South ural state university, Chelyabinsk, Russian federation
# 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/population in cheleabinsk.xlsx",sheet = "Sheet1")
y_lab<- "population in Chelyabinsk" # input name of data
Actual_date_interval <- c("1950/12/31","2020/12/31")
Forecast_date_interval <- c("2021/12/31","2027/12/31")
validation_data_days <-7
frequency<-"years"
# Data Preparation & calculate some of statistics measures
original_data<-Full_original_data$Population
summary(original_data)
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 572873 836315 1080595 981687 1117110 1227795
sd(original_data) # calculate standard deviation
## [1] 190660.9
skewness(original_data) # calculate Cofficient of skewness
## [1] -0.8021526
kurtosis(original_data) # calculate Cofficient of kurtosis
## [1] 2.271894
rows <- NROW(original_data)
training_data<-original_data[1:(rows-validation_data_days)]
testing_data<-original_data[(rows-validation_data_days+1):rows]
AD<-fulldate<-seq(as.Date(Actual_date_interval[1]),as.Date(Actual_date_interval[2]), frequency) #input range for actual date
FD<-seq(as.Date(Forecast_date_interval[1]),as.Date(Forecast_date_interval[2]), frequency) #input range forecasting date
N_forecasting_days<-nrow(data.frame(FD))
validation_dates<-tail(AD,validation_data_days)
validation_data_by_name<-weekdays(validation_dates)
forecasting_data_by_name<-weekdays(FD)
##bats model
# Data Modeling
data_series<-ts(training_data)
autoplot(data_series ,xlab=paste ("Time in ", frequency, sep=" "), ylab = y_lab, main=paste ("Actual Data :", y_lab, sep=" "))
model_bats<-bats(data_series)
accuracy(model_bats) # accuracy on training data
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 169.4551 1643.034 1057 0.02326991 0.1130982 0.09539695 -0.01629944
# Print Model Parameters
model_bats
## BATS(1, {0,0}, 0.97, -)
##
## Call: bats(y = data_series)
##
## Parameters
## Alpha: 1.078879
## Beta: 1.564508
## Damping Parameter: 0.97007
##
## Seed States:
## [,1]
## [1,] 562192.05
## [2,] 12593.13
##
## Sigma: 1643.034
## AIC: 1223.919
plot(model_bats,xlab = paste ("Time in ", frequency ,y_lab , sep=" "), col.main="black", col.lab="blue", col.sub="black", cex.main=1, cex.lab=1, cex.sub=1,font.main=4, font.lab=4)
# Testing Data Evaluation
forecasting_bats <- predict(model_bats, h=N_forecasting_days+validation_data_days)
validation_forecast<-head(forecasting_bats$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 years by using bats Model for ==> population in Chelyabinsk"
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 ==> population in Chelyabinsk"
paste(MAPE_Mean_All,"%")
## [1] "0.13 % MAPE 7 years population 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 7 days in bats Model for ==> population in Chelyabinsk"
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 2014-12-31 среда 1170685 1170834
## 2 2015-12-31 четверг 1181855 1181644
## 3 2016-12-31 суббота 1193148 1192130
## 4 2017-12-31 воскресенье 1204517 1202302
## 5 2018-12-31 понедельник 1215994 1212169
## 6 2019-12-31 вторник 1222108 1221742
## 7 2020-12-31 четверг 1227795 1231028
## MAPE_bats_Model
## 1 0.013 %
## 2 0.018 %
## 3 0.085 %
## 4 0.184 %
## 5 0.315 %
## 6 0.03 %
## 7 0.263 %
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 2021-12-31 пятница 1240035
## 2 2022-12-31 суббота 1248774
## 3 2023-12-31 воскресенье 1257251
## 4 2024-12-31 вторник 1265474
## 5 2025-12-31 среда 1273451
## 6 2026-12-31 четверг 1281189
## 7 2027-12-31 пятница 1288695
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="blue", 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
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,RMSE_bats,MAPE_Mean_All,MAD_bats) # analysis of Error by using Bats Model shows result of correlation ,MSE ,MPER
## correlation_bats RMSE_bats MAPE_Mean_All
## 1 0.9949532 2111.977 0.13 % MAPE 7 years population in Chelyabinsk
## MAD_bats
## 1 607.783
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 2014-12-31 среда 149.1266 0.013 % 0.013 %
## 2 2015-12-31 четверг 211.3941 0.018 % 0.018 %
## 3 2016-12-31 суббота 1018.4379 0.085 % 0.085 %
## 4 2017-12-31 воскресенье 2215.3219 0.184 % 0.184 %
## 5 2018-12-31 понедельник 3824.6530 0.315 % 0.316 %
## 6 2019-12-31 вторник 366.3193 0.03 % 0.03 %
## 7 2020-12-31 четверг 3232.5186 0.263 % 0.263 %
## 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
## Training set -8.676742 1824.837 1294.112 0.002005064 0.1410602 0.1167969
## ACF1
## Training set 0.009651064
# Print Model Parameters
model_TBATS
## TBATS(1, {0,0}, 1, {<6,2>})
##
## Call: NULL
##
## Parameters
## Alpha: 1.044364
## Beta: 1.394573
## Damping Parameter: 1
## Gamma-1 Values: 0.03650915
## Gamma-2 Values: -0.0127078
##
## Seed States:
## [,1]
## [1,] 562497.7030
## [2,] 12578.3278
## [3,] -852.2614
## [4,] 475.7496
## [5,] 991.4793
## [6,] -451.6407
##
## Sigma: 1824.837
## AIC: 1247.352
plot(model_TBATS,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)
# 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 years by using TBATS Model for ==> population in Chelyabinsk"
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 ==> population in Chelyabinsk"
paste(MAPE_Mean_All,"%")
## [1] "0.311 % MAPE 7 years population in Chelyabinsk %"
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 ==> population in Chelyabinsk"
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 2014-12-31 среда 1170685 1170377
## 2 2015-12-31 четверг 1181855 1181124
## 3 2016-12-31 суббота 1193148 1192406
## 4 2017-12-31 воскресенье 1204517 1204960
## 5 2018-12-31 понедельник 1215994 1218493
## 6 2019-12-31 вторник 1222108 1230604
## 7 2020-12-31 четверг 1227795 1241201
## MAPE_TBATS_Model
## 1 0.026 %
## 2 0.062 %
## 3 0.062 %
## 4 0.037 %
## 5 0.206 %
## 6 0.695 %
## 7 1.092 %
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 2021-12-31 пятница 1251947
## 2 2022-12-31 суббота 1263229
## 3 2023-12-31 воскресенье 1275784
## 4 2024-12-31 вторник 1289316
## 5 2025-12-31 среда 1301427
## 6 2026-12-31 четверг 1312024
## 7 2027-12-31 пятница 1322771
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="blue", 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
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,RMSE_tbats,MAPE_Mean_All,MAD_tbats) # analysis of Error by using Holt's linear model shows result of correlation ,MSE ,MPER
## correlation_tbats RMSE_tbats MAPE_Mean_All
## 1 0.9923645 6088.86 0.311 % MAPE 7 years population in Chelyabinsk
## MAD_tbats
## 1 3294.776
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 2014-12-31 среда 307.5421 0.026 % 0.026 %
## 2 2015-12-31 четверг 731.1978 0.062 % 0.062 %
## 3 2016-12-31 суббота 742.0549 0.062 % 0.062 %
## 4 2017-12-31 воскресенье 443.4422 0.037 % 0.037 %
## 5 2018-12-31 понедельник 2499.0083 0.206 % 0.205 %
## 6 2019-12-31 вторник 8495.8830 0.695 % 0.69 %
## 7 2020-12-31 четверг 13405.8902 1.092 % 1.08 %
## 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
## Training set 35.89561 2057.755 1040.567 0.003636185 0.1131804 0.09391386
## ACF1
## Training set 0.3807252
# 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= 1.9999
##
## Smoothing parameters:
## alpha = 0.9999
## beta = 0.9999
##
## Initial states:
## l = 153115029226.739
## b = 11282238865.7469
##
## sigma: 2213521530
##
## AIC AICc BIC
## 3026.323 3027.357 3037.117
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 35.89561 2057.755 1040.567 0.003636185 0.1131804 0.09391386
## ACF1
## Training set 0.3807252
# Testing Data Evaluation
forecasting_holt <- predict(model_holt, h=N_forecasting_days+validation_data_days,lambda = "auto")
validation_forecast<-head(forecasting_holt$mean,validation_data_days)
MAPE_Per_Day<-round( abs(((testing_data-validation_forecast)/testing_data)*100) ,3)
paste ("MAPE % For ",validation_data_days,frequency,"by using holt Model for ==> ",y_lab, sep=" ")
## [1] "MAPE % For 7 years by using holt Model for ==> population in Chelyabinsk"
MAPE_Mean_All<-paste(round(mean(MAPE_Per_Day),3),"% MAPE ",validation_data_days,frequency,y_lab,sep=" ")
MAPE_holt<-paste(round(MAPE_Per_Day,3),"%")
MAPE_holt_Model<-paste(MAPE_Per_Day ,"%")
paste (" MAPE that's Error of Forecasting for ",validation_data_days," days in holt Model for ==> ",y_lab, sep=" ")
## [1] " MAPE that's Error of Forecasting for 7 days in holt Model for ==> population in Chelyabinsk"
paste(MAPE_Mean_All,"%")
## [1] "0.168 % MAPE 7 years population in Chelyabinsk %"
paste ("MAPE that's Error of Forecasting day by day for ",validation_data_days," days in holt Model for ==> ",y_lab, sep=" ")
## [1] "MAPE that's Error of Forecasting day by day for 7 days in holt Model for ==> population in Chelyabinsk"
data.frame(date_holt=validation_dates,validation_data_by_name,actual_data=testing_data,forecasting_holt=validation_forecast,MAPE_holt_Model)
## date_holt validation_data_by_name actual_data forecasting_holt
## 1 2014-12-31 среда 1170685 1170463
## 2 2015-12-31 четверг 1181855 1181206
## 3 2016-12-31 суббота 1193148 1191852
## 4 2017-12-31 воскресенье 1204517 1202405
## 5 2018-12-31 понедельник 1215994 1212865
## 6 2019-12-31 вторник 1222108 1223236
## 7 2020-12-31 четверг 1227795 1233520
## MAPE_holt_Model
## 1 0.019 %
## 2 0.055 %
## 3 0.109 %
## 4 0.175 %
## 5 0.257 %
## 6 0.092 %
## 7 0.466 %
data.frame(FD,forecating_date=forecasting_data_by_name,forecasting_by_holt=tail(forecasting_holt$mean,N_forecasting_days))
## FD forecating_date forecasting_by_holt
## 1 2021-12-31 пятница 1243718
## 2 2022-12-31 суббота 1253834
## 3 2023-12-31 воскресенье 1263869
## 4 2024-12-31 вторник 1273825
## 5 2025-12-31 среда 1283703
## 6 2026-12-31 четверг 1293507
## 7 2027-12-31 пятница 1303236
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="blue", 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
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,RMSE_Holt,MAPE_Mean_All,MAD_Holt) # analysis of Error by using Holt's linear model shows result of correlation ,MSE ,MPER
## correlation_Holt RMSE_Holt MAPE_Mean_All
## 1 0.9932091 2684.515 0.168 % MAPE 7 years population in Chelyabinsk
## MAD_Holt
## 1 79.33375
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 2014-12-31 среда 222.2340 0.019 % 0.019 %
## 2 2015-12-31 четверг 648.9693 0.055 % 0.055 %
## 3 2016-12-31 суббота 1295.5323 0.109 % 0.109 %
## 4 2017-12-31 воскресенье 2112.3511 0.175 % 0.176 %
## 5 2018-12-31 понедельник 3128.9658 0.257 % 0.258 %
## 6 2019-12-31 вторник 1127.9785 0.092 % 0.092 %
## 7 2020-12-31 четверг 5724.7377 0.466 % 0.464 %
#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 ==> population 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 = 1.5003, Truncation lag parameter = 3, p-value = 0.01
pp.test(data_series) # applay pp test
##
## Phillips-Perron Unit Root Test
##
## data: data_series
## Dickey-Fuller Z(alpha) = -0.96716, Truncation lag parameter = 3,
## p-value = 0.9856
## alternative hypothesis: stationary
adf.test(data_series) # applay adf test
##
## Augmented Dickey-Fuller Test
##
## data: data_series
## Dickey-Fuller = -3.0299, Lag order = 3, p-value = 0.1576
## 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="blue", 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 ==> population 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 = 0.9331, Truncation lag parameter = 3, 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) = -4.3629, Truncation lag parameter = 3, p-value
## = 0.861
## alternative hypothesis: stationary
adf.test(diff1_x1) # applay adf test after taking first differences
##
## Augmented Dickey-Fuller Test
##
## data: diff1_x1
## Dickey-Fuller = -1.634, Lag order = 3, p-value = 0.7231
## 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="blue", 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 population 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 = 0.16883, Truncation lag parameter = 3, 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) = -32.84, Truncation lag parameter = 3, p-value
## = 0.01
## alternative hypothesis: stationary
adf.test(diff2_x1) # applay adf test after taking Second differences
##
## Augmented Dickey-Fuller Test
##
## data: diff2_x1
## Dickey-Fuller = -3.5363, Lag order = 3, p-value = 0.046
## 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) : 1116.368
## ARIMA(0,2,1) : 1094.213
## ARIMA(0,2,2) : 1095.765
## ARIMA(0,2,3) : 1097.968
## ARIMA(0,2,4) : 1100.036
## ARIMA(0,2,5) : 1102.28
## ARIMA(1,2,0) : 1105.957
## ARIMA(1,2,1) : 1095.733
## ARIMA(1,2,2) : 1098.021
## ARIMA(1,2,3) : 1100.257
## ARIMA(1,2,4) : Inf
## ARIMA(2,2,0) : 1104.269
## ARIMA(2,2,1) : 1098.02
## ARIMA(2,2,2) : 1100.25
## ARIMA(2,2,3) : Inf
## ARIMA(3,2,0) : 1103.24
## ARIMA(3,2,1) : 1100.006
## ARIMA(3,2,2) : Inf
## ARIMA(4,2,0) : 1103.592
## ARIMA(4,2,1) : 1102.398
## ARIMA(5,2,0) : 1105.091
##
##
##
## Best model: ARIMA(0,2,1)
model1 # show the result of autoarima
## Series: data_series
## ARIMA(0,2,1)
##
## Coefficients:
## ma1
## 0.7915
## s.e. 0.0868
##
## sigma^2 estimated as 2538995: log likelihood=-545
## AIC=1094.01 AICc=1094.21 BIC=1098.26
#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("[ ]*$", "", result)
}
return(result)
}
source("stringthearima.R")
bestmodel <- arima.string(model1, padding = TRUE)
bestmodel <- substring(bestmodel,7,11)
bestmodel <- gsub(" ", "", bestmodel)
bestmodel <- gsub(")", "", bestmodel)
bestmodel <- strsplit(bestmodel, ",")[[1]]
bestmodel <- c(strtoi(bestmodel[1]),strtoi(bestmodel[2]),strtoi(bestmodel[3]))
bestmodel
## [1] 0 2 1
strtoi(bestmodel[3])
## [1] 1
#2. Using ACF and PACF Function
#par(mfrow=c(1,2)) # Code for making two plot in one graph
acf(diff2_x1,xlab = paste ("Time in ", frequency ,y_lab , sep=" "), col.main="black", col.lab="blue", col.sub="black", cex.main=1, cex.lab=1, cex.sub=1,font.main=4, font.lab=4, ylab=y_lab, main=paste("ACF-2nd differenced series ",y_lab, sep=" ",lag.max=20)) # plot ACF "auto correlation function after taking second diffrences
pacf(diff2_x1,xlab = paste ("Time in ", frequency ,y_lab , sep=" "), col.main="black", col.lab="blue", col.sub="black", cex.main=1, cex.lab=1, cex.sub=1,font.main=4, font.lab=4, ylab=y_lab,main=paste("PACF-2nd differenced series ",y_lab, sep=" ",lag.max=20)) # plot PACF " Partial auto correlation function after taking second diffrences
library(forecast) # install library forecast
x1_model1= arima(data_series, order=c(bestmodel)) # Run Best model of auto arima for forecasting
x1_model1 # Show result of best model of auto arima
##
## Call:
## arima(x = data_series, order = c(bestmodel))
##
## Coefficients:
## ma1
## 0.7915
## s.e. 0.0868
##
## sigma^2 estimated as 2488138: log likelihood = -545, aic = 1094.01
paste ("accuracy of autoarima Model For ==> ",y_lab, sep=" ")
## [1] "accuracy of autoarima Model For ==> population in Chelyabinsk"
accuracy(x1_model1) # aacuracy of best model from auto arima
## ME RMSE MAE MPE MAPE MASE
## Training set -14.35341 1555.628 931.9643 0.0004364618 0.08954136 0.08411218
## ACF1
## Training set -0.07729962
x1_model1$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(0,2,1)
## Q* = 4.7955, df = 9, p-value = 0.8518
##
## Model df: 1. 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 ==> population 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 = 13.47, df = 20, p-value = 0.8563
library(tseries)
jarque.bera.test(x1_model1$residuals) # Do test jarque.bera.test
##
## Jarque Bera Test
##
## data: x1_model1$residuals
## X-squared = 180.09, 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='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$mean,validation_data_days)
MAPE_Per_Day<-round(abs(((testing_data-validation_forecast)/testing_data)*100) ,3)
paste ("MAPE % For ",validation_data_days,frequency,"by using bats Model for ==> ",y_lab, sep=" ")
## [1] "MAPE % For 7 years by using bats Model for ==> population in Chelyabinsk"
MAPE_Mean_All<-paste(round(mean(MAPE_Per_Day),3),"% MAPE ",validation_data_days,frequency,y_lab,sep=" ")
MAPE_auto_arima<-paste(round(MAPE_Per_Day,3),"%")
MAPE_auto.arima_Model<-paste(MAPE_Per_Day ,"%")
paste (" MAPE that's Error of Forecasting for ",validation_data_days," days in bats Model for ==> ",y_lab, sep=" ")
## [1] " MAPE that's Error of Forecasting for 7 days in bats Model for ==> population in Chelyabinsk"
paste(MAPE_Mean_All,"%")
## [1] "0.362 % MAPE 7 years population in Chelyabinsk %"
paste ("MAPE that's Error of Forecasting day by day for ",validation_data_days," days in bats Model for ==> ",y_lab, sep=" ")
## [1] "MAPE that's Error of Forecasting day by day for 7 days in bats Model for ==> population in Chelyabinsk"
data.frame(date_auto.arima=validation_dates,validation_data_by_name,actual_data=testing_data,forecasting_auto.arima=validation_forecast,MAPE_auto.arima_Model)
## date_auto.arima validation_data_by_name actual_data forecasting_auto.arima
## 1 2014-12-31 среда 1170685 1171392
## 2 2015-12-31 четверг 1181855 1183164
## 3 2016-12-31 суббота 1193148 1194936
## 4 2017-12-31 воскресенье 1204517 1206708
## 5 2018-12-31 понедельник 1215994 1218480
## 6 2019-12-31 вторник 1222108 1230253
## 7 2020-12-31 четверг 1227795 1242025
## MAPE_auto.arima_Model
## 1 0.06 %
## 2 0.111 %
## 3 0.15 %
## 4 0.182 %
## 5 0.204 %
## 6 0.666 %
## 7 1.159 %
data.frame(FD,forecating_date=forecasting_data_by_name,forecasting_by_auto.arima=tail(forecasting_auto_arima$mean,N_forecasting_days))
## FD forecating_date forecasting_by_auto.arima
## 1 2021-12-31 пятница 1253797
## 2 2022-12-31 суббота 1265569
## 3 2023-12-31 воскресенье 1277341
## 4 2024-12-31 вторник 1289113
## 5 2025-12-31 среда 1300885
## 6 2026-12-31 четверг 1312657
## 7 2027-12-31 пятница 1324429
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 0.9924041 6383.138
## MAPE_Mean_All MAD_auto.arima
## 1 0.362 % MAPE 7 years population in Chelyabinsk 4408.063
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 2014-12-31 среда 707.0871 0.06 %
## 2 2015-12-31 четверг 1309.1741 0.111 %
## 3 2016-12-31 суббота 1788.2612 0.15 %
## 4 2017-12-31 воскресенье 2191.3483 0.182 %
## 5 2018-12-31 понедельник 2486.4353 0.204 %
## 6 2019-12-31 вторник 8144.5224 0.666 %
## 7 2020-12-31 четверг 14229.6094 1.159 %
## REOF_F_auto.arima
## 1 0.06 %
## 2 0.111 %
## 3 0.15 %
## 4 0.182 %
## 5 0.204 %
## 6 0.662 %
## 7 1.146 %
# 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 ==> population in Chelyabinsk"
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 ==> population in Chelyabinsk"
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 ==> population in Chelyabinsk"
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 ==> population in Chelyabinsk"
M4<-mean(REOF_A_auto.arima)
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 ==> population in Chelyabinsk"
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 2014-12-31 пятница 0.01273841 0.02627027
## 2 2015-12-31 суббота 0.01788663 0.06186866
## 3 2016-12-31 воскресенье 0.08535721 0.06219303
## 4 2017-12-31 вторник 0.18391786 0.03681494
## 5 2018-12-31 среда 0.31452894 0.20551157
## 6 2019-12-31 четверг 0.02997438 0.69518267
## 7 2020-12-31 пятница 0.26327836 1.09186714
## MAPE_Holt_error MAPE_autoarima_error
## 1 0.01898324 0.06039943
## 2 0.05491107 0.11077282
## 3 0.10858102 0.14987757
## 4 0.17536914 0.18192755
## 5 0.25731754 0.20447760
## 6 0.09229778 0.66643229
## 7 0.46626169 1.15895646
recommend_Model<-c(M1,M2,M3,M4)
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 ==> population in Chelyabinsk"
best_recommended_model
## [1] 0.1296688
paste ("Best Model For Forecasting ==> ",y_lab, sep=" ")
## [1] "Best Model For Forecasting ==> population in Chelyabinsk"
if(best_recommended_model >= M1) {paste("System Recommend Bats Model That's better For forecasting==> ",y_lab, sep=" ")}
## [1] "System Recommend Bats Model That's better For forecasting==> population in Chelyabinsk"
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=" ")}
if(best_recommended_model >= M4) {paste("System Recommend auto arima Model That's better For forecasting ==> ",y_lab, sep=" ")}
message("System finished Forecasting by using autoarima and Holt's ,and TBATS Model ==>",y_lab, sep=" ")
## System finished Forecasting by using autoarima and Holt's ,and TBATS Model ==>population in Chelyabinsk
message(" Thank you for using our System For Modelling ==> ",y_lab, sep=" ")
## Thank you for using our System For Modelling ==> population in Chelyabinsk