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 <-11
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 = 4.4678, 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) = -4.3559, Truncation lag parameter = 4, p-value
## = 0.8656
## alternative hypothesis: stationary
adf.test(data_series) # applay adf test
##
## Augmented Dickey-Fuller Test
##
## data: data_series
## Dickey-Fuller = -3.1872, Lag order = 6, p-value = 0.09118
## 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 = 1.5507, 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) = -7.6935, Truncation lag parameter = 4, p-value
## = 0.6768
## alternative hypothesis: stationary
adf.test(diff1_x1) # applay adf test after taking first differences
##
## Augmented Dickey-Fuller Test
##
## data: diff1_x1
## Dickey-Fuller = -1.3882, Lag order = 6, p-value = 0.8327
## 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 = 0.12227, 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) = -240.46, 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 = -8.035, 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) : 1797.84
## ARIMA(0,2,1) : 1783.551
## ARIMA(0,2,2) : 1784.025
## ARIMA(0,2,3) : 1779.896
## ARIMA(0,2,4) : 1780.11
## ARIMA(0,2,5) : 1778.563
## ARIMA(1,2,0) : 1785.471
## ARIMA(1,2,1) : 1781.27
## ARIMA(1,2,2) : 1782.888
## ARIMA(1,2,3) : 1781.297
## ARIMA(1,2,4) : 1778.875
## ARIMA(2,2,0) : 1786.935
## ARIMA(2,2,1) : 1782.608
## ARIMA(2,2,2) : 1784.25
## ARIMA(2,2,3) : 1781.725
## ARIMA(3,2,0) : 1787.066
## ARIMA(3,2,1) : 1782.283
## ARIMA(3,2,2) : 1781.973
## ARIMA(4,2,0) : 1775.63
## ARIMA(4,2,1) : 1777.185
## ARIMA(5,2,0) : 1777.583
##
##
##
## Best model: ARIMA(4,2,0)
model1 # show the result of autoarima
## Series: data_series
## ARIMA(4,2,0)
##
## Coefficients:
## ar1 ar2 ar3 ar4
## -0.2946 -0.0965 -0.1608 -0.2438
## s.e. 0.0656 0.0678 0.0675 0.0652
##
## sigma^2 estimated as 203.2: log likelihood=-882.67
## AIC=1775.35 AICc=1775.63 BIC=1792.24
#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
## -0.2946 -0.0965 -0.1608 -0.2438
## s.e. 0.0656 0.0678 0.0675 0.0652
##
## sigma^2 estimated as 199.5: log likelihood = -882.67, aic = 1775.35
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 0.8067865 14.05949 8.512275 0.7514777 3.240755 0.1083733
## ACF1
## Training set 0.002178755
x1_model1$x # show result of best model from auto arima
## NULL
checkresiduals(x1_model1,xlab = paste ("Time in ", frequency ,y_lab , sep=" "), col.main="black", col.lab="blue", col.sub="black", cex.main=1, cex.lab=1, cex.sub=1,font.main=4, font.lab=4, ylab=y_lab) # checkresiduals from best model from using auto arima
##
## Ljung-Box test
##
## data: Residuals from ARIMA(4,2,0)
## Q* = 2.2434, df = 6, p-value = 0.896
##
## Model df: 4. Total lags used: 10
paste("Box-Ljung test , Ljung-Box test For Modelling for ==> ",y_lab, sep=" ")
## [1] "Box-Ljung test , Ljung-Box test For Modelling for ==> COVID 19 Infection cases in Chelyabinsk "
Box.test(x1_model1$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$residuals^2
## X-squared = 53.207, df = 20, p-value = 7.578e-05
library(tseries)
jarque.bera.test(x1_model1$residuals) # Do test jarque.bera.test
##
## Jarque Bera Test
##
## data: x1_model1$residuals
## X-squared = 553.5, df = 2, p-value < 2.2e-16
#Actual Vs Fitted
par(mfrow=c(1,2))
plot(data_series, col='red',lwd=2, main="Actual vs Fitted Plot", xlab='Timein (days)', ylab=y_lab) # plot actual and Fitted model
lines(fitted(x1_model1), col='blue')
#Test data
x1_test <- ts(testing_data, start =(rows-validation_data_days+1) ) # make testing data in time series and start from rows-6
forecasting_auto_arima <- forecast(x1_model1, h=N_forecasting_days+validation_data_days)
validation_forecast<-head(forecasting_auto_arima$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 11 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 11 days in bats Model for ==> COVID 19 Infection cases in Chelyabinsk "
paste(MAPE_Mean_All,"%")
## [1] "0.31 % MAPE 11 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 11 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 2020-10-17 Saturday 17223 17223.78
## 2 2020-10-18 Sunday 17325 17322.90
## 3 2020-10-19 Monday 17427 17420.61
## 4 2020-10-20 Tuesday 17526 17517.87
## 5 2020-10-21 Wednesday 17630 17615.96
## 6 2020-10-22 Thursday 17740 17714.48
## 7 2020-10-23 Friday 17857 17813.21
## 8 2020-10-24 Saturday 17986 17911.82
## 9 2020-10-25 Sunday 18122 18010.17
## 10 2020-10-26 Monday 18255 18108.47
## 11 2020-10-27 Tuesday 18393 18206.77
## MAPE_auto.arima_Model
## 1 0.005 %
## 2 0.012 %
## 3 0.037 %
## 4 0.046 %
## 5 0.08 %
## 6 0.144 %
## 7 0.245 %
## 8 0.412 %
## 9 0.617 %
## 10 0.803 %
## 11 1.012 %
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 18305.15
## 2 2020-10-29 Thursday 18403.58
## 3 2020-10-30 Friday 18502.00
## 4 2020-10-31 Saturday 18600.40
## 5 2020-11-01 Sunday 18698.79
## 6 2020-11-02 Monday 18797.17
## 7 2020-11-03 Tuesday 18895.56
## 8 2020-11-04 Wednesday 18993.95
## 9 2020-11-05 Thursday 19092.34
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 0.9979072 83.68786
## MAPE_Mean_All MAD_auto.arima
## 1 0.31 % MAPE 11 days COVID 19 Infection cases in Chelyabinsk 56.17795
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 2020-10-17 Saturday 0.7812086 0.005 %
## 2 2020-10-18 Sunday 2.0965450 0.012 %
## 3 2020-10-19 Monday 6.3910309 0.037 %
## 4 2020-10-20 Tuesday 8.1312243 0.046 %
## 5 2020-10-21 Wednesday 14.0394516 0.08 %
## 6 2020-10-22 Thursday 25.5174573 0.144 %
## 7 2020-10-23 Friday 43.7853963 0.245 %
## 8 2020-10-24 Saturday 74.1818364 0.412 %
## 9 2020-10-25 Sunday 111.8327297 0.617 %
## 10 2020-10-26 Monday 146.5349362 0.803 %
## 11 2020-10-27 Tuesday 186.2280264 1.012 %
## REOF_F_auto.arima
## 1 0.005 %
## 2 0.012 %
## 3 0.037 %
## 4 0.046 %
## 5 0.08 %
## 6 0.144 %
## 7 0.246 %
## 8 0.414 %
## 9 0.621 %
## 10 0.809 %
## 11 1.023 %