South ural state university, Chelyabinsk, Russian federation
# Imports
# Imports
library(fpp2)
## Warning: package 'fpp2' was built under R version 4.0.3
## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoo
## -- Attaching packages ------------------------------------------------------------------------------ fpp2 2.4 --
## v ggplot2 3.3.2 v fma 2.4
## v forecast 8.13 v expsmooth 2.3
## Warning: package 'ggplot2' was built under R version 4.0.3
## Warning: package 'forecast' was built under R version 4.0.3
##
library(forecast)
library(ggplot2)
library("readxl")
## Warning: package 'readxl' was built under R version 4.0.3
library(moments)
## Warning: package 'moments' was built under R version 4.0.3
library(forecast)
require(forecast)
require(tseries)
## Loading required package: tseries
## Warning: package 'tseries' was built under R version 4.0.3
require(markovchain)
## Loading required package: markovchain
## Warning: package 'markovchain' was built under R version 4.0.3
## Package: markovchain
## Version: 0.8.5-3
## Date: 2020-12-03
## BugReport: https://github.com/spedygiorgio/markovchain/issues
require(data.table)
## Loading required package: data.table
Full_original_data<-read_excel("F:/Phd/ALL Russia Analysis/covidActualTS.xlsx",sheet = "Chelyabinsk ")
y_lab<- "COVID 19 Infection cases in Chelyabinsk " # input name of data
Actual_date_interval <- c("2020/03/12","2020/10/27")
Forecast_date_interval <- c("2020/10/28","2020/11/5")
validation_data_days <-46
frequency<-"days"
# Data Preparation & calculate some of statistics measures
original_data<-Full_original_data$Infected
summary(original_data)
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0 898.5 7795.5 7720.8 13666.5 18393.0
sd(original_data) # calculate standard deviation
## [1] 6300.531
skewness(original_data) # calculate Cofficient of skewness
## [1] 0.09264073
kurtosis(original_data) # calculate Cofficient of kurtosis
## [1] 1.462645
rows <- NROW(original_data)
training_data<-original_data[1:(rows-validation_data_days)]
testing_data<-original_data[(rows-validation_data_days+1):rows]
AD<-fulldate<-seq(as.Date(Actual_date_interval[1]),as.Date(Actual_date_interval[2]), frequency) #input range for actual date
FD<-seq(as.Date(Forecast_date_interval[1]),as.Date(Forecast_date_interval[2]), frequency) #input range forecasting date
N_forecasting_days<-nrow(data.frame(FD))
validation_dates<-tail(AD,validation_data_days)
validation_data_by_name<-weekdays(validation_dates)
forecasting_data_by_name<-weekdays(FD)
data_series<-ts(training_data)
#plot COVID 19 infection cases in Chelyabinsk
autoplot(data_series ,xlab=paste ("Time in ", frequency, sep=" "), ylab = y_lab, main=paste ("Actual Data :", y_lab, sep=" "))
#Auto arima model
##################
require(tseries) # need to install tseries tj test Stationarity in time series
paste ("tests For Check Stationarity in series ==> ",y_lab, sep=" ")
## [1] "tests For Check Stationarity in series ==> COVID 19 Infection cases in Chelyabinsk "
kpss.test(data_series) # applay kpss test
## Warning in kpss.test(data_series): p-value smaller than printed p-value
##
## KPSS Test for Level Stationarity
##
## data: data_series
## KPSS Level = 3.6994, Truncation lag parameter = 4, p-value = 0.01
pp.test(data_series) # applay pp test
##
## Phillips-Perron Unit Root Test
##
## data: data_series
## Dickey-Fuller Z(alpha) = -3.7369, Truncation lag parameter = 4, p-value
## = 0.9004
## alternative hypothesis: stationary
adf.test(data_series) # applay adf test
##
## Augmented Dickey-Fuller Test
##
## data: data_series
## Dickey-Fuller = -2.8579, Lag order = 5, p-value = 0.2175
## alternative hypothesis: stationary
ndiffs(data_series) # Doing first diffrencing on data
## [1] 2
##Taking the first difference
diff1_x1<-diff(data_series)
autoplot(diff1_x1, xlab = paste ("Time in ", frequency ,y_lab , sep=" "), ylab=y_lab,main = "1nd differenced series")
##Testing the stationary of the first differenced series
paste ("tests For Check Stationarity in series after taking first differences in ==> ",y_lab, sep=" ")
## [1] "tests For Check Stationarity in series after taking first differences in ==> COVID 19 Infection cases in Chelyabinsk "
kpss.test(diff1_x1) # applay kpss test after taking first differences
## Warning in kpss.test(diff1_x1): p-value smaller than printed p-value
##
## KPSS Test for Level Stationarity
##
## data: diff1_x1
## KPSS Level = 1.8662, Truncation lag parameter = 4, p-value = 0.01
pp.test(diff1_x1) # applay pp test after taking first differences
##
## Phillips-Perron Unit Root Test
##
## data: diff1_x1
## Dickey-Fuller Z(alpha) = -6.4645, Truncation lag parameter = 4, p-value
## = 0.7449
## alternative hypothesis: stationary
adf.test(diff1_x1) # applay adf test after taking first differences
##
## Augmented Dickey-Fuller Test
##
## data: diff1_x1
## Dickey-Fuller = -0.92892, Lag order = 5, p-value = 0.9471
## alternative hypothesis: stationary
#Taking the second difference
diff2_x1=diff(diff1_x1)
autoplot(diff2_x1, xlab = paste ("Time in ", frequency ,y_lab , sep=" "), ylab=y_lab ,main = "2nd differenced series")
##Testing the stationary of the first differenced series
paste ("tests For Check Stationarity in series after taking Second differences in",y_lab, sep=" ")
## [1] "tests For Check Stationarity in series after taking Second differences in COVID 19 Infection cases in Chelyabinsk "
kpss.test(diff2_x1) # applay kpss test after taking Second differences
## Warning in kpss.test(diff2_x1): p-value greater than printed p-value
##
## KPSS Test for Level Stationarity
##
## data: diff2_x1
## KPSS Level = 0.18591, Truncation lag parameter = 4, p-value = 0.1
pp.test(diff2_x1) # applay pp test after taking Second differences
## Warning in pp.test(diff2_x1): p-value smaller than printed p-value
##
## Phillips-Perron Unit Root Test
##
## data: diff2_x1
## Dickey-Fuller Z(alpha) = -200.86, Truncation lag parameter = 4, p-value
## = 0.01
## alternative hypothesis: stationary
adf.test(diff2_x1) # applay adf test after taking Second differences
## Warning in adf.test(diff2_x1): p-value smaller than printed p-value
##
## Augmented Dickey-Fuller Test
##
## data: diff2_x1
## Dickey-Fuller = -7.4821, Lag order = 5, p-value = 0.01
## alternative hypothesis: stationary
####Fitting an ARIMA Model
#1. Using auto arima function
model1 <- auto.arima(data_series,stepwise=FALSE, approximation=FALSE, trace=T, test = c("kpss", "adf", "pp")) #applaying auto arima
##
## ARIMA(0,2,0) : 1536.59
## ARIMA(0,2,1) : 1524.691
## ARIMA(0,2,2) : 1525.319
## ARIMA(0,2,3) : 1522.219
## ARIMA(0,2,4) : 1522.73
## ARIMA(0,2,5) : 1521.775
## ARIMA(1,2,0) : 1526.41
## ARIMA(1,2,1) : 1522.937
## ARIMA(1,2,2) : 1524.647
## ARIMA(1,2,3) : 1523.737
## ARIMA(1,2,4) : 1522.043
## ARIMA(2,2,0) : 1527.929
## ARIMA(2,2,1) : 1524.418
## ARIMA(2,2,2) : 1526.162
## ARIMA(2,2,3) : 1524.482
## ARIMA(3,2,0) : 1528.362
## ARIMA(3,2,1) : 1524.545
## ARIMA(3,2,2) : 1524.763
## ARIMA(4,2,0) : 1519.07
## ARIMA(4,2,1) : 1520.767
## ARIMA(5,2,0) : 1521.09
##
##
##
## Best model: ARIMA(4,2,0)
model1 # show the result of autoarima
## Series: data_series
## ARIMA(4,2,0)
##
## Coefficients:
## ar1 ar2 ar3 ar4
## -0.2972 -0.1003 -0.1618 -0.244
## s.e. 0.0716 0.0739 0.0735 0.071
##
## sigma^2 estimated as 238: log likelihood=-754.36
## AIC=1518.73 AICc=1519.07 BIC=1534.75
#Make changes in the source of auto arima to run the best model
arima.string <- function (object, padding = FALSE)
{
order <- object$arma[c(1, 6, 2, 3, 7, 4, 5)]
m <- order[7]
result <- paste("ARIMA(", order[1], ",", order[2], ",",
order[3], ")", sep = "")
if (m > 1 && sum(order[4:6]) > 0) {
result <- paste(result, "(", order[4], ",", order[5],
",", order[6], ")[", m, "]", sep = "")
}
if (padding && m > 1 && sum(order[4:6]) == 0) {
result <- paste(result, " ", sep = "")
if (m <= 9) {
result <- paste(result, " ", sep = "")
}
else if (m <= 99) {
result <- paste(result, " ", sep = "")
}
else {
result <- paste(result, " ", sep = "")
}
}
if (!is.null(object$xreg)) {
if (NCOL(object$xreg) == 1 && is.element("drift", names(object$coef))) {
result <- paste(result, "with drift ")
}
else {
result <- paste("Regression with", result, "errors")
}
}
else {
if (is.element("constant", names(object$coef)) || is.element("intercept",
names(object$coef))) {
result <- paste(result, "with non-zero mean")
}
else if (order[2] == 0 && order[5] == 0) {
result <- paste(result, "with zero mean ")
}
else {
result <- paste(result, " ")
}
}
if (!padding) {
result <- gsub("[ ]*$", "", result)
}
return(result)
}
source("stringthearima.R")
bestmodel <- arima.string(model1, padding = TRUE)
bestmodel <- substring(bestmodel,7,11)
bestmodel <- gsub(" ", "", bestmodel)
bestmodel <- gsub(")", "", bestmodel)
bestmodel <- strsplit(bestmodel, ",")[[1]]
bestmodel <- c(strtoi(bestmodel[1]),strtoi(bestmodel[2]),strtoi(bestmodel[3]))
bestmodel
## [1] 4 2 0
#strtoi(bestmodel[3])
library(forecast) # install library forecast
x1_model1= arima(data_series, order=c(bestmodel)) # Run Best model of auto arima for forecasting
x1_model1 # Show result of best model of auto arima
##
## Call:
## arima(x = data_series, order = c(bestmodel))
##
## Coefficients:
## ar1 ar2 ar3 ar4
## -0.2972 -0.1003 -0.1618 -0.244
## s.e. 0.0716 0.0739 0.0735 0.071
##
## sigma^2 estimated as 232.7: log likelihood = -754.36, aic = 1518.73
paste("accuracy of autoarima Model For ==> ",y_lab, sep=" ")
## [1] "accuracy of autoarima Model For ==> COVID 19 Infection cases in Chelyabinsk "
accuracy(x1_model1) # aacuracy of best model from auto arima
## ME RMSE MAE MPE MAPE MASE
## Training set 0.6393234 15.17283 9.296751 0.9028434 3.87662 0.1182776
## ACF1
## Training set 0.002878065
x1_model1$x # show result of best model from auto arima
## NULL
checkresiduals(x1_model1,xlab = paste ("Time in ", frequency ,y_lab , sep=" "), col.main="black", col.lab="blue", col.sub="black", cex.main=1, cex.lab=1, cex.sub=1,font.main=4, font.lab=4, ylab=y_lab)
##
## Ljung-Box test
##
## data: Residuals from ARIMA(4,2,0)
## Q* = 1.9046, df = 6, p-value = 0.9283
##
## Model df: 4. Total lags used: 10
# checkresiduals from best model from using auto arima
paste("Box-Ljung test , Ljung-Box test For Modelling for ==> ",y_lab, sep=" ")
## [1] "Box-Ljung test , Ljung-Box test For Modelling for ==> COVID 19 Infection cases in Chelyabinsk "
Box.test(x1_model1$residuals^2, lag=20, type="Ljung-Box") # Do test for resdulas by using Box-Ljung test , Ljung-Box test For Modelling
##
## Box-Ljung test
##
## data: x1_model1$residuals^2
## X-squared = 36.352, df = 20, p-value = 0.01398
library(tseries)
jarque.bera.test(x1_model1$residuals) # Do test jarque.bera.test
##
## Jarque Bera Test
##
## data: x1_model1$residuals
## X-squared = 336.99, df = 2, p-value < 2.2e-16
#Actual Vs Fitted
par(mfrow=c(1,2))
plot(data_series, col='red',lwd=2, main="Actual vs Fitted Plot", xlab='Timein (days)', ylab=y_lab) # plot actual and Fitted model
lines(fitted(x1_model1), col='blue')
#Test data
x1_test <- ts(testing_data, start =(rows-validation_data_days+1) ) # make testing data in time series and start from rows-6
forecasting_auto_arima <- forecast(x1_model1, h=N_forecasting_days+validation_data_days)
validation_forecast<-head(forecasting_auto_arima$mean,validation_data_days)
MAPE_Per_Day<-round(abs(((testing_data-validation_forecast)/testing_data)*100) ,3)
paste ("MAPE % For ",validation_data_days,frequency,"by using bats Model for ==> ",y_lab, sep=" ")
## [1] "MAPE % For 46 days by using bats Model for ==> COVID 19 Infection cases in Chelyabinsk "
MAPE_Mean_All<-paste(round(mean(MAPE_Per_Day),3),"% MAPE ",validation_data_days,frequency,y_lab,sep=" ")
MAPE_auto_arima<-paste(round(MAPE_Per_Day,3),"%")
MAPE_auto.arima_Model<-paste(MAPE_Per_Day ,"%")
paste (" MAPE that's Error of Forecasting for ",validation_data_days," days in bats Model for ==> ",y_lab, sep=" ")
## [1] " MAPE that's Error of Forecasting for 46 days in bats Model for ==> COVID 19 Infection cases in Chelyabinsk "
paste(MAPE_Mean_All,"%")
## [1] "1.568 % MAPE 46 days COVID 19 Infection cases in Chelyabinsk %"
paste ("MAPE that's Error of Forecasting day by day for ",validation_data_days," days in bats Model for ==> ",y_lab, sep=" ")
## [1] "MAPE that's Error of Forecasting day by day for 46 days in bats Model for ==> COVID 19 Infection cases in Chelyabinsk "
data.frame(date_auto.arima=validation_dates,validation_data_by_name,actual_data=testing_data,forecasting_auto.arima=validation_forecast,MAPE_auto.arima_Model)
## date_auto.arima validation_data_by_name actual_data forecasting_auto.arima
## 1 2020-09-12 Saturday 14453 14449.21
## 2 2020-09-13 Sunday 14514 14514.56
## 3 2020-09-14 Monday 14581 14578.72
## 4 2020-09-15 Tuesday 14652 14644.08
## 5 2020-09-16 Wednesday 14724 14709.37
## 6 2020-09-17 Thursday 14796 14774.72
## 7 2020-09-18 Friday 14857 14840.16
## 8 2020-09-19 Saturday 14928 14905.28
## 9 2020-09-20 Sunday 14997 14970.50
## 10 2020-09-21 Monday 15065 15035.69
## 11 2020-09-22 Tuesday 15142 15100.91
## 12 2020-09-23 Wednesday 15220 15166.18
## 13 2020-09-24 Thursday 15295 15231.42
## 14 2020-09-25 Friday 15367 15296.67
## 15 2020-09-26 Saturday 15424 15361.90
## 16 2020-09-27 Sunday 15489 15427.12
## 17 2020-09-28 Monday 15551 15492.36
## 18 2020-09-29 Tuesday 15619 15557.59
## 19 2020-09-30 Wednesday 15692 15622.83
## 20 2020-10-01 Thursday 15768 15688.07
## 21 2020-10-02 Friday 15847 15753.30
## 22 2020-10-03 Saturday 15926 15818.54
## 23 2020-10-04 Sunday 16007 15883.77
## 24 2020-10-05 Monday 16093 15949.01
## 25 2020-10-06 Tuesday 16182 16014.24
## 26 2020-10-07 Wednesday 16274 16079.48
## 27 2020-10-08 Thursday 16368 16144.71
## 28 2020-10-09 Friday 16460 16209.95
## 29 2020-10-10 Saturday 16553 16275.18
## 30 2020-10-11 Sunday 16649 16340.42
## 31 2020-10-12 Monday 16743 16405.65
## 32 2020-10-13 Tuesday 16832 16470.89
## 33 2020-10-14 Wednesday 16924 16536.12
## 34 2020-10-15 Thursday 17021 16601.36
## 35 2020-10-16 Friday 17123 16666.59
## 36 2020-10-17 Saturday 17223 16731.83
## 37 2020-10-18 Sunday 17325 16797.06
## 38 2020-10-19 Monday 17427 16862.30
## 39 2020-10-20 Tuesday 17526 16927.53
## 40 2020-10-21 Wednesday 17630 16992.77
## 41 2020-10-22 Thursday 17740 17058.00
## 42 2020-10-23 Friday 17857 17123.24
## 43 2020-10-24 Saturday 17986 17188.47
## 44 2020-10-25 Sunday 18122 17253.71
## 45 2020-10-26 Monday 18255 17318.94
## 46 2020-10-27 Tuesday 18393 17384.18
## MAPE_auto.arima_Model
## 1 0.026 %
## 2 0.004 %
## 3 0.016 %
## 4 0.054 %
## 5 0.099 %
## 6 0.144 %
## 7 0.113 %
## 8 0.152 %
## 9 0.177 %
## 10 0.195 %
## 11 0.271 %
## 12 0.354 %
## 13 0.416 %
## 14 0.458 %
## 15 0.403 %
## 16 0.399 %
## 17 0.377 %
## 18 0.393 %
## 19 0.441 %
## 20 0.507 %
## 21 0.591 %
## 22 0.675 %
## 23 0.77 %
## 24 0.895 %
## 25 1.037 %
## 26 1.195 %
## 27 1.364 %
## 28 1.519 %
## 29 1.678 %
## 30 1.853 %
## 31 2.015 %
## 32 2.145 %
## 33 2.292 %
## 34 2.465 %
## 35 2.665 %
## 36 2.852 %
## 37 3.047 %
## 38 3.24 %
## 39 3.415 %
## 40 3.614 %
## 41 3.844 %
## 42 4.109 %
## 43 4.434 %
## 44 4.791 %
## 45 5.128 %
## 46 5.485 %
data.frame(FD,forecating_date=forecasting_data_by_name,forecasting_by_auto.arima=tail(forecasting_auto_arima$mean,N_forecasting_days))
## FD forecating_date forecasting_by_auto.arima
## 1 2020-10-28 Wednesday 17449.42
## 2 2020-10-29 Thursday 17514.65
## 3 2020-10-30 Friday 17579.89
## 4 2020-10-31 Saturday 17645.12
## 5 2020-11-01 Sunday 17710.36
## 6 2020-11-02 Monday 17775.59
## 7 2020-11-03 Tuesday 17840.83
## 8 2020-11-04 Wednesday 17906.06
## 9 2020-11-05 Thursday 17971.30
plot(forecasting_auto_arima)
x1_test <- ts(testing_data, start =(rows-validation_data_days+1) )
lines(x1_test, col='red',lwd=2)
graph4<-autoplot(forecasting_auto_arima,xlab = paste ("Time in ", frequency ,y_lab , sep=" "), col.main="black", col.lab="blue", col.sub="black", cex.main=1, cex.lab=1, cex.sub=1,font.main=4, font.lab=4, ylab=y_lab)
graph4
## Error of forecasting
Error_auto.arima<-abs(testing_data-validation_forecast) # Absolute error of forecast (AEOF)
REOF_A_auto.arima<-abs(((testing_data-validation_forecast)/testing_data)*100) #Relative error of forecast (divided by actual)(REOF_A)
REOF_F_auto.arima<-abs(((testing_data-validation_forecast)/validation_forecast)*100) #Relative error of forecast (divided by forecast)(REOF_F)
correlation_auto.arima<-cor(testing_data,validation_forecast, method = c("pearson")) # correlation coefficient between predicted and actual values
RMSE_auto.arima<-sqrt(sum((Error_auto.arima^2))/validation_data_days) # Root mean square forecast error
MAD_auto.arima<-abs((sum(testing_data-validation_forecast))/validation_data_days) # average forecast accuracy
AEOF_auto.arima<-c(Error_auto.arima)
REOF_auto.arima1<-c(paste(round(REOF_A_auto.arima,3),"%"))
REOF_auto.arima2<-c(paste(round(REOF_F_auto.arima,3),"%"))
data.frame(correlation_auto.arima,RMSE_auto.arima,MAPE_Mean_All,MAD_auto.arima) # analysis of Error by using Holt's linear model shows result of correlation ,MSE ,MPER
## correlation_auto.arima RMSE_auto.arima
## 1 0.9952468 393.4057
## MAPE_Mean_All
## 1 1.568 % MAPE 46 days COVID 19 Infection cases in Chelyabinsk
## MAD_auto.arima
## 1 270.987
data.frame(validation_dates,Validation_day_name=validation_data_by_name,AEOF_auto.arima,REOF_A_auto.arima=REOF_auto.arima1,REOF_F_auto.arima=REOF_auto.arima2) # Analysis of error shows result AEOF,REOF_A,REOF_F
## validation_dates Validation_day_name AEOF_auto.arima REOF_A_auto.arima
## 1 2020-09-12 Saturday 3.7892379 0.026 %
## 2 2020-09-13 Sunday 0.5566942 0.004 %
## 3 2020-09-14 Monday 2.2807896 0.016 %
## 4 2020-09-15 Tuesday 7.9205882 0.054 %
## 5 2020-09-16 Wednesday 14.6269250 0.099 %
## 6 2020-09-17 Thursday 21.2752135 0.144 %
## 7 2020-09-18 Friday 16.8391063 0.113 %
## 8 2020-09-19 Saturday 22.7153268 0.152 %
## 9 2020-09-20 Sunday 26.5003477 0.177 %
## 10 2020-09-21 Monday 29.3089476 0.195 %
## 11 2020-09-22 Tuesday 41.0897591 0.271 %
## 12 2020-09-23 Wednesday 53.8150210 0.354 %
## 13 2020-09-24 Thursday 63.5780154 0.416 %
## 14 2020-09-25 Friday 70.3341120 0.458 %
## 15 2020-09-26 Saturday 62.1042372 0.403 %
## 16 2020-09-27 Sunday 61.8783334 0.399 %
## 17 2020-09-28 Monday 58.6417525 0.377 %
## 18 2020-09-29 Tuesday 61.4073600 0.393 %
## 19 2020-09-30 Wednesday 69.1693237 0.441 %
## 20 2020-10-01 Thursday 79.9329091 0.507 %
## 21 2020-10-02 Friday 93.6986290 0.591 %
## 22 2020-10-03 Saturday 107.4636072 0.675 %
## 23 2020-10-04 Sunday 123.2292183 0.77 %
## 24 2020-10-05 Monday 143.9939749 0.895 %
## 25 2020-10-06 Tuesday 167.7585212 1.037 %
## 26 2020-10-07 Wednesday 194.5232942 1.195 %
## 27 2020-10-08 Thursday 223.2880048 1.364 %
## 28 2020-10-09 Friday 250.0529537 1.519 %
## 29 2020-10-10 Saturday 277.8178527 1.678 %
## 30 2020-10-11 Sunday 308.5826973 1.853 %
## 31 2020-10-12 Monday 337.3475398 2.015 %
## 32 2020-10-13 Tuesday 361.1123384 2.145 %
## 33 2020-10-14 Wednesday 387.8771712 2.292 %
## 34 2020-10-15 Thursday 419.6420118 2.465 %
## 35 2020-10-16 Friday 456.4068543 2.665 %
## 36 2020-10-17 Saturday 491.1717006 2.852 %
## 37 2020-10-18 Sunday 527.9365360 3.047 %
## 38 2020-10-19 Monday 564.7013720 3.24 %
## 39 2020-10-20 Tuesday 598.4662079 3.415 %
## 40 2020-10-21 Wednesday 637.2310446 3.614 %
## 41 2020-10-22 Thursday 681.9958836 3.844 %
## 42 2020-10-23 Friday 733.7607217 4.109 %
## 43 2020-10-24 Saturday 797.5255598 4.434 %
## 44 2020-10-25 Sunday 868.2903974 4.791 %
## 45 2020-10-26 Monday 936.0552347 5.128 %
## 46 2020-10-27 Tuesday 1008.8200723 5.485 %
## REOF_F_auto.arima
## 1 0.026 %
## 2 0.004 %
## 3 0.016 %
## 4 0.054 %
## 5 0.099 %
## 6 0.144 %
## 7 0.113 %
## 8 0.152 %
## 9 0.177 %
## 10 0.195 %
## 11 0.272 %
## 12 0.355 %
## 13 0.417 %
## 14 0.46 %
## 15 0.404 %
## 16 0.401 %
## 17 0.379 %
## 18 0.395 %
## 19 0.443 %
## 20 0.51 %
## 21 0.595 %
## 22 0.679 %
## 23 0.776 %
## 24 0.903 %
## 25 1.048 %
## 26 1.21 %
## 27 1.383 %
## 28 1.543 %
## 29 1.707 %
## 30 1.888 %
## 31 2.056 %
## 32 2.192 %
## 33 2.346 %
## 34 2.528 %
## 35 2.738 %
## 36 2.936 %
## 37 3.143 %
## 38 3.349 %
## 39 3.535 %
## 40 3.75 %
## 41 3.998 %
## 42 4.285 %
## 43 4.64 %
## 44 5.032 %
## 45 5.405 %
## 46 5.803 %