---|
| 1 | Aoizhongxin |
| 2 | Changping |
| 3 | Dingling |
| 4 | Dongsi |
| 5 | Guanyuan |
| 6 | Gucheng |
| 7 | Huairou |
| 8 | Nongzhanguan |
| 9 | Shunyi |
| 10 | Titantan |
| 11 | Wanliu |
| 12 | Wanshouxigong |
Let’s use data from the city called Dongsi.
3 Analysis of Dataset
3.1 Structure
## 'data.frame': 35064 obs. of 18 variables:
## $ No : int 1 2 3 4 5 6 7 8 9 10 ...
## $ year : int 2013 2013 2013 2013 2013 2013 2013 2013 2013 2013 ...
## $ month : int 3 3 3 3 3 3 3 3 3 3 ...
## $ day : int 1 1 1 1 1 1 1 1 1 1 ...
## $ hour : int 0 1 2 3 4 5 6 7 8 9 ...
## $ PM2.5 : num 9 4 7 3 3 4 5 3 3 3 ...
## $ PM10 : num 9 4 7 3 3 4 5 6 6 6 ...
## $ SO2 : num 3 3 NA 5 7 9 10 12 12 9 ...
## $ NO2 : num 17 16 17 18 NA 25 29 40 41 31 ...
## $ CO : int 300 300 300 NA 200 300 400 400 500 400 ...
## $ O3 : num 89 88 60 NA 84 78 67 52 54 69 ...
## $ TEMP : num -0.5 -0.7 -1.2 -1.4 -1.9 -2.4 -2.5 -1.4 -0.3 0.4 ...
## $ PRES : num 1024 1025 1025 1026 1027 ...
## $ DEWP : num -21.4 -22.1 -24.6 -25.5 -24.5 -21.3 -20.4 -20.4 -21.2 -23.3 ...
## $ RAIN : num 0 0 0 0 0 0 0 0 0 0 ...
## $ wd : Factor w/ 16 levels "E","ENE","ESE",..: 7 8 7 4 7 8 8 7 8 4 ...
## $ WSPM : num 5.7 3.9 5.3 4.9 3.2 2.4 2.2 3 4.6 5.5 ...
## $ station: Factor w/ 1 level "Dongsi": 1 1 1 1 1 1 1 1 1 1 ...Combine Day Month and Year fields into single Date field.
c_city01_ds$Date <- as.Date(paste0(c_city01_ds[,2],
Format(c_city01_ds[,3], digits=0, leading="00"),
Format(c_city01_ds[,4], digits=0, leading="00")),'%Y%m%d')
c_city01_ds <- c_city01_ds[c(1,19,5:18)] %>% arrange(Date)
c_city01_ds3.2 Missing Values
3.2.1 VIM::aggr()
This is aggregration plot for missing values
aggr(c_city01_ds[,c(4:13,15)], numbers=TRUE, sortVars=TRUE, labels=names(c_city01_ds[,c(4:13,15)]), cex.axis=.7, gap=3,
ylab=c("Proportion of missingness","Missingness Pattern"),prop = c(TRUE, FALSE))## Warning in plot.aggr(res, ...): not enough vertical space to display frequencies
## (too many combinations)
##
## Variables sorted by number of missings:
## Variable Count
## CO 0.0911761351
## NO2 0.0456593657
## PM2.5 0.0213894593
## O3 0.0189368013
## SO2 0.0189082820
## PM10 0.0157711613
## TEMP 0.0005703856
## PRES 0.0005703856
## DEWP 0.0005703856
## RAIN 0.0005703856
## WSPM 0.00039926993.2.2 VIM::pbox()
Interpretation
- Missingness is shown in PM2.5 w.r,t other variables.
- All boxplots below Blue and RED are overlapping which indicates Values are completely missing at random
- ImputeTS to be used for imputing missing vales
3.3 Imputations
For imputation we will be using R package imputeTS . It has different methods depending on seasonality and trends to impute values for time series.
So let’s first plot the each series by excluding rows with NA values to check if there is trend or seasonality in the data and apply these methods [1] accordingly.
3.3.1 PM2.5 Series
Last observation carried forward Method used
Notice that the trend-cycle (in red) is smoother than the original data and captures the main movement of the time series without all of the minor fluctuations. The order of the moving average (351-MA) determines the smoothness of the trend-cycle estimate and it shows there is no deterministic trend present in the data. There appears to be wandering behavior in the data.
x <- seq(1, length(c_city01_ds$PM2.5), by = 1)
c_city01_ds$impute_pm25 <- na_locf(c_city01_ds$PM2.5)
c_city01_ds$indpm25 <- ifelse(is.na(c_city01_ds$PM2.5), c_city01_ds$impute_pm25, NA)
fig <- plot_ly(x = ~x, y = ~c_city01_ds$PM2.5, name = 'PM2.5 Original', type = 'scatter', mode = 'lines')
fig <- fig %>% add_trace(y = ~c_city01_ds$indpm25, name = 'PM10 Imputed', type = 'scatter', mode = 'lines+markers',
marker = list(color = 'rgb(240, 144, 6)', size = 4))
#fig <- fig %>% add_trace(y = ~ma(ts(c_city01_ds$impute_pm25),351), name = '351-Moving Average', type = 'scatter', mode = 'lines',line = list(color = 'rgb(205, 12, 24)', width = 2))
fig <- fig %>% layout(title = "PM2.5 Series",
xaxis = list(title = "Time"),
yaxis = list (title = "Particulate Matter"))
fig3.3.2 PM10 Series
Last observation carried forward Method used
x <- seq(1, length(c_city01_ds$PM10), by = 1)
c_city01_ds$impute_pm10 <- na_locf(c_city01_ds$PM10)
c_city01_ds$indpm10 <- ifelse(is.na(c_city01_ds$PM10), c_city01_ds$impute_pm10, NA)
fig <- plot_ly(x = ~x, y = ~c_city01_ds$PM10, name = 'PM10 Original', type = 'scatter', mode = 'lines')
fig <- fig %>% add_trace(y = ~c_city01_ds$indpm10, name = 'PM10 Imputed', type = 'scatter', mode = 'lines+markers',
marker = list(color = 'rgb(240, 144, 6)', size = 4))
#fig <- fig %>% add_trace(y = ~ma(ts(c_city01_ds$impute_pm10),351), name = '351-Moving Average', type = 'scatter', mode = 'lines',line = list(color = 'rgb(205, 12, 24)', width = 2))
fig <- fig %>% layout(title = "PM10 Series",
xaxis = list(title = "Time"),
yaxis = list (title = "Particulate Matter"))
fig3.3.3 SO2 Series
Seasonal Adjustment then LOCF
Removes the seasonal component from the time series, performs imputation on the deseasonalized series and afterwards adds the seasonal component again.
x <- seq(1, length(c_city01_ds$SO2), by = 1)
c_city01_ds$impute_so2 <- na_locf(c_city01_ds$SO2)# na_seadec(c_city01_ds$SO2, algorithm = "locf",find_frequency = TRUE)
c_city01_ds$indso2 <- ifelse(is.na(c_city01_ds$SO2), c_city01_ds$impute_so2, NA)
fig <- plot_ly(x = ~x, y = ~c_city01_ds$SO2, name = 'SO2 Original', type = 'scatter', mode = 'lines')
fig <- fig %>% add_trace(y = ~c_city01_ds$indso2, name = 'CO Imputed', type = 'scatter', mode = 'lines+markers',
marker = list(color = 'rgb(240, 144, 6)', size = 4))
#fig <- fig %>% add_trace(y = ~ma(ts(c_city01_ds$impute_so2),351), name = '351-Moving Average', type = 'scatter', mode = 'lines',line = list(color = 'rgb(205, 12, 24)', width = 2))
fig <- fig %>% layout(title = "Sulphur Dioxide Series",
xaxis = list(title = "Time"),
yaxis = list (title = "Sulphur Dioxide"))
fig3.3.4 NO2 Series
Last observation carried forward Method used
x <- seq(1, length(c_city01_ds$NO2), by = 1)
c_city01_ds$impute_no2 <- na_locf(c_city01_ds$NO2)
c_city01_ds$indno2 <- ifelse(is.na(c_city01_ds$NO2), c_city01_ds$impute_no2, NA)
fig <- plot_ly(x = ~x, y = ~c_city01_ds$NO2, name = 'NO2 Original', type = 'scatter', mode = 'lines')
fig <- fig %>% add_trace(y = ~c_city01_ds$indno2, name = 'CO Imputed', type = 'scatter', mode = 'lines+markers',
marker = list(color = 'rgb(240, 144, 6)', size = 4))
#fig <- fig %>% add_trace(y = ~ma(ts(c_city01_ds$impute_no2),351), name = '351-Moving Average', type = 'scatter', mode = 'lines',line = list(color = 'rgb(205, 12, 24)', width = 2))
fig <- fig %>% layout(title = "Nitrogen Dioxide Series",
xaxis = list(title = "Time"),
yaxis = list (title = "Nitrogen Dioxide"))
fig3.3.5 O3 Series
Seasonal Adjustment then LOCF
x <- seq(1, length(c_city01_ds$O3), by = 1)
c_city01_ds$impute_o3 <- na_locf(c_city01_ds$NO2)
c_city01_ds$indo3 <- ifelse(is.na(c_city01_ds$O3), c_city01_ds$impute_o3, NA)
fig <- plot_ly(x = ~x, y = ~c_city01_ds$O3, name = 'O3 Original', type = 'scatter', mode = 'lines')
fig <- fig %>% add_trace(y = ~c_city01_ds$indo3, name = 'O3 Imputed', type = 'scatter', mode = 'lines+markers',
marker = list(color = 'rgb(240, 144, 6)', size = 3))
#fig <- fig %>% add_trace(y = ~ma(ts(c_city01_ds$impute_o3),351), name = '351-Moving Average', type = 'scatter', mode = 'lines',line = list(color = 'rgb(205, 12, 24)', width = 2))
fig <- fig %>% layout(title = "Ozone Series",
xaxis = list(title = "Time"),
yaxis = list (title = "Ozone"))
fig3.3.6 CO Series
Seasonal Adjustment then LOCF
x <- seq(1, length(c_city01_ds$CO), by = 1)
c_city01_ds$impute_co <- na_locf(c_city01_ds$CO)#na.seadec(c_city01_ds$CO, algorithm = "interpolation")
c_city01_ds$indco <- ifelse(is.na(c_city01_ds$CO), c_city01_ds$impute_co, NA)
fig <- plot_ly(x = ~x, y = ~c_city01_ds$CO, name = 'CO Original', type = 'scatter', mode = 'lines')
fig <- fig %>% add_trace(y = ~c_city01_ds$indco, name = 'CO Imputed', type = 'scatter', mode = 'lines+markers',
marker = list(color = 'rgb(240, 144, 6)', size = 3))
#fig <- fig %>% add_trace(y = ~ma(ts(c_city01_ds$impute_co),351), name = '351-Moving Average', type = 'scatter', mode = 'lines',line = list(color = 'rgb(205, 12, 24)', width = 2))
fig <- fig %>% layout(title = "Carbon Monoxide Series",
xaxis = list(title = "Time"),
yaxis = list (title = "Carbon Monoxide"))
fig3.3.7 Imputed Dataset
3.3.8 Final Dataset
Final dataset for this time series analysis is obtained by converting hourly observations to averaged daily observations. This data will then be used to 10 day forecast.
ts_daily_df <- sqldf("select Date,avg(impute_pm25) PM25,avg(impute_pm10) PM10,
avg(impute_so2) SO2,avg(impute_no2) NO2,avg(impute_co) CO,
avg(impute_o3) O3,avg(TEMP) temp,avg(RAIN) rain,avg(PRES) pres from ts_city01_ds group by Date order by Date")
ts_daily_df4 Smoothed Time series plots
Below smoothed time series (using 21 mmoving average) shows trend cyles. ## PM2.5
4.2 SO2
4.3 NO2
5 PM2.5 Analysis
Following analysis is done by using tswge R package that we have used exstensively this course.
5.1 Stationarity
Condition-1 Mean Does not depend on time
Visual evidence suggests, there is no constant downward or upward trend exist in the realization. Therefore mean appears to be constant.
Condition-2 Variance does not depend on time
Except one spike near 1000, there is no strong evidence of non-constant variance.
Condition-3 Mean Does not depend on time
acfs from two halves of the time series shows no significant difference, correlation of \(X_{t_1}\) and \(X_{t_2}\) depends only on
\(t_{2}\)-\(t_{1}\) and not where they are in time.
There is no evidence against stationarity. Time series is therefore stationary.
Other Observations
- No visible trend and seasonality in realization
- ACF cuts off after 2nd lag. No indication of non-stationarity
- Spectral density plot shows peak at 0, indicating wandering behavior.
- There is no strong visual evidence of any other frequency.
- First and second half of ACFs shows no significant difference.
- This is stationary series.
5.2 Split into Train/Test
5.3 Dicky-Fuller Test
\(H_{0}\) - Unit root present
\(H_{1}\) - Unit root not present. or time series is stationary.
Conclusion
pvalue = 0.01. Reject Ho. Suggests strong evidence against presence of unit root.
No visual evidence of sesonality so time series is stationary.
## Warning in adf.test(tt_pm25_ts$train_ts): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: tt_pm25_ts$train_ts
## Dickey-Fuller = -8.9566, Lag order = 11, p-value = 0.01
## alternative hypothesis: stationary5.4 Model ID
AIC suggests ARMA(5,1) is the best fitted model. Ljung box test was performed for p=5 and q=1.
BIC suggests ARMA(1,1) is the best fitted model. Ljung box test was performed for p=1 and q=1.
## ---------WORKING... PLEASE WAIT...
##
##
## Five Smallest Values of aic## ---------WORKING... PLEASE WAIT...
##
##
## Five Smallest Values of bic5.5 White Noise Test
ljung box test provides strong evidence againts null hypothesis that series is white noise. p-value=0.
## Obs 0.5530452 0.1958933 0.06262632 0.02729153 0.03122277 0.02546675 0.01851925 0.0480942 0.08306008 0.08808314 0.07028984 0.05264368 0.06927908 0.08282109 0.08081704 0.06200663 0.02933554 0.009845052 0.0107568 0.02610704 0.05958953 0.06321152 0.05334019 0.0786623## $test
## [1] "Ljung-Box test"
##
## $K
## [1] 24
##
## $chi.square
## [1] 601.5651
##
## $df
## [1] 18
##
## $pval
## [1] 0## Obs 0.5530452 0.1958933 0.06262632 0.02729153 0.03122277 0.02546675 0.01851925 0.0480942 0.08306008 0.08808314 0.07028984 0.05264368 0.06927908 0.08282109 0.08081704 0.06200663 0.02933554 0.009845052 0.0107568 0.02610704 0.05958953 0.06321152 0.05334019 0.0786623## $test
## [1] "Ljung-Box test"
##
## $K
## [1] 24
##
## $chi.square
## [1] 601.5651
##
## $df
## [1] 22
##
## $pval
## [1] 05.6 ARMA
5.6.1 Build a Model
##
## Coefficients of Original polynomial:
## 1.5757 -0.7831 0.2080 -0.0472 0.0261
##
## Factor Roots Abs Recip System Freq
## 1-0.9653B 1.0360 0.9653 0.0000
## 1-0.8283B+0.2765B^2 1.4981+-1.1717i 0.5258 0.1056
## 1+0.2179B+0.0979B^2 -1.1125+-2.9955i 0.3129 0.3066
##
## ##
## Coefficients of Original polynomial:
## 0.3625
##
## Factor Roots Abs Recip System Freq
## 1-0.3625B 2.7585 0.3625 0.0000
##
## 5.6.2 Residual diagnositcs
Ljung box test suggests that residuals are white noise.So assumptions on residuals are met.
## Obs -0.0001881523 -0.001022132 -0.00171976 -0.001977746 0.008193721 -0.009010721 -0.04192617 -0.009892159 0.02172941 0.01829942 0.009902166 -0.02476423 0.01458044 0.0179067 0.01651688 0.01349824 -0.0140909 -0.02150647 -0.01368855 -0.02030182 0.02597591 0.01940536 -0.02380709 0.0370772## $test
## [1] "Ljung-Box test"
##
## $K
## [1] 24
##
## $chi.square
## [1] 12.79612
##
## $df
## [1] 18
##
## $pval
## [1] 0.8035395## Obs -0.0001881523 -0.001022132 -0.00171976 -0.001977746 0.008193721 -0.009010721 -0.04192617 -0.009892159 0.02172941 0.01829942 0.009902166 -0.02476423 0.01458044 0.0179067 0.01651688 0.01349824 -0.0140909 -0.02150647 -0.01368855 -0.02030182 0.02597591 0.01940536 -0.02380709 0.0370772 0.0429581 -0.02662873 -0.01575458 0.0507779 0.02531561 0.01177715 -0.009671479 0.007327128 -0.02263186 -0.01164259 -0.01866861 0.02395932 -0.0429963 0.03045108 0.01068537 -0.006911302 0.04207008 -0.02639989 -0.03661697 -0.01178131 0.04600438 0.01048906 -0.01378056 -0.02356496## $test
## [1] "Ljung-Box test"
##
## $K
## [1] 48
##
## $chi.square
## [1] 39.04114
##
## $df
## [1] 42
##
## $pval
## [1] 0.6016005
## Obs 0.001234195 0.0009126953 -0.01275297 -0.004962188 0.01939087 0.01393892 -0.01562802 0.01613415 0.04602301 0.04062574 0.03087398 -0.006125254 0.03261754 0.03631462 0.03546942 0.03184008 0.002743961 -0.005825104 0.001725101 -0.004982238 0.04040599 0.03408505 -0.009793972 0.04797018## $test
## [1] "Ljung-Box test"
##
## $K
## [1] 24
##
## $chi.square
## [1] 23.24318
##
## $df
## [1] 22
##
## $pval
## [1] 0.3881086## Obs 0.001234195 0.0009126953 -0.01275297 -0.004962188 0.01939087 0.01393892 -0.01562802 0.01613415 0.04602301 0.04062574 0.03087398 -0.006125254 0.03261754 0.03631462 0.03546942 0.03184008 0.002743961 -0.005825104 0.001725101 -0.004982238 0.04040599 0.03408505 -0.009793972 0.04797018 0.05148732 -0.01758795 -0.006874163 0.05939029 0.03483418 0.02273029 5.01446e-05 0.01640283 -0.01371569 -0.002239651 -0.01026027 0.03213572 -0.03538688 0.03799857 0.01787272 -0.000521831 0.0459199 -0.02343426 -0.03261477 -0.007585106 0.05106236 0.01597669 -0.008663508 -0.02062736## $test
## [1] "Ljung-Box test"
##
## $K
## [1] 48
##
## $chi.square
## [1] 52.86132
##
## $df
## [1] 46
##
## $pval
## [1] 0.2262255
## [1] 86.547165.7 ARMA Appropriateness
5.7.1 Realiaztions
- Realizations
Realization from data has few spikes that does not exist in samples using selected model. But other than that it doesn’t look too bad.
5.7.2 ACFs
ACFs in original data pretty much matches with Samples
par(mfrow=c(1,3))
acf(pm25_gen_sample1$data,main="Sample1")
acf(pm25_gen_sample2$data,main="Sample2")
acf(pm25_gen_sample3$data,main="Sample2")
5.7.3 Spectral Estimates
They all look Same.
## $freq
## [1] 0.0006939625 0.0013879251 0.0020818876 0.0027758501 0.0034698126
## [6] 0.0041637752 0.0048577377 0.0055517002 0.0062456627 0.0069396253
## [11] 0.0076335878 0.0083275503 0.0090215128 0.0097154754 0.0104094379
## [16] 0.0111034004 0.0117973629 0.0124913255 0.0131852880 0.0138792505
## [21] 0.0145732130 0.0152671756 0.0159611381 0.0166551006 0.0173490632
## [26] 0.0180430257 0.0187369882 0.0194309507 0.0201249133 0.0208188758
## [31] 0.0215128383 0.0222068008 0.0229007634 0.0235947259 0.0242886884
## [36] 0.0249826509 0.0256766135 0.0263705760 0.0270645385 0.0277585010
## [41] 0.0284524636 0.0291464261 0.0298403886 0.0305343511 0.0312283137
## [46] 0.0319222762 0.0326162387 0.0333102012 0.0340041638 0.0346981263
## [51] 0.0353920888 0.0360860514 0.0367800139 0.0374739764 0.0381679389
## [56] 0.0388619015 0.0395558640 0.0402498265 0.0409437890 0.0416377516
## [61] 0.0423317141 0.0430256766 0.0437196391 0.0444136017 0.0451075642
## [66] 0.0458015267 0.0464954892 0.0471894518 0.0478834143 0.0485773768
## [71] 0.0492713393 0.0499653019 0.0506592644 0.0513532269 0.0520471895
## [76] 0.0527411520 0.0534351145 0.0541290770 0.0548230396 0.0555170021
## [81] 0.0562109646 0.0569049271 0.0575988897 0.0582928522 0.0589868147
## [86] 0.0596807772 0.0603747398 0.0610687023 0.0617626648 0.0624566273
## [91] 0.0631505899 0.0638445524 0.0645385149 0.0652324774 0.0659264400
## [96] 0.0666204025 0.0673143650 0.0680083276 0.0687022901 0.0693962526
## [101] 0.0700902151 0.0707841777 0.0714781402 0.0721721027 0.0728660652
## [106] 0.0735600278 0.0742539903 0.0749479528 0.0756419153 0.0763358779
## [111] 0.0770298404 0.0777238029 0.0784177654 0.0791117280 0.0798056905
## [116] 0.0804996530 0.0811936155 0.0818875781 0.0825815406 0.0832755031
## [121] 0.0839694656 0.0846634282 0.0853573907 0.0860513532 0.0867453158
## [126] 0.0874392783 0.0881332408 0.0888272033 0.0895211659 0.0902151284
## [131] 0.0909090909 0.0916030534 0.0922970160 0.0929909785 0.0936849410
## [136] 0.0943789035 0.0950728661 0.0957668286 0.0964607911 0.0971547536
## [141] 0.0978487162 0.0985426787 0.0992366412 0.0999306037 0.1006245663
## [146] 0.1013185288 0.1020124913 0.1027064539 0.1034004164 0.1040943789
## [151] 0.1047883414 0.1054823040 0.1061762665 0.1068702290 0.1075641915
## [156] 0.1082581541 0.1089521166 0.1096460791 0.1103400416 0.1110340042
## [161] 0.1117279667 0.1124219292 0.1131158917 0.1138098543 0.1145038168
## [166] 0.1151977793 0.1158917418 0.1165857044 0.1172796669 0.1179736294
## [171] 0.1186675920 0.1193615545 0.1200555170 0.1207494795 0.1214434421
## [176] 0.1221374046 0.1228313671 0.1235253296 0.1242192922 0.1249132547
## [181] 0.1256072172 0.1263011797 0.1269951423 0.1276891048 0.1283830673
## [186] 0.1290770298 0.1297709924 0.1304649549 0.1311589174 0.1318528799
## [191] 0.1325468425 0.1332408050 0.1339347675 0.1346287300 0.1353226926
## [196] 0.1360166551 0.1367106176 0.1374045802 0.1380985427 0.1387925052
## [201] 0.1394864677 0.1401804303 0.1408743928 0.1415683553 0.1422623178
## [206] 0.1429562804 0.1436502429 0.1443442054 0.1450381679 0.1457321305
## [211] 0.1464260930 0.1471200555 0.1478140180 0.1485079806 0.1492019431
## [216] 0.1498959056 0.1505898681 0.1512838307 0.1519777932 0.1526717557
## [221] 0.1533657183 0.1540596808 0.1547536433 0.1554476058 0.1561415684
## [226] 0.1568355309 0.1575294934 0.1582234559 0.1589174185 0.1596113810
## [231] 0.1603053435 0.1609993060 0.1616932686 0.1623872311 0.1630811936
## [236] 0.1637751561 0.1644691187 0.1651630812 0.1658570437 0.1665510062
## [241] 0.1672449688 0.1679389313 0.1686328938 0.1693268563 0.1700208189
## [246] 0.1707147814 0.1714087439 0.1721027065 0.1727966690 0.1734906315
## [251] 0.1741845940 0.1748785566 0.1755725191 0.1762664816 0.1769604441
## [256] 0.1776544067 0.1783483692 0.1790423317 0.1797362942 0.1804302568
## [261] 0.1811242193 0.1818181818 0.1825121443 0.1832061069 0.1839000694
## [266] 0.1845940319 0.1852879944 0.1859819570 0.1866759195 0.1873698820
## [271] 0.1880638446 0.1887578071 0.1894517696 0.1901457321 0.1908396947
## [276] 0.1915336572 0.1922276197 0.1929215822 0.1936155448 0.1943095073
## [281] 0.1950034698 0.1956974323 0.1963913949 0.1970853574 0.1977793199
## [286] 0.1984732824 0.1991672450 0.1998612075 0.2005551700 0.2012491325
## [291] 0.2019430951 0.2026370576 0.2033310201 0.2040249827 0.2047189452
## [296] 0.2054129077 0.2061068702 0.2068008328 0.2074947953 0.2081887578
## [301] 0.2088827203 0.2095766829 0.2102706454 0.2109646079 0.2116585704
## [306] 0.2123525330 0.2130464955 0.2137404580 0.2144344205 0.2151283831
## [311] 0.2158223456 0.2165163081 0.2172102706 0.2179042332 0.2185981957
## [316] 0.2192921582 0.2199861207 0.2206800833 0.2213740458 0.2220680083
## [321] 0.2227619709 0.2234559334 0.2241498959 0.2248438584 0.2255378210
## [326] 0.2262317835 0.2269257460 0.2276197085 0.2283136711 0.2290076336
## [331] 0.2297015961 0.2303955586 0.2310895212 0.2317834837 0.2324774462
## [336] 0.2331714087 0.2338653713 0.2345593338 0.2352532963 0.2359472588
## [341] 0.2366412214 0.2373351839 0.2380291464 0.2387231090 0.2394170715
## [346] 0.2401110340 0.2408049965 0.2414989591 0.2421929216 0.2428868841
## [351] 0.2435808466 0.2442748092 0.2449687717 0.2456627342 0.2463566967
## [356] 0.2470506593 0.2477446218 0.2484385843 0.2491325468 0.2498265094
## [361] 0.2505204719 0.2512144344 0.2519083969 0.2526023595 0.2532963220
## [366] 0.2539902845 0.2546842471 0.2553782096 0.2560721721 0.2567661346
## [371] 0.2574600972 0.2581540597 0.2588480222 0.2595419847 0.2602359473
## [376] 0.2609299098 0.2616238723 0.2623178348 0.2630117974 0.2637057599
## [381] 0.2643997224 0.2650936849 0.2657876475 0.2664816100 0.2671755725
## [386] 0.2678695350 0.2685634976 0.2692574601 0.2699514226 0.2706453851
## [391] 0.2713393477 0.2720333102 0.2727272727 0.2734212353 0.2741151978
## [396] 0.2748091603 0.2755031228 0.2761970854 0.2768910479 0.2775850104
## [401] 0.2782789729 0.2789729355 0.2796668980 0.2803608605 0.2810548230
## [406] 0.2817487856 0.2824427481 0.2831367106 0.2838306731 0.2845246357
## [411] 0.2852185982 0.2859125607 0.2866065232 0.2873004858 0.2879944483
## [416] 0.2886884108 0.2893823734 0.2900763359 0.2907702984 0.2914642609
## [421] 0.2921582235 0.2928521860 0.2935461485 0.2942401110 0.2949340736
## [426] 0.2956280361 0.2963219986 0.2970159611 0.2977099237 0.2984038862
## [431] 0.2990978487 0.2997918112 0.3004857738 0.3011797363 0.3018736988
## [436] 0.3025676613 0.3032616239 0.3039555864 0.3046495489 0.3053435115
## [441] 0.3060374740 0.3067314365 0.3074253990 0.3081193616 0.3088133241
## [446] 0.3095072866 0.3102012491 0.3108952117 0.3115891742 0.3122831367
## [451] 0.3129770992 0.3136710618 0.3143650243 0.3150589868 0.3157529493
## [456] 0.3164469119 0.3171408744 0.3178348369 0.3185287994 0.3192227620
## [461] 0.3199167245 0.3206106870 0.3213046495 0.3219986121 0.3226925746
## [466] 0.3233865371 0.3240804997 0.3247744622 0.3254684247 0.3261623872
## [471] 0.3268563498 0.3275503123 0.3282442748 0.3289382373 0.3296321999
## [476] 0.3303261624 0.3310201249 0.3317140874 0.3324080500 0.3331020125
## [481] 0.3337959750 0.3344899375 0.3351839001 0.3358778626 0.3365718251
## [486] 0.3372657876 0.3379597502 0.3386537127 0.3393476752 0.3400416378
## [491] 0.3407356003 0.3414295628 0.3421235253 0.3428174879 0.3435114504
## [496] 0.3442054129 0.3448993754 0.3455933380 0.3462873005 0.3469812630
## [501] 0.3476752255 0.3483691881 0.3490631506 0.3497571131 0.3504510756
## [506] 0.3511450382 0.3518390007 0.3525329632 0.3532269257 0.3539208883
## [511] 0.3546148508 0.3553088133 0.3560027759 0.3566967384 0.3573907009
## [516] 0.3580846634 0.3587786260 0.3594725885 0.3601665510 0.3608605135
## [521] 0.3615544761 0.3622484386 0.3629424011 0.3636363636 0.3643303262
## [526] 0.3650242887 0.3657182512 0.3664122137 0.3671061763 0.3678001388
## [531] 0.3684941013 0.3691880638 0.3698820264 0.3705759889 0.3712699514
## [536] 0.3719639139 0.3726578765 0.3733518390 0.3740458015 0.3747397641
## [541] 0.3754337266 0.3761276891 0.3768216516 0.3775156142 0.3782095767
## [546] 0.3789035392 0.3795975017 0.3802914643 0.3809854268 0.3816793893
## [551] 0.3823733518 0.3830673144 0.3837612769 0.3844552394 0.3851492019
## [556] 0.3858431645 0.3865371270 0.3872310895 0.3879250520 0.3886190146
## [561] 0.3893129771 0.3900069396 0.3907009022 0.3913948647 0.3920888272
## [566] 0.3927827897 0.3934767523 0.3941707148 0.3948646773 0.3955586398
## [571] 0.3962526024 0.3969465649 0.3976405274 0.3983344899 0.3990284525
## [576] 0.3997224150 0.4004163775 0.4011103400 0.4018043026 0.4024982651
## [581] 0.4031922276 0.4038861901 0.4045801527 0.4052741152 0.4059680777
## [586] 0.4066620402 0.4073560028 0.4080499653 0.4087439278 0.4094378904
## [591] 0.4101318529 0.4108258154 0.4115197779 0.4122137405 0.4129077030
## [596] 0.4136016655 0.4142956280 0.4149895906 0.4156835531 0.4163775156
## [601] 0.4170714781 0.4177654407 0.4184594032 0.4191533657 0.4198473282
## [606] 0.4205412908 0.4212352533 0.4219292158 0.4226231783 0.4233171409
## [611] 0.4240111034 0.4247050659 0.4253990285 0.4260929910 0.4267869535
## [616] 0.4274809160 0.4281748786 0.4288688411 0.4295628036 0.4302567661
## [621] 0.4309507287 0.4316446912 0.4323386537 0.4330326162 0.4337265788
## [626] 0.4344205413 0.4351145038 0.4358084663 0.4365024289 0.4371963914
## [631] 0.4378903539 0.4385843164 0.4392782790 0.4399722415 0.4406662040
## [636] 0.4413601666 0.4420541291 0.4427480916 0.4434420541 0.4441360167
## [641] 0.4448299792 0.4455239417 0.4462179042 0.4469118668 0.4476058293
## [646] 0.4482997918 0.4489937543 0.4496877169 0.4503816794 0.4510756419
## [651] 0.4517696044 0.4524635670 0.4531575295 0.4538514920 0.4545454545
## [656] 0.4552394171 0.4559333796 0.4566273421 0.4573213046 0.4580152672
## [661] 0.4587092297 0.4594031922 0.4600971548 0.4607911173 0.4614850798
## [666] 0.4621790423 0.4628730049 0.4635669674 0.4642609299 0.4649548924
## [671] 0.4656488550 0.4663428175 0.4670367800 0.4677307425 0.4684247051
## [676] 0.4691186676 0.4698126301 0.4705065926 0.4712005552 0.4718945177
## [681] 0.4725884802 0.4732824427 0.4739764053 0.4746703678 0.4753643303
## [686] 0.4760582929 0.4767522554 0.4774462179 0.4781401804 0.4788341430
## [691] 0.4795281055 0.4802220680 0.4809160305 0.4816099931 0.4823039556
## [696] 0.4829979181 0.4836918806 0.4843858432 0.4850798057 0.4857737682
## [701] 0.4864677307 0.4871616933 0.4878556558 0.4885496183 0.4892435808
## [706] 0.4899375434 0.4906315059 0.4913254684 0.4920194310 0.4927133935
## [711] 0.4934073560 0.4941013185 0.4947952811 0.4954892436 0.4961832061
## [716] 0.4968771686 0.4975711312 0.4982650937 0.4989590562 0.4996530187
##
## $pzgram
## [1] 6.8088259683 6.7890882760 6.7562451712 6.7103794865 6.6516136788
## [6] 6.5801162576 6.4961099455 6.3998814708 6.2917928416 6.1722938790
## [11] 6.0419356905 5.9013846403 5.7514362032 5.5930278938 5.4272502271
## [16] 5.2553544128 5.0787552481 4.8990274803 4.7178938420 4.5372030853
## [21] 4.3588967479 4.1849641332 4.0173861193 3.8580698516 3.7087779762
## [26] 3.5710575623 3.4461748967 3.3350625866 3.2382846531 3.1560235440
## [31] 3.0880904780 3.0339577280 2.9928088984 2.9636014290 2.9451347417
## [36] 2.9361176592 2.9352297603 2.9411728749 2.9527106136 2.9686953872
## [41] 2.9880836288 3.0099408078 3.0334383302 3.0578446143 3.0825125809
## [46] 3.1068655851 3.1303835014 3.1525903011 3.1730440784 3.1913301028
## [51] 3.2070571351 3.2198569443 3.2293867270 3.2353339514 3.2374230350
## [56] 3.2354232101 3.2291569232 3.2185081526 3.2034300889 3.1839517040
## [61] 3.1601828261 3.1323174288 3.1006349313 3.0654993963 3.0273565900
## [66] 2.9867289495 2.9442085736 2.9004484232 2.8561519790 2.8120616500
## [71] 2.7689462674 2.7275880050 2.6887690634 2.6532584156 2.6217988590
## [76] 2.5950945436 2.5737990719 2.5585042035 2.5497291488 2.5479104399
## [81] 2.5533923933 2.5664182613 2.5871222678 2.6155228405 2.6515174493
## [86] 2.6948795222 2.7452579114 2.8021793162 2.8650539341 2.9331844215
## [91] 3.0057780285 3.0819615487 3.1607985396 3.2413081246 3.3224846213
## [96] 3.4033172381 3.4828091418 3.5599953118 3.6339587350 3.7038446473
## [101] 3.7688726715 3.8283468265 3.8816634808 3.9283173897 3.9679059939
## [106] 4.0001321640 4.0248055626 4.0418427630 4.0512662195 4.0532021315
## [111] 4.0478771934 4.0356141685 4.0168261907 3.9920096661 3.9617356450
## [116] 3.9266395495 3.8874091909 3.8447710874 3.7994751996 3.7522783332
## [121] 3.7039266082 3.6551375414 3.6065824259 3.5588697871 3.5125307379
## [126] 3.4680070265 3.4256424620 3.3856782211 3.3482522913 3.3134030285
## [131] 3.2810765119 3.2511371213 3.2233805472 3.1975483128 3.1733428331
## [136] 3.1504420738 3.1285129741 3.1072229675 3.0862491333 3.0652847359
## [141] 3.0440431291 3.0222592145 2.9996888292 2.9761065881 2.9513028244
## [146] 2.9250803368 2.8972516794 2.8676377012 2.8360679719 2.8023836140
## [151] 2.7664429025 2.7281298103 2.6873654633 2.6441222431 2.5984400363
## [156] 2.5504438832 2.5003620334 2.4485431580 2.3954712309 2.3417763613
## [161] 2.2882396866 2.2357903572 2.1854927190 2.1385221183 2.0961283741
## [166] 2.0595869419 2.0301391215 2.0089242226 1.9969082068 1.9948146455
## [171] 2.0030645443 2.0217313899 2.0505165558 2.0887480731 2.1354030850
## [176] 2.1891515742 2.2484167019 2.3114457114 2.3763849853 2.4413534086
## [181] 2.5045094101 2.5641085896 2.6185503720 2.6664134266 2.7064805389
## [186] 2.7377541895 2.7594643382 2.7710698914 2.7722551455 2.7629222248
## [191] 2.7431802158 2.7133314177 2.6738548784 2.6253872236 2.5687007062
## [196] 2.5046784252 2.4342867867 2.3585455020 2.2784957090 2.1951671476
## [201] 2.1095456512 2.0225424947 1.9349672926 1.8475061301 1.7607063967
## [206] 1.6749693922 1.5905512156 1.5075718032 1.4260313437 1.3458327397
## [211] 1.2668083877 1.1887493451 1.1114349462 1.0346610985 0.9582657844
## [216] 0.8821506550 0.8062979890 0.7307826424 0.6557789223 0.5815625528
## [221] 0.5085080668 0.4370820609 0.3678327912 0.3013765936 0.2383815850
## [226] 0.1795490632 0.1255929871 0.0772179047 0.0350957105 -0.0001583212
## [231] -0.0280096776 -0.0480282236 -0.0599104355 -0.0634992389 -0.0588003455
## [236] -0.0459939731 -0.0254409473 0.0023175751 0.0365671637 0.0764339231
## [241] 0.1209045647 0.1688518438 0.2190641424 0.2702777989 0.3212107497
## [246] 0.3705961488 0.4172148401 0.4599258282 0.4976941785 0.5296160476
## [251] 0.5549407517 0.5730899195 0.5836738240 0.5865049514 0.5816087436
## [256] 0.5692312580 0.5498432406 0.5241398284 0.4930348252 0.4576482738
## [261] 0.4192859446 0.3794094487 0.3395960355 0.3014878154 0.2667311628
## [266] 0.2369083588 0.2134649708 0.1976377936 0.1903890963 0.1923531351
## [271] 0.2038002124 0.2246219920 0.2543395263 0.2921329261 0.3368892812
## [276] 0.3872637446 0.4417478755 0.4987394290 0.5566086133 0.6137571335
## [281] 0.6686677864 0.7199437224 0.7663375713 0.8067713737 0.8403486779
## [286] 0.8663603012 0.8842851848 0.8937875770 0.8947115036 0.8870731919
## [291] 0.8710518254 0.8469787522 0.8153250574 0.7766872582 0.7317707980
## [296] 0.6813710083 0.6263512889 0.5676184261 0.5060952310 0.4426910056
## [301] 0.3782707250 0.3136241969 0.2494367832 0.1862634754 0.1245081568
## [306] 0.0644097237 0.0060363746 -0.0507111536 -0.1060872683 -0.1604806397
## [311] -0.2143858671 -0.2683692561 -0.3230305365 -0.3789624431 -0.4367100057
## [316] -0.4967312293 -0.5593606300 -0.6247769047 -0.6929758858 -0.7637498971
## [321] -0.8366746732 -0.9111051018 -0.9861811383 -1.0608452325 -1.1338723982
## [326] -1.2039135528 -1.2695518688 -1.3293706175 -1.3820294273 -1.4263442387
## [331] -1.4613648347 -1.4864430388 -1.5012848158 -1.5059807505 -1.5010116123
## [336] -1.4872285705 -1.4658105628 -1.4382037671 -1.4060496441 -1.3711084456
## [341] -1.3351844908 -1.3000582130 -1.2674283186 -1.2388657505 -1.2157797343
## [346] -1.1993951548 -1.1907398677 -1.1906402636 -1.1997233662 -1.2184238889
## [351] -1.2469948976 -1.2855209777 -1.3339330397 -1.3920240849 -1.4594653940
## [356] -1.5358226823 -1.6205718040 -1.7131135879 -1.8127873697 -1.9188827677
## [361] -2.0306492462 -2.1473030487 -2.2680311714 -2.3919921995 -2.5183140551
## [366] -2.6460889927 -2.7743665260 -2.9021453457 -3.0283656671 -3.1519037759
## [371] -3.2715707720 -3.3861175748 -3.4942480821 -3.5946419055 -3.6859873028
## [376] -3.7670238020 -3.8365926419 -3.8936917135 -3.9375304191 -3.9675790527
## [381] -3.9836072141 -3.9857065269 -3.9742945140 -3.9500986425 -3.9141218999
## [386] -3.8675933610 -3.8119086825 -3.7485661211 -3.6791034990 -3.6050407210
## [391] -3.5278312180 -3.4488243449 -3.3692395035 -3.2901517310 -3.2124877544
## [396] -3.1370310361 -3.0644340920 -2.9952362799 -2.9298852830 -2.8687606036
## [401] -2.8121975139 -2.7605100653 -2.7140119390 -2.6730341169 -2.6379385799
## [406] -2.6091274766 -2.5870474516 -2.5721890611 -2.5650814135 -2.5662823416
## [411] -2.5763645270 -2.5958980486 -2.6254298177 -2.6654603050 -2.7164178711
## [416] -2.7786309048 -2.8522978889 -2.9374554706 -3.0339446586 -3.1413754453
## [421] -3.2590904951 -3.3861291090 -3.5211934985 -3.6626204971 -3.8083631538
## [426] -3.9559880510 -4.1026953736 -4.2453692583 -4.3806651422 -4.5051380371
## [431] -4.6154104542 -4.7083712553 -4.7813881811 -4.8325093731 -4.8606256345
## [436] -4.8655676855 -4.8481217674 -4.8099607032 -4.7535021114 -4.6817167070
## [441] -4.5979147288 -4.5055370979 -4.4079716437 -4.3084063272 -4.2097233519
## [446] -4.1144319471 -4.0246339502 -3.9420148582 -3.8678531485 -3.8030417175
## [451] -3.7481167092 -3.7032904491 -3.6684864546 -3.6433754752 -3.6274122050
## [456] -3.6198727253 -3.6198928971 -3.6265078783 -3.6386927151 -3.6554036064
## [461] -3.6756190149 -3.6983793667 -3.7228237245 -3.7482216097 -3.7739981428
## [466] -3.7997509188 -3.8252575017 -3.8504730819 -3.8755185850 -3.9006602549
## [471] -3.9262823472 -3.9528549897 -3.9808994525 -4.0109530274 -4.0435354881
## [476] -4.0791187452 -4.1181008949 -4.1607854399 -4.2073660850 -4.2579171890
## [481] -4.3123897041 -4.3706122395 -4.4322967333 -4.4970480919 -4.5643770475
## [486] -4.6337153855 -4.7044326091 -4.7758530538 -4.8472724529 -4.9179730099
## [491] -4.9872361724 -5.0543525296 -5.1186285694 -5.1793904117 -5.2359850442
## [496] -5.2877799769 -5.3341625521 -5.3745403360 -5.4083440406 -5.4350342374
## [501] -5.4541127380 -5.4651389340 -5.4677506777 -5.4616885214 -5.4468214272
## [506] -5.4231715285 -5.3909352767 -5.3504984094 -5.3024426442 -5.2475427826
## [511] -5.1867538745 -5.1211891040 -5.0520899371 -4.9807907030 -4.9086800962
## [516] -4.8371620670 -4.7676182877 -4.7013739111 -4.6396677850 -4.5836277439
## [521] -4.5342511208 -4.4923902611 -4.4587425737 -4.4338445307 -4.4180689886
## [526] -4.4116252487 -4.4145613599 -4.4267682821 -4.4479856577 -4.4778090591
## [531] -4.5156986903 -4.5609896134 -4.6129036289 -4.6705629727 -4.7330059843
## [536] -4.7992048568 -4.8680854908 -4.9385493536 -5.0094970842 -5.0798534070
## [541] -5.1485927323 -5.2147646488 -5.2775183771 -5.3361251754 -5.3899976834
## [546] -5.4387052731 -5.4819846359 -5.5197450756 -5.5520682543 -5.5792024454
## [551] -5.6015516339 -5.6196600473 -5.6341928870 -5.6459141283 -5.6556622898
## [556] -5.6643250319 -5.6728133475 -5.6820359853 -5.6928746025 -5.7061600030
## [561] -5.7226497032 -5.7430069741 -5.7677814574 -5.7973914502 -5.8321079872
## [566] -5.8720409384 -5.9171274709 -5.9671233903 -6.0215980699 -6.0799338734
## [571] -6.1413311455 -6.2048199432 -6.2692796554 -6.3334674477 -6.3960560279
## [576] -6.4556805138 -6.5109932237 -6.5607240726 -6.6037431000 -6.6391206854
## [581] -6.6661804732 -6.6845401248 -6.6941358412 -6.6952281036 -6.6883880240
## [586] -6.6744657567 -6.6545442153 -6.6298825725 -6.6018545342 -6.5718861857
## [591] -6.5413974662 -6.5117502446 -6.4842048073 -6.4598854913 -6.4397553430
## [596] -6.4245990952 -6.4150134231 -6.4114033284 -6.4139835476 -6.4227840252
## [601] -6.4376586901 -6.4582969774 -6.4842377252 -6.5148852221 -6.5495272775
## [606] -6.5873552247 -6.6274857493 -6.6689843601 -6.7108902036 -6.7522417712
## [611] -6.7921028903 -6.8295882411 -6.8638875296 -6.8942873990 -6.9201901848
## [616] -6.9411287329 -6.9567766914 -6.9669539362 -6.9716270816 -6.9709053129
## [621] -6.9650320330 -6.9543730115 -6.9394018449 -6.9206835719 -6.8988572547
## [626] -6.8746182398 -6.8487006786 -6.8218607395 -6.7948607887 -6.7684546859
## [631] -6.7433742290 -6.7203166979 -6.6999333925 -6.6828190332 -6.6695018798
## [636] -6.6604344420 -6.6559846772 -6.6564276146 -6.6619373993 -6.6725798156
## [641] -6.6883054321 -6.7089436061 -6.7341976911 -6.7636419132 -6.7967205002
## [646] -6.8327497596 -6.8709238909 -6.9103253479 -6.9499405292 -6.9886814173
## [651] -7.0254135001 -7.0589898660 -7.0882907800 -7.1122673609 -7.1299872642
## [656] -7.1406796465 -7.1437762675 -7.1389454954 -7.1261162983 -7.1054900308
## [661] -7.0775388800 -7.0429910676 -7.0028041128 -6.9581284576 -6.9102643968
## [666] -6.8606154777 -6.8106413642 -6.7618126792 -6.7155696751 -6.6732858623
## [671] -6.6362370517 -6.6055757128 -6.5823101533 -6.5672877807 -6.5611816214
## [676] -6.5644792950 -6.5774737751 -6.6002554682 -6.6327054047 -6.6744896456
## [681] -6.7250553493 -6.7836293067 -6.8492201179 -6.9206255038 -6.9964464881
## [686] -7.0751102528 -7.1549032948 -7.2340159915 -7.3105987697 -7.3828287452
## [691] -7.4489840774 -7.5075215568 -7.5571514636 -7.5969028678 -7.6261726255
## [696] -7.6447525364 -7.6528313510 -7.6509712055 -7.6400610465 -7.6212521115
## [701] -7.5958821077 -7.5653951963 -7.5312643336 -7.4949212276 -7.4576975229
## [706] -7.4207791732 -7.3851745454 -7.3516957546 -7.3209520584 -7.2933537979
## [711] -7.2691252500 -7.2483247710 -7.2308706655 -7.2165712565 -7.2051576406
## [716] -7.1963175707 -7.1897288525 -7.1850906004 -7.1821507165 -7.1807280679## $freq
## [1] 0.0006844627 0.0013689254 0.0020533881 0.0027378508 0.0034223135
## [6] 0.0041067762 0.0047912389 0.0054757016 0.0061601643 0.0068446270
## [11] 0.0075290897 0.0082135524 0.0088980151 0.0095824778 0.0102669405
## [16] 0.0109514031 0.0116358658 0.0123203285 0.0130047912 0.0136892539
## [21] 0.0143737166 0.0150581793 0.0157426420 0.0164271047 0.0171115674
## [26] 0.0177960301 0.0184804928 0.0191649555 0.0198494182 0.0205338809
## [31] 0.0212183436 0.0219028063 0.0225872690 0.0232717317 0.0239561944
## [36] 0.0246406571 0.0253251198 0.0260095825 0.0266940452 0.0273785079
## [41] 0.0280629706 0.0287474333 0.0294318960 0.0301163587 0.0308008214
## [46] 0.0314852841 0.0321697467 0.0328542094 0.0335386721 0.0342231348
## [51] 0.0349075975 0.0355920602 0.0362765229 0.0369609856 0.0376454483
## [56] 0.0383299110 0.0390143737 0.0396988364 0.0403832991 0.0410677618
## [61] 0.0417522245 0.0424366872 0.0431211499 0.0438056126 0.0444900753
## [66] 0.0451745380 0.0458590007 0.0465434634 0.0472279261 0.0479123888
## [71] 0.0485968515 0.0492813142 0.0499657769 0.0506502396 0.0513347023
## [76] 0.0520191650 0.0527036277 0.0533880903 0.0540725530 0.0547570157
## [81] 0.0554414784 0.0561259411 0.0568104038 0.0574948665 0.0581793292
## [86] 0.0588637919 0.0595482546 0.0602327173 0.0609171800 0.0616016427
## [91] 0.0622861054 0.0629705681 0.0636550308 0.0643394935 0.0650239562
## [96] 0.0657084189 0.0663928816 0.0670773443 0.0677618070 0.0684462697
## [101] 0.0691307324 0.0698151951 0.0704996578 0.0711841205 0.0718685832
## [106] 0.0725530459 0.0732375086 0.0739219713 0.0746064339 0.0752908966
## [111] 0.0759753593 0.0766598220 0.0773442847 0.0780287474 0.0787132101
## [116] 0.0793976728 0.0800821355 0.0807665982 0.0814510609 0.0821355236
## [121] 0.0828199863 0.0835044490 0.0841889117 0.0848733744 0.0855578371
## [126] 0.0862422998 0.0869267625 0.0876112252 0.0882956879 0.0889801506
## [131] 0.0896646133 0.0903490760 0.0910335387 0.0917180014 0.0924024641
## [136] 0.0930869268 0.0937713895 0.0944558522 0.0951403149 0.0958247775
## [141] 0.0965092402 0.0971937029 0.0978781656 0.0985626283 0.0992470910
## [146] 0.0999315537 0.1006160164 0.1013004791 0.1019849418 0.1026694045
## [151] 0.1033538672 0.1040383299 0.1047227926 0.1054072553 0.1060917180
## [156] 0.1067761807 0.1074606434 0.1081451061 0.1088295688 0.1095140315
## [161] 0.1101984942 0.1108829569 0.1115674196 0.1122518823 0.1129363450
## [166] 0.1136208077 0.1143052704 0.1149897331 0.1156741958 0.1163586585
## [171] 0.1170431211 0.1177275838 0.1184120465 0.1190965092 0.1197809719
## [176] 0.1204654346 0.1211498973 0.1218343600 0.1225188227 0.1232032854
## [181] 0.1238877481 0.1245722108 0.1252566735 0.1259411362 0.1266255989
## [186] 0.1273100616 0.1279945243 0.1286789870 0.1293634497 0.1300479124
## [191] 0.1307323751 0.1314168378 0.1321013005 0.1327857632 0.1334702259
## [196] 0.1341546886 0.1348391513 0.1355236140 0.1362080767 0.1368925394
## [201] 0.1375770021 0.1382614648 0.1389459274 0.1396303901 0.1403148528
## [206] 0.1409993155 0.1416837782 0.1423682409 0.1430527036 0.1437371663
## [211] 0.1444216290 0.1451060917 0.1457905544 0.1464750171 0.1471594798
## [216] 0.1478439425 0.1485284052 0.1492128679 0.1498973306 0.1505817933
## [221] 0.1512662560 0.1519507187 0.1526351814 0.1533196441 0.1540041068
## [226] 0.1546885695 0.1553730322 0.1560574949 0.1567419576 0.1574264203
## [231] 0.1581108830 0.1587953457 0.1594798084 0.1601642710 0.1608487337
## [236] 0.1615331964 0.1622176591 0.1629021218 0.1635865845 0.1642710472
## [241] 0.1649555099 0.1656399726 0.1663244353 0.1670088980 0.1676933607
## [246] 0.1683778234 0.1690622861 0.1697467488 0.1704312115 0.1711156742
## [251] 0.1718001369 0.1724845996 0.1731690623 0.1738535250 0.1745379877
## [256] 0.1752224504 0.1759069131 0.1765913758 0.1772758385 0.1779603012
## [261] 0.1786447639 0.1793292266 0.1800136893 0.1806981520 0.1813826146
## [266] 0.1820670773 0.1827515400 0.1834360027 0.1841204654 0.1848049281
## [271] 0.1854893908 0.1861738535 0.1868583162 0.1875427789 0.1882272416
## [276] 0.1889117043 0.1895961670 0.1902806297 0.1909650924 0.1916495551
## [281] 0.1923340178 0.1930184805 0.1937029432 0.1943874059 0.1950718686
## [286] 0.1957563313 0.1964407940 0.1971252567 0.1978097194 0.1984941821
## [291] 0.1991786448 0.1998631075 0.2005475702 0.2012320329 0.2019164956
## [296] 0.2026009582 0.2032854209 0.2039698836 0.2046543463 0.2053388090
## [301] 0.2060232717 0.2067077344 0.2073921971 0.2080766598 0.2087611225
## [306] 0.2094455852 0.2101300479 0.2108145106 0.2114989733 0.2121834360
## [311] 0.2128678987 0.2135523614 0.2142368241 0.2149212868 0.2156057495
## [316] 0.2162902122 0.2169746749 0.2176591376 0.2183436003 0.2190280630
## [321] 0.2197125257 0.2203969884 0.2210814511 0.2217659138 0.2224503765
## [326] 0.2231348392 0.2238193018 0.2245037645 0.2251882272 0.2258726899
## [331] 0.2265571526 0.2272416153 0.2279260780 0.2286105407 0.2292950034
## [336] 0.2299794661 0.2306639288 0.2313483915 0.2320328542 0.2327173169
## [341] 0.2334017796 0.2340862423 0.2347707050 0.2354551677 0.2361396304
## [346] 0.2368240931 0.2375085558 0.2381930185 0.2388774812 0.2395619439
## [351] 0.2402464066 0.2409308693 0.2416153320 0.2422997947 0.2429842574
## [356] 0.2436687201 0.2443531828 0.2450376454 0.2457221081 0.2464065708
## [361] 0.2470910335 0.2477754962 0.2484599589 0.2491444216 0.2498288843
## [366] 0.2505133470 0.2511978097 0.2518822724 0.2525667351 0.2532511978
## [371] 0.2539356605 0.2546201232 0.2553045859 0.2559890486 0.2566735113
## [376] 0.2573579740 0.2580424367 0.2587268994 0.2594113621 0.2600958248
## [381] 0.2607802875 0.2614647502 0.2621492129 0.2628336756 0.2635181383
## [386] 0.2642026010 0.2648870637 0.2655715264 0.2662559890 0.2669404517
## [391] 0.2676249144 0.2683093771 0.2689938398 0.2696783025 0.2703627652
## [396] 0.2710472279 0.2717316906 0.2724161533 0.2731006160 0.2737850787
## [401] 0.2744695414 0.2751540041 0.2758384668 0.2765229295 0.2772073922
## [406] 0.2778918549 0.2785763176 0.2792607803 0.2799452430 0.2806297057
## [411] 0.2813141684 0.2819986311 0.2826830938 0.2833675565 0.2840520192
## [416] 0.2847364819 0.2854209446 0.2861054073 0.2867898700 0.2874743326
## [421] 0.2881587953 0.2888432580 0.2895277207 0.2902121834 0.2908966461
## [426] 0.2915811088 0.2922655715 0.2929500342 0.2936344969 0.2943189596
## [431] 0.2950034223 0.2956878850 0.2963723477 0.2970568104 0.2977412731
## [436] 0.2984257358 0.2991101985 0.2997946612 0.3004791239 0.3011635866
## [441] 0.3018480493 0.3025325120 0.3032169747 0.3039014374 0.3045859001
## [446] 0.3052703628 0.3059548255 0.3066392882 0.3073237509 0.3080082136
## [451] 0.3086926762 0.3093771389 0.3100616016 0.3107460643 0.3114305270
## [456] 0.3121149897 0.3127994524 0.3134839151 0.3141683778 0.3148528405
## [461] 0.3155373032 0.3162217659 0.3169062286 0.3175906913 0.3182751540
## [466] 0.3189596167 0.3196440794 0.3203285421 0.3210130048 0.3216974675
## [471] 0.3223819302 0.3230663929 0.3237508556 0.3244353183 0.3251197810
## [476] 0.3258042437 0.3264887064 0.3271731691 0.3278576318 0.3285420945
## [481] 0.3292265572 0.3299110198 0.3305954825 0.3312799452 0.3319644079
## [486] 0.3326488706 0.3333333333 0.3340177960 0.3347022587 0.3353867214
## [491] 0.3360711841 0.3367556468 0.3374401095 0.3381245722 0.3388090349
## [496] 0.3394934976 0.3401779603 0.3408624230 0.3415468857 0.3422313484
## [501] 0.3429158111 0.3436002738 0.3442847365 0.3449691992 0.3456536619
## [506] 0.3463381246 0.3470225873 0.3477070500 0.3483915127 0.3490759754
## [511] 0.3497604381 0.3504449008 0.3511293634 0.3518138261 0.3524982888
## [516] 0.3531827515 0.3538672142 0.3545516769 0.3552361396 0.3559206023
## [521] 0.3566050650 0.3572895277 0.3579739904 0.3586584531 0.3593429158
## [526] 0.3600273785 0.3607118412 0.3613963039 0.3620807666 0.3627652293
## [531] 0.3634496920 0.3641341547 0.3648186174 0.3655030801 0.3661875428
## [536] 0.3668720055 0.3675564682 0.3682409309 0.3689253936 0.3696098563
## [541] 0.3702943190 0.3709787817 0.3716632444 0.3723477070 0.3730321697
## [546] 0.3737166324 0.3744010951 0.3750855578 0.3757700205 0.3764544832
## [551] 0.3771389459 0.3778234086 0.3785078713 0.3791923340 0.3798767967
## [556] 0.3805612594 0.3812457221 0.3819301848 0.3826146475 0.3832991102
## [561] 0.3839835729 0.3846680356 0.3853524983 0.3860369610 0.3867214237
## [566] 0.3874058864 0.3880903491 0.3887748118 0.3894592745 0.3901437372
## [571] 0.3908281999 0.3915126626 0.3921971253 0.3928815880 0.3935660507
## [576] 0.3942505133 0.3949349760 0.3956194387 0.3963039014 0.3969883641
## [581] 0.3976728268 0.3983572895 0.3990417522 0.3997262149 0.4004106776
## [586] 0.4010951403 0.4017796030 0.4024640657 0.4031485284 0.4038329911
## [591] 0.4045174538 0.4052019165 0.4058863792 0.4065708419 0.4072553046
## [596] 0.4079397673 0.4086242300 0.4093086927 0.4099931554 0.4106776181
## [601] 0.4113620808 0.4120465435 0.4127310062 0.4134154689 0.4140999316
## [606] 0.4147843943 0.4154688569 0.4161533196 0.4168377823 0.4175222450
## [611] 0.4182067077 0.4188911704 0.4195756331 0.4202600958 0.4209445585
## [616] 0.4216290212 0.4223134839 0.4229979466 0.4236824093 0.4243668720
## [621] 0.4250513347 0.4257357974 0.4264202601 0.4271047228 0.4277891855
## [626] 0.4284736482 0.4291581109 0.4298425736 0.4305270363 0.4312114990
## [631] 0.4318959617 0.4325804244 0.4332648871 0.4339493498 0.4346338125
## [636] 0.4353182752 0.4360027379 0.4366872005 0.4373716632 0.4380561259
## [641] 0.4387405886 0.4394250513 0.4401095140 0.4407939767 0.4414784394
## [646] 0.4421629021 0.4428473648 0.4435318275 0.4442162902 0.4449007529
## [651] 0.4455852156 0.4462696783 0.4469541410 0.4476386037 0.4483230664
## [656] 0.4490075291 0.4496919918 0.4503764545 0.4510609172 0.4517453799
## [661] 0.4524298426 0.4531143053 0.4537987680 0.4544832307 0.4551676934
## [666] 0.4558521561 0.4565366188 0.4572210815 0.4579055441 0.4585900068
## [671] 0.4592744695 0.4599589322 0.4606433949 0.4613278576 0.4620123203
## [676] 0.4626967830 0.4633812457 0.4640657084 0.4647501711 0.4654346338
## [681] 0.4661190965 0.4668035592 0.4674880219 0.4681724846 0.4688569473
## [686] 0.4695414100 0.4702258727 0.4709103354 0.4715947981 0.4722792608
## [691] 0.4729637235 0.4736481862 0.4743326489 0.4750171116 0.4757015743
## [696] 0.4763860370 0.4770704997 0.4777549624 0.4784394251 0.4791238877
## [701] 0.4798083504 0.4804928131 0.4811772758 0.4818617385 0.4825462012
## [706] 0.4832306639 0.4839151266 0.4845995893 0.4852840520 0.4859685147
## [711] 0.4866529774 0.4873374401 0.4880219028 0.4887063655 0.4893908282
## [716] 0.4900752909 0.4907597536 0.4914442163 0.4921286790 0.4928131417
## [721] 0.4934976044 0.4941820671 0.4948665298 0.4955509925 0.4962354552
## [726] 0.4969199179 0.4976043806 0.4982888433 0.4989733060 0.4996577687
##
## $pzgram
## [1] 6.87696871 6.85813317 6.82686038 6.78333615 6.72783290 6.66072139
## [7] 6.58248534 6.49373803 6.39524032 6.28791865 6.17288174 6.05143364
## [13] 5.92508103 5.79553146 5.66467967 5.53457861 5.40739286 5.28533289
## [19] 5.17057109 5.06514290 4.97083961 4.88910205 4.82092657 4.76679454
## [25] 4.72663483 4.69982476 4.68522982 4.68127741 4.68605559 4.69742561
## [31] 4.71313709 4.73093619 4.74865987 4.76431255 4.77612391 4.78258884
## [37] 4.78249176 4.77491786 4.75925407 4.73518214 4.70266562 4.66193193
## [43] 4.61345045 4.55790691 4.49617410 4.42927916 4.35836729 4.28466260
## [49] 4.20942649 4.13391508 4.05933714 3.98681464 3.91734800 3.85178817
## [55] 3.79081722 3.73493851 3.68447655 3.63958602 3.60026827 3.56639315
## [61] 3.53772371 3.51394114 3.49466768 3.47948574 3.46795204 3.45960642
## [67] 3.45397549 3.45057196 3.44889079 3.44840375 3.44855375 3.44875056
## [73] 3.44836889 3.44674993 3.44320682 3.43703401 3.42752062 3.41396710
## [79] 3.39570464 3.37211651 3.34266071 3.30689284 3.26448887 3.21526681
## [85] 3.15920674 3.09646862 3.02740687 2.95258099 2.87276115 2.78892728
## [91] 2.70226055 2.61412560 2.52604246 2.43964730 2.35664208 2.27873419
## [97] 2.20756861 2.14465657 2.09130577 2.04855837 2.01714244 1.99744202
## [103] 1.98948887 1.99297663 2.00729556 2.03158368 2.06478876 2.10573493
## [109] 2.15318800 2.20591465 2.26273210 2.32254622 2.38437761 2.44737611
## [115] 2.51082504 2.57413678 2.63684182 2.69857289 2.75904626 2.81804162
## [121] 2.87538212 2.93091559 2.98449798 3.03597968 3.08519505 3.13195554
## [127] 3.17604620 3.21722550 3.25522813 3.28977005 3.32055570 3.34728629
## [133] 3.36966904 3.38742663 3.40030642 3.40808914 3.41059669 3.40769885
## [139] 3.39931864 3.38543634 3.36609197 3.34138621 3.31147974 3.27659093
## [145] 3.23699199 3.19300344 3.14498718 3.09333819 3.03847509 2.98082988
## [151] 2.92083731 2.85892426 2.79549980 2.73094653 2.66561389 2.59981385
## [157] 2.53381982 2.46786886 2.40216748 2.33690100 2.27224595 2.20838517
## [163] 2.14552434 2.08390915 2.02384152 1.96569341 1.90991669 1.85704734
## [169] 1.80770271 1.76257038 1.72238830 1.68791568 1.65989584 1.63901252
## [175] 1.62584239 1.62080745 1.62413147 1.63580445 1.65555921 1.68286250
## [181] 1.71692203 1.75670909 1.80099471 1.84839634 1.89743120 1.94657234
## [187] 1.99430379 2.03917201 2.07983147 2.11508335 2.14390679 2.16548288
## [193] 2.17921176 2.18472334 2.18188213 2.17078643 2.15176182 2.12534896
## [199] 2.09228511 2.05347903 2.00997893 1.96293339 1.91354579 1.86302333
## [205] 1.81252288 1.76309628 1.71563888 1.67084505 1.62917459 1.59083310
## [211] 1.55576827 1.52368275 1.49406234 1.46621722 1.43933248 1.41252414
## [217] 1.38489638 1.35559654 1.32386467 1.28907570 1.25077253 1.20868928
## [223] 1.16276429 1.11314273 1.06016875 1.00436749 0.94641743 0.88711356
## [229] 0.82732290 0.76793396 0.70980285 0.65369914 0.60025527 0.54992328
## [235] 0.50294267 0.45932212 0.41883691 0.38104228 0.34530169 0.31082751
## [241] 0.27673088 0.24207713 0.20594312 0.16747325 0.12593176 0.08074928
## [247] 0.03156245 -0.02175418 -0.07906483 -0.13995811 -0.20374266 -0.26945373
## [253] -0.33587203 -0.40155602 -0.46488858 -0.52413836 -0.57753443 -0.62335175
## [259] -0.66000238 -0.68612614 -0.70067267 -0.70296700 -0.69275159 -0.67020031
## [265] -0.63590303 -0.59082308 -0.53623301 -0.47363601 -0.40468119 -0.33108031
## [271] -0.25453198 -0.17665771 -0.09895166 -0.02274476 0.05081767 0.12078661
## [277] 0.18640993 0.24712535 0.30254716 0.35244881 0.39674226 0.43545555
## [283] 0.46870926 0.49669282 0.51964115 0.53781245 0.55146753 0.56085148
## [289] 0.56617794 0.56761665 0.56528451 0.55924035 0.54948358 0.53595659
## [295] 0.51855062 0.49711476 0.47146747 0.44141029 0.40674276 0.36727825
## [301] 0.32286000 0.27337680 0.21877785 0.15908650 0.09441228 0.02496119
## [307] -0.04895647 -0.12692065 -0.20839921 -0.29274872 -0.37921911 -0.46696228
## [313] -0.55504517 -0.64246737 -0.72818333 -0.81112892 -0.89025174 -0.96454416
## [319] -1.03307786 -1.09503796 -1.14975504 -1.19673286 -1.23567007 -1.26647426
## [325] -1.28926757 -1.30438336 -1.31235452 -1.31389430 -1.30987120 -1.30127982
## [331] -1.28920935 -1.27481162 -1.25927002 -1.24377060 -1.22947589 -1.21750201
## [337] -1.20889894 -1.20463394 -1.20557781 -1.21249345 -1.22602644 -1.24669729
## [343] -1.27489495 -1.31087146 -1.35473763 -1.40645990 -1.46585839 -1.53260672
## [349] -1.60623377 -1.68612824 -1.77154646 -1.86162406 -1.95539226 -2.05179891
## [355] -2.14973462 -2.24806348 -2.34565764 -2.44143441 -2.53439379 -2.62365428
## [361] -2.70848416 -2.78832597 -2.86281169 -2.93176755 -2.99520767 -3.05331702
## [367] -3.10642538 -3.15497430 -3.19947985 -3.24049414 -3.27856829 -3.31421925
## [373] -3.34790230 -3.37999053 -3.41076196 -3.44039448 -3.46896819 -3.49647456
## [379] -3.52283136 -3.54790210 -3.57151845 -3.59350427 -3.61369938 -3.63198167
## [385] -3.64828611 -3.66261938 -3.67506944 -3.68580955 -3.69509678 -3.70326542
## [391] -3.71071620 -3.71790214 -3.72531232 -3.73345475 -3.74283931 -3.75396169
## [397] -3.76728899 -3.78324740 -3.80221215 -3.82449964 -3.85036164 -3.87998128
## [403] -3.91347036 -3.95086771 -3.99213820 -4.03717210 -4.08578467 -4.13771580
## [409] -4.19262984 -4.25011586 -4.30968860 -4.37079065 -4.43279643 -4.49501854
## [415] -4.55671722 -4.61711318 -4.67540430 -4.73078610 -4.78247555 -4.82973733
## [421] -4.87191139 -4.90843977 -4.93889091 -4.96297931 -4.98057847 -4.99172575
## [427] -4.99661821 -4.99559931 -4.98913724 -4.97779628 -4.96220315 -4.94301076
## [433] -4.92086177 -4.89635416 -4.87001103 -4.84225610 -4.81339630 -4.78361225
## [439] -4.75295704 -4.72136350 -4.68865947 -4.65459061 -4.61884944 -4.58110921
## [445] -4.54106053 -4.49844863 -4.45310854 -4.40499586 -4.35421070 -4.30101306
## [451] -4.24582837 -4.18924299 -4.13199010 -4.07492717 -4.01900689 -3.96524348
## [457] -3.91467663 -3.86833497 -3.82720048 -3.79217525 -3.76405112 -3.74348251
## [463] -3.73096260 -3.72680260 -3.73111394 -3.74379341 -3.76451112 -3.79270193
## [469] -3.82756093 -3.86804417 -3.91287635 -3.96056704 -4.00943768 -4.05766096
## [475] -4.10331379 -4.14444395 -4.17914885 -4.20566299 -4.22244847 -4.22828145
## [481] -4.22232631 -4.20418989 -4.17394961 -4.13215239 -4.07978464 -4.01821719
## [487] -3.94913167 -3.87443631 -3.79617904 -3.71646503 -3.63738369 -3.56094807
## [493] -3.48904799 -3.42341623 -3.36560646 -3.31698073 -3.27870444 -3.25174646
## [499] -3.23688263 -3.23470099 -3.24560771 -3.26983250 -3.30743336 -3.35829993
## [505] -3.42215567 -3.49855884 -3.58690255 -3.68641454 -3.79615719 -3.91502864
## [511] -4.04176618 -4.17495276 -4.31302807 -4.45430524 -4.59699408 -4.73923170
## [517] -4.87912040 -5.01477232 -5.14435927 -5.26616533 -5.37863904 -5.48044131
## [523] -5.57048529 -5.64796487 -5.71236932 -5.76348325 -5.80137237 -5.82635706
## [529] -5.83897717 -5.83995150 -5.83013611 -5.81048449 -5.78201245 -5.74576910
## [535] -5.70281457 -5.65420436 -5.60097933 -5.54416028 -5.48474565 -5.42371109
## [541] -5.36200964 -5.30057170 -5.24030409 -5.18208781 -5.12677432 -5.07518052
## [547] -5.02808239 -4.98620775 -4.95022821 -4.92075074 -4.89830909 -4.88335504
## [553] -4.87625001 -4.87725680 -4.88653178 -4.90411757 -4.92993637 -4.96378417
## [559] -5.00532601 -5.05409283 -5.10948026 -5.17075002 -5.23703464 -5.30734617
## [565] -5.38058979 -5.45558277 -5.53107940 -5.60580185 -5.67847648 -5.74787468
## [571] -5.81285630 -5.87241337 -5.92571120 -5.97212365 -6.01125977 -6.04297912
## [577] -6.06739452 -6.08486158 -6.09595605 -6.10144065 -6.10222427 -6.09931639
## [583] -6.09377995 -6.08668544 -6.07906830 -6.07189131 -6.06601295 -6.06216200
## [589] -6.06091854 -6.06270095 -6.06775873 -6.07617061 -6.08784788 -6.10254252
## [595] -6.11986028 -6.13927856 -6.16016905 -6.18182498 -6.20349248 -6.22440539
## [601] -6.24382234 -6.26106442 -6.27555167 -6.28683603 -6.29462862 -6.29881923
## [607] -6.29948649 -6.29689774 -6.29149860 -6.28389269 -6.27481307 -6.26508733
## [613] -6.25559832 -6.24724296 -6.24089105 -6.23734582 -6.23730766 -6.24134188
## [619] -6.24985134 -6.26305442 -6.28096879 -6.30340144 -6.32994567 -6.35998573
## [625] -6.39270991 -6.42713293 -6.46212824 -6.49647021 -6.52888579 -6.55811374
## [631] -6.58296882 -6.60240668 -6.61558489 -6.62191444 -6.62109705 -6.61314418
## [637] -6.59837566 -6.57739800 -6.55106436 -6.52042033 -6.48664026 -6.45095986
## [643] -6.41460991 -6.37875530 -6.34444257 -6.31255791 -6.28379645 -6.25864321
## [649] -6.23736546 -6.22001587 -6.20644574 -6.19632779 -6.18918726 -6.18444040
## [655] -6.18143880 -6.17951761 -6.17804520 -6.17647145 -6.17437149 -6.17148183
## [661] -6.16772609 -6.16322827 -6.15831259 -6.15348986 -6.14943173 -6.14693485
## [667] -6.14687771 -6.15017326 -6.15772010 -6.17035486 -6.18880789 -6.21366362
## [673] -6.24532682 -6.28399538 -6.32964014 -6.38199235 -6.44053929 -6.50452895
## [679] -6.57298446 -6.64472944 -6.71842458 -6.79261591 -6.86579377 -6.93646063
## [685] -7.00320438 -7.06477231 -7.12014014 -7.16856965 -7.20964951 -7.24331479
## [691] -7.26984313 -7.28982810 -7.30413272 -7.31382809 -7.32012323 -7.32429240
## [697] -7.32760528 -7.33126461 -7.33635390 -7.34379699 -7.35432957 -7.36848236
## [703] -7.38657475 -7.40871788 -7.43482567 -7.46463261 -7.49771709 -7.53352922
## [709] -7.57142186 -7.61068374 -7.65057338 -7.69035246 -7.72931708 -7.76682572
## [715] -7.80232229 -7.83535353 -7.86557981 -7.89277923 -7.91684509 -7.93777747
## [721] -7.95566985 -7.97069209 -7.98307105 -7.99307022 -8.00096962 -8.00704679
## [727] -8.01155987 -8.01473305 -8.01674478 -8.01771893## $freq
## [1] 0.0006844627 0.0013689254 0.0020533881 0.0027378508 0.0034223135
## [6] 0.0041067762 0.0047912389 0.0054757016 0.0061601643 0.0068446270
## [11] 0.0075290897 0.0082135524 0.0088980151 0.0095824778 0.0102669405
## [16] 0.0109514031 0.0116358658 0.0123203285 0.0130047912 0.0136892539
## [21] 0.0143737166 0.0150581793 0.0157426420 0.0164271047 0.0171115674
## [26] 0.0177960301 0.0184804928 0.0191649555 0.0198494182 0.0205338809
## [31] 0.0212183436 0.0219028063 0.0225872690 0.0232717317 0.0239561944
## [36] 0.0246406571 0.0253251198 0.0260095825 0.0266940452 0.0273785079
## [41] 0.0280629706 0.0287474333 0.0294318960 0.0301163587 0.0308008214
## [46] 0.0314852841 0.0321697467 0.0328542094 0.0335386721 0.0342231348
## [51] 0.0349075975 0.0355920602 0.0362765229 0.0369609856 0.0376454483
## [56] 0.0383299110 0.0390143737 0.0396988364 0.0403832991 0.0410677618
## [61] 0.0417522245 0.0424366872 0.0431211499 0.0438056126 0.0444900753
## [66] 0.0451745380 0.0458590007 0.0465434634 0.0472279261 0.0479123888
## [71] 0.0485968515 0.0492813142 0.0499657769 0.0506502396 0.0513347023
## [76] 0.0520191650 0.0527036277 0.0533880903 0.0540725530 0.0547570157
## [81] 0.0554414784 0.0561259411 0.0568104038 0.0574948665 0.0581793292
## [86] 0.0588637919 0.0595482546 0.0602327173 0.0609171800 0.0616016427
## [91] 0.0622861054 0.0629705681 0.0636550308 0.0643394935 0.0650239562
## [96] 0.0657084189 0.0663928816 0.0670773443 0.0677618070 0.0684462697
## [101] 0.0691307324 0.0698151951 0.0704996578 0.0711841205 0.0718685832
## [106] 0.0725530459 0.0732375086 0.0739219713 0.0746064339 0.0752908966
## [111] 0.0759753593 0.0766598220 0.0773442847 0.0780287474 0.0787132101
## [116] 0.0793976728 0.0800821355 0.0807665982 0.0814510609 0.0821355236
## [121] 0.0828199863 0.0835044490 0.0841889117 0.0848733744 0.0855578371
## [126] 0.0862422998 0.0869267625 0.0876112252 0.0882956879 0.0889801506
## [131] 0.0896646133 0.0903490760 0.0910335387 0.0917180014 0.0924024641
## [136] 0.0930869268 0.0937713895 0.0944558522 0.0951403149 0.0958247775
## [141] 0.0965092402 0.0971937029 0.0978781656 0.0985626283 0.0992470910
## [146] 0.0999315537 0.1006160164 0.1013004791 0.1019849418 0.1026694045
## [151] 0.1033538672 0.1040383299 0.1047227926 0.1054072553 0.1060917180
## [156] 0.1067761807 0.1074606434 0.1081451061 0.1088295688 0.1095140315
## [161] 0.1101984942 0.1108829569 0.1115674196 0.1122518823 0.1129363450
## [166] 0.1136208077 0.1143052704 0.1149897331 0.1156741958 0.1163586585
## [171] 0.1170431211 0.1177275838 0.1184120465 0.1190965092 0.1197809719
## [176] 0.1204654346 0.1211498973 0.1218343600 0.1225188227 0.1232032854
## [181] 0.1238877481 0.1245722108 0.1252566735 0.1259411362 0.1266255989
## [186] 0.1273100616 0.1279945243 0.1286789870 0.1293634497 0.1300479124
## [191] 0.1307323751 0.1314168378 0.1321013005 0.1327857632 0.1334702259
## [196] 0.1341546886 0.1348391513 0.1355236140 0.1362080767 0.1368925394
## [201] 0.1375770021 0.1382614648 0.1389459274 0.1396303901 0.1403148528
## [206] 0.1409993155 0.1416837782 0.1423682409 0.1430527036 0.1437371663
## [211] 0.1444216290 0.1451060917 0.1457905544 0.1464750171 0.1471594798
## [216] 0.1478439425 0.1485284052 0.1492128679 0.1498973306 0.1505817933
## [221] 0.1512662560 0.1519507187 0.1526351814 0.1533196441 0.1540041068
## [226] 0.1546885695 0.1553730322 0.1560574949 0.1567419576 0.1574264203
## [231] 0.1581108830 0.1587953457 0.1594798084 0.1601642710 0.1608487337
## [236] 0.1615331964 0.1622176591 0.1629021218 0.1635865845 0.1642710472
## [241] 0.1649555099 0.1656399726 0.1663244353 0.1670088980 0.1676933607
## [246] 0.1683778234 0.1690622861 0.1697467488 0.1704312115 0.1711156742
## [251] 0.1718001369 0.1724845996 0.1731690623 0.1738535250 0.1745379877
## [256] 0.1752224504 0.1759069131 0.1765913758 0.1772758385 0.1779603012
## [261] 0.1786447639 0.1793292266 0.1800136893 0.1806981520 0.1813826146
## [266] 0.1820670773 0.1827515400 0.1834360027 0.1841204654 0.1848049281
## [271] 0.1854893908 0.1861738535 0.1868583162 0.1875427789 0.1882272416
## [276] 0.1889117043 0.1895961670 0.1902806297 0.1909650924 0.1916495551
## [281] 0.1923340178 0.1930184805 0.1937029432 0.1943874059 0.1950718686
## [286] 0.1957563313 0.1964407940 0.1971252567 0.1978097194 0.1984941821
## [291] 0.1991786448 0.1998631075 0.2005475702 0.2012320329 0.2019164956
## [296] 0.2026009582 0.2032854209 0.2039698836 0.2046543463 0.2053388090
## [301] 0.2060232717 0.2067077344 0.2073921971 0.2080766598 0.2087611225
## [306] 0.2094455852 0.2101300479 0.2108145106 0.2114989733 0.2121834360
## [311] 0.2128678987 0.2135523614 0.2142368241 0.2149212868 0.2156057495
## [316] 0.2162902122 0.2169746749 0.2176591376 0.2183436003 0.2190280630
## [321] 0.2197125257 0.2203969884 0.2210814511 0.2217659138 0.2224503765
## [326] 0.2231348392 0.2238193018 0.2245037645 0.2251882272 0.2258726899
## [331] 0.2265571526 0.2272416153 0.2279260780 0.2286105407 0.2292950034
## [336] 0.2299794661 0.2306639288 0.2313483915 0.2320328542 0.2327173169
## [341] 0.2334017796 0.2340862423 0.2347707050 0.2354551677 0.2361396304
## [346] 0.2368240931 0.2375085558 0.2381930185 0.2388774812 0.2395619439
## [351] 0.2402464066 0.2409308693 0.2416153320 0.2422997947 0.2429842574
## [356] 0.2436687201 0.2443531828 0.2450376454 0.2457221081 0.2464065708
## [361] 0.2470910335 0.2477754962 0.2484599589 0.2491444216 0.2498288843
## [366] 0.2505133470 0.2511978097 0.2518822724 0.2525667351 0.2532511978
## [371] 0.2539356605 0.2546201232 0.2553045859 0.2559890486 0.2566735113
## [376] 0.2573579740 0.2580424367 0.2587268994 0.2594113621 0.2600958248
## [381] 0.2607802875 0.2614647502 0.2621492129 0.2628336756 0.2635181383
## [386] 0.2642026010 0.2648870637 0.2655715264 0.2662559890 0.2669404517
## [391] 0.2676249144 0.2683093771 0.2689938398 0.2696783025 0.2703627652
## [396] 0.2710472279 0.2717316906 0.2724161533 0.2731006160 0.2737850787
## [401] 0.2744695414 0.2751540041 0.2758384668 0.2765229295 0.2772073922
## [406] 0.2778918549 0.2785763176 0.2792607803 0.2799452430 0.2806297057
## [411] 0.2813141684 0.2819986311 0.2826830938 0.2833675565 0.2840520192
## [416] 0.2847364819 0.2854209446 0.2861054073 0.2867898700 0.2874743326
## [421] 0.2881587953 0.2888432580 0.2895277207 0.2902121834 0.2908966461
## [426] 0.2915811088 0.2922655715 0.2929500342 0.2936344969 0.2943189596
## [431] 0.2950034223 0.2956878850 0.2963723477 0.2970568104 0.2977412731
## [436] 0.2984257358 0.2991101985 0.2997946612 0.3004791239 0.3011635866
## [441] 0.3018480493 0.3025325120 0.3032169747 0.3039014374 0.3045859001
## [446] 0.3052703628 0.3059548255 0.3066392882 0.3073237509 0.3080082136
## [451] 0.3086926762 0.3093771389 0.3100616016 0.3107460643 0.3114305270
## [456] 0.3121149897 0.3127994524 0.3134839151 0.3141683778 0.3148528405
## [461] 0.3155373032 0.3162217659 0.3169062286 0.3175906913 0.3182751540
## [466] 0.3189596167 0.3196440794 0.3203285421 0.3210130048 0.3216974675
## [471] 0.3223819302 0.3230663929 0.3237508556 0.3244353183 0.3251197810
## [476] 0.3258042437 0.3264887064 0.3271731691 0.3278576318 0.3285420945
## [481] 0.3292265572 0.3299110198 0.3305954825 0.3312799452 0.3319644079
## [486] 0.3326488706 0.3333333333 0.3340177960 0.3347022587 0.3353867214
## [491] 0.3360711841 0.3367556468 0.3374401095 0.3381245722 0.3388090349
## [496] 0.3394934976 0.3401779603 0.3408624230 0.3415468857 0.3422313484
## [501] 0.3429158111 0.3436002738 0.3442847365 0.3449691992 0.3456536619
## [506] 0.3463381246 0.3470225873 0.3477070500 0.3483915127 0.3490759754
## [511] 0.3497604381 0.3504449008 0.3511293634 0.3518138261 0.3524982888
## [516] 0.3531827515 0.3538672142 0.3545516769 0.3552361396 0.3559206023
## [521] 0.3566050650 0.3572895277 0.3579739904 0.3586584531 0.3593429158
## [526] 0.3600273785 0.3607118412 0.3613963039 0.3620807666 0.3627652293
## [531] 0.3634496920 0.3641341547 0.3648186174 0.3655030801 0.3661875428
## [536] 0.3668720055 0.3675564682 0.3682409309 0.3689253936 0.3696098563
## [541] 0.3702943190 0.3709787817 0.3716632444 0.3723477070 0.3730321697
## [546] 0.3737166324 0.3744010951 0.3750855578 0.3757700205 0.3764544832
## [551] 0.3771389459 0.3778234086 0.3785078713 0.3791923340 0.3798767967
## [556] 0.3805612594 0.3812457221 0.3819301848 0.3826146475 0.3832991102
## [561] 0.3839835729 0.3846680356 0.3853524983 0.3860369610 0.3867214237
## [566] 0.3874058864 0.3880903491 0.3887748118 0.3894592745 0.3901437372
## [571] 0.3908281999 0.3915126626 0.3921971253 0.3928815880 0.3935660507
## [576] 0.3942505133 0.3949349760 0.3956194387 0.3963039014 0.3969883641
## [581] 0.3976728268 0.3983572895 0.3990417522 0.3997262149 0.4004106776
## [586] 0.4010951403 0.4017796030 0.4024640657 0.4031485284 0.4038329911
## [591] 0.4045174538 0.4052019165 0.4058863792 0.4065708419 0.4072553046
## [596] 0.4079397673 0.4086242300 0.4093086927 0.4099931554 0.4106776181
## [601] 0.4113620808 0.4120465435 0.4127310062 0.4134154689 0.4140999316
## [606] 0.4147843943 0.4154688569 0.4161533196 0.4168377823 0.4175222450
## [611] 0.4182067077 0.4188911704 0.4195756331 0.4202600958 0.4209445585
## [616] 0.4216290212 0.4223134839 0.4229979466 0.4236824093 0.4243668720
## [621] 0.4250513347 0.4257357974 0.4264202601 0.4271047228 0.4277891855
## [626] 0.4284736482 0.4291581109 0.4298425736 0.4305270363 0.4312114990
## [631] 0.4318959617 0.4325804244 0.4332648871 0.4339493498 0.4346338125
## [636] 0.4353182752 0.4360027379 0.4366872005 0.4373716632 0.4380561259
## [641] 0.4387405886 0.4394250513 0.4401095140 0.4407939767 0.4414784394
## [646] 0.4421629021 0.4428473648 0.4435318275 0.4442162902 0.4449007529
## [651] 0.4455852156 0.4462696783 0.4469541410 0.4476386037 0.4483230664
## [656] 0.4490075291 0.4496919918 0.4503764545 0.4510609172 0.4517453799
## [661] 0.4524298426 0.4531143053 0.4537987680 0.4544832307 0.4551676934
## [666] 0.4558521561 0.4565366188 0.4572210815 0.4579055441 0.4585900068
## [671] 0.4592744695 0.4599589322 0.4606433949 0.4613278576 0.4620123203
## [676] 0.4626967830 0.4633812457 0.4640657084 0.4647501711 0.4654346338
## [681] 0.4661190965 0.4668035592 0.4674880219 0.4681724846 0.4688569473
## [686] 0.4695414100 0.4702258727 0.4709103354 0.4715947981 0.4722792608
## [691] 0.4729637235 0.4736481862 0.4743326489 0.4750171116 0.4757015743
## [696] 0.4763860370 0.4770704997 0.4777549624 0.4784394251 0.4791238877
## [701] 0.4798083504 0.4804928131 0.4811772758 0.4818617385 0.4825462012
## [706] 0.4832306639 0.4839151266 0.4845995893 0.4852840520 0.4859685147
## [711] 0.4866529774 0.4873374401 0.4880219028 0.4887063655 0.4893908282
## [716] 0.4900752909 0.4907597536 0.4914442163 0.4921286790 0.4928131417
## [721] 0.4934976044 0.4941820671 0.4948665298 0.4955509925 0.4962354552
## [726] 0.4969199179 0.4976043806 0.4982888433 0.4989733060 0.4996577687
##
## $pzgram
## [1] 6.18098050 6.17069544 6.15374237 6.13039933 6.10104570 6.06615240
## [7] 6.02626951 5.98201147 5.93404033 5.88304741 5.82973409 5.77479258
## [13] 5.71888745 5.66263907 5.60660967 5.55129314 5.49710892 5.44440044
## [19] 5.39343802 5.34442590 5.29751243 5.25280267 5.21037179 5.17027824
## [25] 5.13257509 5.09731863 5.06457296 5.03441039 5.00690700 4.98213389
## [31] 4.96014454 4.94095937 4.92454881 4.91081629 4.89958290 4.89057515
## [37] 4.88341709 4.87762785 4.87262495 4.86773352 4.86220088 4.85521581
## [43] 4.84593122 4.83348920 4.81704712 4.79580369 4.76902396 4.73606261
## [49] 4.69638489 4.64958484 4.59540053 4.53372607 4.46462024 4.38831137
## [55] 4.30519823 4.21584648 4.12098020 4.02146799 3.91830336 3.81257916
## [61] 3.70545626 3.59812720 3.49177599 3.38753608 3.28644887 3.18942573
## [67] 3.09721643 3.01038677 2.92930758 2.85415607 2.78492964 2.72147079
## [73] 2.66350101 2.61066039 2.56254969 2.51877144 2.47896689 2.44284628
## [79] 2.41021068 2.38096414 2.35511587 2.33277264 2.31412224 2.29940957
## [85] 2.28890715 2.28288236 2.28156401 2.28511078 2.29358404 2.30692721
## [91] 2.32495303 2.34733967 2.37363558 2.40327229 2.43558373 2.46983018
## [97] 2.50522478 2.54096064 2.57623668 2.61028090 2.64236998 2.67184482
## [103] 2.69812184 2.72070015 2.73916529 2.75318985 2.76253197 2.76703231
## [109] 2.76661026 2.76126010 2.75104762 2.73610767 2.71664294 2.69292412
## [115] 2.66529123 2.63415597 2.60000448 2.56339980 2.52498309 2.48547256
## [121] 2.44565883 2.40639568 2.36858499 2.33315542 2.30103422 2.27311298
## [127] 2.25020829 2.23301954 2.22208659 2.21775083 2.22012314 2.22906205
## [133] 2.24416479 2.26477244 2.28998956 2.31871675 2.34969396 2.38155133
## [139] 2.41286426 2.44220950 2.46821962 2.48963387 2.50534424 2.51443611
## [145] 2.51622341 2.51027845 2.49645642 2.47491461 2.44612571 2.41088405
## [151] 2.37030319 2.32580225 2.27907857 2.23206368 2.18686059 2.14566162
## [157] 2.11064816 2.08387681 2.06715940 2.06194748 2.06923312 2.08947778
## [163] 2.12257818 2.16787368 2.22419368 2.28993889 2.36318562 2.44180124
## [169] 2.52355917 2.60624380 2.68773937 2.76609938 2.83959650 2.90675417
## [175] 2.96636284 3.01748370 3.05944287 3.09181876 3.11442441 3.12728649
## [181] 3.13062195 3.12481287 3.11038018 3.08795622 3.05825671 3.02205225
## [187] 2.98013989 2.93331541 2.88234711 2.82795189 2.77077485 2.71137311
## [193] 2.65020481 2.58762380 2.52388032 2.45912751 2.39343327 2.32679672
## [199] 2.25916804 2.19047060 2.12062397 2.04956642 1.97727601 1.90378885
## [205] 1.82921394 1.75374376 1.67766021 1.60133544 1.52522765 1.44987172
## [211] 1.37586509 1.30384941 1.23448867 1.16844486 1.10635258 1.04879386
## [217] 0.99627490 0.94920622 0.90788753 0.87249840 0.84309525 0.81961482
## [223] 0.80188358 0.78963224 0.78251397 0.78012495 0.78202555 0.78776069
## [229] 0.79687812 0.80894344 0.82355138 0.84033269 0.85895684 0.87913046
## [235] 0.90059226 0.92310484 0.94644425 0.97038817 0.99470370 1.01913540
## [241] 1.04339478 1.06715163 1.09002790 1.11159460 1.13137165 1.14883097
## [247] 1.16340230 1.17448168 1.18144195 1.18364481 1.18045395 1.17124868
## [253] 1.15543774 1.13247282 1.10186153 1.06317984 1.01608360 0.96031934
## [259] 0.89573433 0.82228573 0.74004915 0.64922630 0.55015187 0.44329925
## [265] 0.32928503 0.20887164 0.08296777 -0.04737426 -0.18096485 -0.31648816
## [271] -0.45251859 -0.58754344 -0.71999228 -0.84827297 -0.97081390 -1.08611101
## [277] -1.19277783 -1.28959552 -1.37555986 -1.44992160 -1.51221686 -1.56228519
## [283] -1.60027343 -1.62662537 -1.64205806 -1.64752710 -1.64418380 -1.63332752
## [289] -1.61635662 -1.59472060 -1.56987590 -1.54324657 -1.51619070 -1.48997276
## [295] -1.46574140 -1.44451237 -1.42715569 -1.41438635 -1.40675801 -1.40465913
## [301] -1.40831115 -1.41776873 -1.43292192 -1.45350060 -1.47908152 -1.50909835
## [307] -1.54285534 -1.57954496 -1.61826990 -1.65806951 -1.69795046 -1.73692085
## [313] -1.77402671 -1.80838906 -1.83923958 -1.86595259 -1.88807080 -1.90532308
## [319] -1.91763253 -1.92511411 -1.92806213 -1.92692835 -1.92229258 -1.91482789
## [325] -1.90526286 -1.89434324 -1.88279515 -1.87129146 -1.86042271 -1.85067325
## [331] -1.84240308 -1.83583564 -1.83105129 -1.82798654 -1.82643873 -1.82607592
## [337] -1.82645189 -1.82702569 -1.82718564 -1.82627687 -1.82363183 -1.81860260
## [343] -1.81059360 -1.79909322 -1.78370270 -1.76416046 -1.74036048 -1.71236362
## [349] -1.68040105 -1.64486982 -1.60632099 -1.56544127 -1.52302964 -1.47997034
## [355] -1.43720409 -1.39569887 -1.35642171 -1.32031226 -1.28825904 -1.26107857
## [361] -1.23949757 -1.22413806 -1.21550530 -1.21397820 -1.21980206 -1.23308344
## [367] -1.25378704 -1.28173483 -1.31660731 -1.35794752 -1.40516799 -1.45756118
## [373] -1.51431393 -1.57452616 -1.63723419 -1.70143849 -1.76613544 -1.83035212
## [379] -1.89318279 -1.95382498 -2.01161323 -2.06604784 -2.11681662 -2.16380761
## [385] -2.20711173 -2.24701500 -2.28398070 -2.31862307 -2.35167423 -2.38394682
## [391] -2.41629466 -2.44957364 -2.48460477 -2.52214080 -2.56283726 -2.60722866
## [397] -2.65570983 -2.70852252 -2.76574699 -2.82729842 -2.89292791 -2.96222793
## [403] -3.03464204 -3.10947886 -3.18593013 -3.26309270 -3.33999399 -3.41562050
## [409] -3.48894851 -3.55897589 -3.62475368 -3.68541616 -3.74020775 -3.78850547
## [415] -3.82983593 -3.86388602 -3.89050730 -3.90971424 -3.92167710 -3.92671064
## [421] -3.92525988 -3.91788419 -3.90524095 -3.88806970 -3.86717743 -3.84342527
## [427] -3.81771669 -3.79098704 -3.76419391 -3.73830818 -3.71430502 -3.69315468
## [433] -3.67581259 -3.66320860 -3.65623504 -3.65573367 -3.66248116 -3.67717319
## [439] -3.70040697 -3.73266205 -3.77427948 -3.82543891 -3.88613399 -3.95614605
## [445] -4.03501651 -4.12201898 -4.21613249 -4.31601818 -4.42000257 -4.52607188
## [451] -4.63188249 -4.73479358 -4.83192755 -4.92026237 -4.99675731 -5.05850807
## [457] -5.10292153 -5.12789398 -5.13197194 -5.11447357 -5.07555250 -5.01619392
## [463] -4.93814378 -4.84378261 -4.73596296 -4.61783264 -4.49266391 -4.36370392
## [469] -4.23405529 -4.10658967 -3.98389227 -3.86823294 -3.76155774 -3.66549502
## [475] -3.58137092 -3.51022980 -3.45285648 -3.40979802 -3.38138354 -3.36774113
## [481] -3.36881171 -3.38435945 -3.41397923 -3.45710149 -3.51299501 -3.58076847
## [487] -3.65937183 -3.74759844 -3.84408958 -3.94734251 -4.05572382 -4.16748950
## [493] -4.28081287 -4.39382123 -4.50464116 -4.61145142 -4.71254132 -4.80637095
## [499] -4.89162899 -4.96728285 -5.03261637 -5.08725083 -5.13114671 -5.16458578
## [505] -5.18813505 -5.20259610 -5.20894463 -5.20826550 -5.20168854 -5.19032946
## [511] -5.17523920 -5.15736384 -5.13751612 -5.11635848 -5.09439727 -5.07198688
## [517] -5.04934259 -5.02656055 -5.00364338 -4.98052967 -4.95712558 -4.93333688
## [523] -4.90909949 -4.88440697 -4.85933325 -4.83404955 -4.80883444 -4.78407690
## [529] -4.76027215 -4.73801114 -4.71796438 -4.70086157 -4.68746802 -4.67855957
## [535] -4.67489686 -4.67720014 -4.68612521 -4.70224106 -4.72600934 -4.75776579
## [541] -4.79770346 -4.84585772 -4.90209273 -4.96608969 -5.03733662 -5.11512035
## [547] -5.19852114 -5.28641102 -5.37745689 -5.47013006 -5.56272378 -5.65338047
## [553] -5.74013003 -5.82093993 -5.89377698 -5.95667899 -6.00783322 -6.04565673
## [559] -6.06887237 -6.07657387 -6.06827339 -6.04392693 -6.00393461 -5.94911651
## [565] -5.88066674 -5.80009127 -5.70913577 -5.60971020 -5.50381599 -5.39348035
## [571] -5.28070052 -5.16739918 -5.05539117 -4.94636042 -4.84184567 -4.74323319
## [577] -4.65175489 -4.56849009 -4.49436982 -4.43018244 -4.37657992 -4.33408405
## [583] -4.30309241 -4.28388366 -4.27662224 -4.28136242 -4.29805175 -4.32653408
## [589] -4.36655239 -4.41775155 -4.47968133 -4.55180002 -4.63347869 -4.72400670
## [595] -4.82259842 -4.92840153 -5.04050699 -5.15796063 -5.27977639 -5.40495085
## [601] -5.53247869 -5.66136847 -5.79065811 -5.91942915 -6.04681914 -6.17203127
## [607] -6.29434058 -6.41309649 -6.52772122 -6.63770446 -6.74259449 -6.84198661
## [613] -6.93550955 -7.02281105 -7.10354358 -7.17735122 -7.24385876 -7.30266376
## [619] -7.35333230 -7.39539898 -7.42837140 -7.45173936 -7.46498859 -7.46761881
## [625] -7.45916545 -7.43922424 -7.40747741 -7.36372025 -7.30788631 -7.24006950
## [631] -7.16054155 -7.06976334 -6.96838905 -6.85726256 -6.73740635 -6.61000330
## [637] -6.47637283 -6.33794272 -6.19621849 -6.05275204 -5.90911109 -5.76685077
## [643] -5.62748842 -5.49248194 -5.36321222 -5.24096946 -5.12694315 -5.02221537
## [649] -4.92775686 -4.84442535 -4.77296565 -4.71401103 -4.66808540 -4.63560595
## [655] -4.61688596 -4.61213739 -4.62147319 -4.64490898 -4.68236408 -4.73366166
## [661] -4.79852808 -4.87659113 -4.96737742 -5.07030874 -5.18469758 -5.30974208
## [667] -5.44452057 -5.58798632 -5.73896307 -5.89614230 -6.05808329 -6.22321740
## [673] -6.38985790 -6.55621716 -6.72043244 -6.88060130 -7.03482711 -7.18127370
## [679] -7.31822727 -7.44416188 -7.55780383 -7.65818893 -7.74470694 -7.81712777
## [685] -7.87560589 -7.92066164 -7.95314073 -7.97415536 -7.98501253 -7.98713555
## [691] -7.98198473 -7.97098285 -7.95544902 -7.93654401 -7.91522817 -7.89223263
## [697] -7.86804333 -7.84289750 -7.81679176 -7.78950112 -7.76060791 -7.72953979
## [703] -7.69561550 -7.65809658 -7.61624281 -7.56936866 -7.51689717 -7.45840815
## [709] -7.39367695 -7.32270142 -7.24571499 -7.16318555 -7.07580068 -6.98444151
## [715] -6.89014784 -6.79407791 -6.69746639 -6.60158326 -6.50769618 -6.41703803
## [721] -6.33078015 -6.25001188 -6.17572579 -6.10880798 -6.05003264 -6.00005975
## [727] -5.95943513 -5.92859175 -5.90785180 -5.89742868
## $freq
## [1] 0.0006844627 0.0013689254 0.0020533881 0.0027378508 0.0034223135
## [6] 0.0041067762 0.0047912389 0.0054757016 0.0061601643 0.0068446270
## [11] 0.0075290897 0.0082135524 0.0088980151 0.0095824778 0.0102669405
## [16] 0.0109514031 0.0116358658 0.0123203285 0.0130047912 0.0136892539
## [21] 0.0143737166 0.0150581793 0.0157426420 0.0164271047 0.0171115674
## [26] 0.0177960301 0.0184804928 0.0191649555 0.0198494182 0.0205338809
## [31] 0.0212183436 0.0219028063 0.0225872690 0.0232717317 0.0239561944
## [36] 0.0246406571 0.0253251198 0.0260095825 0.0266940452 0.0273785079
## [41] 0.0280629706 0.0287474333 0.0294318960 0.0301163587 0.0308008214
## [46] 0.0314852841 0.0321697467 0.0328542094 0.0335386721 0.0342231348
## [51] 0.0349075975 0.0355920602 0.0362765229 0.0369609856 0.0376454483
## [56] 0.0383299110 0.0390143737 0.0396988364 0.0403832991 0.0410677618
## [61] 0.0417522245 0.0424366872 0.0431211499 0.0438056126 0.0444900753
## [66] 0.0451745380 0.0458590007 0.0465434634 0.0472279261 0.0479123888
## [71] 0.0485968515 0.0492813142 0.0499657769 0.0506502396 0.0513347023
## [76] 0.0520191650 0.0527036277 0.0533880903 0.0540725530 0.0547570157
## [81] 0.0554414784 0.0561259411 0.0568104038 0.0574948665 0.0581793292
## [86] 0.0588637919 0.0595482546 0.0602327173 0.0609171800 0.0616016427
## [91] 0.0622861054 0.0629705681 0.0636550308 0.0643394935 0.0650239562
## [96] 0.0657084189 0.0663928816 0.0670773443 0.0677618070 0.0684462697
## [101] 0.0691307324 0.0698151951 0.0704996578 0.0711841205 0.0718685832
## [106] 0.0725530459 0.0732375086 0.0739219713 0.0746064339 0.0752908966
## [111] 0.0759753593 0.0766598220 0.0773442847 0.0780287474 0.0787132101
## [116] 0.0793976728 0.0800821355 0.0807665982 0.0814510609 0.0821355236
## [121] 0.0828199863 0.0835044490 0.0841889117 0.0848733744 0.0855578371
## [126] 0.0862422998 0.0869267625 0.0876112252 0.0882956879 0.0889801506
## [131] 0.0896646133 0.0903490760 0.0910335387 0.0917180014 0.0924024641
## [136] 0.0930869268 0.0937713895 0.0944558522 0.0951403149 0.0958247775
## [141] 0.0965092402 0.0971937029 0.0978781656 0.0985626283 0.0992470910
## [146] 0.0999315537 0.1006160164 0.1013004791 0.1019849418 0.1026694045
## [151] 0.1033538672 0.1040383299 0.1047227926 0.1054072553 0.1060917180
## [156] 0.1067761807 0.1074606434 0.1081451061 0.1088295688 0.1095140315
## [161] 0.1101984942 0.1108829569 0.1115674196 0.1122518823 0.1129363450
## [166] 0.1136208077 0.1143052704 0.1149897331 0.1156741958 0.1163586585
## [171] 0.1170431211 0.1177275838 0.1184120465 0.1190965092 0.1197809719
## [176] 0.1204654346 0.1211498973 0.1218343600 0.1225188227 0.1232032854
## [181] 0.1238877481 0.1245722108 0.1252566735 0.1259411362 0.1266255989
## [186] 0.1273100616 0.1279945243 0.1286789870 0.1293634497 0.1300479124
## [191] 0.1307323751 0.1314168378 0.1321013005 0.1327857632 0.1334702259
## [196] 0.1341546886 0.1348391513 0.1355236140 0.1362080767 0.1368925394
## [201] 0.1375770021 0.1382614648 0.1389459274 0.1396303901 0.1403148528
## [206] 0.1409993155 0.1416837782 0.1423682409 0.1430527036 0.1437371663
## [211] 0.1444216290 0.1451060917 0.1457905544 0.1464750171 0.1471594798
## [216] 0.1478439425 0.1485284052 0.1492128679 0.1498973306 0.1505817933
## [221] 0.1512662560 0.1519507187 0.1526351814 0.1533196441 0.1540041068
## [226] 0.1546885695 0.1553730322 0.1560574949 0.1567419576 0.1574264203
## [231] 0.1581108830 0.1587953457 0.1594798084 0.1601642710 0.1608487337
## [236] 0.1615331964 0.1622176591 0.1629021218 0.1635865845 0.1642710472
## [241] 0.1649555099 0.1656399726 0.1663244353 0.1670088980 0.1676933607
## [246] 0.1683778234 0.1690622861 0.1697467488 0.1704312115 0.1711156742
## [251] 0.1718001369 0.1724845996 0.1731690623 0.1738535250 0.1745379877
## [256] 0.1752224504 0.1759069131 0.1765913758 0.1772758385 0.1779603012
## [261] 0.1786447639 0.1793292266 0.1800136893 0.1806981520 0.1813826146
## [266] 0.1820670773 0.1827515400 0.1834360027 0.1841204654 0.1848049281
## [271] 0.1854893908 0.1861738535 0.1868583162 0.1875427789 0.1882272416
## [276] 0.1889117043 0.1895961670 0.1902806297 0.1909650924 0.1916495551
## [281] 0.1923340178 0.1930184805 0.1937029432 0.1943874059 0.1950718686
## [286] 0.1957563313 0.1964407940 0.1971252567 0.1978097194 0.1984941821
## [291] 0.1991786448 0.1998631075 0.2005475702 0.2012320329 0.2019164956
## [296] 0.2026009582 0.2032854209 0.2039698836 0.2046543463 0.2053388090
## [301] 0.2060232717 0.2067077344 0.2073921971 0.2080766598 0.2087611225
## [306] 0.2094455852 0.2101300479 0.2108145106 0.2114989733 0.2121834360
## [311] 0.2128678987 0.2135523614 0.2142368241 0.2149212868 0.2156057495
## [316] 0.2162902122 0.2169746749 0.2176591376 0.2183436003 0.2190280630
## [321] 0.2197125257 0.2203969884 0.2210814511 0.2217659138 0.2224503765
## [326] 0.2231348392 0.2238193018 0.2245037645 0.2251882272 0.2258726899
## [331] 0.2265571526 0.2272416153 0.2279260780 0.2286105407 0.2292950034
## [336] 0.2299794661 0.2306639288 0.2313483915 0.2320328542 0.2327173169
## [341] 0.2334017796 0.2340862423 0.2347707050 0.2354551677 0.2361396304
## [346] 0.2368240931 0.2375085558 0.2381930185 0.2388774812 0.2395619439
## [351] 0.2402464066 0.2409308693 0.2416153320 0.2422997947 0.2429842574
## [356] 0.2436687201 0.2443531828 0.2450376454 0.2457221081 0.2464065708
## [361] 0.2470910335 0.2477754962 0.2484599589 0.2491444216 0.2498288843
## [366] 0.2505133470 0.2511978097 0.2518822724 0.2525667351 0.2532511978
## [371] 0.2539356605 0.2546201232 0.2553045859 0.2559890486 0.2566735113
## [376] 0.2573579740 0.2580424367 0.2587268994 0.2594113621 0.2600958248
## [381] 0.2607802875 0.2614647502 0.2621492129 0.2628336756 0.2635181383
## [386] 0.2642026010 0.2648870637 0.2655715264 0.2662559890 0.2669404517
## [391] 0.2676249144 0.2683093771 0.2689938398 0.2696783025 0.2703627652
## [396] 0.2710472279 0.2717316906 0.2724161533 0.2731006160 0.2737850787
## [401] 0.2744695414 0.2751540041 0.2758384668 0.2765229295 0.2772073922
## [406] 0.2778918549 0.2785763176 0.2792607803 0.2799452430 0.2806297057
## [411] 0.2813141684 0.2819986311 0.2826830938 0.2833675565 0.2840520192
## [416] 0.2847364819 0.2854209446 0.2861054073 0.2867898700 0.2874743326
## [421] 0.2881587953 0.2888432580 0.2895277207 0.2902121834 0.2908966461
## [426] 0.2915811088 0.2922655715 0.2929500342 0.2936344969 0.2943189596
## [431] 0.2950034223 0.2956878850 0.2963723477 0.2970568104 0.2977412731
## [436] 0.2984257358 0.2991101985 0.2997946612 0.3004791239 0.3011635866
## [441] 0.3018480493 0.3025325120 0.3032169747 0.3039014374 0.3045859001
## [446] 0.3052703628 0.3059548255 0.3066392882 0.3073237509 0.3080082136
## [451] 0.3086926762 0.3093771389 0.3100616016 0.3107460643 0.3114305270
## [456] 0.3121149897 0.3127994524 0.3134839151 0.3141683778 0.3148528405
## [461] 0.3155373032 0.3162217659 0.3169062286 0.3175906913 0.3182751540
## [466] 0.3189596167 0.3196440794 0.3203285421 0.3210130048 0.3216974675
## [471] 0.3223819302 0.3230663929 0.3237508556 0.3244353183 0.3251197810
## [476] 0.3258042437 0.3264887064 0.3271731691 0.3278576318 0.3285420945
## [481] 0.3292265572 0.3299110198 0.3305954825 0.3312799452 0.3319644079
## [486] 0.3326488706 0.3333333333 0.3340177960 0.3347022587 0.3353867214
## [491] 0.3360711841 0.3367556468 0.3374401095 0.3381245722 0.3388090349
## [496] 0.3394934976 0.3401779603 0.3408624230 0.3415468857 0.3422313484
## [501] 0.3429158111 0.3436002738 0.3442847365 0.3449691992 0.3456536619
## [506] 0.3463381246 0.3470225873 0.3477070500 0.3483915127 0.3490759754
## [511] 0.3497604381 0.3504449008 0.3511293634 0.3518138261 0.3524982888
## [516] 0.3531827515 0.3538672142 0.3545516769 0.3552361396 0.3559206023
## [521] 0.3566050650 0.3572895277 0.3579739904 0.3586584531 0.3593429158
## [526] 0.3600273785 0.3607118412 0.3613963039 0.3620807666 0.3627652293
## [531] 0.3634496920 0.3641341547 0.3648186174 0.3655030801 0.3661875428
## [536] 0.3668720055 0.3675564682 0.3682409309 0.3689253936 0.3696098563
## [541] 0.3702943190 0.3709787817 0.3716632444 0.3723477070 0.3730321697
## [546] 0.3737166324 0.3744010951 0.3750855578 0.3757700205 0.3764544832
## [551] 0.3771389459 0.3778234086 0.3785078713 0.3791923340 0.3798767967
## [556] 0.3805612594 0.3812457221 0.3819301848 0.3826146475 0.3832991102
## [561] 0.3839835729 0.3846680356 0.3853524983 0.3860369610 0.3867214237
## [566] 0.3874058864 0.3880903491 0.3887748118 0.3894592745 0.3901437372
## [571] 0.3908281999 0.3915126626 0.3921971253 0.3928815880 0.3935660507
## [576] 0.3942505133 0.3949349760 0.3956194387 0.3963039014 0.3969883641
## [581] 0.3976728268 0.3983572895 0.3990417522 0.3997262149 0.4004106776
## [586] 0.4010951403 0.4017796030 0.4024640657 0.4031485284 0.4038329911
## [591] 0.4045174538 0.4052019165 0.4058863792 0.4065708419 0.4072553046
## [596] 0.4079397673 0.4086242300 0.4093086927 0.4099931554 0.4106776181
## [601] 0.4113620808 0.4120465435 0.4127310062 0.4134154689 0.4140999316
## [606] 0.4147843943 0.4154688569 0.4161533196 0.4168377823 0.4175222450
## [611] 0.4182067077 0.4188911704 0.4195756331 0.4202600958 0.4209445585
## [616] 0.4216290212 0.4223134839 0.4229979466 0.4236824093 0.4243668720
## [621] 0.4250513347 0.4257357974 0.4264202601 0.4271047228 0.4277891855
## [626] 0.4284736482 0.4291581109 0.4298425736 0.4305270363 0.4312114990
## [631] 0.4318959617 0.4325804244 0.4332648871 0.4339493498 0.4346338125
## [636] 0.4353182752 0.4360027379 0.4366872005 0.4373716632 0.4380561259
## [641] 0.4387405886 0.4394250513 0.4401095140 0.4407939767 0.4414784394
## [646] 0.4421629021 0.4428473648 0.4435318275 0.4442162902 0.4449007529
## [651] 0.4455852156 0.4462696783 0.4469541410 0.4476386037 0.4483230664
## [656] 0.4490075291 0.4496919918 0.4503764545 0.4510609172 0.4517453799
## [661] 0.4524298426 0.4531143053 0.4537987680 0.4544832307 0.4551676934
## [666] 0.4558521561 0.4565366188 0.4572210815 0.4579055441 0.4585900068
## [671] 0.4592744695 0.4599589322 0.4606433949 0.4613278576 0.4620123203
## [676] 0.4626967830 0.4633812457 0.4640657084 0.4647501711 0.4654346338
## [681] 0.4661190965 0.4668035592 0.4674880219 0.4681724846 0.4688569473
## [686] 0.4695414100 0.4702258727 0.4709103354 0.4715947981 0.4722792608
## [691] 0.4729637235 0.4736481862 0.4743326489 0.4750171116 0.4757015743
## [696] 0.4763860370 0.4770704997 0.4777549624 0.4784394251 0.4791238877
## [701] 0.4798083504 0.4804928131 0.4811772758 0.4818617385 0.4825462012
## [706] 0.4832306639 0.4839151266 0.4845995893 0.4852840520 0.4859685147
## [711] 0.4866529774 0.4873374401 0.4880219028 0.4887063655 0.4893908282
## [716] 0.4900752909 0.4907597536 0.4914442163 0.4921286790 0.4928131417
## [721] 0.4934976044 0.4941820671 0.4948665298 0.4955509925 0.4962354552
## [726] 0.4969199179 0.4976043806 0.4982888433 0.4989733060 0.4996577687
##
## $pzgram
## [1] 5.1288685560 5.1172615722 5.0979072986 5.0707971643 5.0359297770
## [6] 4.9933211766 4.9430175260 4.8851098812 4.8197506117 4.7471709617
## [11] 4.6676991474 4.5817782656 4.4899831416 4.3930350617 4.2918131276
## [16] 4.1873607566 4.0808856653 3.9737515704 3.8674598980 3.7636200890
## [21] 3.6639077094 3.5700105705 3.4835644094 3.4060812734 3.3388753562
## [26] 3.2829923282 3.2391488382 3.2076885546 3.1885597522 3.1813171902
## [31] 3.1851482493 3.1989205397 3.2212459904 3.2505551432 3.2851751450
## [36] 3.3234056128 3.3635878613 3.4041645665 3.4437284681 3.4810599750
## [41] 3.5151544152 3.5452401500 3.5707889071 3.5915195592 3.6073962695
## [46] 3.6186215301 3.6256242102 3.6290423619 3.6297002681 3.6285791120
## [51] 3.6267807394 3.6254843228 3.6258963127 3.6291948628 3.6364708445
## [56] 3.6486684978 3.6665295069 3.6905446570 3.7209170661 3.7575402332
## [61] 3.7999928605 3.8475507852 3.8992146686 3.9537506295 4.0097400115
## [66] 4.0656340592 4.1198094395 4.1706211660 4.2164503837 4.2557454695
## [71] 4.2870558331 4.3090585725 4.3205786919 4.3206039257 4.3082953698
## [76] 4.2829951176 4.2442320040 4.1917263831 4.1253946547 4.0453540044
## [81] 3.9519275576 3.8456498526 3.7272722201 3.5977673092 3.4583316295
## [86] 3.3103846023 3.1555622652 2.9957035198 2.8328267228 2.6690946231
## [91] 2.5067662288 2.3481352485 2.1954562911 2.0508619209 1.9162756967
## [96] 1.7933280683 1.6832830063 1.5869830869 1.5048192647 1.4367288754
## [101] 1.3822219960 1.3404328445 1.3101901256 1.2900986270 1.2786241061
## [106] 1.2741744245 1.2751715876 1.2801113696 1.2876091406 1.2964320943
## [111] 1.3055191727 1.3139906191 1.3211493192 1.3264760422 1.3296204628
## [116] 1.3303895316 1.3287344195 1.3247369391 1.3185960569 1.3106148682
## [121] 1.3011882009 1.2907908540 1.2799663418 1.2693159130 1.2594875333
## [126] 1.2511644751 1.2450531342 1.2418697192 1.2423255216 1.2471105953
## [131] 1.2568758429 1.2722137269 1.2936380779 1.3215637354 1.3562869986
## [136] 1.3979680451 1.4466165592 1.5020817647 1.5640478773 1.6320356789
## [141] 1.7054105117 1.7833965320 1.8650966305 1.9495170475 2.0355954588
## [146] 2.1222311786 2.2083161362 2.2927654020 2.3745462370 2.4527048730
## [151] 2.5263904664 2.5948758679 2.6575749996 2.7140567109 2.7640550088
## [156] 2.8074755129 2.8443979134 2.8750741068 2.8999215963 2.9195116889
## [161] 2.9345520441 2.9458632508 2.9543493691 2.9609627697 2.9666641382
## [166] 2.9723791289 2.9789537909 2.9871114283 2.9974138946 3.0102303395
## [171] 3.0257160741 3.0438034956 3.0642059988 3.0864346412 3.1098262112
## [176] 3.1335804269 3.1568034047 3.1785543252 3.1978923547 3.2139212873
## [181] 3.2258299221 3.2329267825 3.2346683201 3.2306801629 3.2207712491
## [186] 3.2049408377 3.1833784354 3.1564566745 3.1247171689 3.0888494176
## [191] 3.0496629637 3.0080532777 2.9649622276 2.9213345036 2.8780719203
## [196] 2.8359880397 2.7957659395 2.7579220727 2.7227789686 2.6904489587
## [201] 2.6608302524 2.6336156126 2.6083127862 2.5842748660 2.5607380592
## [206] 2.5368639822 2.5117835993 2.4846402228 2.4546294930 2.4210348477
## [211] 2.3832575721 2.3408410197 2.2934889627 2.2410782638 2.1836661590
## [216] 2.1214924375 2.0549767302 1.9847110046 1.9114472585 1.8360803314
## [221] 1.7596257505 1.6831926138 1.6079517139 1.5350994055 1.4658181212
## [226] 1.4012348816 1.3423795858 1.2901452157 1.2452522726 1.2082197133
## [231] 1.1793443437 1.1586900612 1.1460875846 1.1411444723 1.1432644220
## [236] 1.1516741908 1.1654560437 1.1835834777 1.2049580587 1.2284454944
## [241] 1.2529094851 1.2772423576 1.3003919340 1.3213844692 1.3393437781
## [246] 1.3535068611 1.3632364258 1.3680307145 1.3675309835 1.3615268800
## [251] 1.3499598191 1.3329243055 1.3106669856 1.2835830583 1.2522095492
## [256] 1.2172148689 1.1793840629 1.1395992336 1.0988148062 1.0580276188
## [261] 1.0182422563 0.9804325795 0.9455009886 0.9142375072 0.8872812038
## [266] 0.8650866606 0.8478980992 0.8357333328 0.8283789722 0.8253973653
## [271] 0.8261447314 0.8297990206 0.8353953268 0.8418662842 0.8480848191
## [276] 0.8529068563 0.8552120268 0.8539409736 0.8481284222 0.8369316830
## [281] 0.8196546410 0.7957675355 0.7649229356 0.7269682963 0.6819553365
## [286] 0.6301462594 0.5720165479 0.5082537462 0.4397513306 0.3675965051
## [291] 0.2930506096 0.2175208545 0.1425223732 0.0696301753 0.0004215058
## [296] -0.0635896671 -0.1210229270 -0.1707020020 -0.2117202318 -0.2434950900
## [301] -0.2658064028 -0.2788142530 -0.2830546131 -0.2794131806 -0.2690802124
## [306] -0.2534909616 -0.2342573137 -0.2130963192 -0.1917606557 -0.1719748662
## [311] -0.1553798184 -0.1434864741 -0.1376389293 -0.1389858841 -0.1484592327
## [316] -0.1667582989 -0.1943383205 -0.2314020435 -0.2778936628 -0.3334948196
## [321] -0.3976228731 -0.4694322123 -0.5478199002 -0.6314374199 -0.7187106346
## [326] -0.8078701926 -0.8969943835 -0.9840657586 -1.0670416051 -1.1439366001
## [331] -1.2129138366 -1.2723782286 -1.3210645300 -1.3581113745 -1.3831132572
## [336] -1.3961443687 -1.3977514047 -1.3889162840 -1.3709933349 -1.3456281917
## [341] -1.3146669423 -1.2800639064 -1.2437950894 -1.2077823256 -1.1738309158
## [346] -1.1435815824 -1.1184760465 -1.0997345659 -1.0883433051 -1.0850493441
## [351] -1.0903613419 -1.1045542399 -1.1276768307 -1.1595614679 -1.1998356063
## [356] -1.2479352238 -1.3031204505 -1.3644939199 -1.4310224320 -1.5015624613
## [361] -1.5748898534 -1.6497337083 -1.7248139717 -1.7988816791 -1.8707601860
## [366] -1.9393851625 -2.0038407387 -2.0633890661 -2.1174907722 -2.1658143584
## [371] -2.2082334669 -2.2448120109 -2.2757782640 -2.3014899831 -2.3223933570
## [376] -2.3389789627 -2.3517379669 -2.3611215784 -2.3675063264 -2.3711671998
## [381] -2.3722601067 -2.3708145516 -2.3667368843 -2.3598239540 -2.3497864651
## [386] -2.3362807776 -2.3189473178 -2.2974532068 -2.2715362323 -2.2410469840
## [391] -2.2059859176 -2.1665323894 -2.1230633155 -2.0761600122 -2.0266028570
## [396] -1.9753545067 -1.9235333710 -1.8723797270 -1.8232171934 -1.7774122747
## [401] -1.7363343671 -1.7013181068 -1.6736293315 -1.6544353178 -1.6447794308
## [406] -1.6455599111 -1.6575122562 -1.6811945170 -1.7169748178 -1.7650204963
## [411] -1.8252884376 -1.8975164186 -1.9812155864 -2.0756645478 -2.1799059331
## [416] -2.2927466997 -2.4127638067 -2.5383171809 -2.6675719990 -2.7985321506
## [421] -2.9290862037 -3.0570661870 -3.1803180357 -3.2967807061 -3.4045690075
## [426] -3.5020535094 -3.5879299095 -3.6612703837 -3.7215508876 -3.7686510056
## [431] -3.8028263062 -3.8246565957 -3.8349762712 -3.8347946632 -3.8252146312
## [436] -3.8073568667 -3.7822957257 -3.7510103876 -3.7143531157 -3.6730346336
## [441] -3.6276252574 -3.5785694510 -3.5262108484 -3.4708244484 -3.4126525771
## [446] -3.3519412959 -3.2889741982 -3.2241009809 -3.1577587744 -3.0904849363
## [451] -3.0229208022 -2.9558066525 -2.8899688314 -2.8263004626 -2.7657375136
## [456] -2.7092320495 -2.6577244211 -2.6121158885 -2.5732428569 -2.5418535464
## [461] -2.5185875900 -2.5039587836 -2.4983410146 -2.5019572911 -2.5148717581
## [466] -2.5369846264 -2.5680300261 -2.6075769151 -2.6550333026 -2.7096541607
## [471] -2.7705534772 -2.8367209104 -2.9070434287 -2.9803321252 -3.0553540804
## [476] -3.1308687131 -3.2056675402 -3.2786157100 -3.3486931751 -3.4150330124
## [481] -3.4769542773 -3.5339869636 -3.5858871395 -3.6326411027 -3.6744583417
## [486] -3.7117540583 -3.7451228624 -3.7753058551 -3.8031536193 -3.8295876163
## [491] -3.8555622003 -3.8820289851 -3.9099047373 -3.9400434060 -3.9732124101
## [496] -4.0100729249 -4.0511636619 -4.0968875204 -4.1475004871 -4.2031022640
## [501] -4.2636282820 -4.3288430096 -4.3983347548 -4.4715124914 -4.5476055731
## [506] -4.6256675204 -4.7045853240 -4.7830958472 -4.8598108632 -4.9332519541
## [511] -5.0018958594 -5.0642298770 -5.1188156157 -5.1643579257 -5.1997744073
## [516] -5.2242598370 -5.2373394523 -5.2389055320 -5.2292331400 -5.2089730854
## [521] -5.1791227085 -5.1409775395 -5.0960687436 -5.0460922608 -4.9928356096
## [526] -4.9381075780 -4.8836747559 -4.8312073907 -4.7822356361 -4.7381161109
## [531] -4.7000078701 -4.6688564227 -4.6453842629 -4.6300864392 -4.6232299056
## [536] -4.6248557041 -4.6347833801 -4.6526173912 -4.6777556133 -4.7094003486
## [541] -4.7465724855 -4.7881296170 -4.8327889728 -4.8791559172 -4.9257584927
## [546] -4.9710880290 -5.0136451890 -5.0519900529 -5.0847940145 -5.1108905331
## [551] -5.1293212855 -5.1393741482 -5.1406098021 -5.1328745890 -5.1162984797
## [556] -5.0912784345 -5.0584488292 -5.0186417469 -4.9728406407 -4.9221310897
## [561] -4.8676521378 -4.8105511250 -4.7519441562 -4.6928835308 -4.6343327141
## [566] -4.5771488204 -4.5220721398 -4.4697219674 -4.4205978508 -4.3750853333
## [571] -4.3334652916 -4.2959260173 -4.2625772553 -4.2334654618 -4.2085895929
## [576] -4.1879167630 -4.1713971422 -4.1589774931 -4.1506127893 -4.1462754247
## [581] -4.1459616049 -4.1496946284 -4.1575248966 -4.1695266411 -4.1857915094
## [586] -4.2064193015 -4.2315062878 -4.2611316580 -4.2953427480 -4.3341397691
## [591] -4.3774608247 -4.4251680411 -4.4770356748 -4.5327410767 -4.5918594000
## [596] -4.6538629084 -4.7181256697 -4.7839342746 -4.8505049825 -4.9170073463
## [601] -4.9825939033 -5.0464349514 -5.1077568140 -5.1658813967 -5.2202643657
## [606] -5.2705290224 -5.3164930112 -5.3581854092 -5.3958524928 -5.4299514628
## [611] -5.4611324950 -5.4902105008 -5.5181287797 -5.5459172276 -5.5746478913
## [616] -5.6053904703 -5.6391699465 -5.6769279649 -5.7194890205 -5.7675319863
## [621] -5.8215671220 -5.8819184450 -5.9487112224 -6.0218643461 -6.1010874443
## [626] -6.1858827255 -6.2755517144 -6.3692071757 -6.4657905929 -6.5640955454
## [631] -6.6627971653 -6.7604875536 -6.8557165787 -6.9470369195 -7.0330515982
## [636] -7.1124616779 -7.1841113922 -7.2470278279 -7.3004524911 -7.3438626496
## [641] -7.3769812259 -7.3997750759 -7.4124425645 -7.4153922660 -7.4092152406
## [646] -7.3946535881 -7.3725678683 -7.3439055775 -7.3096722749 -7.2709063056
## [651] -7.2286574465 -7.1839693040 -7.1378649349 -7.0913349689 -7.0453274457
## [656] -7.0007386310 -6.9584041869 -6.9190902266 -6.8834839493 -6.8521837041
## [661] -6.8256884710 -6.8043868719 -6.7885459261 -6.7782998677 -6.7736394471
## [666] -6.7744022486 -6.7802646916 -6.7907365280 -6.8051587986 -6.8227063584
## [671] -6.8423961724 -6.8631025962 -6.8835807249 -6.9024985712 -6.9184782738
## [676] -6.9301457337 -6.9361870633 -6.9354091165 -6.9268003199 -6.9095872603
## [681] -6.8832822134 -6.8477171863 -6.8030611142 -6.7498184801 -6.6888095530
## [686] -6.6211343240 -6.5481237243 -6.4712826116 -6.3922292197 -6.3126353510
## [691] -6.2341707289 -6.1584538362 -6.0870104622 -6.0212402118 -5.9623904999
## [696] -5.9115370742 -5.8695698774 -5.8371830277 -5.8148678120 -5.8029078123
## [701] -5.8013755759 -5.8101305759 -5.8288185631 -5.8568727744 -5.8935178153
## [706] -5.9377773478 -5.9884869612 -6.0443137188 -6.1037838070 -6.1653193881
## [711] -6.2272851229 -6.2880438663 -6.3460197849 -6.3997657469 -6.4480304938
## [716] -6.4898201258 -6.5244480878 -6.5515683501 -6.5711878589 -6.5836564203
## [721] -6.5896346138 -6.5900426566 -6.5859949283 -6.5787258599 -6.5695130082
## [726] -6.5596025298 -6.5501411761 -6.5421176730 -6.5363151671 -6.53327550245.7.4 ARMA Forecast
PM2.5 Model: As per BIC
(1 - 0.36B ) (\(X_t\) - 86.54) = (1 + 0.28B) (\(a_t\))
Final PM2.5 Model: As per AIC
(1 - 1.58B + 0.79\(B^2\) - 0.22\(B^3\) + 0.06\(B^4\) - 0.03\(B^5\)) (\(X_t\) - 86.54) = (1 - 0.93B) (\(a_t\))
| Method | ASE | Window ASE | \(\sigma^2\) | AIC |
|---|---|---|---|---|
| AIC | 3547.03 | 5654.971 | 3659.862 | 8.21 |
| BIC | 3388.96 | 5764.101 | 3685.13 | 8.21 |
From the above AIC method seems appropriate for forecasting due to lowest windowed ASE.
fore_pm25 <- fore.aruma.wge(tt_pm25_ts$train_ts,phi=e_pm25_aic$phi,theta=e_pm25_aic$theta, n.ahead=20,plot=FALSE)
plotts.fore.wge(tt_pm25_ts$test_ts,fore_pm25,plot=TRUE)window_arma_pm25<-window.ase.wge(ts_daily_df$PM25,phi=e_pm25_aic$phi,theta=e_pm25_aic$theta,trainingSize = 500,horizon = 20)
mean(window_arma_pm25)## [1] 5654.9715.8 ARIMA
From Realization,ACFs and spectral analysis we determined that Time series is stationary. But if we look at full length time series we see that there is pattern repeats every year. PM2.5 leves are lowest and then it peaks up and then goes down again. So lets take into account that sesonality. i.e. let’s model assuming that time series is not a stationary.
5.8.1 Build a Model
## ---------WORKING... PLEASE WAIT...
##
##
## Five Smallest Values of aic##
## Coefficients of Original polynomial:
## 0.6744 -0.2332
##
## Factor Roots Abs Recip System Freq
## 1-0.6744B+0.2332B^2 1.4457+-1.4824i 0.4829 0.1270
##
## ## [1] 0.6743537 -0.2332345## [1] 6900.42## [1] 8.844924## [1] 8.8588325.8.2 Residual diagnositcs
Ljung box test suggests that residuals are white noise.So assumptions on residuals are met.
## Obs 0.00206265 -0.01316139 0.01587749 0.003910852 -0.02350566 -0.05181897 0.009743788 -0.03212132 0.0180016 0.009887968 -0.03121394 0.005250239 0.04500742 -0.04326562 0.02648095 0.01425458 0.002183925 -0.0219641 0.01378266 -0.01134597 0.05572137 -0.03172055 -0.01145437 0.03961221## $test
## [1] "Ljung-Box test"
##
## $K
## [1] 24
##
## $chi.square
## [1] 19.25168
##
## $df
## [1] 22
##
## $pval
## [1] 0.6297398## Obs 0.00206265 -0.01316139 0.01587749 0.003910852 -0.02350566 -0.05181897 0.009743788 -0.03212132 0.0180016 0.009887968 -0.03121394 0.005250239 0.04500742 -0.04326562 0.02648095 0.01425458 0.002183925 -0.0219641 0.01378266 -0.01134597 0.05572137 -0.03172055 -0.01145437 0.03961221 0.05927395 0.009596647 -0.02991589 0.03361678 0.06780902 -0.02902656 -0.04745418 0.05185142 -0.0542475 -0.003233708 -0.02039587 -0.01524462 -0.009991874 0.05167393 0.0007471355 -0.02078971 0.03198034 -0.006700283 0.0002952841 0.001895108 0.04857626 -0.03239038 -0.03342724 0.0004926967## $test
## [1] "Ljung-Box test"
##
## $K
## [1] 48
##
## $chi.square
## [1] 50.81318
##
## $df
## [1] 46
##
## $pval
## [1] 0.28968855.8.3 ARIMA Forecast
Final ARIMA(2,367,0) PM2.5 Model: As per AIC
(1 - 0.67B + 0.23\(B^2\)) (1- \(B^{367}\)) (\(X_t\) - 86.54) = \(a_t\)
| Method | ASE | Window ASE | \(\sigma^2\) | AIC |
|---|---|---|---|---|
| AIC | 6971.01 | 10640.08 | 6900.42 | 8.84 |
fore_pm25 <- fore.aruma.wge(tt_pm25_ts$train_ts,phi=est_d1_pm25$phi,theta=0, s=367 ,n.ahead=20,plot=FALSE)
plotts.fore.wge(tt_pm25_ts$test_ts,fore_pm25,plot=TRUE)window_arima_pm25<-window.ase.wge(ts_daily_df$PM25,phi=est_d1_pm25$phi,theta=0,trainingSize = 500,horizon = 20,s=367)
mean(window_arima_pm25)## [1] 10640.08## [1] 6971.0115.9 ARIMA Appropriateness
5.9.1 Realiaztions
- Realizations
Realization from data has few spikes that does not exist in samples using selected model. But other than that it doesn’t look too bad.
5.9.2 ACFs
ACFs in original data pretty much matches with Samples
par(mfrow=c(1,3))
acf(pm25_gen_arima_sample1$data,main="Sample1")
acf(pm25_gen_arima_sample2$data,main="Sample2")
acf(pm25_gen_arima_sample3$data,main="Sample2")
5.9.3 Spectral Estimates
They all look Same.Realization in sample2 matches with original data and others are somewhat close. Model looks good on these paramters
## $freq
## [1] 0.0006939625 0.0013879251 0.0020818876 0.0027758501 0.0034698126
## [6] 0.0041637752 0.0048577377 0.0055517002 0.0062456627 0.0069396253
## [11] 0.0076335878 0.0083275503 0.0090215128 0.0097154754 0.0104094379
## [16] 0.0111034004 0.0117973629 0.0124913255 0.0131852880 0.0138792505
## [21] 0.0145732130 0.0152671756 0.0159611381 0.0166551006 0.0173490632
## [26] 0.0180430257 0.0187369882 0.0194309507 0.0201249133 0.0208188758
## [31] 0.0215128383 0.0222068008 0.0229007634 0.0235947259 0.0242886884
## [36] 0.0249826509 0.0256766135 0.0263705760 0.0270645385 0.0277585010
## [41] 0.0284524636 0.0291464261 0.0298403886 0.0305343511 0.0312283137
## [46] 0.0319222762 0.0326162387 0.0333102012 0.0340041638 0.0346981263
## [51] 0.0353920888 0.0360860514 0.0367800139 0.0374739764 0.0381679389
## [56] 0.0388619015 0.0395558640 0.0402498265 0.0409437890 0.0416377516
## [61] 0.0423317141 0.0430256766 0.0437196391 0.0444136017 0.0451075642
## [66] 0.0458015267 0.0464954892 0.0471894518 0.0478834143 0.0485773768
## [71] 0.0492713393 0.0499653019 0.0506592644 0.0513532269 0.0520471895
## [76] 0.0527411520 0.0534351145 0.0541290770 0.0548230396 0.0555170021
## [81] 0.0562109646 0.0569049271 0.0575988897 0.0582928522 0.0589868147
## [86] 0.0596807772 0.0603747398 0.0610687023 0.0617626648 0.0624566273
## [91] 0.0631505899 0.0638445524 0.0645385149 0.0652324774 0.0659264400
## [96] 0.0666204025 0.0673143650 0.0680083276 0.0687022901 0.0693962526
## [101] 0.0700902151 0.0707841777 0.0714781402 0.0721721027 0.0728660652
## [106] 0.0735600278 0.0742539903 0.0749479528 0.0756419153 0.0763358779
## [111] 0.0770298404 0.0777238029 0.0784177654 0.0791117280 0.0798056905
## [116] 0.0804996530 0.0811936155 0.0818875781 0.0825815406 0.0832755031
## [121] 0.0839694656 0.0846634282 0.0853573907 0.0860513532 0.0867453158
## [126] 0.0874392783 0.0881332408 0.0888272033 0.0895211659 0.0902151284
## [131] 0.0909090909 0.0916030534 0.0922970160 0.0929909785 0.0936849410
## [136] 0.0943789035 0.0950728661 0.0957668286 0.0964607911 0.0971547536
## [141] 0.0978487162 0.0985426787 0.0992366412 0.0999306037 0.1006245663
## [146] 0.1013185288 0.1020124913 0.1027064539 0.1034004164 0.1040943789
## [151] 0.1047883414 0.1054823040 0.1061762665 0.1068702290 0.1075641915
## [156] 0.1082581541 0.1089521166 0.1096460791 0.1103400416 0.1110340042
## [161] 0.1117279667 0.1124219292 0.1131158917 0.1138098543 0.1145038168
## [166] 0.1151977793 0.1158917418 0.1165857044 0.1172796669 0.1179736294
## [171] 0.1186675920 0.1193615545 0.1200555170 0.1207494795 0.1214434421
## [176] 0.1221374046 0.1228313671 0.1235253296 0.1242192922 0.1249132547
## [181] 0.1256072172 0.1263011797 0.1269951423 0.1276891048 0.1283830673
## [186] 0.1290770298 0.1297709924 0.1304649549 0.1311589174 0.1318528799
## [191] 0.1325468425 0.1332408050 0.1339347675 0.1346287300 0.1353226926
## [196] 0.1360166551 0.1367106176 0.1374045802 0.1380985427 0.1387925052
## [201] 0.1394864677 0.1401804303 0.1408743928 0.1415683553 0.1422623178
## [206] 0.1429562804 0.1436502429 0.1443442054 0.1450381679 0.1457321305
## [211] 0.1464260930 0.1471200555 0.1478140180 0.1485079806 0.1492019431
## [216] 0.1498959056 0.1505898681 0.1512838307 0.1519777932 0.1526717557
## [221] 0.1533657183 0.1540596808 0.1547536433 0.1554476058 0.1561415684
## [226] 0.1568355309 0.1575294934 0.1582234559 0.1589174185 0.1596113810
## [231] 0.1603053435 0.1609993060 0.1616932686 0.1623872311 0.1630811936
## [236] 0.1637751561 0.1644691187 0.1651630812 0.1658570437 0.1665510062
## [241] 0.1672449688 0.1679389313 0.1686328938 0.1693268563 0.1700208189
## [246] 0.1707147814 0.1714087439 0.1721027065 0.1727966690 0.1734906315
## [251] 0.1741845940 0.1748785566 0.1755725191 0.1762664816 0.1769604441
## [256] 0.1776544067 0.1783483692 0.1790423317 0.1797362942 0.1804302568
## [261] 0.1811242193 0.1818181818 0.1825121443 0.1832061069 0.1839000694
## [266] 0.1845940319 0.1852879944 0.1859819570 0.1866759195 0.1873698820
## [271] 0.1880638446 0.1887578071 0.1894517696 0.1901457321 0.1908396947
## [276] 0.1915336572 0.1922276197 0.1929215822 0.1936155448 0.1943095073
## [281] 0.1950034698 0.1956974323 0.1963913949 0.1970853574 0.1977793199
## [286] 0.1984732824 0.1991672450 0.1998612075 0.2005551700 0.2012491325
## [291] 0.2019430951 0.2026370576 0.2033310201 0.2040249827 0.2047189452
## [296] 0.2054129077 0.2061068702 0.2068008328 0.2074947953 0.2081887578
## [301] 0.2088827203 0.2095766829 0.2102706454 0.2109646079 0.2116585704
## [306] 0.2123525330 0.2130464955 0.2137404580 0.2144344205 0.2151283831
## [311] 0.2158223456 0.2165163081 0.2172102706 0.2179042332 0.2185981957
## [316] 0.2192921582 0.2199861207 0.2206800833 0.2213740458 0.2220680083
## [321] 0.2227619709 0.2234559334 0.2241498959 0.2248438584 0.2255378210
## [326] 0.2262317835 0.2269257460 0.2276197085 0.2283136711 0.2290076336
## [331] 0.2297015961 0.2303955586 0.2310895212 0.2317834837 0.2324774462
## [336] 0.2331714087 0.2338653713 0.2345593338 0.2352532963 0.2359472588
## [341] 0.2366412214 0.2373351839 0.2380291464 0.2387231090 0.2394170715
## [346] 0.2401110340 0.2408049965 0.2414989591 0.2421929216 0.2428868841
## [351] 0.2435808466 0.2442748092 0.2449687717 0.2456627342 0.2463566967
## [356] 0.2470506593 0.2477446218 0.2484385843 0.2491325468 0.2498265094
## [361] 0.2505204719 0.2512144344 0.2519083969 0.2526023595 0.2532963220
## [366] 0.2539902845 0.2546842471 0.2553782096 0.2560721721 0.2567661346
## [371] 0.2574600972 0.2581540597 0.2588480222 0.2595419847 0.2602359473
## [376] 0.2609299098 0.2616238723 0.2623178348 0.2630117974 0.2637057599
## [381] 0.2643997224 0.2650936849 0.2657876475 0.2664816100 0.2671755725
## [386] 0.2678695350 0.2685634976 0.2692574601 0.2699514226 0.2706453851
## [391] 0.2713393477 0.2720333102 0.2727272727 0.2734212353 0.2741151978
## [396] 0.2748091603 0.2755031228 0.2761970854 0.2768910479 0.2775850104
## [401] 0.2782789729 0.2789729355 0.2796668980 0.2803608605 0.2810548230
## [406] 0.2817487856 0.2824427481 0.2831367106 0.2838306731 0.2845246357
## [411] 0.2852185982 0.2859125607 0.2866065232 0.2873004858 0.2879944483
## [416] 0.2886884108 0.2893823734 0.2900763359 0.2907702984 0.2914642609
## [421] 0.2921582235 0.2928521860 0.2935461485 0.2942401110 0.2949340736
## [426] 0.2956280361 0.2963219986 0.2970159611 0.2977099237 0.2984038862
## [431] 0.2990978487 0.2997918112 0.3004857738 0.3011797363 0.3018736988
## [436] 0.3025676613 0.3032616239 0.3039555864 0.3046495489 0.3053435115
## [441] 0.3060374740 0.3067314365 0.3074253990 0.3081193616 0.3088133241
## [446] 0.3095072866 0.3102012491 0.3108952117 0.3115891742 0.3122831367
## [451] 0.3129770992 0.3136710618 0.3143650243 0.3150589868 0.3157529493
## [456] 0.3164469119 0.3171408744 0.3178348369 0.3185287994 0.3192227620
## [461] 0.3199167245 0.3206106870 0.3213046495 0.3219986121 0.3226925746
## [466] 0.3233865371 0.3240804997 0.3247744622 0.3254684247 0.3261623872
## [471] 0.3268563498 0.3275503123 0.3282442748 0.3289382373 0.3296321999
## [476] 0.3303261624 0.3310201249 0.3317140874 0.3324080500 0.3331020125
## [481] 0.3337959750 0.3344899375 0.3351839001 0.3358778626 0.3365718251
## [486] 0.3372657876 0.3379597502 0.3386537127 0.3393476752 0.3400416378
## [491] 0.3407356003 0.3414295628 0.3421235253 0.3428174879 0.3435114504
## [496] 0.3442054129 0.3448993754 0.3455933380 0.3462873005 0.3469812630
## [501] 0.3476752255 0.3483691881 0.3490631506 0.3497571131 0.3504510756
## [506] 0.3511450382 0.3518390007 0.3525329632 0.3532269257 0.3539208883
## [511] 0.3546148508 0.3553088133 0.3560027759 0.3566967384 0.3573907009
## [516] 0.3580846634 0.3587786260 0.3594725885 0.3601665510 0.3608605135
## [521] 0.3615544761 0.3622484386 0.3629424011 0.3636363636 0.3643303262
## [526] 0.3650242887 0.3657182512 0.3664122137 0.3671061763 0.3678001388
## [531] 0.3684941013 0.3691880638 0.3698820264 0.3705759889 0.3712699514
## [536] 0.3719639139 0.3726578765 0.3733518390 0.3740458015 0.3747397641
## [541] 0.3754337266 0.3761276891 0.3768216516 0.3775156142 0.3782095767
## [546] 0.3789035392 0.3795975017 0.3802914643 0.3809854268 0.3816793893
## [551] 0.3823733518 0.3830673144 0.3837612769 0.3844552394 0.3851492019
## [556] 0.3858431645 0.3865371270 0.3872310895 0.3879250520 0.3886190146
## [561] 0.3893129771 0.3900069396 0.3907009022 0.3913948647 0.3920888272
## [566] 0.3927827897 0.3934767523 0.3941707148 0.3948646773 0.3955586398
## [571] 0.3962526024 0.3969465649 0.3976405274 0.3983344899 0.3990284525
## [576] 0.3997224150 0.4004163775 0.4011103400 0.4018043026 0.4024982651
## [581] 0.4031922276 0.4038861901 0.4045801527 0.4052741152 0.4059680777
## [586] 0.4066620402 0.4073560028 0.4080499653 0.4087439278 0.4094378904
## [591] 0.4101318529 0.4108258154 0.4115197779 0.4122137405 0.4129077030
## [596] 0.4136016655 0.4142956280 0.4149895906 0.4156835531 0.4163775156
## [601] 0.4170714781 0.4177654407 0.4184594032 0.4191533657 0.4198473282
## [606] 0.4205412908 0.4212352533 0.4219292158 0.4226231783 0.4233171409
## [611] 0.4240111034 0.4247050659 0.4253990285 0.4260929910 0.4267869535
## [616] 0.4274809160 0.4281748786 0.4288688411 0.4295628036 0.4302567661
## [621] 0.4309507287 0.4316446912 0.4323386537 0.4330326162 0.4337265788
## [626] 0.4344205413 0.4351145038 0.4358084663 0.4365024289 0.4371963914
## [631] 0.4378903539 0.4385843164 0.4392782790 0.4399722415 0.4406662040
## [636] 0.4413601666 0.4420541291 0.4427480916 0.4434420541 0.4441360167
## [641] 0.4448299792 0.4455239417 0.4462179042 0.4469118668 0.4476058293
## [646] 0.4482997918 0.4489937543 0.4496877169 0.4503816794 0.4510756419
## [651] 0.4517696044 0.4524635670 0.4531575295 0.4538514920 0.4545454545
## [656] 0.4552394171 0.4559333796 0.4566273421 0.4573213046 0.4580152672
## [661] 0.4587092297 0.4594031922 0.4600971548 0.4607911173 0.4614850798
## [666] 0.4621790423 0.4628730049 0.4635669674 0.4642609299 0.4649548924
## [671] 0.4656488550 0.4663428175 0.4670367800 0.4677307425 0.4684247051
## [676] 0.4691186676 0.4698126301 0.4705065926 0.4712005552 0.4718945177
## [681] 0.4725884802 0.4732824427 0.4739764053 0.4746703678 0.4753643303
## [686] 0.4760582929 0.4767522554 0.4774462179 0.4781401804 0.4788341430
## [691] 0.4795281055 0.4802220680 0.4809160305 0.4816099931 0.4823039556
## [696] 0.4829979181 0.4836918806 0.4843858432 0.4850798057 0.4857737682
## [701] 0.4864677307 0.4871616933 0.4878556558 0.4885496183 0.4892435808
## [706] 0.4899375434 0.4906315059 0.4913254684 0.4920194310 0.4927133935
## [711] 0.4934073560 0.4941013185 0.4947952811 0.4954892436 0.4961832061
## [716] 0.4968771686 0.4975711312 0.4982650937 0.4989590562 0.4996530187
##
## $pzgram
## [1] 6.8088259683 6.7890882760 6.7562451712 6.7103794865 6.6516136788
## [6] 6.5801162576 6.4961099455 6.3998814708 6.2917928416 6.1722938790
## [11] 6.0419356905 5.9013846403 5.7514362032 5.5930278938 5.4272502271
## [16] 5.2553544128 5.0787552481 4.8990274803 4.7178938420 4.5372030853
## [21] 4.3588967479 4.1849641332 4.0173861193 3.8580698516 3.7087779762
## [26] 3.5710575623 3.4461748967 3.3350625866 3.2382846531 3.1560235440
## [31] 3.0880904780 3.0339577280 2.9928088984 2.9636014290 2.9451347417
## [36] 2.9361176592 2.9352297603 2.9411728749 2.9527106136 2.9686953872
## [41] 2.9880836288 3.0099408078 3.0334383302 3.0578446143 3.0825125809
## [46] 3.1068655851 3.1303835014 3.1525903011 3.1730440784 3.1913301028
## [51] 3.2070571351 3.2198569443 3.2293867270 3.2353339514 3.2374230350
## [56] 3.2354232101 3.2291569232 3.2185081526 3.2034300889 3.1839517040
## [61] 3.1601828261 3.1323174288 3.1006349313 3.0654993963 3.0273565900
## [66] 2.9867289495 2.9442085736 2.9004484232 2.8561519790 2.8120616500
## [71] 2.7689462674 2.7275880050 2.6887690634 2.6532584156 2.6217988590
## [76] 2.5950945436 2.5737990719 2.5585042035 2.5497291488 2.5479104399
## [81] 2.5533923933 2.5664182613 2.5871222678 2.6155228405 2.6515174493
## [86] 2.6948795222 2.7452579114 2.8021793162 2.8650539341 2.9331844215
## [91] 3.0057780285 3.0819615487 3.1607985396 3.2413081246 3.3224846213
## [96] 3.4033172381 3.4828091418 3.5599953118 3.6339587350 3.7038446473
## [101] 3.7688726715 3.8283468265 3.8816634808 3.9283173897 3.9679059939
## [106] 4.0001321640 4.0248055626 4.0418427630 4.0512662195 4.0532021315
## [111] 4.0478771934 4.0356141685 4.0168261907 3.9920096661 3.9617356450
## [116] 3.9266395495 3.8874091909 3.8447710874 3.7994751996 3.7522783332
## [121] 3.7039266082 3.6551375414 3.6065824259 3.5588697871 3.5125307379
## [126] 3.4680070265 3.4256424620 3.3856782211 3.3482522913 3.3134030285
## [131] 3.2810765119 3.2511371213 3.2233805472 3.1975483128 3.1733428331
## [136] 3.1504420738 3.1285129741 3.1072229675 3.0862491333 3.0652847359
## [141] 3.0440431291 3.0222592145 2.9996888292 2.9761065881 2.9513028244
## [146] 2.9250803368 2.8972516794 2.8676377012 2.8360679719 2.8023836140
## [151] 2.7664429025 2.7281298103 2.6873654633 2.6441222431 2.5984400363
## [156] 2.5504438832 2.5003620334 2.4485431580 2.3954712309 2.3417763613
## [161] 2.2882396866 2.2357903572 2.1854927190 2.1385221183 2.0961283741
## [166] 2.0595869419 2.0301391215 2.0089242226 1.9969082068 1.9948146455
## [171] 2.0030645443 2.0217313899 2.0505165558 2.0887480731 2.1354030850
## [176] 2.1891515742 2.2484167019 2.3114457114 2.3763849853 2.4413534086
## [181] 2.5045094101 2.5641085896 2.6185503720 2.6664134266 2.7064805389
## [186] 2.7377541895 2.7594643382 2.7710698914 2.7722551455 2.7629222248
## [191] 2.7431802158 2.7133314177 2.6738548784 2.6253872236 2.5687007062
## [196] 2.5046784252 2.4342867867 2.3585455020 2.2784957090 2.1951671476
## [201] 2.1095456512 2.0225424947 1.9349672926 1.8475061301 1.7607063967
## [206] 1.6749693922 1.5905512156 1.5075718032 1.4260313437 1.3458327397
## [211] 1.2668083877 1.1887493451 1.1114349462 1.0346610985 0.9582657844
## [216] 0.8821506550 0.8062979890 0.7307826424 0.6557789223 0.5815625528
## [221] 0.5085080668 0.4370820609 0.3678327912 0.3013765936 0.2383815850
## [226] 0.1795490632 0.1255929871 0.0772179047 0.0350957105 -0.0001583212
## [231] -0.0280096776 -0.0480282236 -0.0599104355 -0.0634992389 -0.0588003455
## [236] -0.0459939731 -0.0254409473 0.0023175751 0.0365671637 0.0764339231
## [241] 0.1209045647 0.1688518438 0.2190641424 0.2702777989 0.3212107497
## [246] 0.3705961488 0.4172148401 0.4599258282 0.4976941785 0.5296160476
## [251] 0.5549407517 0.5730899195 0.5836738240 0.5865049514 0.5816087436
## [256] 0.5692312580 0.5498432406 0.5241398284 0.4930348252 0.4576482738
## [261] 0.4192859446 0.3794094487 0.3395960355 0.3014878154 0.2667311628
## [266] 0.2369083588 0.2134649708 0.1976377936 0.1903890963 0.1923531351
## [271] 0.2038002124 0.2246219920 0.2543395263 0.2921329261 0.3368892812
## [276] 0.3872637446 0.4417478755 0.4987394290 0.5566086133 0.6137571335
## [281] 0.6686677864 0.7199437224 0.7663375713 0.8067713737 0.8403486779
## [286] 0.8663603012 0.8842851848 0.8937875770 0.8947115036 0.8870731919
## [291] 0.8710518254 0.8469787522 0.8153250574 0.7766872582 0.7317707980
## [296] 0.6813710083 0.6263512889 0.5676184261 0.5060952310 0.4426910056
## [301] 0.3782707250 0.3136241969 0.2494367832 0.1862634754 0.1245081568
## [306] 0.0644097237 0.0060363746 -0.0507111536 -0.1060872683 -0.1604806397
## [311] -0.2143858671 -0.2683692561 -0.3230305365 -0.3789624431 -0.4367100057
## [316] -0.4967312293 -0.5593606300 -0.6247769047 -0.6929758858 -0.7637498971
## [321] -0.8366746732 -0.9111051018 -0.9861811383 -1.0608452325 -1.1338723982
## [326] -1.2039135528 -1.2695518688 -1.3293706175 -1.3820294273 -1.4263442387
## [331] -1.4613648347 -1.4864430388 -1.5012848158 -1.5059807505 -1.5010116123
## [336] -1.4872285705 -1.4658105628 -1.4382037671 -1.4060496441 -1.3711084456
## [341] -1.3351844908 -1.3000582130 -1.2674283186 -1.2388657505 -1.2157797343
## [346] -1.1993951548 -1.1907398677 -1.1906402636 -1.1997233662 -1.2184238889
## [351] -1.2469948976 -1.2855209777 -1.3339330397 -1.3920240849 -1.4594653940
## [356] -1.5358226823 -1.6205718040 -1.7131135879 -1.8127873697 -1.9188827677
## [361] -2.0306492462 -2.1473030487 -2.2680311714 -2.3919921995 -2.5183140551
## [366] -2.6460889927 -2.7743665260 -2.9021453457 -3.0283656671 -3.1519037759
## [371] -3.2715707720 -3.3861175748 -3.4942480821 -3.5946419055 -3.6859873028
## [376] -3.7670238020 -3.8365926419 -3.8936917135 -3.9375304191 -3.9675790527
## [381] -3.9836072141 -3.9857065269 -3.9742945140 -3.9500986425 -3.9141218999
## [386] -3.8675933610 -3.8119086825 -3.7485661211 -3.6791034990 -3.6050407210
## [391] -3.5278312180 -3.4488243449 -3.3692395035 -3.2901517310 -3.2124877544
## [396] -3.1370310361 -3.0644340920 -2.9952362799 -2.9298852830 -2.8687606036
## [401] -2.8121975139 -2.7605100653 -2.7140119390 -2.6730341169 -2.6379385799
## [406] -2.6091274766 -2.5870474516 -2.5721890611 -2.5650814135 -2.5662823416
## [411] -2.5763645270 -2.5958980486 -2.6254298177 -2.6654603050 -2.7164178711
## [416] -2.7786309048 -2.8522978889 -2.9374554706 -3.0339446586 -3.1413754453
## [421] -3.2590904951 -3.3861291090 -3.5211934985 -3.6626204971 -3.8083631538
## [426] -3.9559880510 -4.1026953736 -4.2453692583 -4.3806651422 -4.5051380371
## [431] -4.6154104542 -4.7083712553 -4.7813881811 -4.8325093731 -4.8606256345
## [436] -4.8655676855 -4.8481217674 -4.8099607032 -4.7535021114 -4.6817167070
## [441] -4.5979147288 -4.5055370979 -4.4079716437 -4.3084063272 -4.2097233519
## [446] -4.1144319471 -4.0246339502 -3.9420148582 -3.8678531485 -3.8030417175
## [451] -3.7481167092 -3.7032904491 -3.6684864546 -3.6433754752 -3.6274122050
## [456] -3.6198727253 -3.6198928971 -3.6265078783 -3.6386927151 -3.6554036064
## [461] -3.6756190149 -3.6983793667 -3.7228237245 -3.7482216097 -3.7739981428
## [466] -3.7997509188 -3.8252575017 -3.8504730819 -3.8755185850 -3.9006602549
## [471] -3.9262823472 -3.9528549897 -3.9808994525 -4.0109530274 -4.0435354881
## [476] -4.0791187452 -4.1181008949 -4.1607854399 -4.2073660850 -4.2579171890
## [481] -4.3123897041 -4.3706122395 -4.4322967333 -4.4970480919 -4.5643770475
## [486] -4.6337153855 -4.7044326091 -4.7758530538 -4.8472724529 -4.9179730099
## [491] -4.9872361724 -5.0543525296 -5.1186285694 -5.1793904117 -5.2359850442
## [496] -5.2877799769 -5.3341625521 -5.3745403360 -5.4083440406 -5.4350342374
## [501] -5.4541127380 -5.4651389340 -5.4677506777 -5.4616885214 -5.4468214272
## [506] -5.4231715285 -5.3909352767 -5.3504984094 -5.3024426442 -5.2475427826
## [511] -5.1867538745 -5.1211891040 -5.0520899371 -4.9807907030 -4.9086800962
## [516] -4.8371620670 -4.7676182877 -4.7013739111 -4.6396677850 -4.5836277439
## [521] -4.5342511208 -4.4923902611 -4.4587425737 -4.4338445307 -4.4180689886
## [526] -4.4116252487 -4.4145613599 -4.4267682821 -4.4479856577 -4.4778090591
## [531] -4.5156986903 -4.5609896134 -4.6129036289 -4.6705629727 -4.7330059843
## [536] -4.7992048568 -4.8680854908 -4.9385493536 -5.0094970842 -5.0798534070
## [541] -5.1485927323 -5.2147646488 -5.2775183771 -5.3361251754 -5.3899976834
## [546] -5.4387052731 -5.4819846359 -5.5197450756 -5.5520682543 -5.5792024454
## [551] -5.6015516339 -5.6196600473 -5.6341928870 -5.6459141283 -5.6556622898
## [556] -5.6643250319 -5.6728133475 -5.6820359853 -5.6928746025 -5.7061600030
## [561] -5.7226497032 -5.7430069741 -5.7677814574 -5.7973914502 -5.8321079872
## [566] -5.8720409384 -5.9171274709 -5.9671233903 -6.0215980699 -6.0799338734
## [571] -6.1413311455 -6.2048199432 -6.2692796554 -6.3334674477 -6.3960560279
## [576] -6.4556805138 -6.5109932237 -6.5607240726 -6.6037431000 -6.6391206854
## [581] -6.6661804732 -6.6845401248 -6.6941358412 -6.6952281036 -6.6883880240
## [586] -6.6744657567 -6.6545442153 -6.6298825725 -6.6018545342 -6.5718861857
## [591] -6.5413974662 -6.5117502446 -6.4842048073 -6.4598854913 -6.4397553430
## [596] -6.4245990952 -6.4150134231 -6.4114033284 -6.4139835476 -6.4227840252
## [601] -6.4376586901 -6.4582969774 -6.4842377252 -6.5148852221 -6.5495272775
## [606] -6.5873552247 -6.6274857493 -6.6689843601 -6.7108902036 -6.7522417712
## [611] -6.7921028903 -6.8295882411 -6.8638875296 -6.8942873990 -6.9201901848
## [616] -6.9411287329 -6.9567766914 -6.9669539362 -6.9716270816 -6.9709053129
## [621] -6.9650320330 -6.9543730115 -6.9394018449 -6.9206835719 -6.8988572547
## [626] -6.8746182398 -6.8487006786 -6.8218607395 -6.7948607887 -6.7684546859
## [631] -6.7433742290 -6.7203166979 -6.6999333925 -6.6828190332 -6.6695018798
## [636] -6.6604344420 -6.6559846772 -6.6564276146 -6.6619373993 -6.6725798156
## [641] -6.6883054321 -6.7089436061 -6.7341976911 -6.7636419132 -6.7967205002
## [646] -6.8327497596 -6.8709238909 -6.9103253479 -6.9499405292 -6.9886814173
## [651] -7.0254135001 -7.0589898660 -7.0882907800 -7.1122673609 -7.1299872642
## [656] -7.1406796465 -7.1437762675 -7.1389454954 -7.1261162983 -7.1054900308
## [661] -7.0775388800 -7.0429910676 -7.0028041128 -6.9581284576 -6.9102643968
## [666] -6.8606154777 -6.8106413642 -6.7618126792 -6.7155696751 -6.6732858623
## [671] -6.6362370517 -6.6055757128 -6.5823101533 -6.5672877807 -6.5611816214
## [676] -6.5644792950 -6.5774737751 -6.6002554682 -6.6327054047 -6.6744896456
## [681] -6.7250553493 -6.7836293067 -6.8492201179 -6.9206255038 -6.9964464881
## [686] -7.0751102528 -7.1549032948 -7.2340159915 -7.3105987697 -7.3828287452
## [691] -7.4489840774 -7.5075215568 -7.5571514636 -7.5969028678 -7.6261726255
## [696] -7.6447525364 -7.6528313510 -7.6509712055 -7.6400610465 -7.6212521115
## [701] -7.5958821077 -7.5653951963 -7.5312643336 -7.4949212276 -7.4576975229
## [706] -7.4207791732 -7.3851745454 -7.3516957546 -7.3209520584 -7.2933537979
## [711] -7.2691252500 -7.2483247710 -7.2308706655 -7.2165712565 -7.2051576406
## [716] -7.1963175707 -7.1897288525 -7.1850906004 -7.1821507165 -7.1807280679## $freq
## [1] 0.0006844627 0.0013689254 0.0020533881 0.0027378508 0.0034223135
## [6] 0.0041067762 0.0047912389 0.0054757016 0.0061601643 0.0068446270
## [11] 0.0075290897 0.0082135524 0.0088980151 0.0095824778 0.0102669405
## [16] 0.0109514031 0.0116358658 0.0123203285 0.0130047912 0.0136892539
## [21] 0.0143737166 0.0150581793 0.0157426420 0.0164271047 0.0171115674
## [26] 0.0177960301 0.0184804928 0.0191649555 0.0198494182 0.0205338809
## [31] 0.0212183436 0.0219028063 0.0225872690 0.0232717317 0.0239561944
## [36] 0.0246406571 0.0253251198 0.0260095825 0.0266940452 0.0273785079
## [41] 0.0280629706 0.0287474333 0.0294318960 0.0301163587 0.0308008214
## [46] 0.0314852841 0.0321697467 0.0328542094 0.0335386721 0.0342231348
## [51] 0.0349075975 0.0355920602 0.0362765229 0.0369609856 0.0376454483
## [56] 0.0383299110 0.0390143737 0.0396988364 0.0403832991 0.0410677618
## [61] 0.0417522245 0.0424366872 0.0431211499 0.0438056126 0.0444900753
## [66] 0.0451745380 0.0458590007 0.0465434634 0.0472279261 0.0479123888
## [71] 0.0485968515 0.0492813142 0.0499657769 0.0506502396 0.0513347023
## [76] 0.0520191650 0.0527036277 0.0533880903 0.0540725530 0.0547570157
## [81] 0.0554414784 0.0561259411 0.0568104038 0.0574948665 0.0581793292
## [86] 0.0588637919 0.0595482546 0.0602327173 0.0609171800 0.0616016427
## [91] 0.0622861054 0.0629705681 0.0636550308 0.0643394935 0.0650239562
## [96] 0.0657084189 0.0663928816 0.0670773443 0.0677618070 0.0684462697
## [101] 0.0691307324 0.0698151951 0.0704996578 0.0711841205 0.0718685832
## [106] 0.0725530459 0.0732375086 0.0739219713 0.0746064339 0.0752908966
## [111] 0.0759753593 0.0766598220 0.0773442847 0.0780287474 0.0787132101
## [116] 0.0793976728 0.0800821355 0.0807665982 0.0814510609 0.0821355236
## [121] 0.0828199863 0.0835044490 0.0841889117 0.0848733744 0.0855578371
## [126] 0.0862422998 0.0869267625 0.0876112252 0.0882956879 0.0889801506
## [131] 0.0896646133 0.0903490760 0.0910335387 0.0917180014 0.0924024641
## [136] 0.0930869268 0.0937713895 0.0944558522 0.0951403149 0.0958247775
## [141] 0.0965092402 0.0971937029 0.0978781656 0.0985626283 0.0992470910
## [146] 0.0999315537 0.1006160164 0.1013004791 0.1019849418 0.1026694045
## [151] 0.1033538672 0.1040383299 0.1047227926 0.1054072553 0.1060917180
## [156] 0.1067761807 0.1074606434 0.1081451061 0.1088295688 0.1095140315
## [161] 0.1101984942 0.1108829569 0.1115674196 0.1122518823 0.1129363450
## [166] 0.1136208077 0.1143052704 0.1149897331 0.1156741958 0.1163586585
## [171] 0.1170431211 0.1177275838 0.1184120465 0.1190965092 0.1197809719
## [176] 0.1204654346 0.1211498973 0.1218343600 0.1225188227 0.1232032854
## [181] 0.1238877481 0.1245722108 0.1252566735 0.1259411362 0.1266255989
## [186] 0.1273100616 0.1279945243 0.1286789870 0.1293634497 0.1300479124
## [191] 0.1307323751 0.1314168378 0.1321013005 0.1327857632 0.1334702259
## [196] 0.1341546886 0.1348391513 0.1355236140 0.1362080767 0.1368925394
## [201] 0.1375770021 0.1382614648 0.1389459274 0.1396303901 0.1403148528
## [206] 0.1409993155 0.1416837782 0.1423682409 0.1430527036 0.1437371663
## [211] 0.1444216290 0.1451060917 0.1457905544 0.1464750171 0.1471594798
## [216] 0.1478439425 0.1485284052 0.1492128679 0.1498973306 0.1505817933
## [221] 0.1512662560 0.1519507187 0.1526351814 0.1533196441 0.1540041068
## [226] 0.1546885695 0.1553730322 0.1560574949 0.1567419576 0.1574264203
## [231] 0.1581108830 0.1587953457 0.1594798084 0.1601642710 0.1608487337
## [236] 0.1615331964 0.1622176591 0.1629021218 0.1635865845 0.1642710472
## [241] 0.1649555099 0.1656399726 0.1663244353 0.1670088980 0.1676933607
## [246] 0.1683778234 0.1690622861 0.1697467488 0.1704312115 0.1711156742
## [251] 0.1718001369 0.1724845996 0.1731690623 0.1738535250 0.1745379877
## [256] 0.1752224504 0.1759069131 0.1765913758 0.1772758385 0.1779603012
## [261] 0.1786447639 0.1793292266 0.1800136893 0.1806981520 0.1813826146
## [266] 0.1820670773 0.1827515400 0.1834360027 0.1841204654 0.1848049281
## [271] 0.1854893908 0.1861738535 0.1868583162 0.1875427789 0.1882272416
## [276] 0.1889117043 0.1895961670 0.1902806297 0.1909650924 0.1916495551
## [281] 0.1923340178 0.1930184805 0.1937029432 0.1943874059 0.1950718686
## [286] 0.1957563313 0.1964407940 0.1971252567 0.1978097194 0.1984941821
## [291] 0.1991786448 0.1998631075 0.2005475702 0.2012320329 0.2019164956
## [296] 0.2026009582 0.2032854209 0.2039698836 0.2046543463 0.2053388090
## [301] 0.2060232717 0.2067077344 0.2073921971 0.2080766598 0.2087611225
## [306] 0.2094455852 0.2101300479 0.2108145106 0.2114989733 0.2121834360
## [311] 0.2128678987 0.2135523614 0.2142368241 0.2149212868 0.2156057495
## [316] 0.2162902122 0.2169746749 0.2176591376 0.2183436003 0.2190280630
## [321] 0.2197125257 0.2203969884 0.2210814511 0.2217659138 0.2224503765
## [326] 0.2231348392 0.2238193018 0.2245037645 0.2251882272 0.2258726899
## [331] 0.2265571526 0.2272416153 0.2279260780 0.2286105407 0.2292950034
## [336] 0.2299794661 0.2306639288 0.2313483915 0.2320328542 0.2327173169
## [341] 0.2334017796 0.2340862423 0.2347707050 0.2354551677 0.2361396304
## [346] 0.2368240931 0.2375085558 0.2381930185 0.2388774812 0.2395619439
## [351] 0.2402464066 0.2409308693 0.2416153320 0.2422997947 0.2429842574
## [356] 0.2436687201 0.2443531828 0.2450376454 0.2457221081 0.2464065708
## [361] 0.2470910335 0.2477754962 0.2484599589 0.2491444216 0.2498288843
## [366] 0.2505133470 0.2511978097 0.2518822724 0.2525667351 0.2532511978
## [371] 0.2539356605 0.2546201232 0.2553045859 0.2559890486 0.2566735113
## [376] 0.2573579740 0.2580424367 0.2587268994 0.2594113621 0.2600958248
## [381] 0.2607802875 0.2614647502 0.2621492129 0.2628336756 0.2635181383
## [386] 0.2642026010 0.2648870637 0.2655715264 0.2662559890 0.2669404517
## [391] 0.2676249144 0.2683093771 0.2689938398 0.2696783025 0.2703627652
## [396] 0.2710472279 0.2717316906 0.2724161533 0.2731006160 0.2737850787
## [401] 0.2744695414 0.2751540041 0.2758384668 0.2765229295 0.2772073922
## [406] 0.2778918549 0.2785763176 0.2792607803 0.2799452430 0.2806297057
## [411] 0.2813141684 0.2819986311 0.2826830938 0.2833675565 0.2840520192
## [416] 0.2847364819 0.2854209446 0.2861054073 0.2867898700 0.2874743326
## [421] 0.2881587953 0.2888432580 0.2895277207 0.2902121834 0.2908966461
## [426] 0.2915811088 0.2922655715 0.2929500342 0.2936344969 0.2943189596
## [431] 0.2950034223 0.2956878850 0.2963723477 0.2970568104 0.2977412731
## [436] 0.2984257358 0.2991101985 0.2997946612 0.3004791239 0.3011635866
## [441] 0.3018480493 0.3025325120 0.3032169747 0.3039014374 0.3045859001
## [446] 0.3052703628 0.3059548255 0.3066392882 0.3073237509 0.3080082136
## [451] 0.3086926762 0.3093771389 0.3100616016 0.3107460643 0.3114305270
## [456] 0.3121149897 0.3127994524 0.3134839151 0.3141683778 0.3148528405
## [461] 0.3155373032 0.3162217659 0.3169062286 0.3175906913 0.3182751540
## [466] 0.3189596167 0.3196440794 0.3203285421 0.3210130048 0.3216974675
## [471] 0.3223819302 0.3230663929 0.3237508556 0.3244353183 0.3251197810
## [476] 0.3258042437 0.3264887064 0.3271731691 0.3278576318 0.3285420945
## [481] 0.3292265572 0.3299110198 0.3305954825 0.3312799452 0.3319644079
## [486] 0.3326488706 0.3333333333 0.3340177960 0.3347022587 0.3353867214
## [491] 0.3360711841 0.3367556468 0.3374401095 0.3381245722 0.3388090349
## [496] 0.3394934976 0.3401779603 0.3408624230 0.3415468857 0.3422313484
## [501] 0.3429158111 0.3436002738 0.3442847365 0.3449691992 0.3456536619
## [506] 0.3463381246 0.3470225873 0.3477070500 0.3483915127 0.3490759754
## [511] 0.3497604381 0.3504449008 0.3511293634 0.3518138261 0.3524982888
## [516] 0.3531827515 0.3538672142 0.3545516769 0.3552361396 0.3559206023
## [521] 0.3566050650 0.3572895277 0.3579739904 0.3586584531 0.3593429158
## [526] 0.3600273785 0.3607118412 0.3613963039 0.3620807666 0.3627652293
## [531] 0.3634496920 0.3641341547 0.3648186174 0.3655030801 0.3661875428
## [536] 0.3668720055 0.3675564682 0.3682409309 0.3689253936 0.3696098563
## [541] 0.3702943190 0.3709787817 0.3716632444 0.3723477070 0.3730321697
## [546] 0.3737166324 0.3744010951 0.3750855578 0.3757700205 0.3764544832
## [551] 0.3771389459 0.3778234086 0.3785078713 0.3791923340 0.3798767967
## [556] 0.3805612594 0.3812457221 0.3819301848 0.3826146475 0.3832991102
## [561] 0.3839835729 0.3846680356 0.3853524983 0.3860369610 0.3867214237
## [566] 0.3874058864 0.3880903491 0.3887748118 0.3894592745 0.3901437372
## [571] 0.3908281999 0.3915126626 0.3921971253 0.3928815880 0.3935660507
## [576] 0.3942505133 0.3949349760 0.3956194387 0.3963039014 0.3969883641
## [581] 0.3976728268 0.3983572895 0.3990417522 0.3997262149 0.4004106776
## [586] 0.4010951403 0.4017796030 0.4024640657 0.4031485284 0.4038329911
## [591] 0.4045174538 0.4052019165 0.4058863792 0.4065708419 0.4072553046
## [596] 0.4079397673 0.4086242300 0.4093086927 0.4099931554 0.4106776181
## [601] 0.4113620808 0.4120465435 0.4127310062 0.4134154689 0.4140999316
## [606] 0.4147843943 0.4154688569 0.4161533196 0.4168377823 0.4175222450
## [611] 0.4182067077 0.4188911704 0.4195756331 0.4202600958 0.4209445585
## [616] 0.4216290212 0.4223134839 0.4229979466 0.4236824093 0.4243668720
## [621] 0.4250513347 0.4257357974 0.4264202601 0.4271047228 0.4277891855
## [626] 0.4284736482 0.4291581109 0.4298425736 0.4305270363 0.4312114990
## [631] 0.4318959617 0.4325804244 0.4332648871 0.4339493498 0.4346338125
## [636] 0.4353182752 0.4360027379 0.4366872005 0.4373716632 0.4380561259
## [641] 0.4387405886 0.4394250513 0.4401095140 0.4407939767 0.4414784394
## [646] 0.4421629021 0.4428473648 0.4435318275 0.4442162902 0.4449007529
## [651] 0.4455852156 0.4462696783 0.4469541410 0.4476386037 0.4483230664
## [656] 0.4490075291 0.4496919918 0.4503764545 0.4510609172 0.4517453799
## [661] 0.4524298426 0.4531143053 0.4537987680 0.4544832307 0.4551676934
## [666] 0.4558521561 0.4565366188 0.4572210815 0.4579055441 0.4585900068
## [671] 0.4592744695 0.4599589322 0.4606433949 0.4613278576 0.4620123203
## [676] 0.4626967830 0.4633812457 0.4640657084 0.4647501711 0.4654346338
## [681] 0.4661190965 0.4668035592 0.4674880219 0.4681724846 0.4688569473
## [686] 0.4695414100 0.4702258727 0.4709103354 0.4715947981 0.4722792608
## [691] 0.4729637235 0.4736481862 0.4743326489 0.4750171116 0.4757015743
## [696] 0.4763860370 0.4770704997 0.4777549624 0.4784394251 0.4791238877
## [701] 0.4798083504 0.4804928131 0.4811772758 0.4818617385 0.4825462012
## [706] 0.4832306639 0.4839151266 0.4845995893 0.4852840520 0.4859685147
## [711] 0.4866529774 0.4873374401 0.4880219028 0.4887063655 0.4893908282
## [716] 0.4900752909 0.4907597536 0.4914442163 0.4921286790 0.4928131417
## [721] 0.4934976044 0.4941820671 0.4948665298 0.4955509925 0.4962354552
## [726] 0.4969199179 0.4976043806 0.4982888433 0.4989733060 0.4996577687
##
## $pzgram
## [1] 2.94923708 2.94909778 2.94891116 2.94874059 2.94866527 2.94877172
## [7] 2.94914403 2.94985396 2.95095153 2.95245716 2.95435614 2.95659600
## [13] 2.95908715 2.96170708 2.96430785 2.96672644 2.96879751 2.97036747
## [19] 2.97130912 2.97153580 2.97101393 2.96977316 2.96791319 2.96560651
## [25] 2.96309671 2.96069196 2.95875368 2.95768076 2.95788992 2.95979311
## [31] 2.96377337 2.97016059 2.97920877 2.99107667 3.00581312 3.02334845
## [37] 3.04349264 3.06594049 3.09028352 3.11602767 3.14261568 3.16945255
## [43] 3.19593258 3.22146635 3.24550627 3.26756958 3.28725782 3.30427213
## [49] 3.31842405 3.32964137 3.33796915 3.34356578 3.34669435 3.34770935
## [55] 3.34703939 3.34516615 3.34260057 3.33985708 3.33742703 3.33575268
## [61] 3.33520328 3.33605445 3.33847244 3.34250423 3.34807420 3.35498765
## [67] 3.36294112 3.37153873 3.38031366 3.38875359 3.39632864 3.40252052
## [73] 3.40685163 3.40891290 3.40838950 3.40508343 3.39893251 3.39002475
## [79] 3.37860777 3.36509230 3.35004912 3.33419863 3.31839246 3.30358650
## [85] 3.29080559 3.28110019 3.27549656 3.27494270 3.28025326 3.29205716
## [91] 3.31075228 3.33647108 3.36906055 3.40807827 3.45280525 3.50227411
## [97] 3.55530999 3.61058057 3.66665109 3.72204043 3.77527486 3.82493687
## [103] 3.86970762 3.90840204 3.93999666 3.96365057 3.97872018 3.98476850
## [109] 3.98156970 3.96910933 3.94758057 3.91737639 3.87907751 3.83343569
## [115] 3.78135199 3.72384954 3.66204087 3.59708987 3.53016952 3.46241670
## [121] 3.39488634 3.32850792 3.26404720 3.20207661 3.14295705 3.08683304
## [127] 3.03364213 2.98313833 2.93492779 2.88851416 2.84335043 2.79889365
## [133] 2.75465904 2.71027022 2.66550305 2.62032046 2.57489667 2.52962905
## [139] 2.48513625 2.44224179 2.40194239 2.36536123 2.33368732 2.30810360
## [145] 2.28970769 2.27943093 2.27796235 2.28568467 2.30262861 2.32845043
## [151] 2.36243477 2.40352224 2.45035834 2.50135807 2.55477988 2.60880231
## [157] 2.66159754 2.71139797 2.75655293 2.79557482 2.82717470 2.85028858
## [163] 2.86409585 2.86803139 2.86179305 2.84534538 2.81892051 2.78301627
## [169] 2.73839129 2.68605638 2.62726077 2.56347203 2.49634782 2.42769823
## [175] 2.35943767 2.29352633 2.23190245 2.17640805 2.12871250 2.09023946
## [181] 2.06210350 2.04506248 2.03949057 2.04537462 2.06233403 2.08966137
## [187] 2.12637908 2.17130583 2.22312618 2.28045749 2.34190954 2.40613363
## [193] 2.47185989 2.53792250 2.60327399 2.66699012 2.72826736 2.78641497
## [199] 2.84084381 2.89105340 2.93661896 2.97717945 3.01242762 3.04210268
## [205] 3.06598594 3.08389959 3.09570848 3.10132481 3.10071501 3.09390864
## [211] 3.08100831 3.06220006 3.03776329 3.00807922 2.97363705 2.93503664
## [217] 2.89298673 2.84829771 2.80186815 2.75466447 2.70769364 2.66196934
## [223] 2.61847257 2.57810870 2.54166331 2.50976012 2.48282415 2.46105349
## [229] 2.44440240 2.43257733 2.42504655 2.42106260 2.41969570 2.41987531
## [235] 2.42043680 2.42016984 2.41786584 2.41236205 2.40258073 2.38756252
## [241] 2.36649374 2.33872758 2.30379976 2.26143893 2.21157250 2.15432805
## [247] 2.09003066 2.01919594 1.94251893 1.86085819 1.77521505 1.68670770
## [253] 1.59654015 1.50596647 1.41625113 1.32862694 1.24425253 1.16417187
## [259] 1.08927843 1.02028665 0.95771294 0.90186767 0.85285868 0.81060572
## [265] 0.77486423 0.74525614 0.72130470 0.70247048 0.68818559 0.67788384
## [271] 0.67102515 0.66711310 0.66570522 0.66641634 0.66891556 0.67291796
## [277] 0.67817240 0.68444672 0.69151205 0.69912742 0.70702626 0.71490562
## [283] 0.72241908 0.72917392 0.73473249 0.73861800 0.74032399 0.73932717
## [289] 0.73510268 0.72714104 0.71496594 0.69815212 0.67634259 0.64926454
## [295] 0.61674355 0.57871539 0.53523520 0.48648345 0.43276847 0.37452491
## [301] 0.31230792 0.24678263 0.17870877 0.10892026 0.03830023 -0.03224806
## [307] -0.10183335 -0.16960990 -0.23481075 -0.29677830 -0.35499003 -0.40907721
## [313] -0.45883497 -0.50422246 -0.54535299 -0.58247481 -0.61594365 -0.64618928
## [319] -0.67367839 -0.69887631 -0.72221004 -0.74403486 -0.76460634 -0.78405913
## [325] -0.80239378 -0.81947208 -0.83502147 -0.84864825 -0.85985936 -0.86809166
## [331] -0.87274754 -0.87323466 -0.86900763 -0.85960879 -0.84470494 -0.82411745
## [337] -0.79784315 -0.76606428 -0.72914699 -0.68762855 -0.64219477 -0.59364961
## [343] -0.54287981 -0.49081723 -0.43840156 -0.38654576 -0.33610594 -0.28785676
## [349] -0.24247325 -0.20051901 -0.16244074 -0.12856867 -0.09912233 -0.07422096
## [355] -0.05389774 -0.03811713 -0.02679436 -0.01981617 -0.01706190 -0.01842379
## [361] -0.02382572 -0.03323920 -0.04669605 -0.06429688 -0.08621498 -0.11269549
## [367] -0.14404972 -0.18064496 -0.22289030 -0.27121893 -0.32606781 -0.38785527
## [373] -0.45695737 -0.53368361 -0.61825265 -0.71076851 -0.81119791 -0.91934918
## [379] -1.03485361 -1.15714987 -1.28547291 -1.41884835 -1.55609428 -1.69583208
## [385] -1.83650809 -1.97642777 -2.11380305 -2.24681278 -2.37367455 -2.49272434
## [391] -2.60249866 -2.70181232 -2.78982444 -2.86608547 -2.93055992 -2.98362203
## [397] -3.02602493 -3.05884690 -3.08342086 -3.10125449 -3.11394842 -3.12311898
## [403] -3.13033043 -3.13703965 -3.14455451 -3.15400576 -3.16633125 -3.18227077
## [409] -3.20236947 -3.22698789 -3.25631684 -3.29039543 -3.32913083 -3.37231872
## [415] -3.41966321 -3.47079562 -3.52529119 -3.58268347 -3.64247576 -3.70414967
## [421] -3.76717078 -3.83099176 -3.89505343 -3.95878467 -4.02160203 -4.08291020
## [427] -4.14210427 -4.19857499 -4.25171753 -4.30094433 -4.34570187 -4.38549080
## [433] -4.41988837 -4.44857140 -4.47133790 -4.48812512 -4.49902202 -4.50427449
## [439] -4.50428208 -4.49958607 -4.49084914 -4.47882792 -4.46434022 -4.44822886
## [445] -4.43132450 -4.41440953 -4.39818494 -4.38324170 -4.37003785 -4.35888218
## [451] -4.34992515 -4.34315723 -4.33841492 -4.33539440 -4.33367231 -4.33273324
## [457] -4.33200281 -4.33088499 -4.32880171 -4.32523259 -4.31975211 -4.31206175
## [463] -4.30201456 -4.28963020 -4.27509935 -4.25877700 -4.24116533 -4.22288758
## [469] -4.20465494 -4.18722903 -4.17138234 -4.15785921 -4.14733917 -4.14040445
## [475] -4.13751271 -4.13897587 -4.14494550 -4.15540510 -4.17016939 -4.18889073
## [481] -4.21107276 -4.23609094 -4.26321994 -4.29166701 -4.32061055 -4.34924225
## [487] -4.37681076 -4.40266461 -4.42629159 -4.44735175 -4.46570175 -4.48140861
## [493] -4.49475173 -4.50621343 -4.51645862 -4.52630574 -4.53669101 -4.54862895
## [499] -4.56317132 -4.58136709 -4.60422483 -4.63267896 -4.66756031 -4.70957119
## [505] -4.75926491 -4.81702926 -4.88307360 -4.95741916 -5.03989234 -5.13012088
## [511] -5.22753324 -5.33136163 -5.44064942 -5.55426378 -5.67091474 -5.78918143
## [517] -5.90754629 -6.02443734 -6.13827794 -6.24754238 -6.35081466 -6.44684688
## [523] -6.53461260 -6.61335081 -6.68259597 -6.74219122 -6.79228300 -6.83329742
## [529] -6.86590065 -6.89094693 -6.90941863 -6.92236351 -6.93083322 -6.93582707
## [535] -6.93824362 -6.93884172 -6.93821206 -6.93675919 -6.93469381 -6.93203488
## [541] -6.92862080 -6.92412898 -6.91810283 -6.90998528 -6.89915749 -6.88498103
## [547] -6.86684175 -6.84419280 -6.81659445 -6.78374810 -6.74552225 -6.70196868
## [553] -6.65332781 -6.60002330 -6.54264642 -6.48193213 -6.41872873 -6.35396370
## [559] -6.28860812 -6.22364186 -6.16002136 -6.09865113 -6.04036003 -5.98588235
## [565] -5.93584380 -5.89075216 -5.85099195 -5.81682286 -5.78838108 -5.76568336
## [571] -5.74863308 -5.73702814 -5.73057029 -5.72887558 -5.73148590 -5.73788114
## [577] -5.74749206 -5.75971353 -5.77391788 -5.78946840 -5.80573244 -5.82209419
## [583] -5.83796664 -5.85280281 -5.86610574 -5.87743744 -5.88642631 -5.89277337
## [589] -5.89625687 -5.89673560 -5.89415068 -5.88852598 -5.87996717 -5.86865939
## [595] -5.85486367 -5.83891200 -5.82120122 -5.80218582 -5.78236953 -5.76229618
## [601] -5.74253964 -5.72369330 -5.70635913 -5.69113662 -5.67861179 -5.66934657
## [607] -5.66386862 -5.66266187 -5.66615794 -5.67472850 -5.68867870 -5.70824182
## [613] -5.73357509 -5.76475684 -5.80178487 -5.84457632 -5.89296868 -5.94672233
## [619] -6.00552430 -6.06899329 -6.13668584 -6.20810353 -6.28270093 -6.35989410
## [625] -6.43906930 -6.51959164 -6.60081321 -6.68208045 -6.76274054 -6.84214646
## [631] -6.91966102 -6.99465961 -7.06653238 -7.13468603 -7.19854605 -7.25755996
## [637] -7.31120237 -7.35898234 -7.40045354 -7.43522718 -7.46298753 -7.48350917
## [643] -7.49667496 -7.50249326 -7.50111273 -7.49283309 -7.47811039 -7.45755567
## [649] -7.43192661 -7.40211196 -7.36910955 -7.33399895 -7.29791004 -7.26198923
## [655] -7.22736473 -7.19511225 -7.16622221 -7.14156940 -7.12188539 -7.10773422
## [661] -7.09949151 -7.09732724 -7.10119239 -7.11081005 -7.12567153 -7.14503870
## [667] -7.16795342 -7.19325573 -7.21961193 -7.24555346 -7.26952722 -7.28995629
## [673] -7.30530934 -7.31417518 -7.31533740 -7.30784329 -7.29106073 -7.26471734
## [679] -7.22891780 -7.18413764 -7.13119445 -7.07119981 -7.00549743 -6.93559361
## [685] -6.86308631 -6.78959825 -6.71671801 -6.64595168 -6.57868605 -6.51616337
## [691] -6.45946653 -6.40951349 -6.36705905 -6.33270262 -6.30690029 -6.28997993
## [697] -6.28215831 -6.28355916 -6.29423146 -6.31416728 -6.34331856 -6.38161229
## [703] -6.42896352 -6.48528586 -6.55049890 -6.62453219 -6.70732541 -6.79882452
## [709] -6.89897351 -7.00770171 -7.12490653 -7.25043156 -7.38404025 -7.52538524
## [715] -7.67397382 -7.82913019 -7.98995566 -8.15528849 -8.32366592 -8.49329225
## [721] -8.66201758 -8.82733382 -8.98639493 -9.13606887 -9.27302766 -9.39387831
## [727] -9.49533217 -9.57440235 -9.62861082 -9.65617975## $freq
## [1] 0.0006844627 0.0013689254 0.0020533881 0.0027378508 0.0034223135
## [6] 0.0041067762 0.0047912389 0.0054757016 0.0061601643 0.0068446270
## [11] 0.0075290897 0.0082135524 0.0088980151 0.0095824778 0.0102669405
## [16] 0.0109514031 0.0116358658 0.0123203285 0.0130047912 0.0136892539
## [21] 0.0143737166 0.0150581793 0.0157426420 0.0164271047 0.0171115674
## [26] 0.0177960301 0.0184804928 0.0191649555 0.0198494182 0.0205338809
## [31] 0.0212183436 0.0219028063 0.0225872690 0.0232717317 0.0239561944
## [36] 0.0246406571 0.0253251198 0.0260095825 0.0266940452 0.0273785079
## [41] 0.0280629706 0.0287474333 0.0294318960 0.0301163587 0.0308008214
## [46] 0.0314852841 0.0321697467 0.0328542094 0.0335386721 0.0342231348
## [51] 0.0349075975 0.0355920602 0.0362765229 0.0369609856 0.0376454483
## [56] 0.0383299110 0.0390143737 0.0396988364 0.0403832991 0.0410677618
## [61] 0.0417522245 0.0424366872 0.0431211499 0.0438056126 0.0444900753
## [66] 0.0451745380 0.0458590007 0.0465434634 0.0472279261 0.0479123888
## [71] 0.0485968515 0.0492813142 0.0499657769 0.0506502396 0.0513347023
## [76] 0.0520191650 0.0527036277 0.0533880903 0.0540725530 0.0547570157
## [81] 0.0554414784 0.0561259411 0.0568104038 0.0574948665 0.0581793292
## [86] 0.0588637919 0.0595482546 0.0602327173 0.0609171800 0.0616016427
## [91] 0.0622861054 0.0629705681 0.0636550308 0.0643394935 0.0650239562
## [96] 0.0657084189 0.0663928816 0.0670773443 0.0677618070 0.0684462697
## [101] 0.0691307324 0.0698151951 0.0704996578 0.0711841205 0.0718685832
## [106] 0.0725530459 0.0732375086 0.0739219713 0.0746064339 0.0752908966
## [111] 0.0759753593 0.0766598220 0.0773442847 0.0780287474 0.0787132101
## [116] 0.0793976728 0.0800821355 0.0807665982 0.0814510609 0.0821355236
## [121] 0.0828199863 0.0835044490 0.0841889117 0.0848733744 0.0855578371
## [126] 0.0862422998 0.0869267625 0.0876112252 0.0882956879 0.0889801506
## [131] 0.0896646133 0.0903490760 0.0910335387 0.0917180014 0.0924024641
## [136] 0.0930869268 0.0937713895 0.0944558522 0.0951403149 0.0958247775
## [141] 0.0965092402 0.0971937029 0.0978781656 0.0985626283 0.0992470910
## [146] 0.0999315537 0.1006160164 0.1013004791 0.1019849418 0.1026694045
## [151] 0.1033538672 0.1040383299 0.1047227926 0.1054072553 0.1060917180
## [156] 0.1067761807 0.1074606434 0.1081451061 0.1088295688 0.1095140315
## [161] 0.1101984942 0.1108829569 0.1115674196 0.1122518823 0.1129363450
## [166] 0.1136208077 0.1143052704 0.1149897331 0.1156741958 0.1163586585
## [171] 0.1170431211 0.1177275838 0.1184120465 0.1190965092 0.1197809719
## [176] 0.1204654346 0.1211498973 0.1218343600 0.1225188227 0.1232032854
## [181] 0.1238877481 0.1245722108 0.1252566735 0.1259411362 0.1266255989
## [186] 0.1273100616 0.1279945243 0.1286789870 0.1293634497 0.1300479124
## [191] 0.1307323751 0.1314168378 0.1321013005 0.1327857632 0.1334702259
## [196] 0.1341546886 0.1348391513 0.1355236140 0.1362080767 0.1368925394
## [201] 0.1375770021 0.1382614648 0.1389459274 0.1396303901 0.1403148528
## [206] 0.1409993155 0.1416837782 0.1423682409 0.1430527036 0.1437371663
## [211] 0.1444216290 0.1451060917 0.1457905544 0.1464750171 0.1471594798
## [216] 0.1478439425 0.1485284052 0.1492128679 0.1498973306 0.1505817933
## [221] 0.1512662560 0.1519507187 0.1526351814 0.1533196441 0.1540041068
## [226] 0.1546885695 0.1553730322 0.1560574949 0.1567419576 0.1574264203
## [231] 0.1581108830 0.1587953457 0.1594798084 0.1601642710 0.1608487337
## [236] 0.1615331964 0.1622176591 0.1629021218 0.1635865845 0.1642710472
## [241] 0.1649555099 0.1656399726 0.1663244353 0.1670088980 0.1676933607
## [246] 0.1683778234 0.1690622861 0.1697467488 0.1704312115 0.1711156742
## [251] 0.1718001369 0.1724845996 0.1731690623 0.1738535250 0.1745379877
## [256] 0.1752224504 0.1759069131 0.1765913758 0.1772758385 0.1779603012
## [261] 0.1786447639 0.1793292266 0.1800136893 0.1806981520 0.1813826146
## [266] 0.1820670773 0.1827515400 0.1834360027 0.1841204654 0.1848049281
## [271] 0.1854893908 0.1861738535 0.1868583162 0.1875427789 0.1882272416
## [276] 0.1889117043 0.1895961670 0.1902806297 0.1909650924 0.1916495551
## [281] 0.1923340178 0.1930184805 0.1937029432 0.1943874059 0.1950718686
## [286] 0.1957563313 0.1964407940 0.1971252567 0.1978097194 0.1984941821
## [291] 0.1991786448 0.1998631075 0.2005475702 0.2012320329 0.2019164956
## [296] 0.2026009582 0.2032854209 0.2039698836 0.2046543463 0.2053388090
## [301] 0.2060232717 0.2067077344 0.2073921971 0.2080766598 0.2087611225
## [306] 0.2094455852 0.2101300479 0.2108145106 0.2114989733 0.2121834360
## [311] 0.2128678987 0.2135523614 0.2142368241 0.2149212868 0.2156057495
## [316] 0.2162902122 0.2169746749 0.2176591376 0.2183436003 0.2190280630
## [321] 0.2197125257 0.2203969884 0.2210814511 0.2217659138 0.2224503765
## [326] 0.2231348392 0.2238193018 0.2245037645 0.2251882272 0.2258726899
## [331] 0.2265571526 0.2272416153 0.2279260780 0.2286105407 0.2292950034
## [336] 0.2299794661 0.2306639288 0.2313483915 0.2320328542 0.2327173169
## [341] 0.2334017796 0.2340862423 0.2347707050 0.2354551677 0.2361396304
## [346] 0.2368240931 0.2375085558 0.2381930185 0.2388774812 0.2395619439
## [351] 0.2402464066 0.2409308693 0.2416153320 0.2422997947 0.2429842574
## [356] 0.2436687201 0.2443531828 0.2450376454 0.2457221081 0.2464065708
## [361] 0.2470910335 0.2477754962 0.2484599589 0.2491444216 0.2498288843
## [366] 0.2505133470 0.2511978097 0.2518822724 0.2525667351 0.2532511978
## [371] 0.2539356605 0.2546201232 0.2553045859 0.2559890486 0.2566735113
## [376] 0.2573579740 0.2580424367 0.2587268994 0.2594113621 0.2600958248
## [381] 0.2607802875 0.2614647502 0.2621492129 0.2628336756 0.2635181383
## [386] 0.2642026010 0.2648870637 0.2655715264 0.2662559890 0.2669404517
## [391] 0.2676249144 0.2683093771 0.2689938398 0.2696783025 0.2703627652
## [396] 0.2710472279 0.2717316906 0.2724161533 0.2731006160 0.2737850787
## [401] 0.2744695414 0.2751540041 0.2758384668 0.2765229295 0.2772073922
## [406] 0.2778918549 0.2785763176 0.2792607803 0.2799452430 0.2806297057
## [411] 0.2813141684 0.2819986311 0.2826830938 0.2833675565 0.2840520192
## [416] 0.2847364819 0.2854209446 0.2861054073 0.2867898700 0.2874743326
## [421] 0.2881587953 0.2888432580 0.2895277207 0.2902121834 0.2908966461
## [426] 0.2915811088 0.2922655715 0.2929500342 0.2936344969 0.2943189596
## [431] 0.2950034223 0.2956878850 0.2963723477 0.2970568104 0.2977412731
## [436] 0.2984257358 0.2991101985 0.2997946612 0.3004791239 0.3011635866
## [441] 0.3018480493 0.3025325120 0.3032169747 0.3039014374 0.3045859001
## [446] 0.3052703628 0.3059548255 0.3066392882 0.3073237509 0.3080082136
## [451] 0.3086926762 0.3093771389 0.3100616016 0.3107460643 0.3114305270
## [456] 0.3121149897 0.3127994524 0.3134839151 0.3141683778 0.3148528405
## [461] 0.3155373032 0.3162217659 0.3169062286 0.3175906913 0.3182751540
## [466] 0.3189596167 0.3196440794 0.3203285421 0.3210130048 0.3216974675
## [471] 0.3223819302 0.3230663929 0.3237508556 0.3244353183 0.3251197810
## [476] 0.3258042437 0.3264887064 0.3271731691 0.3278576318 0.3285420945
## [481] 0.3292265572 0.3299110198 0.3305954825 0.3312799452 0.3319644079
## [486] 0.3326488706 0.3333333333 0.3340177960 0.3347022587 0.3353867214
## [491] 0.3360711841 0.3367556468 0.3374401095 0.3381245722 0.3388090349
## [496] 0.3394934976 0.3401779603 0.3408624230 0.3415468857 0.3422313484
## [501] 0.3429158111 0.3436002738 0.3442847365 0.3449691992 0.3456536619
## [506] 0.3463381246 0.3470225873 0.3477070500 0.3483915127 0.3490759754
## [511] 0.3497604381 0.3504449008 0.3511293634 0.3518138261 0.3524982888
## [516] 0.3531827515 0.3538672142 0.3545516769 0.3552361396 0.3559206023
## [521] 0.3566050650 0.3572895277 0.3579739904 0.3586584531 0.3593429158
## [526] 0.3600273785 0.3607118412 0.3613963039 0.3620807666 0.3627652293
## [531] 0.3634496920 0.3641341547 0.3648186174 0.3655030801 0.3661875428
## [536] 0.3668720055 0.3675564682 0.3682409309 0.3689253936 0.3696098563
## [541] 0.3702943190 0.3709787817 0.3716632444 0.3723477070 0.3730321697
## [546] 0.3737166324 0.3744010951 0.3750855578 0.3757700205 0.3764544832
## [551] 0.3771389459 0.3778234086 0.3785078713 0.3791923340 0.3798767967
## [556] 0.3805612594 0.3812457221 0.3819301848 0.3826146475 0.3832991102
## [561] 0.3839835729 0.3846680356 0.3853524983 0.3860369610 0.3867214237
## [566] 0.3874058864 0.3880903491 0.3887748118 0.3894592745 0.3901437372
## [571] 0.3908281999 0.3915126626 0.3921971253 0.3928815880 0.3935660507
## [576] 0.3942505133 0.3949349760 0.3956194387 0.3963039014 0.3969883641
## [581] 0.3976728268 0.3983572895 0.3990417522 0.3997262149 0.4004106776
## [586] 0.4010951403 0.4017796030 0.4024640657 0.4031485284 0.4038329911
## [591] 0.4045174538 0.4052019165 0.4058863792 0.4065708419 0.4072553046
## [596] 0.4079397673 0.4086242300 0.4093086927 0.4099931554 0.4106776181
## [601] 0.4113620808 0.4120465435 0.4127310062 0.4134154689 0.4140999316
## [606] 0.4147843943 0.4154688569 0.4161533196 0.4168377823 0.4175222450
## [611] 0.4182067077 0.4188911704 0.4195756331 0.4202600958 0.4209445585
## [616] 0.4216290212 0.4223134839 0.4229979466 0.4236824093 0.4243668720
## [621] 0.4250513347 0.4257357974 0.4264202601 0.4271047228 0.4277891855
## [626] 0.4284736482 0.4291581109 0.4298425736 0.4305270363 0.4312114990
## [631] 0.4318959617 0.4325804244 0.4332648871 0.4339493498 0.4346338125
## [636] 0.4353182752 0.4360027379 0.4366872005 0.4373716632 0.4380561259
## [641] 0.4387405886 0.4394250513 0.4401095140 0.4407939767 0.4414784394
## [646] 0.4421629021 0.4428473648 0.4435318275 0.4442162902 0.4449007529
## [651] 0.4455852156 0.4462696783 0.4469541410 0.4476386037 0.4483230664
## [656] 0.4490075291 0.4496919918 0.4503764545 0.4510609172 0.4517453799
## [661] 0.4524298426 0.4531143053 0.4537987680 0.4544832307 0.4551676934
## [666] 0.4558521561 0.4565366188 0.4572210815 0.4579055441 0.4585900068
## [671] 0.4592744695 0.4599589322 0.4606433949 0.4613278576 0.4620123203
## [676] 0.4626967830 0.4633812457 0.4640657084 0.4647501711 0.4654346338
## [681] 0.4661190965 0.4668035592 0.4674880219 0.4681724846 0.4688569473
## [686] 0.4695414100 0.4702258727 0.4709103354 0.4715947981 0.4722792608
## [691] 0.4729637235 0.4736481862 0.4743326489 0.4750171116 0.4757015743
## [696] 0.4763860370 0.4770704997 0.4777549624 0.4784394251 0.4791238877
## [701] 0.4798083504 0.4804928131 0.4811772758 0.4818617385 0.4825462012
## [706] 0.4832306639 0.4839151266 0.4845995893 0.4852840520 0.4859685147
## [711] 0.4866529774 0.4873374401 0.4880219028 0.4887063655 0.4893908282
## [716] 0.4900752909 0.4907597536 0.4914442163 0.4921286790 0.4928131417
## [721] 0.4934976044 0.4941820671 0.4948665298 0.4955509925 0.4962354552
## [726] 0.4969199179 0.4976043806 0.4982888433 0.4989733060 0.4996577687
##
## $pzgram
## [1] 3.48266507 3.46613652 3.43893111 3.40156178 3.35474523 3.29939903
## [7] 3.23663585 3.16775318 3.09421700 3.01763768 2.93973664 2.86230275
## [13] 2.78713846 2.71599682 2.65051182 2.59212628 2.54202271 2.50106310
## [19] 2.46974377 2.44817016 2.43605450 2.43273685 2.43722730 2.44826497
## [25] 2.46438810 2.48400887 2.50548731 2.52719954 2.54759725 2.56525646
## [31] 2.57891533 2.58750141 2.59014939 2.58621079 2.57525698 2.55707667
## [37] 2.53166910 2.49923348 2.46015532 2.41499003 2.36444383 2.30935242
## [43] 2.25065756 2.18938193 2.12660305 2.06342692 2.00096255 1.94029841
## [49] 1.88248204 1.82850381 1.77928544 1.73567362 1.69843835 1.66827503
## [55] 1.64580901 1.63160052 1.62614828 1.62988973 1.64319664 1.66636514
## [61] 1.69960040 1.74299702 1.79651692 1.85996731 1.93298177 2.01500714
## [67] 2.10529864 2.20292471 2.30678211 2.41562054 2.52807535 2.64270599
## [73] 2.75803771 2.87260397 2.98498728 3.09385683 3.19800168 3.29635894
## [79] 3.38803672 3.47233204 3.54874394 3.61698223 3.67697200 3.72885406
## [85] 3.77298122 3.80990986 3.84038654 3.86532869 3.88579892 3.90297245
## [91] 3.91809737 3.93244848 3.94727565 3.96374889 3.98290314 4.00558608
## [97] 4.03241311 4.06373304 4.09960752 4.13980598 4.18381662 4.23087194
## [103] 4.27998651 4.33000350 4.37964588 4.42756889 4.47241016 4.51283532
## [109] 4.54757749 4.57546977 4.59547096 4.60668488 4.60837406 4.59996874
## [115] 4.58107223 4.55146321 4.51109584 4.46009789 4.39876729 4.32756691
## [121] 4.24711758 4.15818880 4.06168678 3.95863942 3.85017762 3.73751278
## [127] 3.62191048 3.50466059 3.38704472 3.27030218 3.15559632 3.04398306
## [133] 2.93638414 2.83356695 2.73613284 2.64451491 2.55898562 2.47967363
## [139] 2.40658824 2.33964951 2.27872117 2.22364378 2.17426536 2.13046719
## [145] 2.09218302 2.05941045 2.03221387 2.01071909 1.99510034 1.98556076
## [151] 1.98230824 1.98552849 1.99535772 2.01185713 2.03499137 2.06461265
## [157] 2.10045166 2.14211596 2.18909549 2.24077448 2.29644859 2.35534537
## [163] 2.41664663 2.47951084 2.54309428 2.60656975 2.66914217 2.73006075
## [169] 2.78862768 2.84420349 2.89620974 2.94412936 2.98750541 3.02593882
## [175] 3.05908578 3.08665517 3.10840664 3.12414961 3.13374343 3.13709894
## [181] 3.13418140 3.12501479 3.10968733 3.08835793 3.06126333 3.02872546
## [187] 2.99115843 2.94907463 2.90308920 2.85392206 2.80239668 2.74943479
## [193] 2.69604614 2.64331283 2.59236764 2.54436653 2.50045571 2.46173443
## [199] 2.42921518 2.40378360 2.38616076 2.37687075 2.37621611 2.38426340
## [205] 2.40083995 2.42554237 2.45775578 2.49668211 2.54137509 2.59077923
## [211] 2.64376998 2.69919271 2.75589866 2.81277641 2.86877832 2.92294161
## [217] 2.97440430 3.02241656 3.06634798 3.10569146 3.14006440 3.16920757
## [223] 3.19298228 3.21136587 3.22444600 3.23241347 3.23555375 3.23423704
## [229] 3.22890678 3.22006652 3.20826510 3.19408036 3.17810142 3.16091013
## [235] 3.14306204 3.12506786 3.10737607 3.09035756 3.07429336 3.05936606
## [241] 3.04565548 3.03313901 3.02169656 3.01111969 3.00112454 2.99136744
## [247] 2.98146248 2.97099978 2.95956349 2.94674864 2.93217594 2.91550402
## [253] 2.89643865 2.87473870 2.85021882 2.82274909 2.79225166 2.75869504
## [259] 2.72208646 2.68246276 2.63988071 2.59440724 2.54611029 2.49505102
## [265] 2.44127781 2.38482255 2.32569956 2.26390716 2.19943205 2.13225600
## [271] 2.06236471 1.98975821 1.91446212 1.83653921 1.75610035 1.67331425
## [277] 1.58841524 1.50170848 1.41357204 1.32445550 1.23487486 1.14540370
## [283] 1.05666073 0.96929435 0.88396471 0.80132425 0.72199784 0.64656361
## [289] 0.57553573 0.50935003 0.44835346 0.39279760 0.34283653 0.29852846
## [295] 0.25984061 0.22665621 0.19878272 0.17595988 0.15786696 0.14412832
## [301] 0.13431700 0.12795639 0.12452015 0.12343119 0.12406040 0.12572584
## [307] 0.12769352 0.12918012 0.12935842 0.12736548 0.12231379 0.11330500
## [313] 0.09944603 0.07986706 0.05374081 0.02030281 -0.02112804 -0.07112866
## [319] -0.13015131 -0.19850353 -0.27632868 -0.36358756 -0.46004127 -0.56523602
## [325] -0.67849068 -0.79888820 -0.92527228 -1.05625128 -1.19021104 -1.32533892
## [331] -1.45966051 -1.59109016 -1.71749507 -1.83677110 -1.94692639 -2.04616722
## [337] -2.13297864 -2.20619209 -2.26503273 -2.30914116 -2.33856739 -2.35373815
## [343] -2.35540231 -2.34456133 -2.32239284 -2.29017526 -2.24922007 -2.20081625
## [349] -2.14618944 -2.08647609 -2.02271157 -1.95582999 -1.88667286 -1.81600358
## [355] -1.74452486 -1.67289663 -1.60175215 -1.53171107 -1.46338826 -1.39739819
## [361] -1.33435481 -1.27486734 -1.21953287 -1.16892641 -1.12358956 -1.08401862
## [367] -1.05065304 -1.02386496 -1.00395039 -0.99112259 -0.98550783 -0.98714387
## [373] -0.99598113 -1.01188661 -1.03465040 -1.06399458 -1.09958418 -1.14103984
## [379] -1.18795145 -1.23989236 -1.29643322 -1.35715480 -1.42165885 -1.48957640
## [385] -1.56057276 -1.63434878 -1.71063818 -1.78920108 -1.86981395 -1.95225685
## [391] -2.03629884 -2.12168274 -2.20811070 -2.29523195 -2.38263430 -2.46984074
## [397] -2.55631230 -2.64145805 -2.72465249 -2.80526032 -2.88266738 -2.95631621
## [403] -3.02574373 -3.09061809 -3.15077131 -3.20622441 -3.25720238 -3.30413673
## [409] -3.34765504 -3.38855756 -3.42778239 -3.46636163 -3.50537108 -3.54587657
## [415] -3.58887966 -3.63526496 -3.68575088 -3.74084527 -3.80080687 -3.86561370
## [421] -3.93493914 -4.00813746 -4.08424027 -4.16196648 -4.23974820 -4.31577506
## [427] -4.38805901 -4.45451977 -4.51308905 -4.56182856 -4.59905361 -4.62345108
## [433] -4.63417950 -4.63093937 -4.61400480 -4.58421285 -4.54291215 -4.49187838
## [439] -4.43320741 -4.36919851 -4.30223929 -4.23470146 -4.16885315 -4.10679052
## [445] -4.05038851 -4.00126857 -3.96078076 -3.92999646 -3.90970895 -3.90043891
## [451] -3.90244292 -3.91572350 -3.94003996 -3.97491965 -4.01967005 -4.07339221
## [457] -4.13499648 -4.20322172 -4.27665928 -4.35378263 -4.43298370 -4.51261599
## [463] -4.59104403 -4.66669767 -4.73812875 -4.80406662 -4.86346859 -4.91556092
## [469] -4.95986658 -4.99621703 -5.02474658 -5.04586998 -5.06024529 -5.06872557
## [475] -5.07230373 -5.07205491 -5.06908073 -5.06445851 -5.05919810 -5.05420753
## [481] -5.05026814 -5.04801896 -5.04794967 -5.05040128 -5.05557335 -5.06353658
## [487] -5.07424962 -5.08757892 -5.10332039 -5.12122183 -5.14100485 -5.16238529
## [493] -5.18509095 -5.20887598 -5.23353092 -5.25888821 -5.28482284 -5.31124843
## [499] -5.33810916 -5.36536841 -5.39299523 -5.42094960 -5.44916812 -5.47755113
## [505] -5.50595269 -5.53417437 -5.56196373 -5.58901811 -5.61499386 -5.63952097
## [511] -5.66222240 -5.68273730 -5.70074647 -5.71599851 -5.72833470 -5.73771055
## [517] -5.74421214 -5.74806592 -5.74964076 -5.74944206 -5.74809800 -5.74633892
## [523] -5.74497104 -5.74484608 -5.74682851 -5.75176181 -5.76043537 -5.77355277
## [529] -5.79170242 -5.81533099 -5.84471991 -5.87996523 -5.92096108 -5.96738701
## [535] -6.01869998 -6.07413172 -6.13269268 -6.19318395 -6.25421869 -6.31425433
## [541] -6.37163672 -6.42465625 -6.47161487 -6.51090157 -6.54107196 -6.56092626
## [547] -6.56957899 -6.56651367 -6.55161667 -6.52518652 -6.48791781 -6.44086148
## [553] -6.38536608 -6.32300620 -6.25550487 -6.18465632 -6.11225436 -6.04002985
## [559] -5.96959939 -5.90242551 -5.83978786 -5.78276410 -5.73221879 -5.68879881
## [565] -5.65293360 -5.62483934 -5.60452599 -5.59180693 -5.58631097 -5.58749683
## [571] -5.59467055 -5.60700608 -5.62356959 -5.64334766 -5.66527949 -5.68829266
## [577] -5.71134163 -5.73344766 -5.75373823 -5.77148391 -5.78613014 -5.79732183
## [583] -5.80491885 -5.80900119 -5.80986347 -5.80799941 -5.80407750 -5.79890999
## [589] -5.79341753 -5.78859191 -5.78545917 -5.78504496 -5.78834353 -5.79629118
## [595] -5.80974465 -5.82946434 -5.85610206 -5.89019283 -5.93215018 -5.98226415
## [601] -6.04070154 -6.10750776 -6.18260983 -6.26582012 -6.35684046 -6.45526648
## [607] -6.56059190 -6.67221277 -6.78943155 -6.91146125 -7.03742962 -7.16638385
## [613] -7.29729609 -7.42907022 -7.56055062 -7.69053355 -7.81778172 -7.94104274
## [619] -8.05907178 -8.17065850 -8.27465780 -8.37002346 -8.45584309 -8.53137223
## [625] -8.59606511 -8.64959932 -8.69189202 -8.72310566 -8.74364215 -8.75412539
## [631] -8.75537325 -8.74836086 -8.73417788 -8.71398269 -8.68895632 -8.66025895
## [637] -8.62899079 -8.59615918 -8.56265257 -8.52922211 -8.49647058 -8.46484868
## [643] -8.43465810 -8.40606082 -8.37909398 -8.35368963 -8.32969838 -8.30691601
## [649] -8.28511200 -8.26405853 -8.24355865 -8.22347219 -8.20373804 -8.18439145
## [655] -8.16557550 -8.14754617 -8.13067070 -8.11541968 -8.10235349 -8.09210403
## [661] -8.08535302 -8.08280808 -8.08517783 -8.09314705 -8.10735281 -8.12836216
## [667] -8.15665184 -8.19259016 -8.23642122 -8.28825131 -8.34803764 -8.41557919
## [673] -8.49050995 -8.57229455 -8.66022677 -8.75343129 -8.85086941 -8.95134933
## [679] -9.05354180 -9.15600170 -9.25719606 -9.35553844 -9.44942927 -9.53730077
## [685] -9.61766489 -9.68916132 -9.75060277 -9.80101424 -9.83966312 -9.86607788
## [691] -9.88005382 -9.88164588 -9.87114935 -9.84907093 -9.81609276 -9.77303284
## [697] -9.72080492 -9.66038043 -9.59275472 -9.51891867 -9.43983642 -9.35642917
## [703] -9.26956455 -9.18005083 -9.08863503 -8.99600415 -8.90278833 -8.80956551
## [709] -8.71686664 -8.62518116 -8.53496219 -8.44663132 -8.36058268 -8.27718641
## [715] -8.19679127 -8.11972668 -8.04630401 -7.97681738 -7.91154390 -7.85074358
## [721] -7.79465893 -7.74351428 -7.69751503 -7.65684673 -7.62167422 -7.59214068
## [727] -7.56836685 -7.55045028 -7.53846468 -7.53245948
## $freq
## [1] 0.0006844627 0.0013689254 0.0020533881 0.0027378508 0.0034223135
## [6] 0.0041067762 0.0047912389 0.0054757016 0.0061601643 0.0068446270
## [11] 0.0075290897 0.0082135524 0.0088980151 0.0095824778 0.0102669405
## [16] 0.0109514031 0.0116358658 0.0123203285 0.0130047912 0.0136892539
## [21] 0.0143737166 0.0150581793 0.0157426420 0.0164271047 0.0171115674
## [26] 0.0177960301 0.0184804928 0.0191649555 0.0198494182 0.0205338809
## [31] 0.0212183436 0.0219028063 0.0225872690 0.0232717317 0.0239561944
## [36] 0.0246406571 0.0253251198 0.0260095825 0.0266940452 0.0273785079
## [41] 0.0280629706 0.0287474333 0.0294318960 0.0301163587 0.0308008214
## [46] 0.0314852841 0.0321697467 0.0328542094 0.0335386721 0.0342231348
## [51] 0.0349075975 0.0355920602 0.0362765229 0.0369609856 0.0376454483
## [56] 0.0383299110 0.0390143737 0.0396988364 0.0403832991 0.0410677618
## [61] 0.0417522245 0.0424366872 0.0431211499 0.0438056126 0.0444900753
## [66] 0.0451745380 0.0458590007 0.0465434634 0.0472279261 0.0479123888
## [71] 0.0485968515 0.0492813142 0.0499657769 0.0506502396 0.0513347023
## [76] 0.0520191650 0.0527036277 0.0533880903 0.0540725530 0.0547570157
## [81] 0.0554414784 0.0561259411 0.0568104038 0.0574948665 0.0581793292
## [86] 0.0588637919 0.0595482546 0.0602327173 0.0609171800 0.0616016427
## [91] 0.0622861054 0.0629705681 0.0636550308 0.0643394935 0.0650239562
## [96] 0.0657084189 0.0663928816 0.0670773443 0.0677618070 0.0684462697
## [101] 0.0691307324 0.0698151951 0.0704996578 0.0711841205 0.0718685832
## [106] 0.0725530459 0.0732375086 0.0739219713 0.0746064339 0.0752908966
## [111] 0.0759753593 0.0766598220 0.0773442847 0.0780287474 0.0787132101
## [116] 0.0793976728 0.0800821355 0.0807665982 0.0814510609 0.0821355236
## [121] 0.0828199863 0.0835044490 0.0841889117 0.0848733744 0.0855578371
## [126] 0.0862422998 0.0869267625 0.0876112252 0.0882956879 0.0889801506
## [131] 0.0896646133 0.0903490760 0.0910335387 0.0917180014 0.0924024641
## [136] 0.0930869268 0.0937713895 0.0944558522 0.0951403149 0.0958247775
## [141] 0.0965092402 0.0971937029 0.0978781656 0.0985626283 0.0992470910
## [146] 0.0999315537 0.1006160164 0.1013004791 0.1019849418 0.1026694045
## [151] 0.1033538672 0.1040383299 0.1047227926 0.1054072553 0.1060917180
## [156] 0.1067761807 0.1074606434 0.1081451061 0.1088295688 0.1095140315
## [161] 0.1101984942 0.1108829569 0.1115674196 0.1122518823 0.1129363450
## [166] 0.1136208077 0.1143052704 0.1149897331 0.1156741958 0.1163586585
## [171] 0.1170431211 0.1177275838 0.1184120465 0.1190965092 0.1197809719
## [176] 0.1204654346 0.1211498973 0.1218343600 0.1225188227 0.1232032854
## [181] 0.1238877481 0.1245722108 0.1252566735 0.1259411362 0.1266255989
## [186] 0.1273100616 0.1279945243 0.1286789870 0.1293634497 0.1300479124
## [191] 0.1307323751 0.1314168378 0.1321013005 0.1327857632 0.1334702259
## [196] 0.1341546886 0.1348391513 0.1355236140 0.1362080767 0.1368925394
## [201] 0.1375770021 0.1382614648 0.1389459274 0.1396303901 0.1403148528
## [206] 0.1409993155 0.1416837782 0.1423682409 0.1430527036 0.1437371663
## [211] 0.1444216290 0.1451060917 0.1457905544 0.1464750171 0.1471594798
## [216] 0.1478439425 0.1485284052 0.1492128679 0.1498973306 0.1505817933
## [221] 0.1512662560 0.1519507187 0.1526351814 0.1533196441 0.1540041068
## [226] 0.1546885695 0.1553730322 0.1560574949 0.1567419576 0.1574264203
## [231] 0.1581108830 0.1587953457 0.1594798084 0.1601642710 0.1608487337
## [236] 0.1615331964 0.1622176591 0.1629021218 0.1635865845 0.1642710472
## [241] 0.1649555099 0.1656399726 0.1663244353 0.1670088980 0.1676933607
## [246] 0.1683778234 0.1690622861 0.1697467488 0.1704312115 0.1711156742
## [251] 0.1718001369 0.1724845996 0.1731690623 0.1738535250 0.1745379877
## [256] 0.1752224504 0.1759069131 0.1765913758 0.1772758385 0.1779603012
## [261] 0.1786447639 0.1793292266 0.1800136893 0.1806981520 0.1813826146
## [266] 0.1820670773 0.1827515400 0.1834360027 0.1841204654 0.1848049281
## [271] 0.1854893908 0.1861738535 0.1868583162 0.1875427789 0.1882272416
## [276] 0.1889117043 0.1895961670 0.1902806297 0.1909650924 0.1916495551
## [281] 0.1923340178 0.1930184805 0.1937029432 0.1943874059 0.1950718686
## [286] 0.1957563313 0.1964407940 0.1971252567 0.1978097194 0.1984941821
## [291] 0.1991786448 0.1998631075 0.2005475702 0.2012320329 0.2019164956
## [296] 0.2026009582 0.2032854209 0.2039698836 0.2046543463 0.2053388090
## [301] 0.2060232717 0.2067077344 0.2073921971 0.2080766598 0.2087611225
## [306] 0.2094455852 0.2101300479 0.2108145106 0.2114989733 0.2121834360
## [311] 0.2128678987 0.2135523614 0.2142368241 0.2149212868 0.2156057495
## [316] 0.2162902122 0.2169746749 0.2176591376 0.2183436003 0.2190280630
## [321] 0.2197125257 0.2203969884 0.2210814511 0.2217659138 0.2224503765
## [326] 0.2231348392 0.2238193018 0.2245037645 0.2251882272 0.2258726899
## [331] 0.2265571526 0.2272416153 0.2279260780 0.2286105407 0.2292950034
## [336] 0.2299794661 0.2306639288 0.2313483915 0.2320328542 0.2327173169
## [341] 0.2334017796 0.2340862423 0.2347707050 0.2354551677 0.2361396304
## [346] 0.2368240931 0.2375085558 0.2381930185 0.2388774812 0.2395619439
## [351] 0.2402464066 0.2409308693 0.2416153320 0.2422997947 0.2429842574
## [356] 0.2436687201 0.2443531828 0.2450376454 0.2457221081 0.2464065708
## [361] 0.2470910335 0.2477754962 0.2484599589 0.2491444216 0.2498288843
## [366] 0.2505133470 0.2511978097 0.2518822724 0.2525667351 0.2532511978
## [371] 0.2539356605 0.2546201232 0.2553045859 0.2559890486 0.2566735113
## [376] 0.2573579740 0.2580424367 0.2587268994 0.2594113621 0.2600958248
## [381] 0.2607802875 0.2614647502 0.2621492129 0.2628336756 0.2635181383
## [386] 0.2642026010 0.2648870637 0.2655715264 0.2662559890 0.2669404517
## [391] 0.2676249144 0.2683093771 0.2689938398 0.2696783025 0.2703627652
## [396] 0.2710472279 0.2717316906 0.2724161533 0.2731006160 0.2737850787
## [401] 0.2744695414 0.2751540041 0.2758384668 0.2765229295 0.2772073922
## [406] 0.2778918549 0.2785763176 0.2792607803 0.2799452430 0.2806297057
## [411] 0.2813141684 0.2819986311 0.2826830938 0.2833675565 0.2840520192
## [416] 0.2847364819 0.2854209446 0.2861054073 0.2867898700 0.2874743326
## [421] 0.2881587953 0.2888432580 0.2895277207 0.2902121834 0.2908966461
## [426] 0.2915811088 0.2922655715 0.2929500342 0.2936344969 0.2943189596
## [431] 0.2950034223 0.2956878850 0.2963723477 0.2970568104 0.2977412731
## [436] 0.2984257358 0.2991101985 0.2997946612 0.3004791239 0.3011635866
## [441] 0.3018480493 0.3025325120 0.3032169747 0.3039014374 0.3045859001
## [446] 0.3052703628 0.3059548255 0.3066392882 0.3073237509 0.3080082136
## [451] 0.3086926762 0.3093771389 0.3100616016 0.3107460643 0.3114305270
## [456] 0.3121149897 0.3127994524 0.3134839151 0.3141683778 0.3148528405
## [461] 0.3155373032 0.3162217659 0.3169062286 0.3175906913 0.3182751540
## [466] 0.3189596167 0.3196440794 0.3203285421 0.3210130048 0.3216974675
## [471] 0.3223819302 0.3230663929 0.3237508556 0.3244353183 0.3251197810
## [476] 0.3258042437 0.3264887064 0.3271731691 0.3278576318 0.3285420945
## [481] 0.3292265572 0.3299110198 0.3305954825 0.3312799452 0.3319644079
## [486] 0.3326488706 0.3333333333 0.3340177960 0.3347022587 0.3353867214
## [491] 0.3360711841 0.3367556468 0.3374401095 0.3381245722 0.3388090349
## [496] 0.3394934976 0.3401779603 0.3408624230 0.3415468857 0.3422313484
## [501] 0.3429158111 0.3436002738 0.3442847365 0.3449691992 0.3456536619
## [506] 0.3463381246 0.3470225873 0.3477070500 0.3483915127 0.3490759754
## [511] 0.3497604381 0.3504449008 0.3511293634 0.3518138261 0.3524982888
## [516] 0.3531827515 0.3538672142 0.3545516769 0.3552361396 0.3559206023
## [521] 0.3566050650 0.3572895277 0.3579739904 0.3586584531 0.3593429158
## [526] 0.3600273785 0.3607118412 0.3613963039 0.3620807666 0.3627652293
## [531] 0.3634496920 0.3641341547 0.3648186174 0.3655030801 0.3661875428
## [536] 0.3668720055 0.3675564682 0.3682409309 0.3689253936 0.3696098563
## [541] 0.3702943190 0.3709787817 0.3716632444 0.3723477070 0.3730321697
## [546] 0.3737166324 0.3744010951 0.3750855578 0.3757700205 0.3764544832
## [551] 0.3771389459 0.3778234086 0.3785078713 0.3791923340 0.3798767967
## [556] 0.3805612594 0.3812457221 0.3819301848 0.3826146475 0.3832991102
## [561] 0.3839835729 0.3846680356 0.3853524983 0.3860369610 0.3867214237
## [566] 0.3874058864 0.3880903491 0.3887748118 0.3894592745 0.3901437372
## [571] 0.3908281999 0.3915126626 0.3921971253 0.3928815880 0.3935660507
## [576] 0.3942505133 0.3949349760 0.3956194387 0.3963039014 0.3969883641
## [581] 0.3976728268 0.3983572895 0.3990417522 0.3997262149 0.4004106776
## [586] 0.4010951403 0.4017796030 0.4024640657 0.4031485284 0.4038329911
## [591] 0.4045174538 0.4052019165 0.4058863792 0.4065708419 0.4072553046
## [596] 0.4079397673 0.4086242300 0.4093086927 0.4099931554 0.4106776181
## [601] 0.4113620808 0.4120465435 0.4127310062 0.4134154689 0.4140999316
## [606] 0.4147843943 0.4154688569 0.4161533196 0.4168377823 0.4175222450
## [611] 0.4182067077 0.4188911704 0.4195756331 0.4202600958 0.4209445585
## [616] 0.4216290212 0.4223134839 0.4229979466 0.4236824093 0.4243668720
## [621] 0.4250513347 0.4257357974 0.4264202601 0.4271047228 0.4277891855
## [626] 0.4284736482 0.4291581109 0.4298425736 0.4305270363 0.4312114990
## [631] 0.4318959617 0.4325804244 0.4332648871 0.4339493498 0.4346338125
## [636] 0.4353182752 0.4360027379 0.4366872005 0.4373716632 0.4380561259
## [641] 0.4387405886 0.4394250513 0.4401095140 0.4407939767 0.4414784394
## [646] 0.4421629021 0.4428473648 0.4435318275 0.4442162902 0.4449007529
## [651] 0.4455852156 0.4462696783 0.4469541410 0.4476386037 0.4483230664
## [656] 0.4490075291 0.4496919918 0.4503764545 0.4510609172 0.4517453799
## [661] 0.4524298426 0.4531143053 0.4537987680 0.4544832307 0.4551676934
## [666] 0.4558521561 0.4565366188 0.4572210815 0.4579055441 0.4585900068
## [671] 0.4592744695 0.4599589322 0.4606433949 0.4613278576 0.4620123203
## [676] 0.4626967830 0.4633812457 0.4640657084 0.4647501711 0.4654346338
## [681] 0.4661190965 0.4668035592 0.4674880219 0.4681724846 0.4688569473
## [686] 0.4695414100 0.4702258727 0.4709103354 0.4715947981 0.4722792608
## [691] 0.4729637235 0.4736481862 0.4743326489 0.4750171116 0.4757015743
## [696] 0.4763860370 0.4770704997 0.4777549624 0.4784394251 0.4791238877
## [701] 0.4798083504 0.4804928131 0.4811772758 0.4818617385 0.4825462012
## [706] 0.4832306639 0.4839151266 0.4845995893 0.4852840520 0.4859685147
## [711] 0.4866529774 0.4873374401 0.4880219028 0.4887063655 0.4893908282
## [716] 0.4900752909 0.4907597536 0.4914442163 0.4921286790 0.4928131417
## [721] 0.4934976044 0.4941820671 0.4948665298 0.4955509925 0.4962354552
## [726] 0.4969199179 0.4976043806 0.4982888433 0.4989733060 0.4996577687
##
## $pzgram
## [1] 3.223319365 3.233308965 3.249506427 3.271262149 3.297720212
## [6] 3.327865684 3.360577661 3.394683892 3.429013280 3.462443330
## [11] 3.493940505 3.522592391 3.547631296 3.568449512 3.584606842
## [16] 3.595831195 3.602013138 3.603195298 3.599557459 3.591398172
## [21] 3.579113701 3.563175106 3.544104370 3.522450466 3.498766391
## [26] 3.473588124 3.447416515 3.420702910 3.393839196 3.367152616
## [31] 3.340905419 3.315299047 3.290482207 3.266561922 3.243616410
## [36] 3.221708557 3.200898742 3.181255847 3.162865514 3.145834939
## [41] 3.130293806 3.116391318 3.104289582 3.094153954 3.086141206
## [46] 3.080386559 3.076990797 3.076008606 3.077439262 3.081220517
## [51] 3.087226301 3.095268439 3.105102281 3.116435732 3.128940926
## [56] 3.142267549 3.156056746 3.169954512 3.183623592 3.196753079
## [61] 3.209065107 3.220318311 3.230307988 3.238863145 3.245840870
## [66] 3.251118670 3.254585562 3.256132853 3.255645537 3.252995285
## [71] 3.248035855 3.240601658 3.230509983 3.217567167 3.201578744
## [76] 3.182363333 3.159769830 3.133697202 3.104116033 3.071090791
## [81] 3.034801645 2.995564502 2.953847832 2.910284666 2.865678109
## [86] 2.820998615 2.777371405 2.736052665 2.698393735 2.665793345
## [91] 2.639639169 2.621241377 2.611762334 2.612147854 2.623066164
## [96] 2.644860611 2.677521162 2.720677844 2.773616811 2.835317147
## [101] 2.904504316 2.979714721 3.059365301 3.141822495 3.225465939
## [106] 3.308743656 3.390216988 3.468594783 3.542757347 3.611771307
## [111] 3.674896855 3.731588879 3.781493415 3.824440587 3.860434948
## [116] 3.889643867 3.912384307 3.929108171 3.940386165 3.946890084
## [121] 3.949373323 3.948649504 3.945569151 3.940994575 3.935773316
## [126] 3.930710769 3.926542905 3.923910223 3.923334247 3.925197955
## [131] 3.929731423 3.937003763 3.946922053 3.959237522 3.973558738
## [136] 3.989371079 4.006061384 4.022946399 4.039303511 4.054402273
## [141] 4.067535334 4.078047584 4.085362559 4.089005328 4.088621314
## [146] 4.083990599 4.075037329 4.061833916 4.044599686 4.023693688
## [151] 3.999601399 3.972915199 3.944308720 3.914505475 3.884242641
## [156] 3.854231364 3.825115496 3.797431108 3.771569467 3.747746156
## [161] 3.725978806 3.706075296 3.687633488 3.670052559 3.652555067
## [166] 3.634218021 3.614010727 3.590836890 3.563578561 3.531139820
## [171] 3.492488595 3.446695559 3.392969570 3.330689546 3.259432966
## [176] 3.179001277 3.089442468 2.991070819 2.884483480 2.770573010
## [181] 2.650534371 2.525864264 2.398350009 2.270044844 2.143226425
## [186] 2.020335958 1.903896751 1.796413343 1.700255418 1.617534234
## [191] 1.549982412 1.498849771 1.464827710 1.448011931 1.447908468
## [196] 1.463481990 1.493239556 1.535338801 1.587707643 1.648163088
## [201] 1.714519153 1.784677190 1.856695398 1.928837188 1.999600131
## [206] 2.067728406 2.132212106 2.192276686 2.247365459 2.297117542
## [211] 2.341343138 2.379997594 2.413155312 2.440984339 2.463722274
## [216] 2.481654019 2.495091834 2.504358089 2.509771036 2.511633870
## [221] 2.510227219 2.505805079 2.498594069 2.488795717 2.476591299
## [226] 2.462148656 2.445630236 2.427201574 2.407039350 2.385338195
## [231] 2.362315458 2.338213282 2.313297486 2.287852944 2.262175424
## [236] 2.236560110 2.211287327 2.186606246 2.162717641 2.139756899
## [241] 2.117778617 2.096744069 2.076512696 2.056838502 2.037371904
## [246] 2.017667146 1.997195020 1.975360248 1.951522606 1.925020695
## [251] 1.895197197 1.861424507 1.823129708 1.779818053 1.731094252
## [256] 1.676681040 1.616434607 1.550356544 1.478602008 1.401483767
## [261] 1.319471812 1.233188132 1.143396325 1.050985723 0.956949904
## [266] 0.862359687 0.768331094 0.675989221 0.586429448 0.500677970
## [271] 0.419653973 0.344136026 0.274735189 0.211876960 0.155793532
## [276] 0.106526958 0.063942823 0.027753132 -0.002453663 -0.027178031
## [281] -0.046972833 -0.062407501 -0.074035156 -0.082364454 -0.087837390
## [286] -0.090813788 -0.091562736 -0.090260892 -0.086997247 -0.081783753
## [291] -0.074571018 -0.065268125 -0.053765494 -0.039959598 -0.023778219
## [296] -0.005204923 0.015698621 0.038773689 0.063750015 0.090236562
## [301] 0.117718021 0.145557139 0.173002614 0.199201960 0.223218509
## [306] 0.244051575 0.260658763 0.271979468 0.276958788 0.274571240
## [311] 0.263843943 0.243879140 0.213876177 0.173153231 0.121169205
## [316] 0.057546310 -0.017906151 -0.105166383 -0.203972797 -0.313794929
## [321] -0.433803133 -0.562838376 -0.699384416 -0.841545875 -0.987037112
## [326] -1.133188247 -1.276975720 -1.415085031 -1.544011976 -1.660205239
## [331] -1.760247283 -1.841062452 -1.900132651 -1.935694251 -1.946887780
## [336] -1.933836409 -1.897640030 -1.840286302 -1.764494159 -1.673515071
## [341] -1.570920515 -1.460400992 -1.345594657 -1.229955010 -1.116659352
## [346] -1.008554251 -0.908131170 -0.817524444 -0.738524188 -0.672597850
## [351] -0.620915521 -0.584375493 -0.563627667 -0.559093361 -0.570980709
## [356] -0.599295359 -0.643846524 -0.704248694 -0.779919608 -0.870075268
## [361] -0.973723081 -1.089654548 -1.216439272 -1.352422460 -1.495728533
## [366] -1.644273693 -1.795790411 -1.947866487 -2.098000501 -2.243673969
## [371] -2.382438400 -2.512012820 -2.630384638 -2.735904581 -2.827365453
## [376] -2.904055248 -2.965777722 -3.012837632 -3.045992571 -3.066377663
## [381] -3.075412407 -3.074700185 -3.065930373 -3.050791025 -3.030897516
## [386] -3.007739714 -2.982647968 -2.956776428 -2.931101112 -2.906429636
## [391] -2.883419392 -2.862601135 -2.844405238 -2.829188274 -2.817257993
## [396] -2.808895157 -2.804371108 -2.803960343 -2.807947694 -2.816630091
## [401] -2.830313135 -2.849302973 -2.873894109 -2.904353936 -2.940904777
## [406] -2.983704280 -3.032824940 -3.088233551 -3.149771326 -3.217135488
## [411] -3.289863169 -3.367318610 -3.448684769 -3.532960663 -3.618965896
## [416] -3.705353918 -3.790635453 -3.873213228 -3.951428469 -4.023618646
## [421] -4.088184639 -4.143663965 -4.188805269 -4.222638146 -4.244531948
## [426] -4.254237686 -4.251908627 -4.238097413 -4.213730203 -4.180060896
## [431] -4.138610432 -4.091097282 -4.039365260 -3.985314097 -3.930836859
## [436] -3.877766759 -3.827834475 -3.782635815 -3.743608743 -3.712018252
## [441] -3.688947325 -3.675292281 -3.671760915 -3.678872124 -3.696955963
## [446] -3.726153397 -3.766415303 -3.817500556 -3.878973350 -3.950200200
## [451] -4.030347383 -4.118379920 -4.213063536 -4.312971327 -4.416497114
## [456] -4.521877505 -4.627224533 -4.730570221 -4.829923475 -4.923338344
## [461] -5.008990990 -5.085260829 -5.150809712 -5.204651949 -5.246208007
## [466] -5.275335817 -5.292335946 -5.297929883 -5.293213853 -5.279593243
## [471] -5.258704460 -5.232331502 -5.202323969 -5.170521829 -5.138690486
## [476] -5.108467894 -5.081323966 -5.058531414 -5.041146502 -5.029997928
## [481] -5.025682082 -5.028563166 -5.038776988 -5.056237667 -5.080646837
## [486] -5.111505324 -5.148127512 -5.189658859 -5.235097081 -5.283317508
## [491] -5.333102948 -5.383178054 -5.432247792 -5.479038982 -5.522343382
## [496] -5.561060171 -5.594235362 -5.621095515 -5.641073319 -5.653823152
## [501] -5.659225538 -5.657380426 -5.648590246 -5.633334579 -5.612238928
## [506] -5.586040340 -5.555552621 -5.521633511 -5.485155682 -5.446982747
## [511] -5.407950901 -5.368856182 -5.330446954 -5.293420859 -5.258425328
## [516] -5.226060655 -5.196884643 -5.171417945 -5.150149328 -5.133540237
## [521] -5.122028236 -5.116029015 -5.115936851 -5.122123511 -5.134935708
## [526] -5.154691307 -5.181674534 -5.216130478 -5.258259210 -5.308209817
## [531] -5.366074664 -5.431884179 -5.505602401 -5.587123561 -5.676269869
## [536] -5.772790705 -5.876363314 -5.986595105 -6.103027535 -6.225141530
## [541] -6.352364247 -6.484076916 -6.619623348 -6.758318601 -6.899457180
## [546] -7.042320092 -7.186180039 -7.330304097 -7.473953339 -7.616379094
## [551] -7.756815834 -7.894471052 -8.028512977 -8.158057418 -8.282155507
## [556] -8.399784546 -8.509844400 -8.611161990 -8.702506217 -8.782615053
## [561] -8.850235551 -8.904176087 -8.943368415 -8.966935291 -8.974257831
## [566] -8.965035745 -8.939333600 -8.897607320 -8.840707251 -8.769856972
## [571] -8.686609930 -8.592788540 -8.490411957 -8.381619264 -8.268594291
## [576] -8.153497012 -8.038404819 -7.925265294 -7.815860669 -7.711783095
## [581] -7.614419153 -7.524941750 -7.444307502 -7.373257852 -7.312322436
## [586] -7.261823511 -7.221880609 -7.192414854 -7.173152713 -7.163629213
## [591] -7.163190933 -7.170999359 -7.186035426 -7.207106333 -7.232855925
## [596] -7.261780073 -7.292248518 -7.322534506 -7.350853159 -7.375408883
## [601] -7.394451185 -7.406337095 -7.409597121 -7.403000487 -7.385614583
## [606] -7.356853341 -7.316509780 -7.264769294 -7.202202171 -7.129736017
## [611] -7.048610857 -6.960321207 -6.866550295 -6.769101618 -6.669832434
## [616] -6.570592672 -6.473171529 -6.379252757 -6.290378672 -6.207922149
## [621] -6.133065439 -6.066784476 -6.009837373 -5.962755996 -5.925839813
## [626] -5.899151571 -5.882514793 -5.875513474 -5.877494852 -5.887576470
## [631] -5.904659130 -5.927447521 -5.954480269 -5.984170897 -6.014860411
## [636] -6.044881169 -6.072630138 -6.096647944 -6.115698403 -6.128841969
## [641] -6.135496046 -6.135475723 -6.129010202 -6.116732785 -6.099645358
## [646] -6.079061072 -6.056531083 -6.033762200 -6.012532205 -5.994608635
## [651] -5.981675217 -5.975268486 -5.976725504 -5.987142470 -6.007343222
## [656] -6.037856370 -6.078899888 -6.130372335 -6.191850480 -6.262593724
## [661] -6.341556387 -6.427409451 -6.518573645 -6.613265666 -6.709558735
## [666] -6.805457444 -6.898985066 -6.988279208 -7.071689358 -7.147867963
## [671] -7.215845751 -7.275082591 -7.325487348 -7.367403689 -7.401562990
## [676] -7.429009396 -7.451005034 -7.468924785 -7.484149894 -7.497968327
## [681] -7.511487823 -7.525565440 -7.540755599 -7.557277408 -7.575001304
## [686] -7.593454832 -7.611847275 -7.629112780 -7.643971207 -7.655005104
## [691] -7.660749840 -7.659792243 -7.650871241 -7.632972583 -7.605409124
## [696] -7.567878800 -7.520494472 -7.463782919 -7.398654016 -7.326344554
## [701] -7.248343816 -7.166309220 -7.081980250 -6.997097610 -6.913332628
## [706] -6.832229780 -6.755163342 -6.683307656 -6.617619546 -6.558830946
## [711] -6.507449657 -6.463766327 -6.427865999 -6.399642943 -6.378817803
## [716] -6.364956414 -6.357489846 -6.355735437 -6.358918661 -6.366195726
## [721] -6.376676799 -6.389449705 -6.403603857 -6.418254113 -6.432564121
## [726] -6.445768655 -6.457194353 -6.466278258 -6.472583541 -6.4758118565.10 STL Decompostion
5.10.1 Decompose Time Series
5.10.2 Forecast
fit <- ts(tt_pm25_ts$train_ts,frequency=365) %>% stl( t.window=13, s.window="periodic", robust=TRUE)
pred <- fit %>% forecast(method="naive",h=20)
summary(fit)## Call:
## stl(x = ., s.window = "periodic", t.window = 13, robust = TRUE)
##
## Time.series components:
## seasonal trend remainder
## Min. :-79.81691 Min. : -8.52031 Min. :-275.6153
## 1st Qu.:-28.96175 1st Qu.: 59.00506 1st Qu.: -16.9382
## Median : -9.34930 Median : 76.28166 Median : 0.5996
## Mean : -0.51636 Mean : 77.22246 Mean : 9.8411
## 3rd Qu.: 13.15090 3rd Qu.: 93.35446 3rd Qu.: 22.5770
## Max. :282.15023 Max. :215.74722 Max. : 476.4804
## IQR:
## STL.seasonal STL.trend STL.remainder data
## 42.11 34.35 39.52 81.17
## % 51.9 42.3 48.7 100.0
##
## Weights:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.8041 0.9452 0.8053 0.9875 1.0000
##
## Other components: List of 5
## $ win : Named num [1:3] 14411 13 365
## $ deg : Named int [1:3] 0 1 1
## $ jump : Named num [1:3] 1442 2 37
## $ inner: int 1
## $ outer: int 15##
## Forecast method: STL + Random walk
##
## Model Information:
## Call: rwf(y = x, h = h, drift = FALSE, level = level)
##
## Residual sd: 72.5797
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -0.001751472 72.5545 47.36104 -48.22309 119.3651 0.65097
## ACF1
## Training set -0.1872593
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 4.947945 106.142765 13.160429 199.1251 -36.06145 248.3470
## 4.950685 4.587500 -126.909381 136.0844 -196.51962 205.6946
## 4.953425 -5.846121 -166.896252 155.2040 -252.15104 240.4588
## 4.956164 72.643006 -113.321667 258.6077 -211.76542 357.0514
## 4.958904 211.714023 3.799199 419.6288 -106.26426 529.6923
## 4.961644 304.579923 76.820644 532.3392 -43.74784 652.9077
## 4.964384 48.052450 -197.955688 294.0606 -328.18453 424.2894
## 4.967123 -11.308621 -274.302383 251.6851 -413.52287 390.9056
## 4.969863 -6.072099 -285.019107 272.8749 -432.68473 420.5405
## 4.972603 -11.530115 -305.566080 282.5058 -461.21932 438.1591
## 4.975342 78.591514 -229.796007 386.9790 -393.04650 550.2295
## 4.978082 116.861087 -205.239174 438.9613 -375.74875 609.4709
## 4.980822 96.755437 -238.497143 432.0080 -415.96914 609.4800
## 4.983562 33.244152 -314.663893 381.1522 -498.83529 565.3236
## 4.986301 138.491927 -221.627112 498.6110 -412.26262 689.2465
## 4.989041 25.291170 -346.638174 397.2205 -543.52568 594.1080
## 4.991781 -3.334701 -386.710694 380.0413 -589.65769 582.9883
## 4.994521 -25.964137 -420.454780 368.5265 -629.28551 577.3572
## 4.997260 -2.664316 -407.964923 402.6363 -622.51810 617.1895
## 5.000000 3.894547 -411.935102 419.7242 -632.06202 639.85115.11 VAR Model
5.11.1 Build a Model
s=365 which is 12 months or annual for dataset with daily observations.
ASE - 7.8
Window ASE - 243.682
lag_so2 <- dplyr::lag(x_so2,1)
lag_no2 <- dplyr::lag(x_no2,1)
lag_co <- dplyr::lag(x_co ,1)
X <- cbind(pm25=x_pm25, so2=lag_so2, no2=lag_no2,co=lag_co)
VARselect(X[2:dim(X)[1]],lag.max = 20, type = "const")## $selection
## AIC(n) HQ(n) SC(n) FPE(n)
## 2 2 2 2
##
## $criteria
## 1 2 3 4 5 6
## AIC(n) 8.923034 8.886223 8.888011 8.889527 8.890296 8.891640
## HQ(n) 8.926543 8.891486 8.895029 8.898299 8.900823 8.903922
## SC(n) 8.932300 8.900121 8.906542 8.912690 8.918092 8.924068
## FPE(n) 7502.821527 7231.652947 7244.598101 7255.583721 7261.167943 7270.935661
## 7 8 9 10 11 12
## AIC(n) 8.890998 8.892857 8.894391 8.894176 8.895989 8.895835
## HQ(n) 8.905034 8.908648 8.911936 8.913476 8.917044 8.918644
## SC(n) 8.928059 8.934551 8.940717 8.945135 8.951581 8.956059
## FPE(n) 7266.270294 7279.790086 7290.968817 7289.399782 7302.633220 7301.508788
## 13 14 15 16 17 18
## AIC(n) 8.896537 8.897703 8.896333 8.897489 8.899132 8.900951
## HQ(n) 8.921101 8.924021 8.924406 8.927316 8.930713 8.934287
## SC(n) 8.961394 8.967193 8.970455 8.976243 8.982519 8.988971
## FPE(n) 7306.640639 7315.166735 7305.154733 7313.602659 7325.633236 7338.975985
## 19 20
## AIC(n) 8.902811 8.903420
## HQ(n) 8.937902 8.940265
## SC(n) 8.995464 9.000705
## FPE(n) 7352.645625 7357.128467##
## VAR Estimation Results:
## =========================
## Endogenous variables: pm25, so2, no2, co
## Deterministic variables: const
## Sample size: 1093
## Log Likelihood: -8727.368
## Roots of the characteristic polynomial:
## 0.8748 0.5171 0.4971 0.4971 0.322 0.1459 0.1459 0.01251
## Call:
## VAR(y = X[2:1096, ], p = 2, type = "const")
##
##
## Estimation results for equation pm25:
## =====================================
## pm25 = pm25.l1 + so2.l1 + no2.l1 + co.l1 + pm25.l2 + so2.l2 + no2.l2 + co.l2 + const
##
## Estimate Std. Error t value Pr(>|t|)
## pm25.l1 0.60100 0.03079 19.520 < 2e-16 ***
## so2.l1 7.16300 3.95296 1.812 0.0703 .
## no2.l1 -3.94513 6.61785 -0.596 0.5512
## co.l1 -3.64588 6.72920 -0.542 0.5881
## pm25.l2 -0.18898 0.04378 -4.317 1.73e-05 ***
## so2.l2 -1.17589 3.96590 -0.297 0.7669
## no2.l2 0.90441 6.58957 0.137 0.8909
## co.l2 -1.50716 6.23070 -0.242 0.8089
## const 0.46170 2.83800 0.163 0.8708
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 84.98 on 1084 degrees of freedom
## Multiple R-Squared: 0.2926, Adjusted R-squared: 0.2873
## F-statistic: 56.03 on 8 and 1084 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation so2:
## ====================================
## so2 = pm25.l1 + so2.l1 + no2.l1 + co.l1 + pm25.l2 + so2.l2 + no2.l2 + co.l2 + const
##
## Estimate Std. Error t value Pr(>|t|)
## pm25.l1 0.0068701 0.0002814 24.412 < 2e-16 ***
## so2.l1 0.4825323 0.0361318 13.355 < 2e-16 ***
## no2.l1 0.2541764 0.0604901 4.202 2.86e-05 ***
## co.l1 -0.3208801 0.0615078 -5.217 2.18e-07 ***
## pm25.l2 -0.0037479 0.0004001 -9.366 < 2e-16 ***
## so2.l2 0.0060907 0.0362501 0.168 0.866600
## no2.l2 -0.2245443 0.0602315 -3.728 0.000203 ***
## co.l2 0.0827338 0.0569514 1.453 0.146594
## const -0.1784372 0.0259406 -6.879 1.02e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.7768 on 1084 degrees of freedom
## Multiple R-Squared: 0.5265, Adjusted R-squared: 0.523
## F-statistic: 150.7 on 8 and 1084 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation no2:
## ====================================
## no2 = pm25.l1 + so2.l1 + no2.l1 + co.l1 + pm25.l2 + so2.l2 + no2.l2 + co.l2 + const
##
## Estimate Std. Error t value Pr(>|t|)
## pm25.l1 0.0045265 0.0001668 27.136 < 2e-16 ***
## so2.l1 -0.0819185 0.0214166 -3.825 0.000138 ***
## no2.l1 0.8000967 0.0358545 22.315 < 2e-16 ***
## co.l1 -0.2416602 0.0364578 -6.628 5.34e-11 ***
## pm25.l2 -0.0025530 0.0002372 -10.764 < 2e-16 ***
## so2.l2 -0.0178499 0.0214866 -0.831 0.406302
## no2.l2 0.0635419 0.0357013 1.780 0.075385 .
## co.l2 0.0077309 0.0337570 0.229 0.818900
## const -0.0358008 0.0153759 -2.328 0.020075 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.4604 on 1084 degrees of freedom
## Multiple R-Squared: 0.7258, Adjusted R-squared: 0.7238
## F-statistic: 358.7 on 8 and 1084 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation co:
## ===================================
## co = pm25.l1 + so2.l1 + no2.l1 + co.l1 + pm25.l2 + so2.l2 + no2.l2 + co.l2 + const
##
## Estimate Std. Error t value Pr(>|t|)
## pm25.l1 0.0065955 0.0001755 37.579 < 2e-16 ***
## so2.l1 -0.0265487 0.0225342 -1.178 0.238993
## no2.l1 0.0912619 0.0377256 2.419 0.015723 *
## co.l1 0.3763464 0.0383603 9.811 < 2e-16 ***
## pm25.l2 -0.0031176 0.0002496 -12.493 < 2e-16 ***
## so2.l2 -0.0132235 0.0226079 -0.585 0.558731
## no2.l2 -0.0822854 0.0375643 -2.191 0.028699 *
## co.l2 0.0009036 0.0355186 0.025 0.979708
## const -0.0564721 0.0161782 -3.491 0.000501 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.4845 on 1084 degrees of freedom
## Multiple R-Squared: 0.6748, Adjusted R-squared: 0.6724
## F-statistic: 281.2 on 8 and 1084 DF, p-value: < 2.2e-16
##
##
##
## Covariance matrix of residuals:
## pm25 so2 no2 co
## pm25 7222.133 9.6062 5.9590 4.7220
## so2 9.606 0.6034 0.1397 0.2005
## no2 5.959 0.1397 0.2120 0.1197
## co 4.722 0.2005 0.1197 0.2347
##
## Correlation matrix of residuals:
## pm25 so2 no2 co
## pm25 1.0000 0.1455 0.1523 0.1147
## so2 0.1455 1.0000 0.3907 0.5329
## no2 0.1523 0.3907 1.0000 0.5367
## co 0.1147 0.5329 0.5367 1.0000##
## Call:
## lm(formula = y ~ -1 + ., data = datamat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -430.82 -43.71 -1.24 45.54 537.74
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## pm25.l1 0.60100 0.03079 19.520 < 2e-16 ***
## so2.l1 7.16300 3.95296 1.812 0.0703 .
## no2.l1 -3.94513 6.61785 -0.596 0.5512
## co.l1 -3.64588 6.72920 -0.542 0.5881
## pm25.l2 -0.18898 0.04378 -4.317 1.73e-05 ***
## so2.l2 -1.17589 3.96590 -0.297 0.7669
## no2.l2 0.90441 6.58957 0.137 0.8909
## co.l2 -1.50716 6.23070 -0.242 0.8089
## const 0.46170 2.83800 0.163 0.8708
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 84.98 on 1084 degrees of freedom
## Multiple R-squared: 0.2926, Adjusted R-squared: 0.2873
## F-statistic: 56.03 on 8 and 1084 DF, p-value: < 2.2e-165.11.2 Residual diagnositcs
##
## VAR Estimation Results:
## =======================
##
## Estimated coefficients for equation pm25:
## =========================================
## Call:
## pm25 = pm25.l1 + so2.l1 + no2.l1 + co.l1 + pm25.l2 + so2.l2 + no2.l2 + co.l2 + const
##
## pm25.l1 so2.l1 no2.l1 co.l1 pm25.l2 so2.l2 no2.l2
## 0.6009966 7.1629968 -3.9451272 -3.6458849 -0.1889785 -1.1758934 0.9044107
## co.l2 const
## -1.5071645 0.4617003
##
##
## Estimated coefficients for equation so2:
## ========================================
## Call:
## so2 = pm25.l1 + so2.l1 + no2.l1 + co.l1 + pm25.l2 + so2.l2 + no2.l2 + co.l2 + const
##
## pm25.l1 so2.l1 no2.l1 co.l1 pm25.l2 so2.l2
## 0.006870089 0.482532295 0.254176396 -0.320880117 -0.003747859 0.006090706
## no2.l2 co.l2 const
## -0.224544329 0.082733788 -0.178437175
##
##
## Estimated coefficients for equation no2:
## ========================================
## Call:
## no2 = pm25.l1 + so2.l1 + no2.l1 + co.l1 + pm25.l2 + so2.l2 + no2.l2 + co.l2 + const
##
## pm25.l1 so2.l1 no2.l1 co.l1 pm25.l2 so2.l2
## 0.004526537 -0.081918482 0.800096692 -0.241660244 -0.002553024 -0.017849861
## no2.l2 co.l2 const
## 0.063541900 0.007730899 -0.035800769
##
##
## Estimated coefficients for equation co:
## =======================================
## Call:
## co = pm25.l1 + so2.l1 + no2.l1 + co.l1 + pm25.l2 + so2.l2 + no2.l2 + co.l2 + const
##
## pm25.l1 so2.l1 no2.l1 co.l1 pm25.l2
## 0.0065954713 -0.0265487340 0.0912619438 0.3763463606 -0.0031175859
## so2.l2 no2.l2 co.l2 const
## -0.0132235428 -0.0822853512 0.0009036387 -0.0564721133## $pm25
## fcst lower upper CI
## [1,] -17.588924 -184.1528 148.9750 166.5639
## [2,] -10.674280 -205.5597 184.2111 194.8854
## [3,] -5.087442 -202.5318 192.3569 197.4444
## [4,] -2.435193 -200.0108 195.1404 197.5756
## [5,] -2.020706 -199.8784 195.8370 197.8577
## [6,] -2.387554 -200.3523 195.5772 197.9648
## [7,] -2.719783 -200.7004 195.2608 197.9806
## [8,] -2.827869 -200.8138 195.1581 197.9859
## [9,] -2.787196 -200.7779 195.2035 197.9907
## [10,] -2.699788 -200.6939 195.2943 197.9941
## [11,] -2.618860 -200.6154 195.3776 197.9965
## [12,] -2.556531 -200.5549 195.4418 197.9983
## [13,] -2.508228 -200.5080 195.4915 197.9997
## [14,] -2.467785 -200.4686 195.5330 198.0008
## [15,] -2.431924 -200.4335 195.5697 198.0016
## [16,] -2.399677 -200.4019 195.6026 198.0022
## [17,] -2.370938 -200.3737 195.6318 198.0027
## [18,] -2.345624 -200.3487 195.6575 198.0031
## [19,] -2.323478 -200.3268 195.6799 198.0034
## [20,] -2.304139 -200.3077 195.6994 198.0036
## [21,] -2.287243 -200.2910 195.7165 198.0037
## [22,] -2.272465 -200.2763 195.7314 198.0039
## [23,] -2.259534 -200.2635 195.7444 198.0040
## [24,] -2.248218 -200.2523 195.7558 198.0040
## [25,] -2.238317 -200.2424 195.7658 198.0041
## [26,] -2.229654 -200.2338 195.7745 198.0041
## [27,] -2.222076 -200.2262 195.7821 198.0042
## [28,] -2.215448 -200.2196 195.7887 198.0042
## [29,] -2.209649 -200.2139 195.7946 198.0042
## [30,] -2.204576 -200.2088 195.7996 198.0042
##
## $so2
## fcst lower upper CI
## [1,] -0.07540616 -1.597874 1.447061 1.522467
## [2,] -0.35016317 -2.441867 1.741541 2.091704
## [3,] -0.30499131 -2.483754 1.873771 2.178762
## [4,] -0.28393199 -2.475655 1.907791 2.191723
## [5,] -0.28973443 -2.496095 1.916626 2.206361
## [6,] -0.30432824 -2.515769 1.907112 2.211440
## [7,] -0.31558777 -2.527615 1.896440 2.212028
## [8,] -0.32080883 -2.532857 1.891239 2.212048
## [9,] -0.32207432 -2.534155 1.890006 2.212080
## [10,] -0.32182154 -2.533916 1.890273 2.212095
## [11,] -0.32136477 -2.533464 1.890735 2.212099
## [12,] -0.32107999 -2.533182 1.891022 2.212102
## [13,] -0.32093747 -2.533042 1.891167 2.212105
## [14,] -0.32084201 -2.532948 1.891264 2.212106
## [15,] -0.32074427 -2.532851 1.891363 2.212107
## [16,] -0.32063773 -2.532746 1.891470 2.212108
## [17,] -0.32053155 -2.532640 1.891577 2.212109
## [18,] -0.32043388 -2.532543 1.891675 2.212109
## [19,] -0.32034782 -2.532457 1.891762 2.212110
## [20,] -0.32027300 -2.532383 1.891837 2.212110
## [21,] -0.32020787 -2.532318 1.891902 2.212110
## [22,] -0.32015093 -2.532261 1.891959 2.212110
## [23,] -0.32010101 -2.532211 1.892009 2.212110
## [24,] -0.32005725 -2.532168 1.892053 2.212110
## [25,] -0.32001892 -2.532129 1.892092 2.212110
## [26,] -0.31998536 -2.532096 1.892125 2.212111
## [27,] -0.31995601 -2.532067 1.892155 2.212111
## [28,] -0.31993033 -2.532041 1.892180 2.212111
## [29,] -0.31990787 -2.532019 1.892203 2.212111
## [30,] -0.31988822 -2.531999 1.892222 2.212111
##
## $no2
## fcst lower upper CI
## [1,] 0.37901644 -0.5234018 1.281435 0.9024182
## [2,] 0.28915445 -1.0722406 1.650549 1.3613950
## [3,] 0.29412286 -1.1841947 1.772440 1.4783176
## [4,] 0.28598933 -1.2516224 1.823601 1.5376117
## [5,] 0.26408712 -1.3203241 1.848498 1.5844112
## [6,] 0.23786404 -1.3808420 1.856570 1.6187061
## [7,] 0.21445888 -1.4294318 1.858350 1.6438907
## [8,] 0.19569806 -1.4671844 1.858581 1.6628825
## [9,] 0.18067015 -1.4966103 1.857951 1.6772805
## [10,] 0.16809027 -1.5201085 1.856289 1.6881988
## [11,] 0.15715019 -1.5393467 1.853647 1.6964969
## [12,] 0.14749281 -1.5553256 1.850311 1.7028184
## [13,] 0.13897305 -1.5686680 1.846614 1.7076410
## [14,] 0.13149452 -1.5798288 1.842818 1.7113233
## [15,] 0.12495353 -1.5891827 1.839090 1.7141363
## [16,] 0.11923911 -1.5970469 1.835525 1.7162860
## [17,] 0.11424530 -1.6036840 1.832175 1.7179293
## [18,] 0.10987850 -1.6093073 1.829064 1.7191858
## [19,] 0.10605843 -1.6140883 1.826205 1.7201467
## [20,] 0.10271621 -1.6181655 1.823598 1.7208817
## [21,] 0.09979217 -1.6216519 1.821236 1.7214440
## [22,] 0.09723414 -1.6246401 1.819108 1.7218742
## [23,] 0.09499638 -1.6272070 1.817200 1.7222034
## [24,] 0.09303882 -1.6294164 1.815494 1.7224552
## [25,] 0.09132636 -1.6313216 1.813974 1.7226479
## [26,] 0.08982829 -1.6329671 1.812624 1.7227954
## [27,] 0.08851777 -1.6343905 1.811426 1.7229083
## [28,] 0.08737133 -1.6356233 1.810366 1.7229946
## [29,] 0.08636841 -1.6366923 1.809429 1.7230607
## [30,] 0.08549106 -1.6376202 1.808602 1.7231113
##
## $co
## fcst lower upper CI
## [1,] -0.02305443 -0.9725642 0.9264553 0.9495098
## [2,] -0.19882738 -1.7359708 1.3383160 1.5371434
## [3,] -0.14139301 -1.7958963 1.5131103 1.6545033
## [4,] -0.09436404 -1.7496767 1.5609486 1.6553126
## [5,] -0.07884511 -1.7392083 1.5815181 1.6603632
## [6,] -0.07995100 -1.7435290 1.5836271 1.6635781
## [7,] -0.08419167 -1.7482756 1.5798922 1.6640839
## [8,] -0.08632248 -1.7504395 1.5777946 1.6641171
## [9,] -0.08630409 -1.7504669 1.5778587 1.6641628
## [10,] -0.08541898 -1.7496108 1.5787729 1.6641919
## [11,] -0.08453761 -1.7487438 1.5796686 1.6642062
## [12,] -0.08392261 -1.7481386 1.5802934 1.6642160
## [13,] -0.08352631 -1.7477500 1.5806974 1.6642237
## [14,] -0.08324276 -1.7474722 1.5809867 1.6642295
## [15,] -0.08300541 -1.7472393 1.5812285 1.6642339
## [16,] -0.08279082 -1.7470281 1.5814464 1.6642372
## [17,] -0.08259637 -1.7468362 1.5816434 1.6642398
## [18,] -0.08242373 -1.7466655 1.5818180 1.6642418
## [19,] -0.08227283 -1.7465161 1.5819705 1.6642433
## [20,] -0.08214161 -1.7463861 1.5821028 1.6642444
## [21,] -0.08202740 -1.7462727 1.5822179 1.6642453
## [22,] -0.08192770 -1.7461737 1.5823183 1.6642460
## [23,] -0.08184050 -1.7460870 1.5824060 1.6642465
## [24,] -0.08176419 -1.7460111 1.5824827 1.6642469
## [25,] -0.08169741 -1.7459446 1.5825498 1.6642472
## [26,] -0.08163899 -1.7458864 1.5826085 1.6642475
## [27,] -0.08158788 -1.7458355 1.5826598 1.6642476
## [28,] -0.08154317 -1.7457909 1.5827046 1.6642478
## [29,] -0.08150406 -1.7457519 1.5827438 1.6642479
## [30,] -0.08146985 -1.7457178 1.5827781 1.66424805.11.3 VAR Rolling Window ASE
VARWindowASE <- var.window.ase.wge(X,x=ts_daily_df$PM25,x_p=2,trainingSize =600,horizon = 20,x_lag=1)
mean(VARWindowASE)## [1] 243.62826 PM10 Analysis
6.1 Stationarity
First plot the data.
- No visible trend and seasonality in realization
- ACF cuts off after 2nd lag. No indication of non-stationarity
- Spectral density plot shows peak at 0, indicating wandering behavior
- First and second half of ACFs shows no significant difference.
- This is stationary series.
6.2 Split into Train/Test
6.3 Dicky-Fuller Test
\(H_{0}\) - Unit root present
\(H_{1}\) - Unit root not present. or time series is stationary.
Conclusion
pvalue = 0.01. Reject Ho. Suggests strong evidence against presence of unit root.
No visual evidence of sesonality so time series is stationary.
## Warning in adf.test(tt_pm10_ts$train_ts): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: tt_pm10_ts$train_ts
## Dickey-Fuller = -8.3554, Lag order = 11, p-value = 0.01
## alternative hypothesis: stationary6.4 Model ID
## ---------WORKING... PLEASE WAIT...
##
##
## Five Smallest Values of aicAR(6,1) turns out to be the best mode as per AIC function.
6.5 Estimate Parameters
6.5.1 Estimation
##
## Coefficients of Original polynomial:
## 1.5367 -0.7205 0.1915 -0.0362 -0.0335 0.0448
##
## Factor Roots Abs Recip System Freq
## 1-0.9732B 1.0275 0.9732 0.0000
## 1-1.0842B+0.4133B^2 1.3116+-0.8362i 0.6429 0.0903
## 1+0.1172B+0.2761B^2 -0.2123+-1.8911i 0.5255 0.2678
## 1+0.4035B -2.4781 0.4035 0.5000
##
## ## [1] 1.53668859 -0.72052819 0.19154879 -0.03623074 -0.03352981 0.04482387## [1] 0.934823## [1] 110.7666
6.5.2 Residual diagnositcs
## Obs -0.0008863582 -0.002190483 0.0003356206 -0.0007377869 0.004632811 0.006264072 -0.03551586 -0.006313547 0.008870318 0.04421087 -0.01537624 -0.01848627 0.01012401 0.01853992 -0.001019537 0.0108432 -0.02166164 -0.05481056 -0.01098366 -0.0226941 0.004681241 0.02812718 -0.007642125 0.04244317## $test
## [1] "Ljung-Box test"
##
## $K
## [1] 24
##
## $chi.square
## [1] 16.5282
##
## $df
## [1] 17
##
## $pval
## [1] 0.4867469## Obs -0.0008863582 -0.002190483 0.0003356206 -0.0007377869 0.004632811 0.006264072 -0.03551586 -0.006313547 0.008870318 0.04421087 -0.01537624 -0.01848627 0.01012401 0.01853992 -0.001019537 0.0108432 -0.02166164 -0.05481056 -0.01098366 -0.0226941 0.004681241 0.02812718 -0.007642125 0.04244317 0.0167526 -0.004519385 -0.002749417 0.05848487 0.03145483 0.01406576 -0.01134344 0.02369013 -0.008841578 -0.03103694 -0.01795642 0.02331942 -0.03734233 0.0400099 0.03467535 -0.02300888 0.0487511 -0.02985911 -0.00898804 -0.03063819 0.04447112 -0.004417444 -0.00231838 -0.008690027## $test
## [1] "Ljung-Box test"
##
## $K
## [1] 48
##
## $chi.square
## [1] 44.07721
##
## $df
## [1] 41
##
## $pval
## [1] 0.34276426.6 Final Model
Final PM10 Model:
(1 - 1.53B + 0.72\(B^2\) - 0.18\(B^3\) + 0.04\(B^4\) + 0.03\(B^5\) - 0.05\(B^5\) ) (\(X_t\) - 110.111) = (1 - 0.93B) (\(a_t\))
6.7 ARMA
6.7.2 Rolling Window ASE
window_arma_pm10<-window.ase.wge(ts_daily_df$PM10,phi=e_pm10$phi,theta=e_pm10$theta,trainingSize = 500,horizon = 20)
mean(window_arma_pm10)## [1] 7008.496.8 VAR Model
6.8.1 Build a Model
s=365 which is 12 months or annual for dataset with daily observations.
ASE - 7.8
Window ASE - 243.682
lag10_so2 <- dplyr::lag(x10_so2,1)
lag10_no2 <- dplyr::lag(x10_no2,1)
lag10_co <- dplyr::lag(x10_co ,1)
X10 <- cbind(pm10=x10_pm10, so2=lag10_so2, no2=lag10_no2,co=lag10_co)
VARselect(X10[2:dim(X10)[1]],lag.max = 20, type = "const")## $selection
## AIC(n) HQ(n) SC(n) FPE(n)
## 2 2 2 2
##
## $criteria
## 1 2 3 4 5 6
## AIC(n) 9.079985 9.058256 9.059817 9.061190 9.062818 9.064636
## HQ(n) 9.083494 9.063520 9.066835 9.069963 9.073345 9.076917
## SC(n) 9.089250 9.072154 9.078347 9.084354 9.090614 9.097064
## FPE(n) 8777.833573 8589.158332 8602.576288 8614.399600 8628.433481 8644.131333
## 7 8 9 10 11 12
## AIC(n) 9.064204 9.065835 9.067367 9.066914 9.068012 9.069628
## HQ(n) 9.078240 9.081625 9.084912 9.086214 9.089066 9.092437
## SC(n) 9.101265 9.107528 9.113693 9.117873 9.123603 9.129853
## FPE(n) 8640.399726 8654.502003 8667.772137 8663.850815 8673.367941 8687.405296
## 13 14 15 16 17 18
## AIC(n) 9.070992 9.072346 9.073854 9.074456 9.076291 9.077695
## HQ(n) 9.095555 9.098664 9.101927 9.104283 9.107872 9.111031
## SC(n) 9.135848 9.141835 9.147976 9.153211 9.159678 9.165715
## FPE(n) 8699.258473 8711.049117 8724.204033 8729.457924 8745.492101 8757.784866
## 19 20
## AIC(n) 9.079035 9.080093
## HQ(n) 9.114126 9.116938
## SC(n) 9.171688 9.177378
## FPE(n) 8769.541413 8778.822332##
## VAR Estimation Results:
## =========================
## Endogenous variables: pm10, so2, no2, co
## Deterministic variables: const
## Sample size: 1093
## Log Likelihood: -8899.737
## Roots of the characteristic polynomial:
## 0.8755 0.524 0.4413 0.4413 0.4216 0.1343 0.1343 0.02679
## Call:
## VAR(y = X10[2:1096, ], p = 2, type = "const")
##
##
## Estimation results for equation pm10:
## =====================================
## pm10 = pm10.l1 + so2.l1 + no2.l1 + co.l1 + pm10.l2 + so2.l2 + no2.l2 + co.l2 + const
##
## Estimate Std. Error t value Pr(>|t|)
## pm10.l1 0.56499 0.03086 18.307 < 2e-16 ***
## so2.l1 7.57059 4.31061 1.756 0.07933 .
## no2.l1 -6.12900 7.21680 -0.849 0.39592
## co.l1 -11.48029 6.99430 -1.641 0.10101
## pm10.l2 -0.10999 0.04040 -2.722 0.00658 **
## so2.l2 -0.65644 4.30394 -0.153 0.87880
## no2.l2 4.71072 7.15757 0.658 0.51059
## co.l2 -2.82618 6.72999 -0.420 0.67461
## const 1.59865 3.09312 0.517 0.60537
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 92.33 on 1084 degrees of freedom
## Multiple R-Squared: 0.2656, Adjusted R-squared: 0.2602
## F-statistic: 49 on 8 and 1084 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation so2:
## ====================================
## so2 = pm10.l1 + so2.l1 + no2.l1 + co.l1 + pm10.l2 + so2.l2 + no2.l2 + co.l2 + const
##
## Estimate Std. Error t value Pr(>|t|)
## pm10.l1 0.0062081 0.0002626 23.638 < 2e-16 ***
## so2.l1 0.4949450 0.0366824 13.493 < 2e-16 ***
## no2.l1 0.2138302 0.0614133 3.482 0.000518 ***
## co.l1 -0.2582311 0.0595199 -4.339 1.57e-05 ***
## pm10.l2 -0.0034857 0.0003438 -10.139 < 2e-16 ***
## so2.l2 0.0017399 0.0366256 0.048 0.962118
## no2.l2 -0.1966254 0.0609094 -3.228 0.001283 **
## co.l2 0.0565979 0.0572708 0.988 0.323251
## const -0.1807393 0.0263217 -6.867 1.11e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.7857 on 1084 degrees of freedom
## Multiple R-Squared: 0.5156, Adjusted R-squared: 0.512
## F-statistic: 144.2 on 8 and 1084 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation no2:
## ====================================
## no2 = pm10.l1 + so2.l1 + no2.l1 + co.l1 + pm10.l2 + so2.l2 + no2.l2 + co.l2 + const
##
## Estimate Std. Error t value Pr(>|t|)
## pm10.l1 0.0040068 0.0001578 25.395 < 2e-16 ***
## so2.l1 -0.0710187 0.0220378 -3.223 0.00131 **
## no2.l1 0.7783590 0.0368956 21.096 < 2e-16 ***
## co.l1 -0.1963243 0.0357580 -5.490 4.99e-08 ***
## pm10.l2 -0.0024410 0.0002065 -11.818 < 2e-16 ***
## so2.l2 -0.0191667 0.0220037 -0.871 0.38391
## no2.l2 0.0789267 0.0365928 2.157 0.03123 *
## co.l2 -0.0094685 0.0344068 -0.275 0.78322
## const -0.0353871 0.0158134 -2.238 0.02544 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.472 on 1084 degrees of freedom
## Multiple R-Squared: 0.7118, Adjusted R-squared: 0.7097
## F-statistic: 334.7 on 8 and 1084 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation co:
## ===================================
## co = pm10.l1 + so2.l1 + no2.l1 + co.l1 + pm10.l2 + so2.l2 + no2.l2 + co.l2 + const
##
## Estimate Std. Error t value Pr(>|t|)
## pm10.l1 0.0056389 0.0001758 32.073 < 2e-16 ***
## so2.l1 -0.0103322 0.0245567 -0.421 0.67402
## no2.l1 0.0671079 0.0411127 1.632 0.10291
## co.l1 0.4562014 0.0398451 11.449 < 2e-16 ***
## pm10.l2 -0.0030460 0.0002302 -13.235 < 2e-16 ***
## so2.l2 -0.0128658 0.0245187 -0.525 0.59987
## no2.l2 -0.0694859 0.0407753 -1.704 0.08865 .
## co.l2 -0.0178999 0.0383395 -0.467 0.64068
## const -0.0547419 0.0176209 -3.107 0.00194 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.526 on 1084 degrees of freedom
## Multiple R-Squared: 0.6167, Adjusted R-squared: 0.6139
## F-statistic: 218 on 8 and 1084 DF, p-value: < 2.2e-16
##
##
##
## Covariance matrix of residuals:
## pm10 so2 no2 co
## pm10 8524.639 9.2868 6.9361 3.9347
## so2 9.287 0.6173 0.1527 0.2291
## no2 6.936 0.1527 0.2228 0.1420
## co 3.935 0.2291 0.1420 0.2767
##
## Correlation matrix of residuals:
## pm10 so2 no2 co
## pm10 1.00000 0.1280 0.1592 0.08102
## so2 0.12802 1.0000 0.4117 0.55446
## no2 0.15915 0.4117 1.0000 0.57184
## co 0.08102 0.5545 0.5718 1.00000ccf(ts_daily_df$PM10,ts_daily_df$SO2)
ccf(ts_daily_df$PM10,ts_daily_df$NO2)
ccf(ts_daily_df$PM10,ts_daily_df$CO)
6.8.2 VAR forecast
##
## VAR Estimation Results:
## =======================
##
## Estimated coefficients for equation pm10:
## =========================================
## Call:
## pm10 = pm10.l1 + so2.l1 + no2.l1 + co.l1 + pm10.l2 + so2.l2 + no2.l2 + co.l2 + const
##
## pm10.l1 so2.l1 no2.l1 co.l1 pm10.l2 so2.l2
## 0.5649867 7.5705851 -6.1290000 -11.4802918 -0.1099856 -0.6564396
## no2.l2 co.l2 const
## 4.7107155 -2.8261769 1.5986505
##
##
## Estimated coefficients for equation so2:
## ========================================
## Call:
## so2 = pm10.l1 + so2.l1 + no2.l1 + co.l1 + pm10.l2 + so2.l2 + no2.l2 + co.l2 + const
##
## pm10.l1 so2.l1 no2.l1 co.l1 pm10.l2 so2.l2
## 0.006208144 0.494944997 0.213830182 -0.258231074 -0.003485726 0.001739946
## no2.l2 co.l2 const
## -0.196625409 0.056597862 -0.180739315
##
##
## Estimated coefficients for equation no2:
## ========================================
## Call:
## no2 = pm10.l1 + so2.l1 + no2.l1 + co.l1 + pm10.l2 + so2.l2 + no2.l2 + co.l2 + const
##
## pm10.l1 so2.l1 no2.l1 co.l1 pm10.l2 so2.l2
## 0.004006803 -0.071018687 0.778358996 -0.196324294 -0.002440981 -0.019166749
## no2.l2 co.l2 const
## 0.078926700 -0.009468485 -0.035387057
##
##
## Estimated coefficients for equation co:
## =======================================
## Call:
## co = pm10.l1 + so2.l1 + no2.l1 + co.l1 + pm10.l2 + so2.l2 + no2.l2 + co.l2 + const
##
## pm10.l1 so2.l1 no2.l1 co.l1 pm10.l2 so2.l2
## 0.005638880 -0.010332235 0.067107904 0.456201352 -0.003046009 -0.012865848
## no2.l2 co.l2 const
## -0.069485927 -0.017899889 -0.054741856## $pm10
## fcst lower upper CI
## [1,] -13.5041490 -194.4656 167.4573 180.9615
## [2,] -6.7159086 -214.8079 201.3761 208.0920
## [3,] -1.3132034 -212.0697 209.4433 210.7565
## [4,] 1.0505905 -209.9582 212.0594 211.0088
## [5,] 1.4071181 -209.7122 212.5264 211.1193
## [6,] 1.1251378 -210.0220 212.2723 211.1471
## [7,] 0.8643657 -210.2867 212.0154 211.1511
## [8,] 0.7489306 -210.4033 211.9012 211.1523
## [9,] 0.7230406 -210.4298 211.8758 211.1528
## [10,] 0.7271724 -210.4258 211.8802 211.1530
## [11,] 0.7348587 -210.4182 211.8880 211.1531
## [12,] 0.7405423 -210.4126 211.8937 211.1532
## [13,] 0.7451434 -210.4081 211.8983 211.1532
## [14,] 0.7497379 -210.4035 211.9030 211.1532
## [15,] 0.7544900 -210.3988 211.9077 211.1532
## [16,] 0.7591461 -210.3941 211.9124 211.1533
## [17,] 0.7634681 -210.3898 211.9167 211.1533
## [18,] 0.7673514 -210.3859 211.9206 211.1533
## [19,] 0.7707896 -210.3825 211.9241 211.1533
## [20,] 0.7738177 -210.3795 211.9271 211.1533
## [21,] 0.7764797 -210.3768 211.9298 211.1533
## [22,] 0.7788175 -210.3745 211.9321 211.1533
## [23,] 0.7808684 -210.3724 211.9342 211.1533
## [24,] 0.7826663 -210.3706 211.9360 211.1533
## [25,] 0.7842416 -210.3691 211.9375 211.1533
## [26,] 0.7856213 -210.3677 211.9389 211.1533
## [27,] 0.7868296 -210.3665 211.9401 211.1533
## [28,] 0.7878876 -210.3654 211.9412 211.1533
## [29,] 0.7888140 -210.3645 211.9421 211.1533
## [30,] 0.7896252 -210.3637 211.9429 211.1533
##
## $so2
## fcst lower upper CI
## [1,] -0.05407453 -1.594017 1.485867 1.539942
## [2,] -0.37114473 -2.469927 1.727637 2.098782
## [3,] -0.32889045 -2.513849 1.856068 2.184958
## [4,] -0.29775626 -2.496339 1.900827 2.198583
## [5,] -0.29737269 -2.506438 1.911692 2.209065
## [6,] -0.30775681 -2.519756 1.904242 2.211999
## [7,] -0.31620420 -2.528527 1.896119 2.212323
## [8,] -0.32010372 -2.532449 1.892241 2.212345
## [9,] -0.32113302 -2.533489 1.891223 2.212356
## [10,] -0.32110351 -2.533465 1.891258 2.212362
## [11,] -0.32089555 -2.533260 1.891469 2.212364
## [12,] -0.32074028 -2.533106 1.891626 2.212366
## [13,] -0.32063275 -2.533000 1.891735 2.212368
## [14,] -0.32053896 -2.532907 1.891830 2.212368
## [15,] -0.32044603 -2.532815 1.891923 2.212369
## [16,] -0.32035550 -2.532725 1.892014 2.212370
## [17,] -0.32027158 -2.532642 1.892099 2.212370
## [18,] -0.32019647 -2.532567 1.892174 2.212370
## [19,] -0.32013030 -2.532501 1.892240 2.212371
## [20,] -0.32007225 -2.532443 1.892299 2.212371
## [21,] -0.32002137 -2.532392 1.892350 2.212371
## [22,] -0.31997674 -2.532348 1.892394 2.212371
## [23,] -0.31993762 -2.532309 1.892434 2.212371
## [24,] -0.31990334 -2.532275 1.892468 2.212371
## [25,] -0.31987331 -2.532245 1.892498 2.212371
## [26,] -0.31984701 -2.532218 1.892524 2.212371
## [27,] -0.31982398 -2.532195 1.892547 2.212371
## [28,] -0.31980382 -2.532175 1.892568 2.212371
## [29,] -0.31978617 -2.532158 1.892585 2.212371
## [30,] -0.31977071 -2.532142 1.892601 2.212371
##
## $no2
## fcst lower upper CI
## [1,] 0.39723773 -0.5279198 1.322395 0.9251575
## [2,] 0.27915314 -1.0876899 1.645996 1.3668431
## [3,] 0.28220112 -1.1964246 1.760827 1.4786257
## [4,] 0.27700738 -1.2606740 1.814689 1.5376813
## [5,] 0.25720357 -1.3261951 1.840602 1.5833986
## [6,] 0.23298910 -1.3843086 1.850287 1.6172977
## [7,] 0.21124345 -1.4314039 1.853891 1.6426473
## [8,] 0.19349667 -1.4683392 1.855333 1.6618359
## [9,] 0.17893618 -1.4974587 1.855331 1.6763949
## [10,] 0.16656038 -1.5209008 1.854022 1.6874612
## [11,] 0.15577574 -1.5401165 1.851668 1.6958922
## [12,] 0.14630069 -1.5560265 1.848628 1.7023272
## [13,] 0.13798032 -1.5692640 1.845225 1.7072443
## [14,] 0.13068952 -1.5803149 1.841694 1.7110044
## [15,] 0.12430838 -1.5895730 1.838190 1.7138814
## [16,] 0.11872440 -1.5973590 1.834808 1.7160834
## [17,] 0.11383694 -1.6039326 1.831607 1.7177696
## [18,] 0.10955823 -1.6095027 1.828619 1.7190610
## [19,] 0.10581207 -1.6142382 1.825862 1.7200503
## [20,] 0.10253213 -1.6182761 1.823340 1.7208082
## [21,] 0.09966040 -1.6217286 1.821049 1.7213890
## [22,] 0.09714611 -1.6246880 1.818980 1.7218341
## [23,] 0.09494477 -1.6272304 1.817120 1.7221752
## [24,] 0.09301743 -1.6294192 1.815454 1.7224366
## [25,] 0.09132998 -1.6313070 1.813967 1.7226370
## [26,] 0.08985255 -1.6329380 1.812643 1.7227906
## [27,] 0.08855902 -1.6343493 1.811467 1.7229083
## [28,] 0.08742649 -1.6355721 1.810425 1.7229986
## [29,] 0.08643491 -1.6366328 1.809503 1.7230677
## [30,] 0.08556676 -1.6375540 1.808688 1.7231207
##
## $co
## fcst lower upper CI
## [1,] 0.02860225 -1.002299 1.059504 1.030902
## [2,] -0.18223053 -1.746634 1.382173 1.564404
## [3,] -0.13946266 -1.797922 1.518996 1.658459
## [4,] -0.09433731 -1.753286 1.564611 1.658949
## [5,] -0.07906979 -1.741568 1.583428 1.662498
## [6,] -0.07947482 -1.743759 1.584809 1.664284
## [7,] -0.08275547 -1.747299 1.581788 1.664543
## [8,] -0.08441226 -1.748981 1.580156 1.664569
## [9,] -0.08449693 -1.749084 1.580090 1.664587
## [10,] -0.08398345 -1.748583 1.580616 1.664599
## [11,] -0.08345134 -1.748059 1.581157 1.664608
## [12,] -0.08305335 -1.747668 1.581561 1.664614
## [13,] -0.08276343 -1.747383 1.581856 1.664619
## [14,] -0.08253274 -1.747156 1.582090 1.664623
## [15,] -0.08233427 -1.746960 1.582292 1.664626
## [16,] -0.08215884 -1.746787 1.582469 1.664628
## [17,] -0.08200405 -1.746634 1.582626 1.664630
## [18,] -0.08186839 -1.746500 1.582763 1.664631
## [19,] -0.08174992 -1.746382 1.582882 1.664632
## [20,] -0.08164648 -1.746280 1.582987 1.664633
## [21,] -0.08155607 -1.746190 1.583078 1.664634
## [22,] -0.08147696 -1.746111 1.583157 1.664634
## [23,] -0.08140772 -1.746042 1.583227 1.664634
## [24,] -0.08134710 -1.745982 1.583288 1.664635
## [25,] -0.08129403 -1.745929 1.583341 1.664635
## [26,] -0.08124757 -1.745883 1.583388 1.664635
## [27,] -0.08120689 -1.745842 1.583428 1.664635
## [28,] -0.08117128 -1.745807 1.583464 1.664635
## [29,] -0.08114010 -1.745775 1.583495 1.664635
## [30,] -0.08111280 -1.745748 1.583523 1.6646356.8.3 VAR Rolling Window ASE
VARWindowASE <- var10.window.ase.wge(X10,x=ts_daily_df$PM10,x_p=2,trainingSize =600,horizon = 20,x_lag=1)
mean(VARWindowASE)## [1] 249.6376.9 MLR
tt_pm10_ts <- ttsplit.wge(ts_daily_df$PM10,h=20)
tt_so2_ts <- ttsplit.wge(ts_daily_df$SO2,h=20)
tt_no2_ts <- ttsplit.wge(ts_daily_df$NO2,h=20)
tt_co_ts <- ttsplit.wge(ts_daily_df$CO,h=20)
tt_o3_ts <- ttsplit.wge(ts_daily_df$O3,h=20)
lag10_so2 <- dplyr::lag(tt_so2_ts$train_ts,1)
lag10_no2 <- dplyr::lag(tt_no2_ts$train_ts,1)
lag10_co <- dplyr::lag(tt_co_ts$train_ts ,1)
tt_o3_ts$train_ts## [1] 28.083333 58.500000 93.458333 47.671642 137.458333 161.500000
## [7] 174.000000 151.764150 58.208333 53.208333 83.125000 88.875000
## [13] 36.000000 71.750000 115.500000 122.625000 140.500000 55.333333
## [19] 53.583333 39.000000 76.625000 52.250000 46.125000 23.958333
## [25] 65.119783 81.041667 46.166667 29.125000 69.833333 50.833333
## [31] 71.000000 45.625000 55.166667 79.750000 65.250000 38.000000
## [37] 19.291667 59.375000 15.458333 9.541667 23.886321 51.105962
## [43] 71.368387 57.166667 26.708333 44.791667 34.833333 27.791667
## [49] 16.082508 36.615367 38.666667 49.844692 52.208333 61.666667
## [55] 49.041667 32.916667 67.458333 55.958333 58.625000 31.982625
## [61] 38.333333 38.500000 55.458333 31.375000 33.625000 77.166667
## [67] 67.250000 74.416667 60.000000 43.000000 44.083333 57.541667
## [73] 50.416667 70.333333 43.750000 56.708333 50.083333 51.250000
## [79] 44.083333 23.583333 38.250000 70.791667 61.208333 79.875000
## [85] 77.208333 53.000000 48.541667 41.833333 56.375000 19.041667
## [91] 64.708333 33.541667 49.000000 51.041667 52.041667 64.875000
## [97] 96.541667 109.041667 66.250000 64.875000 47.687925 43.958333
## [103] 49.109996 28.904529 45.183108 45.516721 64.173358 73.044029
## [109] 32.172221 63.309387 38.100258 79.836037 62.077587 44.712629
## [115] 38.314113 35.534008 53.591854 49.451638 51.615842 63.172521
## [121] 55.182929 62.992883 64.874800 20.213496 51.205242 51.128254
## [127] 37.441587 52.086321 29.469104 45.379854 62.308550 55.294133
## [133] 56.508825 42.693846 34.575942 36.107138 43.497938 35.747862
## [139] 53.600408 39.212300 43.746008 47.809238 36.842796 46.799846
## [145] 25.422983 40.649400 45.157446 63.069871 74.609442 47.638154
## [151] 80.049892 58.578933 31.915596 59.571217 59.023750 40.187475
## [157] 39.571575 39.879525 56.217983 38.048933 47.270325 43.925646
## [163] 47.090687 21.556500 25.080817 37.185433 40.708333 49.208333
## [169] 63.416667 40.000000 18.416667 36.333333 43.791667 54.125000
## [175] 72.333333 44.083333 45.875000 39.708333 38.750000 47.875000
## [181] 51.708333 32.000000 21.708333 42.791667 45.250000 44.458333
## [187] 56.000000 49.916667 29.625000 47.958333 57.375000 79.375000
## [193] 41.958333 51.735883 61.067908 80.250000 54.666667 53.416667
## [199] 46.166667 61.291667 82.583333 61.541667 42.833333 51.791667
## [205] 53.083333 87.500000 43.583333 38.875000 43.916667 76.916667
## [211] 108.791667 82.086033 86.750000 79.958333 56.500000 31.000000
## [217] 56.833333 78.083333 112.875000 101.166667 56.166667 50.416667
## [223] 99.250000 45.291667 67.916667 85.625000 57.291667 47.458333
## [229] 39.583333 68.166667 96.750000 132.500000 51.000000 60.583333
## [235] 90.465833 89.333333 34.833333 18.625000 55.259671 71.666667
## [241] 124.125000 118.875000 47.708333 75.541667 104.875000 107.291667
## [247] 100.458333 45.041667 57.333333 112.291667 62.625000 42.333333
## [253] 99.583333 70.500000 27.916667 47.041667 62.625000 104.833333
## [259] 56.000000 94.541667 36.291667 25.833333 22.750000 41.958333
## [265] 56.166667 73.666667 86.833333 109.250000 48.958333 27.041667
## [271] 25.916667 11.500000 28.625000 66.750000 44.916667 67.333333
## [277] 86.416667 62.916667 74.250000 43.541667 76.583333 108.208333
## [283] 70.625000 24.916667 35.333333 33.083333 31.416667 48.666667
## [289] 54.958333 52.250000 81.250000 53.608587 39.333333 49.845617
## [295] 55.875000 68.041667 92.541667 89.833333 140.208333 92.500000
## [301] 23.791667 47.886333 32.386979 76.166667 53.541667 49.416667
## [307] 64.916667 92.500000 49.083333 76.333333 64.791667 62.541667
## [313] 62.375000 23.625000 34.083333 72.166667 100.208333 36.541667
## [319] 86.416667 75.666667 93.291667 117.666667 67.791667 15.416667
## [325] 2.041667 2.000000 2.000000 2.000000 2.000000 2.000000
## [331] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000
## [337] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000
## [343] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000
## [349] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000
## [355] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000
## [361] 2.000000 20.333333 124.416667 22.708333 55.083333 54.041667
## [367] 81.500000 91.333333 34.000000 35.208333 28.458333 49.375000
## [373] 85.666667 65.833333 66.958333 81.208333 22.833333 37.583333
## [379] 27.791667 87.458333 67.250000 52.375000 36.125000 44.916667
## [385] 24.041667 41.250000 57.291667 66.375000 97.333333 92.708333
## [391] 114.333333 115.666667 66.625000 36.208333 55.208333 80.333333
## [397] 60.916667 52.958333 30.208333 35.625000 26.875000 52.333333
## [403] 55.541667 81.541667 46.250000 46.291667 59.458333 77.583333
## [409] 68.625000 85.250000 57.500000 67.041667 81.625000 63.750000
## [415] 65.500000 55.750000 57.458333 56.458333 38.000000 40.250000
## [421] 78.833333 29.041667 43.000000 45.208333 62.916667 50.916667
## [427] 32.000000 20.625000 41.916667 14.000000 32.333333 58.541667
## [433] 46.541667 41.750000 52.166667 49.416667 25.750000 41.791667
## [439] 55.041667 25.708333 63.000000 58.916667 67.958333 67.208333
## [445] 50.541667 53.208333 58.958333 67.741667 42.041667 45.083333
## [451] 27.333333 57.166667 49.375000 58.291667 48.500000 23.000000
## [457] 23.000000 23.000000 23.000000 27.333333 52.583333 51.625000
## [463] 51.083333 34.333333 69.916667 51.958333 50.875000 35.916667
## [469] 43.583333 50.291667 44.416667 59.083333 43.083333 43.041667
## [475] 42.125000 51.791667 31.208333 38.416667 39.041667 42.458333
## [481] 62.666667 52.250000 38.500000 37.833333 43.208333 56.958333
## [487] 56.875000 46.541667 34.291667 47.750000 55.250000 42.083333
## [493] 40.833333 48.041667 48.791667 55.875000 49.875000 45.708333
## [499] 37.291667 47.125000 52.500000 26.125000 53.583333 51.416667
## [505] 45.958333 33.125000 28.000000 24.333333 43.375000 70.333333
## [511] 79.583333 45.625000 34.541667 31.875000 25.875000 26.083333
## [517] 32.041667 38.041667 39.250000 43.500000 49.916667 45.916667
## [523] 38.708333 54.750000 51.750000 46.041667 42.750000 33.791667
## [529] 44.829167 50.000000 33.783333 39.875000 54.458333 56.916667
## [535] 32.875000 40.458333 41.583333 35.962500 51.541667 66.212500
## [541] 68.916667 31.300000 43.041667 63.875000 71.625000 59.395833
## [547] 40.416667 70.583333 34.500000 61.366667 37.416667 39.041667
## [553] 68.083333 62.583333 42.791667 48.416667 26.291667 56.250000
## [559] 61.625000 70.875000 55.750000 52.250000 58.708333 40.125000
## [565] 75.750000 58.083333 66.791667 61.375000 68.000000 52.833333
## [571] 76.000000 53.125000 41.208333 62.875000 66.416667 49.291667
## [577] 71.083333 48.416667 42.333333 63.500000 55.666667 69.458333
## [583] 43.541667 40.000000 54.625000 96.000000 133.166667 124.833333
## [589] 94.291667 64.541667 24.916667 31.833333 61.916667 48.000000
## [595] 53.916667 113.208333 138.583333 124.416667 81.375000 56.125000
## [601] 71.083333 93.125000 106.833333 109.041667 48.708333 53.125000
## [607] 60.875000 94.666667 111.708333 92.291667 48.125000 19.566667
## [613] 45.416667 81.708333 49.666667 24.041667 40.750000 54.458333
## [619] 53.625000 70.708333 33.541667 9.000000 44.208333 58.041667
## [625] 99.750000 55.125000 45.625000 77.625000 126.791667 138.791667
## [631] 108.666667 50.666667 83.583333 70.166667 87.250000 115.166667
## [637] 56.125000 64.666667 88.833333 38.625000 17.333333 44.416667
## [643] 25.125000 16.250000 39.791667 64.583333 45.083333 54.208333
## [649] 108.333333 58.000000 33.583333 18.583333 44.625000 74.208333
## [655] 44.208333 11.958333 56.375000 79.583333 43.625000 25.583333
## [661] 27.500000 73.416667 76.833333 29.875000 50.750000 97.291667
## [667] 118.466667 101.916667 91.625000 60.208333 14.083333 39.041667
## [673] 56.833333 97.125000 105.583333 58.375000 38.000000 65.291667
## [679] 95.250000 70.083333 106.291667 42.000000 62.666667 88.625000
## [685] 97.000000 106.333333 37.750000 54.416667 55.316667 61.583333
## [691] 81.416667 37.000000 84.166667 116.958333 62.875000 52.291667
## [697] 40.916667 30.304167 47.083333 34.625000 27.125000 42.583333
## [703] 69.083333 84.000000 55.416667 18.500000 37.916667 78.333333
## [709] 24.375000 25.583333 71.791667 97.208333 46.916667 44.500000
## [715] 82.541667 116.166667 124.541667 52.041667 33.208333 34.000000
## [721] 37.500000 40.500000 26.166667 16.458333 32.208333 57.791667
## [727] 30.125000 21.208333 30.208333 35.791667 38.125000 49.875000
## [733] 11.333333 34.166667 55.000000 74.208333 82.625000 53.375000
## [739] 13.875000 33.875000 50.125000 63.791667 60.291667 78.541667
## [745] 71.125000 84.125000 58.833333 40.083333 75.166667 47.625000
## [751] 29.083333 16.625000 36.375000 60.166667 75.083333 47.250000
## [757] 45.250000 30.250000 74.416667 67.708333 51.820833 46.458333
## [763] 24.791667 28.791667 32.791667 31.916667 16.958333 24.375000
## [769] 44.375000 60.083333 71.125000 49.583333 14.833333 19.666667
## [775] 56.041667 78.291667 14.541667 55.291667 102.083333 36.250000
## [781] 44.416667 43.958333 49.541667 57.250000 45.250000 73.125000
## [787] 51.750000 49.250000 43.791667 50.916667 44.208333 52.958333
## [793] 31.250000 25.250000 22.000000 41.791667 30.375000 33.916667
## [799] 41.375000 33.791667 48.000000 33.083333 50.250000 80.708333
## [805] 33.250000 27.541667 45.500000 45.125000 33.833333 12.458333
## [811] 17.916667 43.208333 47.291667 57.333333 60.875000 45.458333
## [817] 32.708333 31.791667 35.333333 48.916667 44.208333 45.875000
## [823] 51.166667 30.208333 39.375000 54.875000 51.291667 31.416667
## [829] 36.333333 32.375000 60.125000 41.208333 30.958333 24.166667
## [835] 21.125000 33.083333 40.750000 53.791667 41.250000 26.375000
## [841] 29.250000 33.000000 41.000000 36.416667 41.291667 45.916667
## [847] 64.916667 27.750000 34.666667 38.916667 51.750000 28.541667
## [853] 38.500000 32.041667 37.000000 47.791667 43.583333 35.166667
## [859] 38.291667 38.541667 41.791667 30.083333 38.125000 48.125000
## [865] 44.333333 32.833333 48.625000 31.625000 39.625000 31.333333
## [871] 39.416667 36.791667 33.416667 37.208333 54.250000 54.916667
## [877] 44.458333 30.916667 34.291667 29.083333 39.333333 25.250000
## [883] 17.083333 41.916667 16.083333 29.541667 36.916667 49.833333
## [889] 30.500000 35.791667 36.208333 38.833333 45.041667 35.083333
## [895] 34.708333 41.291667 34.958333 39.666667 44.375000 38.625000
## [901] 35.166667 40.750000 27.375000 26.125000 24.333333 25.666667
## [907] 23.791667 28.666667 27.083333 26.583333 28.625000 23.958333
## [913] 29.208333 31.333333 23.333333 20.916667 22.875000 30.750000
## [919] 24.125000 30.041667 45.416667 59.291667 51.333333 31.916667
## [925] 26.416667 25.458333 44.208333 62.666667 72.166667 80.041667
## [931] 83.250000 60.916667 51.458333 52.000000 70.000000 65.541667
## [937] 49.541667 67.583333 44.833333 50.291667 63.833333 49.041667
## [943] 51.041667 41.166667 25.541667 54.291667 48.750000 86.541667
## [949] 90.291667 92.625000 99.375000 20.583333 33.041667 11.750000
## [955] 15.958333 39.791667 76.750000 110.625000 98.916667 134.375000
## [961] 149.041667 46.333333 54.625000 72.375000 43.791667 49.000000
## [967] 55.125000 54.125000 54.791667 26.375000 37.583333 22.000000
## [973] 20.000000 44.625000 57.041667 74.916667 86.083333 97.250000
## [979] 120.416667 65.375000 27.833333 28.083333 44.958333 54.750000
## [985] 53.416667 55.875000 56.166667 67.791667 75.416667 71.125000
## [991] 46.625000 46.791667 42.458333 42.375000 41.666667 40.958333
## [997] 33.958333 31.541667 50.375000 39.958333 26.541667 90.791667
## [1003] 100.500000 107.958333 134.458333 123.083333 17.375000 15.666667
## [1009] 39.375000 58.125000 89.250000 89.750000 89.625000 96.708333
## [1015] 53.291667 54.416667 82.375000 93.583333 59.583333 14.708333
## [1021] 16.291667 50.791667 60.208333 84.375000 99.708333 102.791667
## [1027] 107.083333 103.291667 69.083333 133.625000 92.500000 50.750000
## [1033] 71.416667 115.916667 51.416667 67.208333 100.625000 117.750000
## [1039] 70.000000 30.000000 39.125000 38.291667 24.291667 36.791667
## [1045] 66.375000 55.250000 23.791667 27.750000 46.166667 68.333333
## [1051] 84.458333 67.666667 29.875000 16.541667 55.083333 76.541667
## [1057] 63.041667 17.041667 12.333333 24.041667 46.625000 38.583333
## [1063] 71.083333 82.791667 44.375000 44.416667 25.166667 28.583333
## [1069] 48.750000 60.708333 20.625000 16.375000 37.916667 42.500000
## [1075] 38.458333 39.166667 62.625000 63.666667 53.708333 10.166667
## [1081] 11.125000 13.541667 15.208333 43.125000 23.958333 27.625000
## [1087] 16.083333 54.416667 39.958333 17.333333 34.750000 34.958333
## [1093] 31.958333 31.166667 31.125000 33.708333 72.708333 113.125000
## [1099] 121.583333 115.416667 16.666667 69.875000 45.625000 12.666667
## [1105] 11.083333 15.833333 33.583333 76.541667 28.208333 57.083333
## [1111] 71.916667 93.000000 101.875000 82.333333 39.375000 55.083333
## [1117] 75.416667 78.791667 28.833333 28.666667 36.791667 24.541667
## [1123] 33.708333 70.083333 26.833333 40.208333 80.500000 26.750000
## [1129] 16.625000 17.833333 40.625000 52.916667 66.000000 52.000000
## [1135] 31.166667 63.083333 39.166667 39.250000 58.083333 75.500000
## [1141] 33.791667 63.583333 9.041667 7.750000 18.916667 63.958333
## [1147] 38.958333 72.750000 30.208333 32.666667 44.791667 31.166667
## [1153] 35.166667 57.125000 55.333333 67.916667 76.541667 52.583333
## [1159] 35.291667 37.291667 54.750000 48.375000 9.375000 26.000000
## [1165] 42.666667 39.875000 40.875000 35.750000 25.791667 49.041667
## [1171] 33.541667 37.375000 51.333333 66.958333 38.583333 40.791667
## [1177] 46.500000 48.291667 58.750000 50.458333 29.375000 22.208333
## [1183] 30.041667 59.541667 35.000000 30.333333 57.958333 35.375000
## [1189] 43.125000 55.416667 63.458333 38.416667 42.000000 38.125000
## [1195] 23.916667 34.708333 38.083333 24.041667 23.750000 36.750000
## [1201] 34.416667 31.041667 33.958333 56.625000 54.875000 56.208333
## [1207] 38.958333 32.125000 36.125000 35.166667 41.250000 31.958333
## [1213] 29.833333 46.958333 40.541667 40.083333 38.708333 38.958333
## [1219] 25.000000 42.875000 44.000000 36.750000 32.083333 35.166667
## [1225] 33.000000 38.250000 37.333333 27.833333 26.291667 31.916667
## [1231] 35.291667 41.333333 34.416667 44.041667 48.125000 52.458333
## [1237] 49.750000 24.000000 31.125000 35.250000 72.166667 78.000000
## [1243] 67.833333 25.291667 37.083333 18.000000 20.375000 25.666667
## [1249] 31.875000 32.583333 37.708333 37.916667 36.125000 27.750000
## [1255] 35.666667 23.208333 29.708333 43.291667 25.375000 26.666667
## [1261] 38.375000 45.458333 46.750000 34.875000 38.541667 74.458333
## [1267] 50.250000 28.541667 42.083333 51.250000 31.083333 44.416667
## [1273] 55.791667 29.083333 37.500000 45.041667 36.125000 47.333333
## [1279] 56.250000 43.416667 33.833333 36.250000 37.250000 51.250000
## [1285] 30.791667 40.708333 33.916667 50.833333 45.958333 46.541667
## [1291] 32.541667 43.333333 57.333333 56.333333 50.125000 47.833333
## [1297] 40.833333 41.416667 39.916667 50.458333 102.458333 75.333333
## [1303] 54.541667 69.958333 57.333333 58.833333 34.875000 37.625000
## [1309] 64.083333 85.625000 110.541667 87.291667 60.541667 40.791667
## [1315] 50.333333 47.125000 40.833333 23.208333 54.416667 92.833333
## [1321] 81.083333 64.500000 92.375000 114.833333 114.333333 61.250000
## [1327] 59.916667 77.416667 96.416667 62.458333 47.750000 39.500000
## [1333] 51.458333 71.875000 90.500000 58.458333 58.166667 34.458333
## [1339] 51.250000 51.541667 20.458333 53.875000 93.875000 113.875000
## [1345] 127.875000 97.958333 50.916667 45.375000 54.000000 94.958333
## [1351] 66.541667 77.291667 68.416667 72.750000 42.375000 64.500000
## [1357] 84.250000 70.041667 89.791667 64.333333 44.291667 18.666667
## [1363] 25.958333 34.041667 65.000000 92.833333 113.875000 48.041667
## [1369] 76.041667 82.291667 67.208333 45.375000 89.500000 127.458333
## [1375] 127.666667 40.750000 95.166667 101.250000 57.583333 36.083333
## [1381] 61.166667 102.458333 100.125000 49.000000 43.916667 42.458333
## [1387] 95.916667 109.333333 109.375000 121.291667 135.458333 157.166667
## [1393] 42.666667 42.375000 90.916667 95.875000 52.500000 39.625000
## [1399] 38.666667 47.125000 101.000000 129.541667 128.875000 97.000000
## [1405] 147.708333 144.291667 111.791667 109.916667 86.666667 29.041667
## [1411] 33.208333 37.875000 79.833333 56.416667 24.291667 44.000000
## [1417] 45.750000 70.125000 75.958333 51.458333 40.083333 34.750000
## [1423] 30.666667 44.250000 50.666667 74.125000 93.416667 63.041667
## [1429] 28.916667 68.000000 18.583333 18.875000 38.083333 22.375000
## [1435] 58.041667 82.375000 96.875000 23.625000 23.791667 53.333333
## [1441] 30.500000## [1] 1440t=1:1440
pm10_ds <- data.frame(tx=t,pm10=tt_pm10_ts$train_ts[2:1441],so2=lag10_so2[2:1441],no2=lag10_no2[2:1441],co=lag10_co[2:1441])
ksfit=lm(pm10~t+so2+co+no2, data = pm10_ds)
summary(ksfit)##
## Call:
## lm(formula = pm10 ~ t + so2 + co + no2, data = pm10_ds)
##
## Residuals:
## Min 1Q Median 3Q Max
## -253.87 -45.53 -8.65 35.53 450.30
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 30.566914 5.384896 5.676 1.66e-08 ***
## t 0.011179 0.004761 2.348 0.01901 *
## so2 0.732379 0.124796 5.869 5.45e-09 ***
## co 0.008635 0.002765 3.123 0.00183 **
## no2 0.897182 0.091561 9.799 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 70.19 on 1435 degrees of freedom
## Multiple R-squared: 0.2416, Adjusted R-squared: 0.2395
## F-statistic: 114.3 on 4 and 1435 DF, p-value: < 2.2e-16## $type
## [1] "aic"
##
## $value
## [1] 8.42777
##
## $p
## [1] 3
##
## $q
## [1] 0
##
## $phi
## [1] 0.26780468 -0.02631680 0.06233021
##
## $theta
## [1] 0
##
## $vara
## [1] 4546.961##
## Call:
## arima(x = pm10_ds$pm10, order = c(3, 0, 0), xreg = cbind(t, pm10_ds[, 3:5]))
##
## Coefficients:
## ar1 ar2 ar3 intercept t so2 no2 co
## 0.4480 -0.0587 0.0461 70.6644 0.0057 0.3581 0.6151 -0.0020
## s.e. 0.0394 0.0308 0.0264 8.7782 0.0077 0.1556 0.1185 0.0036
##
## sigma^2 estimated as 4377: log likelihood = -8079.89, aic = 16177.77
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.00843641 66.15625 48.50201 -52.29227 77.27025 0.900903
## ACF1
## Training set -0.0006143675## Obs -0.0006143675 0.0008850667 -0.006499312 0.0120486 -0.009477762 0.02075855 -0.01600055 0.01649633 0.02980983 0.07527094 0.01436253 0.007164722 0.03740712 0.03480053 0.02950447 0.03191777 -0.0004911441 -0.0369363 0.007768205 -0.003195674 0.02261731 0.03955085 0.007572953 0.05448263## $test
## [1] "Ljung-Box test"
##
## $K
## [1] 24
##
## $chi.square
## [1] 27.8087
##
## $df
## [1] 21
##
## $pval
## [1] 0.14566067 SO2 Analysis
7.1 Stationarity
First plot the data.
- Visual evidence of trend in some part of realization.
- Realization show periodic behavior but that is not reflected in spectral density.
- Peak at 0 and 0.13. But that’s not a strong evidence.
- ACFs are not damping but has very mild evidence of sinusoidal behavior
- Mild visual evidence that first half and second half of ACFs are different.
- Variance is not constant
- Time series not a stationary time series.
7.2 Split into Train/Test
## [1] 18.534437.3 Dicky-Fuller Test
\(H_{0}\) - Unit root present
\(H_{1}\) - Unit root not present. or time series is stationary.
Conclusion
pvalue = 0.01. Reject Ho. Suggests strong evidence against presence of unit root.
No visual evidence of sesonality so time series is stationary.
## Warning in adf.test(tt_so2_ts$train_ts): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: tt_so2_ts$train_ts
## Dickey-Fuller = -4.4819, Lag order = 11, p-value = 0.01
## alternative hypothesis: stationaryp-value < 0.05 (using Cochrane-Orcutt Test) provides strong evidence that trend exist.
7.4 Model ID
## ---------WORKING... PLEASE WAIT...
##
##
## Five Smallest Values of aicAR(9,1) turns out to be the best mode as per AIC function.
7.5 ARIMA
7.5.1 Estimate Parameters
- Ljung Box test fails for K-48. So Resudial assumptions are not met.
- This series has seasonality and needs to be tranformed.
- Since these are daily observations annual seasonal component can be removed. But for this s=365 instead of s=12 due to daily observations,
##
## Coefficients of Original polynomial:
## 1.4019 -0.5427 0.0695 -0.0085 0.1103 -0.0890 -0.0217 0.2146 -0.1417
##
## Factor Roots Abs Recip System Freq
## 1-0.9910B 1.0091 0.9910 0.0000
## 1-1.2524B+0.6857B^2 0.9132+-0.7902i 0.8281 0.1135
## 1-0.2339B+0.6427B^2 0.1819+-1.2340i 0.8017 0.2267
## 1+1.0616B+0.6205B^2 -0.8554+-0.9380i 0.7877 0.3677
## 1+0.7299B -1.3700 0.7299 0.5000
## 1-0.7162B 1.3962 0.7162 0.0000
##
## ## Obs -0.003580677 0.002833143 0.002257824 -0.005517542 -0.009865679 0.005111487 -0.002897585 -0.006194265 0.01766593 -0.01688836 -0.01550766 -0.02972703 0.06277652 -0.02684268 0.0332766 -0.03924441 -0.04111658 0.003847906 -0.04384311 0.001074689 -0.01060938 0.01340347 0.04754802 -0.003186641## $test
## [1] "Ljung-Box test"
##
## $K
## [1] 24
##
## $chi.square
## [1] 22.54284
##
## $df
## [1] 14
##
## $pval
## [1] 0.06812726## Obs -0.003580677 0.002833143 0.002257824 -0.005517542 -0.009865679 0.005111487 -0.002897585 -0.006194265 0.01766593 -0.01688836 -0.01550766 -0.02972703 0.06277652 -0.02684268 0.0332766 -0.03924441 -0.04111658 0.003847906 -0.04384311 0.001074689 -0.01060938 0.01340347 0.04754802 -0.003186641 0.07430912 -0.03239694 -0.03565677 0.009201041 0.03763682 0.03658233 0.05640757 0.005842422 0.02560925 -0.08143254 -0.04506584 0.03648348 -0.03431505 0.1014667 0.02766491 0.04083197 0.01856858 -0.04007335 0.02468095 0.01147331 -0.04269395 0.06958953 -0.03259759 -0.01289488## $test
## [1] "Ljung-Box test"
##
## $K
## [1] 48
##
## $chi.square
## [1] 95.11124
##
## $df
## [1] 38
##
## $pval
## [1] 8.485069e-077.5.2 Stationarize Series
- Variance is not constant. log transformation needs to be applied before stationarzing series.
- Remove seasonal component from time series.
- Since these are daily observations annual seasonal component can be removed. But for this s=365 instead of s=12 due to daily observations,
## ---------WORKING... PLEASE WAIT...
##
##
## Five Smallest Values of aic## [1] 0## [1] -0.5099709 -0.1575921
7.5.3 Residual diagnositcs
## Obs -0.005321354 -0.01329364 -0.03335789 -0.01055413 -0.04111233 0.05351165 -0.0001298941 0.02857394 0.05623655 -0.00998852 0.01547255 0.02172267 -0.01797564 0.0007880134 0.03221442 -0.03517078 -0.03782465 0.009073126 0.01982738 0.04748212 0.01776086 0.01671858 0.04213314 -0.01442289## $test
## [1] "Ljung-Box test"
##
## $K
## [1] 24
##
## $chi.square
## [1] 21.93383
##
## $df
## [1] 22
##
## $pval
## [1] 0.4638443## Obs -0.005321354 -0.01329364 -0.03335789 -0.01055413 -0.04111233 0.05351165 -0.0001298941 0.02857394 0.05623655 -0.00998852 0.01547255 0.02172267 -0.01797564 0.0007880134 0.03221442 -0.03517078 -0.03782465 0.009073126 0.01982738 0.04748212 0.01776086 0.01671858 0.04213314 -0.01442289 -0.0006400589 -0.02179692 0.01077715 0.03591711 0.05877482 0.008174739 -0.02892867 0.01619192 -0.0102592 -0.04084097 -0.01085419 0.02591362 -0.03114494 0.06780411 0.03229151 0.06085462 -0.004367754 0.009638848 -0.005170652 -0.02481785 -0.04617717 0.02349056 0.03670645 -0.01562068## $test
## [1] "Ljung-Box test"
##
## $K
## [1] 48
##
## $chi.square
## [1] 49.18616
##
## $df
## [1] 46
##
## $pval
## [1] 0.34679377.6 Final Model
Final SO2 Model:
(1 - 1.40B + 0.558\(B^2\) - 0.088\(B^3\) + 0.028\(B^4\) - 0.118\(B^5\) + 0.088\(B^6\) + 0.028\(B^7\) - 0.218\(B^8\) + 0.148\(B^2\))(\(X_t\) - 18.53) = (1 - 0.89B) (\(a_t\))
7.7 SO2 Forecasting
Final SO2 Model:
(1 - 5\(B^{365}\) ) (\(X_t\) - 18.53) = (1 + 0.5B + 0.16\(B^2\) ) (\(a_t\))
- Hypothesis test on residual passed with strong evidence indicating residuals are white p-value > 0.05 for both K=24 and K=48
window_arima_so2<-window.ase.wge(tt_so2_ts$train_ts,phi=est_d1_so2$phi,theta=est_d1_so2$theta,trainingSize = 500,horizon = 20,s=365)
mean(window_arima_so2)## [1] 477.55337.8 Neural Network
7.8.1 Build a Model
Commented because it runs slow. But have obtained the model offline and its in powerpoint.
8 Conclusion
- For PM2.5 which is correlated to other variables, VAR Model found to be useful.
- Neural Network takes time but returns best results for univariate.
- Similar techniques can be used for CO,NO2 forecasting.
9 References
- Applied Time Series with R, Dr.Wayne Woodword,Henry L. Gray,Alan C. Elliott
- An article on ENVIRONMENTAL EFFECTS OF AIR POLLUTION AND APPLICATION OF ENGINEERED METHODS TO COMBAT THE PROBLEM, Journal of Industrial Pollution Control.
- Middle Tennessee Air Quality Forecast , Nashville Health Department.
- An article on Major Air pollutants.
- An article on common Air Pollutants and its effects on human beings.
- World Air Quality Forecast
- Kaggle article on handling Time series missing values. [1]