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 Deaths 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$Death
summary(original_data)
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.00 3.00 88.00 65.16 104.00 159.00
sd(original_data) # calculate standard deviation
## [1] 53.3066
skewness(original_data) # calculate Cofficient of skewness
## [1] 0.003022683
kurtosis(original_data) # calculate Cofficient of kurtosis
## [1] 1.522648
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 Deaths 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 = 4.3275, 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) = -4.3196, Truncation lag parameter = 4, p-value
## = 0.8677
## alternative hypothesis: stationary
adf.test(data_series) # applay adf test
##
## Augmented Dickey-Fuller Test
##
## data: data_series
## Dickey-Fuller = -2.1162, Lag order = 6, p-value = 0.5272
## alternative hypothesis: stationary
ndiffs(data_series) # Doing first diffrencing on data
## [1] 1
##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 Deaths cases in Chelyabinsk "
kpss.test(diff1_x1) # applay kpss test after taking first differences
##
## KPSS Test for Level Stationarity
##
## data: diff1_x1
## KPSS Level = 0.39903, Truncation lag parameter = 4, p-value = 0.07757
pp.test(diff1_x1) # applay pp test after taking first differences
## Warning in pp.test(diff1_x1): p-value smaller than printed p-value
##
## Phillips-Perron Unit Root Test
##
## data: diff1_x1
## Dickey-Fuller Z(alpha) = -217.64, Truncation lag parameter = 4, p-value
## = 0.01
## alternative hypothesis: stationary
adf.test(diff1_x1) # applay adf test after taking first differences
##
## Augmented Dickey-Fuller Test
##
## data: diff1_x1
## Dickey-Fuller = -3.1228, Lag order = 6, p-value = 0.1049
## 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 Deaths 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 = 0.01464, 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) = -248.63, 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 = -11.461, 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(0,1,0) : 894.3487
## ARIMA(0,1,0) with drift : 866.1337
## ARIMA(0,1,1) : 887.0238
## ARIMA(0,1,1) with drift : 866.0249
## ARIMA(0,1,2) : 889.0688
## ARIMA(0,1,2) with drift : 867.4303
## ARIMA(0,1,3) : 874.5235
## ARIMA(0,1,3) with drift : 858.353
## ARIMA(0,1,4) : 875.1446
## ARIMA(0,1,4) with drift : 860.029
## ARIMA(0,1,5) : 875.2645
## ARIMA(0,1,5) with drift : 860.8915
## ARIMA(1,1,0) : 886.3034
## ARIMA(1,1,0) with drift : 866.2097
## ARIMA(1,1,1) : 855.8743
## ARIMA(1,1,1) with drift : 862.1341
## ARIMA(1,1,2) : 856.5271
## ARIMA(1,1,2) with drift : 864.2279
## ARIMA(1,1,3) : 847.323
## ARIMA(1,1,3) with drift : 846.1526
## ARIMA(1,1,4) : 849.1109
## ARIMA(1,1,4) with drift : 848.1202
## ARIMA(2,1,0) : 887.1569
## ARIMA(2,1,0) with drift : 868.1613
## ARIMA(2,1,1) : 857.1905
## ARIMA(2,1,1) with drift : 864.2275
## ARIMA(2,1,2) : 854.7511
## ARIMA(2,1,2) with drift : 854.1882
## ARIMA(2,1,3) : 848.1552
## ARIMA(2,1,3) with drift : 847.3552
## ARIMA(3,1,0) : 867.0422
## ARIMA(3,1,0) with drift : 857.01
## ARIMA(3,1,1) : 847.9264
## ARIMA(3,1,1) with drift : 847.1016
## ARIMA(3,1,2) : 849.8887
## ARIMA(3,1,2) with drift : 849.1264
## ARIMA(4,1,0) : 861.7194
## ARIMA(4,1,0) with drift : 854.9335
## ARIMA(4,1,1) : 849.9125
## ARIMA(4,1,1) with drift : 849.1459
## ARIMA(5,1,0) : 860.2871
## ARIMA(5,1,0) with drift : 855.1957
##
##
##
## Best model: ARIMA(1,1,3) with drift
model1 # show the result of autoarima
## Series: data_series
## ARIMA(1,1,3) with drift
##
## Coefficients:
## ar1 ma1 ma2 ma3 drift
## 0.9213 -0.8964 -0.1518 0.2557 0.6597
## s.e. 0.0443 0.0781 0.0959 0.0726 0.2800
##
## sigma^2 estimated as 2.738: log likelihood=-416.88
## AIC=845.75 AICc=846.15 BIC=866.06
#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] 1 1 3
#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 ma1 ma2 ma3
## 0.9604 -0.9205 -0.1491 0.2491
## s.e. 0.0249 0.0708 0.0975 0.0719
##
## sigma^2 estimated as 2.711: log likelihood = -418.52, aic = 847.04
paste("accuracy of autoarima Model For ==> ",y_lab, sep=" ")
## [1] "accuracy of autoarima Model For ==> COVID 19 Deaths cases in Chelyabinsk "
accuracy(x1_model1) # aacuracy of best model from auto arima
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 0.169536 1.642885 0.8083013 1.535179 3.611448 1.198705 -0.01944753
x1_model1$x # show result of best model from auto arima
## NULL
checkresiduals(x1_model1,xlab = paste ("Time in ", frequency ,y_lab , sep=" "), col.main="black", col.lab="blue", col.sub="black", cex.main=1, cex.lab=1, cex.sub=1,font.main=4, font.lab=4, ylab=y_lab) # checkresiduals from best model from using auto arima
##
## Ljung-Box test
##
## data: Residuals from ARIMA(1,1,3)
## Q* = 11.098, df = 6, p-value = 0.0854
##
## Model df: 4. Total lags used: 10
paste("Box-Ljung test , Ljung-Box test For Modelling for ==> ",y_lab, sep=" ")
## [1] "Box-Ljung test , Ljung-Box test For Modelling for ==> COVID 19 Deaths cases in Chelyabinsk "
Box.test(x1_model1$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$residuals^2
## X-squared = 21.665, df = 20, p-value = 0.359
library(tseries)
jarque.bera.test(x1_model1$residuals) # Do test jarque.bera.test
##
## Jarque Bera Test
##
## data: x1_model1$residuals
## X-squared = 10483, df = 2, p-value < 2.2e-16
#Actual Vs Fitted
par(mfrow=c(1,2))
plot(data_series, col='red',lwd=2, main="Actual vs Fitted Plot", xlab='Timein (days)', ylab=y_lab) # plot actual and Fitted model
lines(fitted(x1_model1), col='blue')
#Test data
x1_test <- ts(testing_data, start =(rows-validation_data_days+1) ) # make testing data in time series and start from rows-6
forecasting_auto_arima <- forecast(x1_model1, h=N_forecasting_days+validation_data_days)
validation_forecast<-head(forecasting_auto_arima$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 Deaths 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 Deaths cases in Chelyabinsk "
paste(MAPE_Mean_All,"%")
## [1] "2.305 % MAPE 11 days COVID 19 Deaths 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 Deaths 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 147 147.5308
## 2 2020-10-18 Sunday 147 148.3946
## 3 2020-10-19 Monday 152 149.1766
## 4 2020-10-20 Tuesday 152 149.9276
## 5 2020-10-21 Wednesday 155 150.6489
## 6 2020-10-22 Thursday 155 151.3416
## 7 2020-10-23 Friday 155 152.0068
## 8 2020-10-24 Saturday 159 152.6457
## 9 2020-10-25 Sunday 159 153.2592
## 10 2020-10-26 Monday 159 153.8484
## 11 2020-10-27 Tuesday 159 154.4143
## MAPE_auto.arima_Model
## 1 0.361 %
## 2 0.949 %
## 3 1.857 %
## 4 1.363 %
## 5 2.807 %
## 6 2.36 %
## 7 1.931 %
## 8 3.996 %
## 9 3.611 %
## 10 3.24 %
## 11 2.884 %
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 154.9577
## 2 2020-10-29 Thursday 155.4796
## 3 2020-10-30 Friday 155.9809
## 4 2020-10-31 Saturday 156.4622
## 5 2020-11-01 Sunday 156.9245
## 6 2020-11-02 Monday 157.3685
## 7 2020-11-03 Tuesday 157.7948
## 8 2020-11-04 Wednesday 158.2043
## 9 2020-11-05 Thursday 158.5975
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 0.9591067 4.007607
## MAPE_Mean_All MAD_auto.arima
## 1 2.305 % MAPE 11 days COVID 19 Deaths cases in Chelyabinsk 3.255041
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 0.5308279 0.361 %
## 2 2020-10-18 Sunday 1.3946037 0.949 %
## 3 2020-10-19 Monday 2.8233790 1.857 %
## 4 2020-10-20 Tuesday 2.0723547 1.363 %
## 5 2020-10-21 Wednesday 4.3510952 2.807 %
## 6 2020-10-22 Thursday 3.6584208 2.36 %
## 7 2020-10-23 Friday 2.9931985 1.931 %
## 8 2020-10-24 Saturday 6.3543405 3.996 %
## 9 2020-10-25 Sunday 5.7408019 3.611 %
## 10 2020-10-26 Monday 5.1515791 3.24 %
## 11 2020-10-27 Tuesday 4.5857085 2.884 %
## REOF_F_auto.arima
## 1 0.36 %
## 2 0.94 %
## 3 1.893 %
## 4 1.382 %
## 5 2.888 %
## 6 2.417 %
## 7 1.969 %
## 8 4.163 %
## 9 3.746 %
## 10 3.348 %
## 11 2.97 %