South ural state university, Chelyabinsk, Russian federation
# Imports
library(fpp2)
## 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
##
library(forecast)
library(ggplot2)
library("readxl")
library(moments)
library(forecast)
require(forecast)
require(tseries)
## Loading required package: tseries
require(markovchain)
## Loading required package: markovchain
## 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
#population in turkey = 84833098
#WHO COVID-19 global table data January 11th 2021 at 11.53.00 AM.csv
Full_original_data<-read.csv("F:/Phd/COVID 19 in 2021/WHO_data.csv")
View(Full_original_data)
y_lab<- "Covid 19 Infection cases in Turkey " # input name of data
Actual_date_interval <- c("2020/01/03","2021/01/10")
Forecast_date_interval <- c("2021/01/11","2021/01/17")
validation_data_days <-7
frequency <-"days"
Population <-84833098 # population in Turkey
# Data Preparation & calculate some of statistics measures
Covid_data<-Full_original_data[Full_original_data$Country == "Turkey", ]
original_data<-Covid_data$Cumulative_cases
View(original_data)
summary(original_data)
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0 24718 207371 282555 330349 1502780
sd(original_data) # calculate standard deviation
## [1] 343734.1
skewness(original_data) # calculate Cofficient of skewness
## [1] 2.079256
kurtosis(original_data) # calculate Cofficient of kurtosis
## [1] 6.916256
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)
##bats model
# Data Modeling
data_series<-ts(training_data)
autoplot(data_series ,xlab=paste ("Time in ", frequency, sep=" "), ylab = y_lab, main=paste ("Actual Data :", y_lab, sep=" "))
model_bats<-bats(data_series)
accuracy(model_bats) # accuracy on training data
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 88.14223 1281.37 295.18 NaN Inf 0.07620521 -0.024969
# Print Model Parameters
model_bats
## BATS(1, {2,0}, -, -)
##
## Call: bats(y = data_series)
##
## Parameters
## Alpha: 0.0910748
## AR coefficients: 1.948154 -0.950386
##
## Seed States:
## [,1]
## [1,] 12.12023
## [2,] 0.00000
## [3,] 0.00000
##
## Sigma: 1281.37
## AIC: 7431.541
plot(model_bats,xlab = paste ("Time in ", frequency ,y_lab , sep=" "), col.main="black", col.lab="black", col.sub="black", cex.main=1, cex.lab=1, cex.sub=1,font.main=4, font.lab=4)
# Testing Data Evaluation
forecasting_bats <- predict(model_bats, h=N_forecasting_days+validation_data_days)
validation_forecast<-head(forecasting_bats$mean,validation_data_days)
MAPE_Per_Day<-round( abs(((testing_data-validation_forecast)/testing_data)*100) ,3)
paste ("MAPE % For ",validation_data_days,frequency,"by using bats Model for ==> ",y_lab, sep=" ")
## [1] "MAPE % For 7 days by using bats Model for ==> Covid 19 Infection cases in Turkey "
MAPE_Mean_All<-paste(round(mean(MAPE_Per_Day),3),"% MAPE ",validation_data_days,frequency,y_lab,sep=" ")
MAPE_bats<-paste(round(MAPE_Per_Day,3),"%")
MAPE_bats_Model<-paste(MAPE_Per_Day ,"%")
paste (" MAPE that's Error of Forecasting for ",validation_data_days," days in bats Model for ==> ",y_lab, sep=" ")
## [1] " MAPE that's Error of Forecasting for 7 days in bats Model for ==> Covid 19 Infection cases in Turkey "
paste(MAPE_Mean_All,"%")
## [1] "0.953 % MAPE 7 days Covid 19 Infection cases in Turkey %"
paste ("MAPE that's Error of Forecasting day by day for ",validation_data_days," days in bats Model for ==> ",y_lab, sep=" ")
## [1] "MAPE that's Error of Forecasting day by day for 7 days in bats Model for ==> Covid 19 Infection cases in Turkey "
data.frame(date_bats=validation_dates,validation_data_by_name,actual_data=testing_data,forecasting_bats=validation_forecast,MAPE_bats_Model)
## date_bats validation_data_by_name actual_data forecasting_bats
## 1 2021-01-04 Monday 1427574 1428104
## 2 2021-01-05 Tuesday 1441269 1437754
## 3 2021-01-06 Wednesday 1455763 1446663
## 4 2021-01-07 Thursday 1469593 1454847
## 5 2021-01-08 Friday 1481764 1462325
## 6 2021-01-09 Saturday 1493243 1469113
## 7 2021-01-10 Sunday 1502780 1475233
## MAPE_bats_Model
## 1 0.037 %
## 2 0.244 %
## 3 0.625 %
## 4 1.003 %
## 5 1.312 %
## 6 1.616 %
## 7 1.833 %
data.frame(FD,forecating_date=forecasting_data_by_name,forecasting_by_bats=tail(forecasting_bats$mean,N_forecasting_days))
## FD forecating_date forecasting_by_bats
## 1 2021-01-11 Monday 1480702
## 2 2021-01-12 Tuesday 1485542
## 3 2021-01-13 Wednesday 1489773
## 4 2021-01-14 Thursday 1493414
## 5 2021-01-15 Friday 1496489
## 6 2021-01-16 Saturday 1499017
## 7 2021-01-17 Sunday 1501020
plot(forecasting_bats)
x1_test <- ts(testing_data, start =(rows-validation_data_days+1) )
lines(x1_test, col='red',lwd=2)
graph1<-autoplot(forecasting_bats,xlab = paste ("Time in ", frequency ,y_lab , sep=" "), col.main="black", col.lab="black", col.sub="black", cex.main=1, cex.lab=1, cex.sub=1,font.main=4, font.lab=4, ylab=y_lab)
graph1
## Error of forecasting
Error_bats<-abs(testing_data-validation_forecast) # Absolute error of forecast (AEOF)
REOF_A_bats<-abs(((testing_data-validation_forecast)/testing_data)*100) #Relative error of forecast (divided by actual)(REOF_A)
REOF_F_bats<-abs(((testing_data-validation_forecast)/validation_forecast)*100) #Relative error of forecast (divided by forecast)(REOF_F)
correlation_bats<-cor(testing_data,validation_forecast, method = c("pearson")) # correlation coefficient between predicted and actual values
RMSE_bats<-sqrt(sum((Error_bats^2))/validation_data_days) # Root mean square forecast error
MSE_bats<-(sum((Error_bats^2))/validation_data_days) # Root mean square forecast error
MAD_bats<-abs((sum(testing_data-validation_forecast))/validation_data_days) # average forecast accuracy
AEOF_bats<-c(Error_bats)
REOF_Abats<-c(paste(round(REOF_A_bats,3),"%"))
REOF_Fbats<-c(paste(round(REOF_F_bats,3),"%"))
data.frame(correlation_bats,MSE_bats,RMSE_bats,MAPE_Mean_All,MAD_bats) # analysis of Error by using Bats Model shows result of correlation ,MSE ,MPER
## correlation_bats MSE_bats RMSE_bats
## 1 0.9997113 290264237 17037.14
## MAPE_Mean_All MAD_bats
## 1 0.953 % MAPE 7 days Covid 19 Infection cases in Turkey 13992.26
data.frame(validation_dates,Validation_day_name=validation_data_by_name,AEOF_bats,REOF_Abats,REOF_Fbats) # Analysis of error shows result AEOF,REOF_A,REOF_F
## validation_dates Validation_day_name AEOF_bats REOF_Abats REOF_Fbats
## 1 2021-01-04 Monday 530.4594 0.037 % 0.037 %
## 2 2021-01-05 Tuesday 3514.5199 0.244 % 0.244 %
## 3 2021-01-06 Wednesday 9099.8948 0.625 % 0.629 %
## 4 2021-01-07 Thursday 14745.7634 1.003 % 1.014 %
## 5 2021-01-08 Friday 19439.4458 1.312 % 1.329 %
## 6 2021-01-09 Saturday 24129.5614 1.616 % 1.642 %
## 7 2021-01-10 Sunday 27547.1054 1.833 % 1.867 %
## TBATS Model
# Data Modeling
data_series<-ts(training_data)
model_TBATS<-forecast:::fitSpecificTBATS(data_series,use.box.cox=FALSE, use.beta=TRUE, seasonal.periods=c(6),use.damping=FALSE,k.vector=c(2))
accuracy(model_TBATS) # accuracy on training data
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 28.17034 1285.432 425.7613 NaN Inf 0.1099168 -0.000599952
# Print Model Parameters
model_TBATS
## TBATS(1, {0,0}, 1, {<6,2>})
##
## Call: NULL
##
## Parameters
## Alpha: 0.9881883
## Beta: 1.058792
## Damping Parameter: 1
## Gamma-1 Values: -0.0007540652
## Gamma-2 Values: 0.001468062
##
## Seed States:
## [,1]
## [1,] -80.46459
## [2,] 17.85984
## [3,] 26.85867
## [4,] 42.36763
## [5,] 105.24842
## [6,] -51.13483
##
## Sigma: 1285.432
## AIC: 7441.864
plot(model_TBATS,xlab = paste ("Time in ", frequency ,y_lab , sep=" "), col.main="black", col.lab="black", col.sub="black", cex.main=1, cex.lab=1, cex.sub=1,font.main=4, font.lab=4, ylab=y_lab)
# Testing Data Evaluation
forecasting_tbats <- predict(model_TBATS, h=N_forecasting_days+validation_data_days)
validation_forecast<-head(forecasting_tbats$mean,validation_data_days)
MAPE_Per_Day<-round( abs(((testing_data-validation_forecast)/testing_data)*100) ,3)
paste ("MAPE % For ",validation_data_days,frequency,"by using TBATS Model for ==> ",y_lab, sep=" ")
## [1] "MAPE % For 7 days by using TBATS Model for ==> Covid 19 Infection cases in Turkey "
MAPE_Mean_All<-paste(round(mean(MAPE_Per_Day),3),"% MAPE ",validation_data_days,frequency,y_lab,sep=" ")
MAPE_TBATS<-paste(round(MAPE_Per_Day,3),"%")
MAPE_TBATS_Model<-paste(MAPE_Per_Day ,"%")
paste (" MAPE that's Error of Forecasting for ",validation_data_days," days in TBATS Model for ==> ",y_lab, sep=" ")
## [1] " MAPE that's Error of Forecasting for 7 days in TBATS Model for ==> Covid 19 Infection cases in Turkey "
paste(MAPE_Mean_All,"%")
## [1] "0.421 % MAPE 7 days Covid 19 Infection cases in Turkey %"
paste ("MAPE that's Error of Forecasting day by day for ",validation_data_days," days in TBATS Model for ==> ",y_lab, sep=" ")
## [1] "MAPE that's Error of Forecasting day by day for 7 days in TBATS Model for ==> Covid 19 Infection cases in Turkey "
data.frame(date_TBATS=validation_dates,validation_data_by_name,actual_data=testing_data,forecasting_TBATS=validation_forecast,MAPE_TBATS_Model)
## date_TBATS validation_data_by_name actual_data forecasting_TBATS
## 1 2021-01-04 Monday 1427574 1428572
## 2 2021-01-05 Tuesday 1441269 1439674
## 3 2021-01-06 Wednesday 1455763 1450568
## 4 2021-01-07 Thursday 1469593 1461264
## 5 2021-01-08 Friday 1481764 1472399
## 6 2021-01-09 Saturday 1493243 1483495
## 7 2021-01-10 Sunday 1502780 1494358
## MAPE_TBATS_Model
## 1 0.07 %
## 2 0.111 %
## 3 0.357 %
## 4 0.567 %
## 5 0.632 %
## 6 0.653 %
## 7 0.56 %
data.frame(FD,forecating_date=forecasting_data_by_name,forecasting_by_TBATS=tail(forecasting_tbats$mean,N_forecasting_days))
## FD forecating_date forecasting_by_TBATS
## 1 2021-01-11 Monday 1505459
## 2 2021-01-12 Tuesday 1516353
## 3 2021-01-13 Wednesday 1527049
## 4 2021-01-14 Thursday 1538184
## 5 2021-01-15 Friday 1549280
## 6 2021-01-16 Saturday 1560143
## 7 2021-01-17 Sunday 1571244
plot(forecasting_tbats)
x1_test <- ts(testing_data, start =(rows-validation_data_days+1) )
lines(x1_test, col='red',lwd=2)
graph2<-autoplot(forecasting_tbats,xlab = paste ("Time in ", frequency ,y_lab , sep=" "), col.main="black", col.lab="black", col.sub="black", cex.main=1, cex.lab=1, cex.sub=1,font.main=4, font.lab=4, ylab=y_lab)
graph2
## Error of forecasting TBATS Model
Error_tbats<-abs(testing_data-validation_forecast) # Absolute error of forecast (AEOF)
REOF_A_tbats1<-abs(((testing_data-validation_forecast)/testing_data)*100) #Relative error of forecast (divided by actual)(REOF_A)
REOF_F_tbats<-abs(((testing_data-validation_forecast)/validation_forecast)*100) #Relative error of forecast (divided by forecast)(REOF_F)
correlation_tbats<-cor(testing_data,validation_forecast, method = c("pearson")) # correlation coefficient between predicted and actual values
RMSE_tbats<-sqrt(sum((Error_tbats^2))/validation_data_days) # Root mean square forecast error
MSE_tbats<-(sum((Error_tbats^2))/validation_data_days) # Root mean square forecast error
MAD_tbats<-abs((sum(testing_data-validation_forecast))/validation_data_days) # average forecast accuracy
AEOF_tbats<-c(Error_tbats)
REOF_A_tbats<-c(paste(round(REOF_A_tbats1,3),"%"))
REOF_F_tbats<-c(paste(round(REOF_F_tbats,3),"%"))
data.frame(correlation_tbats,MSE_tbats,RMSE_tbats,MAPE_Mean_All,MAD_tbats) # analysis of Error by using Holt's linear model shows result of correlation ,MSE ,MPER
## correlation_tbats MSE_tbats RMSE_tbats
## 1 0.9976799 50511728 7107.16
## MAPE_Mean_All MAD_tbats
## 1 0.421 % MAPE 7 days Covid 19 Infection cases in Turkey 5950.992
data.frame(validation_dates,Validation_day_name=validation_data_by_name,AEOF_tbats,REOF_A_tbats,REOF_F_tbats) # Analysis of error shows result AEOF,REOF_A,REOF_F
## validation_dates Validation_day_name AEOF_tbats REOF_A_tbats REOF_F_tbats
## 1 2021-01-04 Monday 998.4443 0.07 % 0.07 %
## 2 2021-01-05 Tuesday 1595.0433 0.111 % 0.111 %
## 3 2021-01-06 Wednesday 5194.9870 0.357 % 0.358 %
## 4 2021-01-07 Thursday 8329.4838 0.567 % 0.57 %
## 5 2021-01-08 Friday 9365.0595 0.632 % 0.636 %
## 6 2021-01-09 Saturday 9748.4690 0.653 % 0.657 %
## 7 2021-01-10 Sunday 8422.3432 0.56 % 0.564 %
## Holt's linear trend
# Data Modeling
data_series<-ts(training_data)
model_holt<-holt(data_series,h=N_forecasting_days+validation_data_days,lambda = "auto")
accuracy(model_holt) # accuracy on training data
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set -60.48006 1453.343 378.9246 Inf Inf 0.09782514 0.4216128
# Print Model Parameters
summary(model_holt$model)
## Holt's method
##
## Call:
## holt(y = data_series, h = N_forecasting_days + validation_data_days,
##
## Call:
## lambda = "auto")
##
## Box-Cox transformation: lambda= 0.429
##
## Smoothing parameters:
## alpha = 0.9702
## beta = 0.6349
##
## Initial states:
## l = -2.1606
## b = -0.3631
##
## sigma: 1.3156
##
## AIC AICc BIC
## 2374.587 2374.753 2394.114
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set -60.48006 1453.343 378.9246 Inf Inf 0.09782514 0.4216128
# Testing Data Evaluation
forecasting_holt <- predict(model_holt, h=N_forecasting_days+validation_data_days,lambda = "auto")
validation_forecast<-head(forecasting_holt$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 holt Model for ==> ",y_lab, sep=" ")
## [1] "MAPE % For 7 days by using holt Model for ==> Covid 19 Infection cases in Turkey "
MAPE_Mean_All<-paste(round(mean(MAPE_Per_Day),3),"% MAPE ",validation_data_days,frequency,y_lab,sep=" ")
MAPE_holt<-paste(round(MAPE_Per_Day,3),"%")
MAPE_holt_Model<-paste(MAPE_Per_Day ,"%")
paste (" MAPE that's Error of Forecasting for ",validation_data_days," days in holt Model for ==> ",y_lab, sep=" ")
## [1] " MAPE that's Error of Forecasting for 7 days in holt Model for ==> Covid 19 Infection cases in Turkey "
paste(MAPE_Mean_All,"%")
## [1] "0.148 % MAPE 7 days Covid 19 Infection cases in Turkey %"
paste ("MAPE that's Error of Forecasting day by day for ",validation_data_days," days in holt Model for ==> ",y_lab, sep=" ")
## [1] "MAPE that's Error of Forecasting day by day for 7 days in holt Model for ==> Covid 19 Infection cases in Turkey "
data.frame(date_holt=validation_dates,validation_data_by_name,actual_data=testing_data,forecasting_holt=validation_forecast,MAPE_holt_Model)
## date_holt validation_data_by_name actual_data forecasting_holt
## 1 2021-01-04 Monday 1427574 1429700
## 2 2021-01-05 Tuesday 1441269 1441696
## 3 2021-01-06 Wednesday 1455763 1453749
## 4 2021-01-07 Thursday 1469593 1465860
## 5 2021-01-08 Friday 1481764 1478028
## 6 2021-01-09 Saturday 1493243 1490253
## 7 2021-01-10 Sunday 1502780 1502536
## MAPE_holt_Model
## 1 0.149 %
## 2 0.03 %
## 3 0.138 %
## 4 0.254 %
## 5 0.252 %
## 6 0.2 %
## 7 0.016 %
data.frame(FD,forecating_date=forecasting_data_by_name,forecasting_by_holt=tail(forecasting_holt$mean,N_forecasting_days))
## FD forecating_date forecasting_by_holt
## 1 2021-01-11 Monday 1514877
## 2 2021-01-12 Tuesday 1527275
## 3 2021-01-13 Wednesday 1539731
## 4 2021-01-14 Thursday 1552245
## 5 2021-01-15 Friday 1564816
## 6 2021-01-16 Saturday 1577446
## 7 2021-01-17 Sunday 1590133
plot(forecasting_holt)
x1_test <- ts(testing_data, start =(rows-validation_data_days+1) )
lines(x1_test, col='red',lwd=2)
graph3<-autoplot(forecasting_holt,xlab = paste ("Time in ", frequency ,y_lab , sep=" "), col.main="black", col.lab="black", col.sub="black", cex.main=1, cex.lab=1, cex.sub=1,font.main=4, font.lab=4, ylab=y_lab)
graph3
## Error of forecasting by using Holt's linear model
Error_Holt<-abs(testing_data-validation_forecast) # Absolute error of forecast (AEOF)
REOF_A_Holt1<-abs(((testing_data-validation_forecast)/testing_data)*100) #Relative error of forecast (divided by actual)(REOF_A)
REOF_F_Holt<-abs(((testing_data-validation_forecast)/validation_forecast)*100) #Relative error of forecast (divided by forecast)(REOF_F)
correlation_Holt<-cor(testing_data,validation_forecast, method = c("pearson")) # correlation coefficient between predicted and actual values
RMSE_Holt<-sqrt(sum((Error_Holt^2))/validation_data_days) # Root mean square forecast error
MSE_Holt<-(sum((Error_Holt^2))/validation_data_days) # Root mean square forecast error
MAD_Holt<-abs((sum(testing_data-validation_forecast))/validation_data_days) # average forecast accuracy
AEOF_Holt<-c(Error_Holt)
REOF_A_Holt<-c(paste(round(REOF_A_Holt1,3),"%"))
REOF_F_Holt<-c(paste(round(REOF_F_Holt,3),"%"))
REOF_A_Holt11<-mean(abs(((testing_data-validation_forecast)/testing_data)*100))
data.frame(correlation_Holt,MSE_Holt,RMSE_Holt,MAPE_Mean_All,MAD_Holt) # analysis of Error by using Holt's linear model shows result of correlation ,MSE ,MPER
## correlation_Holt MSE_Holt RMSE_Holt
## 1 0.997545 6521742 2553.77
## MAPE_Mean_All MAD_Holt
## 1 0.148 % MAPE 7 days Covid 19 Infection cases in Turkey 1451.935
data.frame(validation_dates,Validation_day_name=validation_data_by_name,AEOF_Holt,REOF_A_Holt,REOF_F_Holt) # Analysis of error shows result AEOF,REOF_A,REOF_F
## validation_dates Validation_day_name AEOF_Holt REOF_A_Holt REOF_F_Holt
## 1 2021-01-04 Monday 2126.1368 0.149 % 0.149 %
## 2 2021-01-05 Tuesday 427.0793 0.03 % 0.03 %
## 3 2021-01-06 Wednesday 2013.7114 0.138 % 0.139 %
## 4 2021-01-07 Thursday 3733.1677 0.254 % 0.255 %
## 5 2021-01-08 Friday 3736.2219 0.252 % 0.253 %
## 6 2021-01-09 Saturday 2989.8066 0.2 % 0.201 %
## 7 2021-01-10 Sunday 243.8546 0.016 % 0.016 %
#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 Turkey "
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 = 4.1387, Truncation lag parameter = 5, p-value = 0.01
pp.test(data_series) # applay pp test
## Warning in pp.test(data_series): p-value greater than printed p-value
##
## Phillips-Perron Unit Root Test
##
## data: data_series
## Dickey-Fuller Z(alpha) = 7.6255, Truncation lag parameter = 5, p-value
## = 0.99
## alternative hypothesis: stationary
adf.test(data_series) # applay adf test
##
## Augmented Dickey-Fuller Test
##
## data: data_series
## Dickey-Fuller = -1.8175, Lag order = 7, p-value = 0.654
## 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=" "), col.main="black", col.lab="black", col.sub="black", cex.main=1, cex.lab=1, cex.sub=1,font.main=4, font.lab=4, ylab=y_lab,main = "1nd differenced series")
## Warning: Ignoring unknown parameters: col.main, col.lab, col.sub, cex.main,
## cex.lab, cex.sub, font.main, font.lab
##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 Turkey "
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 = 2.0918, Truncation lag parameter = 5, 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) = -9.37, Truncation lag parameter = 5, p-value =
## 0.5857
## alternative hypothesis: stationary
adf.test(diff1_x1) # applay adf test after taking first differences
##
## Augmented Dickey-Fuller Test
##
## data: diff1_x1
## Dickey-Fuller = -2.9286, Lag order = 7, p-value = 0.185
## alternative hypothesis: stationary
#Taking the second difference
diff2_x1=diff(diff1_x1)
autoplot(diff2_x1, xlab = paste ("Time in ", frequency ,y_lab , sep=" "), col.main="black", col.lab="black", col.sub="black", cex.main=1, cex.lab=1, cex.sub=1,font.main=4, font.lab=4, ylab=y_lab ,main = "2nd differenced series")
## Warning: Ignoring unknown parameters: col.main, col.lab, col.sub, cex.main,
## cex.lab, cex.sub, font.main, font.lab
##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 Turkey "
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 = 0.061882, Truncation lag parameter = 5, 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) = -364.1, Truncation lag parameter = 5, 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 = -4.9047, Lag order = 7, 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) : 6272.345
## ARIMA(0,2,1) : 6273.742
## ARIMA(0,2,2) : 6275.726
## ARIMA(0,2,3) : 6277.755
## ARIMA(0,2,4) : 6275.208
## ARIMA(0,2,5) : 6276.566
## ARIMA(1,2,0) : 6273.726
## ARIMA(1,2,1) : 6272.805
## ARIMA(1,2,2) : Inf
## ARIMA(1,2,3) : 6275.636
## ARIMA(1,2,4) : 6275.3
## ARIMA(2,2,0) : 6275.709
## ARIMA(2,2,1) : Inf
## ARIMA(2,2,2) : 6276.234
## ARIMA(2,2,3) : 6278.252
## ARIMA(3,2,0) : 6277.677
## ARIMA(3,2,1) : 6275.775
## ARIMA(3,2,2) : Inf
## ARIMA(4,2,0) : 6275.123
## ARIMA(4,2,1) : 6275.58
## ARIMA(5,2,0) : 6276.527
##
##
##
## Best model: ARIMA(0,2,0)
model1 # show the result of autoarima
## Series: data_series
## ARIMA(0,2,0)
##
## sigma^2 estimated as 1691490: log likelihood=-3135.17
## AIC=6272.33 AICc=6272.35 BIC=6276.23
#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] 0 2 0
strtoi(bestmodel[3])
## [1] 0
#2. Using ACF and PACF Function
#par(mfrow=c(1,2)) # Code for making two plot in one graph
acf(diff2_x1,xlab = paste ("Time in ", frequency ,y_lab , sep=" "), col.main="black", col.lab="black", col.sub="black", cex.main=1, cex.lab=1, cex.sub=1,font.main=4, font.lab=4, ylab=y_lab, main=paste("ACF-2nd differenced series ",y_lab, sep=" ",lag.max=20)) # plot ACF "auto correlation function after taking second diffrences
pacf(diff2_x1,xlab = paste ("Time in ", frequency ,y_lab , sep=" "), col.main="black", col.lab="black", col.sub="black", cex.main=1, cex.lab=1, cex.sub=1,font.main=4, font.lab=4, ylab=y_lab,main=paste("PACF-2nd differenced series ",y_lab, sep=" ",lag.max=20)) # plot PACF " Partial auto correlation function after taking second diffrences
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))
##
##
## sigma^2 estimated as 1691490: log likelihood = -3135.17, aic = 6272.33
paste ("accuracy of autoarima Model For ==> ",y_lab, sep=" ")
## [1] "accuracy of autoarima Model For ==> Covid 19 Infection cases in Turkey "
accuracy(x1_model1) # aacuracy of best model from auto arima
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 30.46322 1297.024 305.3025 0.7617839 2.400916 0.07881846 0.0413259
x1_model1$x # show result of best model from auto arima
## NULL
checkresiduals(x1_model1,xlab = paste ("Time in ", frequency ,y_lab , sep=" "), col.main="black", col.lab="black", col.sub="black", cex.main=1, cex.lab=1, cex.sub=1,font.main=4, font.lab=4, ylab=y_lab) # checkresiduals from best model from using auto arima
##
## Ljung-Box test
##
## data: Residuals from ARIMA(0,2,0)
## Q* = 18.011, df = 10, p-value = 0.05478
##
## Model df: 0. Total lags used: 10
paste("Box-Ljung test , Ljung-Box test For Modelling for ==> ",y_lab, sep=" ")
## [1] "Box-Ljung test , Ljung-Box test For Modelling for ==> Covid 19 Infection cases in Turkey "
Box.test(x1_model1$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$residuals^2
## X-squared = 0.22673, df = 20, p-value = 1
library(tseries)
jarque.bera.test(x1_model1$residuals) # Do test jarque.bera.test
##
## Jarque Bera Test
##
## data: x1_model1$residuals
## X-squared = 859785, df = 2, p-value < 2.2e-16
#Actual Vs Fitted
plot(data_series, col='red',lwd=2, main="Actual vs Fitted Plot", xlab='Time in (days)', ylab=y_lab) # plot actual and Fitted model
lines(fitted(x1_model1), col='black')
#Test data
x1_test <- ts(testing_data, start =(rows-validation_data_days+1) ) # make testing data in time series and start from rows-6
forecasting_auto_arima <- forecast(x1_model1, h=N_forecasting_days+validation_data_days)
validation_forecast<-head(forecasting_auto_arima$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 7 days by using bats Model for ==> Covid 19 Infection cases in Turkey "
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 7 days in bats Model for ==> Covid 19 Infection cases in Turkey "
paste(MAPE_Mean_All,"%")
## [1] "0.364 % MAPE 7 days Covid 19 Infection cases in Turkey %"
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 7 days in bats Model for ==> Covid 19 Infection cases in Turkey "
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 2021-01-04 Monday 1427574 1428877
## 2 2021-01-05 Tuesday 1441269 1440057
## 3 2021-01-06 Wednesday 1455763 1451237
## 4 2021-01-07 Thursday 1469593 1462417
## 5 2021-01-08 Friday 1481764 1473597
## 6 2021-01-09 Saturday 1493243 1484777
## 7 2021-01-10 Sunday 1502780 1495957
## MAPE_auto.arima_Model
## 1 0.091 %
## 2 0.084 %
## 3 0.311 %
## 4 0.488 %
## 5 0.551 %
## 6 0.567 %
## 7 0.454 %
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 2021-01-11 Monday 1507137
## 2 2021-01-12 Tuesday 1518317
## 3 2021-01-13 Wednesday 1529497
## 4 2021-01-14 Thursday 1540677
## 5 2021-01-15 Friday 1551857
## 6 2021-01-16 Saturday 1563037
## 7 2021-01-17 Sunday 1574217
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="black", 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
MSE_auto.arima<-(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,MSE_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 MSE_auto.arima RMSE_auto.arima
## 1 0.9978134 37153254 6095.347
## MAPE_Mean_All MAD_auto.arima
## 1 0.364 % MAPE 7 days Covid 19 Infection cases in Turkey 5009.571
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 2021-01-04 Monday 1303 0.091 %
## 2 2021-01-05 Tuesday 1212 0.084 %
## 3 2021-01-06 Wednesday 4526 0.311 %
## 4 2021-01-07 Thursday 7176 0.488 %
## 5 2021-01-08 Friday 8167 0.551 %
## 6 2021-01-09 Saturday 8466 0.567 %
## 7 2021-01-10 Sunday 6823 0.454 %
## REOF_F_auto.arima
## 1 0.091 %
## 2 0.084 %
## 3 0.312 %
## 4 0.491 %
## 5 0.554 %
## 6 0.57 %
## 7 0.456 %
# SIR Model
#install.packages("dplyr")
library(deSolve)
first<-rows-13
secondr<-rows-7
vector_SIR<-original_data[first:secondr]
Infected <- c(vector_SIR)
Day <- 1:(length(Infected))
N <- Population # population of the us
SIR <- function(time, state, parameters) {
par <- as.list(c(state, parameters))
with(par, {
dS <- -beta/N * I * S
dI <- beta/N * I * S - gamma * I
dR <- gamma * I
list(c(dS, dI, dR))
})
}
init <- c(S = N-Infected[1], I = Infected[1], R = 0)
RSS <- function(parameters) {
names(parameters) <- c("beta", "gamma")
out <- ode(y = init, times = Day, func = SIR, parms = parameters)
fit <- out[ , 3]
sum((Infected - fit)^2)
}
# optimize with some sensible conditions
Opt <- optim(c(0.5, 0.5), RSS, method = "L-BFGS-B",
lower = c(0, 0), upper = c(10, 10))
Opt$message
## [1] "CONVERGENCE: REL_REDUCTION_OF_F <= FACTR*EPSMCH"
Opt_par <- setNames(Opt$par, c("beta", "gamma"))
Opt_par
## beta gamma
## 0.2277098 0.2114388
# beta gamma
# 0.6512503 0.4920399
out <- ode(y = init, times = Day, func = SIR, parms = Opt_par)
plot(out)
plot(out, obs=data.frame(time=Day, I=Infected))
result_SIR<-data.frame(out)
validation_forecast<-result_SIR$I
## Error of forecasting
Error_SIR<-abs(testing_data-validation_forecast) # Absolute error of forecast (AEOF)
REOF_A_SIR<-abs(((testing_data-validation_forecast)/testing_data)*100) #Relative error of forecast (divided by actual)(REOF_A)
REOF_F_SIR<-abs(((testing_data-validation_forecast)/validation_forecast)*100) #Relative error of forecast (divided by forecast)(REOF_F)
correlation_SIR<-cor(testing_data,validation_forecast, method = c("pearson")) # correlation coefficient between predicted and actual values
RMSE_SIR<-sqrt(sum((Error_SIR^2))/validation_data_days) # Root mean square forecast error
MSE_SIR<-(sum((Error_SIR^2))/validation_data_days) # Root mean square forecast error
MAD_SIR<-abs((sum(testing_data-validation_forecast))/validation_data_days) # average forecast accuracy
AEOF_SIR<-c(Error_SIR)
REOF_A_SIR<-c(paste(round(REOF_A_SIR,3),"%"))
REOF_A_SIR1<-mean(abs(((testing_data-validation_forecast)/testing_data)*100))
REOF_F_SIR<-c(paste(round(REOF_F_SIR,3),"%"))
MAPE_Mean_All<-paste(round(mean(abs(((testing_data-validation_forecast)/testing_data)*100)),3),"% MAPE ",validation_data_days,frequency,y_lab,sep=" ")
data.frame(correlation_SIR,MSE_SIR,RMSE_SIR,MAPE_Mean_All,MAD_SIR) # analysis of Error by using SIR's linear model shows result of correlation ,MSE ,MPER
## correlation_SIR MSE_SIR RMSE_SIR
## 1 0.9998478 8008897684 89492.44
## MAPE_Mean_All MAD_SIR
## 1 6.101 % MAPE 7 days Covid 19 Infection cases in Turkey 89447.12
data.frame(validation_dates,Validation_day_name=validation_data_by_name,AEOF_SIR,REOF_A_SIR,REOF_F_SIR,validation_forecast,testing_data) # Analysis of error shows result AEOF,REOF_A,REOF_F
## validation_dates Validation_day_name AEOF_SIR REOF_A_SIR REOF_F_SIR
## 1 2021-01-04 Monday 94334.00 6.608 % 7.076 %
## 2 2021-01-05 Tuesday 91542.20 6.351 % 6.782 %
## 3 2021-01-06 Wednesday 90450.57 6.213 % 6.625 %
## 4 2021-01-07 Thursday 89640.65 6.1 % 6.496 %
## 5 2021-01-08 Friday 88159.54 5.95 % 6.326 %
## 6 2021-01-09 Saturday 87013.61 5.827 % 6.188 %
## 7 2021-01-10 Sunday 84989.30 5.655 % 5.994 %
## validation_forecast testing_data
## 1 1333240 1427574
## 2 1349727 1441269
## 3 1365312 1455763
## 4 1379952 1469593
## 5 1393604 1481764
## 6 1406229 1493243
## 7 1417791 1502780
## forecasting by SIR model
Infected <- c(tail(original_data,validation_data_days))
Day <- 1:(length(Infected))
N <- Population # population of the us
SIR <- function(time, state, parameters) {
par <- as.list(c(state, parameters))
with(par, {
dS <- -beta/N * I * S
dI <- beta/N * I * S - gamma * I
dR <- gamma * I
list(c(dS, dI, dR))
})
}
init <- c(S = N-Infected[1], I = Infected[1], R = 0)
RSS <- function(parameters) {
names(parameters) <- c("beta", "gamma")
out <- ode(y = init, times = Day, func = SIR, parms = parameters)
fit <- out[ , 3]
sum((Infected - fit)^2)
}
# optimize with some sensible conditions
Opt <- optim(c(0.5, 0.5), RSS, method = "L-BFGS-B",
lower = c(0, 0), upper = c(10, 10))
Opt$message
## [1] "CONVERGENCE: REL_REDUCTION_OF_F <= FACTR*EPSMCH"
Opt_par <- setNames(Opt$par, c("beta", "gamma"))
Opt_par
## beta gamma
## 0.2097288 0.1954336
# beta gamma
# 0.6512503 0.4920399
out <- ode(y = init, times = Day, func = SIR, parms = Opt_par)
plot(out)
plot(out, obs=data.frame(time=Day, I=Infected))
result_SIR <-data.frame(out)
data.frame(FD,forecating_date=forecasting_data_by_name,forecasting_by_SIR=result_SIR$I)
## FD forecating_date forecasting_by_SIR
## 1 2021-01-11 Monday 1427574
## 2 2021-01-12 Tuesday 1442500
## 3 2021-01-13 Wednesday 1456514
## 4 2021-01-14 Thursday 1469581
## 5 2021-01-15 Friday 1481667
## 6 2021-01-16 Saturday 1492740
## 7 2021-01-17 Sunday 1502772
# Choose Best model by least error
paste("System Summarizes Error ==> ( MAPE ) of Forecasting by using bats model and BATS Model, Holt's Linear Models , and autoarima for ==> ", y_lab , sep=" ")
## [1] "System Summarizes Error ==> ( MAPE ) of Forecasting by using bats model and BATS Model, Holt's Linear Models , and autoarima for ==> Covid 19 Infection cases in Turkey "
M1<-mean(REOF_A_bats)
paste("System Summarizes Error ==> ( MAPE ) of Forecasting by using TBATS Model For ==> ", y_lab , sep=" ")
## [1] "System Summarizes Error ==> ( MAPE ) of Forecasting by using TBATS Model For ==> Covid 19 Infection cases in Turkey "
M2<-mean(REOF_A_tbats1)
paste("System Summarizes Error ==> ( MAPE ) of Forecasting by using Holt's Linear << Exponential Smoothing >> For ==> ", y_lab , sep=" ")
## [1] "System Summarizes Error ==> ( MAPE ) of Forecasting by using Holt's Linear << Exponential Smoothing >> For ==> Covid 19 Infection cases in Turkey "
M3<-REOF_A_Holt11
paste("System Summarizes Error ==> ( MAPE ) of Forecasting by using auto arima Model For ==> ", y_lab , sep=" ")
## [1] "System Summarizes Error ==> ( MAPE ) of Forecasting by using auto arima Model For ==> Covid 19 Infection cases in Turkey "
M4<-mean(REOF_A_auto.arima)
paste("System Summarizes Error ==> ( MAPE ) of Forecasting by using SIR Model For ==> ", y_lab , sep=" ")
## [1] "System Summarizes Error ==> ( MAPE ) of Forecasting by using SIR Model For ==> Covid 19 Infection cases in Turkey "
M5<-REOF_A_SIR1
paste("System Summarizes Error ==> ( MAPE ) of Forecasting by using autoarima Model For ==> ", y_lab , sep=" ")
## [1] "System Summarizes Error ==> ( MAPE ) of Forecasting by using autoarima Model For ==> Covid 19 Infection cases in Turkey "
data.frame(validation_dates,forecating_date=forecasting_data_by_name,MAPE_bats_error=REOF_A_bats,MAPE_TBATS_error=REOF_A_tbats1,MAPE_Holt_error=REOF_A_Holt1,MAPE_autoarima_error = REOF_A_auto.arima)
## validation_dates forecating_date MAPE_bats_error MAPE_TBATS_error
## 1 2021-01-04 Monday 0.0371581 0.06993993
## 2 2021-01-05 Tuesday 0.2438490 0.11066937
## 3 2021-01-06 Wednesday 0.6250945 0.35685665
## 4 2021-01-07 Thursday 1.0033910 0.56678848
## 5 2021-01-08 Friday 1.3119124 0.63202099
## 6 2021-01-09 Saturday 1.6159166 0.65283875
## 7 2021-01-10 Sunday 1.8330764 0.56045085
## MAPE_Holt_error MAPE_autoarima_error
## 1 0.14893356 0.09127373
## 2 0.02963217 0.08409256
## 3 0.13832687 0.31090226
## 4 0.25402732 0.48829846
## 5 0.25214689 0.55116739
## 6 0.20022238 0.56695394
## 7 0.01622690 0.45402521
recommend_Model<-c(M1,M2,M3,M4,M5)
best_recommended_model<-min(recommend_Model)
paste ("lodaing ..... ... . .Select Minimum MAPE from Models for select best Model ==> ", y_lab , sep=" ")
## [1] "lodaing ..... ... . .Select Minimum MAPE from Models for select best Model ==> Covid 19 Infection cases in Turkey "
best_recommended_model
## [1] 0.1485023
paste ("Best Model For Forecasting ==> ",y_lab, sep=" ")
## [1] "Best Model For Forecasting ==> Covid 19 Infection cases in Turkey "
if(best_recommended_model >= M1) {paste("System Recommend Bats Model That's better For forecasting==> ",y_lab, sep=" ")}
if(best_recommended_model >= M2) {paste("System Recommend That's better TBATS For forecasting ==> ",y_lab, sep=" ")}
if(best_recommended_model >= M3) {paste("System Recommend Holt's Linear Model < Exponential Smoothing Model > That's better For forecasting ==> ",y_lab, sep=" ")}
## [1] "System Recommend Holt's Linear Model < Exponential Smoothing Model > That's better For forecasting ==> Covid 19 Infection cases in Turkey "
if(best_recommended_model >= M4) {paste("System Recommend auto arima Model That's better For forecasting ==> ",y_lab, sep=" ")}
if(best_recommended_model >= M5) {paste("System Recommend SIR Model That's better For forecasting ==> ",y_lab, sep=" ")}
message("System finished Forecasting by using autoarima and Holt's ,TBATS, and SIR Model ==>",y_lab, sep=" ")
## System finished Forecasting by using autoarima and Holt's ,TBATS, and SIR Model ==>Covid 19 Infection cases in Turkey
message(" Thank you for using our System For Modelling ==> ",y_lab, sep=" ")
## Thank you for using our System For Modelling ==> Covid 19 Infection cases in Turkey