R Markdown : Looks into all methods employed to forecast mobile APP visits for next 4 months. Essence of this excercise captured in our project final report.
if (!require("pacman")) install.packages("pacman")
## Loading required package: pacman
pacman::p_load("moments","extRemes","stringi", "ggplot2", "TTR", "forecast", "zoo", "rts", "xts")
setwd("D:\\Google Drive\\FA\\_FAProject")
##### Step 1 Load Organised Data
eCommerceTrafficData <- read.csv("eCommerceTrafficDataMonthly.csv")
##### Visualise data
#par(mfrow=c(2, 2))
appVisits.ts <- ts(eCommerceTrafficData$Mapp, start = c(2014,1), freq = 12)
plot(appVisits.ts, xlab = "", ylab = "Mobile App Visits", bty = "l", col = 'blue')
## Partition of data of appVisits.ts
totalRecords <- length(appVisits.ts)
nValid <- 4
nTrain <- totalRecords - nValid
train.ts <- window(appVisits.ts,start = c (2014,1), end = c(2014,nTrain))
train.ts
## Jan Feb Mar Apr May Jun Jul
## 2014 2869521 3229880 4684880 8562608 12314911 14010825 16747894
## 2015 59594456 63379983 88641001 95791175 110746932 120414108 142212584
## 2016 164123172 163592403 146220981 234314655 318261988 180043453 164339827
## Aug Sep Oct Nov Dec
## 2014 21862188 27516353 45598418 38226274 62578002
## 2015 149697399 120369528 215154440 159265899 153666682
## 2016 170356543 156311048 257243995 143502849 178127276
valid.ts <- window(appVisits.ts, start = c (2014,nTrain+1), end = c(2014,totalRecords))
valid.ts
## Jan Feb Mar Apr
## 2017 182350139 153941816 172790970 165909172
### Fist lets do the seasonal naive forecast
naive.pred <- snaive(train.ts, h = 4)
naive.pred
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Jan 2017 164123172 38714513 289531831 -27672821 355919165
## Feb 2017 163592403 38183744 289001062 -28203590 355388396
## Mar 2017 146220981 20812322 271629640 -45575012 338016974
## Apr 2017 234314655 108905996 359723314 42518662 426110648
valid.ts
## Jan Feb Mar Apr
## 2017 182350139 153941816 172790970 165909172
plot(train.ts, xlab = "Monthly", ylab = "Mobile App Visits", bty = "l", col = 'blue')
lines(naive.pred$fitted, lwd = 2, col = "green")
lines(valid.ts, lwd = 2, col = "blue")
accuracy(naive.pred, valid.ts)
## ME RMSE MAE MPE MAPE MASE
## Training set 84093185 97856897 85406772 58.642109 59.55748 1.0000000
## Test set -8314779 38113746 30713257 -5.531781 18.21805 0.3596115
## ACF1 Theil's U
## Training set 0.5203794 NA
## Test set -0.3936157 2.152795
# ############ Linear Trend Model
train.lm <- tslm(train.ts ~ trend)
# now train a model and plot it
train.lm.pred <- forecast(train.lm, h = nValid, level = 0)
plot(train.lm.pred, xlab = "Monthly", ylab = "Mobile App Visits", bty = "l", col = 'black',flty = 2)
lines(train.lm.pred$fitted, lwd = 2, col = "blue")
lines(valid.ts)
summary(train.lm)
##
## Call:
## tslm(formula = train.ts ~ trend)
##
## Residuals:
## Min 1Q Median 3Q Max
## -80040587 -17740267 -4609969 9641164 135465921
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -14149552 12948413 -1.093 0.282
## trend 6791228 610281 11.128 7.05e-13 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 38040000 on 34 degrees of freedom
## Multiple R-squared: 0.7846, Adjusted R-squared: 0.7782
## F-statistic: 123.8 on 1 and 34 DF, p-value: 7.05e-13
accuracy(train.lm.pred,valid.ts)
## ME RMSE MAE MPE MAPE MASE
## Training set 8.536113e-10 36966949 24400937 -3.835911 41.23754 0.2857026
## Test set -7.856471e+07 79931000 78564711 -47.196241 47.19624 0.9198885
## ACF1 Theil's U
## Training set 0.2191944 NA
## Test set -0.3316361 4.371476
############# Exponential Trend Moedl #####
train.lm.expo.trend <- tslm(train.ts ~ trend, lambda = 0)
train.lm.expo.trend.pred <- forecast(train.lm.expo.trend, h = nValid, level = 0)
plot(train.lm.expo.trend.pred, xlab = "Monthly", ylab = "Mobile App Visits", bty = "l", col = 'black',flty = 2)
lines(train.lm.expo.trend.pred$fitted, lwd = 2, col = "blue")
lines(valid.ts)
summary(train.lm.expo.trend)
##
## Call:
## tslm(formula = train.ts ~ trend, lambda = 0)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.2543 -0.3711 0.1625 0.4911 0.7834
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 16.015356 0.206429 77.58 < 2e-16 ***
## trend 0.108549 0.009729 11.16 6.58e-13 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.6064 on 34 degrees of freedom
## Multiple R-squared: 1, Adjusted R-squared: 1
## F-statistic: 1.428e+18 on 1 and 34 DF, p-value: < 2.2e-16
accuracy(train.lm.expo.trend.pred, valid.ts)
## ME RMSE MAE MPE MAPE MASE
## Training set -7393697 85434099 55662537 -20.90667 59.56623 0.6517345
## Test set -424921938 431610594 424921938 -253.82955 253.82955 4.9752722
## ACF1 Theil's U
## Training set 0.7047932 NA
## Test set 0.1836972 23.60487
################## Quadratic or polynomial trend model
train.lm.poly.trend <- tslm(train.ts ~ (trend + I(trend^2)))
train.lm.poly.trend.pred <- forecast(train.lm.poly.trend, h = nValid, level = 0)
plot(train.lm.poly.trend.pred, xlab = "Monthly", ylab = "Mobile App Visits", bty = "l", col = 'black',flty = 2)
lines(train.lm.poly.trend.pred$fitted, lwd = 2, col = "blue")
lines(valid.ts)
summary(train.lm.poly.trend)
##
## Call:
## tslm(formula = train.ts ~ (trend + I(trend^2)))
##
## Residuals:
## Min 1Q Median 3Q Max
## -60213621 -22151192 -6681768 9505072 135747440
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -42422082 19392079 -2.188 0.0359 *
## trend 11255312 2416781 4.657 5.05e-05 ***
## I(trend^2) -120651 63356 -1.904 0.0656 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 36650000 on 33 degrees of freedom
## Multiple R-squared: 0.8059, Adjusted R-squared: 0.7941
## F-statistic: 68.51 on 2 and 33 DF, p-value: 1.786e-12
accuracy(train.lm.poly.trend.pred, valid.ts)
## ME RMSE MAE MPE MAPE MASE
## Training set -1.189215e-09 35089145 24174358 49.8526 87.8712 0.2830497
## Test set -4.317378e+07 44634615 43173778 -26.0901 26.0901 0.5055077
## ACF1 Theil's U
## Training set 0.1449017 NA
## Test set -0.5654056 2.455583
### check seasoanlity in this Monthly time series
fit <- tbats(appVisits.ts)
fit
## BATS(0, {0,0}, 0.947, -)
##
## Call: tbats(y = appVisits.ts)
##
## Parameters
## Lambda: 0
## Alpha: 0.2481564
## Beta: 0.06782401
## Damping Parameter: 0.946678
##
## Seed States:
## [,1]
## [1,] 14.8142911
## [2,] 0.3440338
##
## Sigma: 0.2198401
## AIC: 1487.595
seasonal <- !is.null(fit$seasonal)
# this shows there is seasonlity present
seasonal
## [1] FALSE
## Season is false but lets check model with seasonality
# Only Season
train.lm.season <- tslm(train.ts ~ season)
train.lm.season.pred <- forecast(train.lm.season, h = nValid, level = 0)
plot(train.lm.season.pred, xlab = "Monthly", ylab = "Mobile App Visits", bty = "l", col = 'black',flty = 2)
lines(train.lm.season.pred$fitted, lwd = 2, col = "blue")
lines(valid.ts)
summary(train.lm.season)
##
## Call:
## tslm(formula = train.ts ~ season)
##
## Residuals:
## Min 1Q Median 3Q Max
## -134793033 -73598812 17280932 55280179 171154044
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 75529050 52851985 1.429 0.166
## season2 1205039 74743994 0.016 0.987
## season3 4319904 74743994 0.058 0.954
## season4 37360430 74743994 0.500 0.622
## season5 71578894 74743994 0.958 0.348
## season6 29293746 74743994 0.392 0.699
## season7 32237719 74743994 0.431 0.670
## season8 38442994 74743994 0.514 0.612
## season9 25869927 74743994 0.346 0.732
## season10 97136568 74743994 1.300 0.206
## season11 38135958 74743994 0.510 0.615
## season12 55928270 74743994 0.748 0.462
##
## Residual standard error: 91540000 on 24 degrees of freedom
## Multiple R-squared: 0.1193, Adjusted R-squared: -0.2843
## F-statistic: 0.2957 on 11 and 24 DF, p-value: 0.9803
accuracy(train.lm.season.pred, valid.ts)
## ME RMSE MAE MPE MAPE MASE
## Training set 0 74743994 64578017 -296.09231 331.19869 0.7561229
## Test set 82497631 84883868 82497631 48.61995 48.61995 0.9659378
## ACF1 Theil's U
## Training set 0.8759955 NA
## Test set -0.3078260 3.951817
### Additive Seasonal Model with trend
train.lm.tns <- tslm(train.ts ~ trend + season)
train.lm.tns.pred <- forecast(train.lm.tns, h = nValid, level = 0)
plot(train.lm.tns.pred, xlab = "Monthly", ylab = "Mobile App Visits", bty = "l", col = 'black',flty = 2)
lines(train.lm.tns.pred$fitted, lwd = 2, col = "blue")
lines(valid.ts)
summary(train.lm.tns)
##
## Call:
## tslm(formula = train.ts ~ trend + season)
##
## Residuals:
## Min 1Q Median 3Q Max
## -54255343 -18349290 1625161 15308228 87060860
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -15571901 22843647 -0.682 0.502
## trend 7007765 628513 11.150 9.44e-11 ***
## season2 -5802726 30175153 -0.192 0.849
## season3 -9695627 30194784 -0.321 0.751
## season4 16337134 30227473 0.540 0.594
## season5 43547832 30273178 1.438 0.164
## season6 -5745081 30331841 -0.189 0.851
## season7 -9808874 30403386 -0.323 0.750
## season8 -10611364 30487723 -0.348 0.731
## season9 -30192197 30584746 -0.987 0.334
## season10 34066679 30694334 1.110 0.279
## season11 -31941696 30816354 -1.037 0.311
## season12 -21157149 30950659 -0.684 0.501
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 36950000 on 23 degrees of freedom
## Multiple R-squared: 0.8625, Adjusted R-squared: 0.7908
## F-statistic: 12.02 on 12 and 23 DF, p-value: 3.269e-07
accuracy(train.lm.tns.pred, valid.ts)
## ME RMSE MAE MPE MAPE MASE
## Training set -9.832648e-10 29533407 23388827 2.893108 63.68492 0.2738521
## Test set -8.568874e+07 87988475 85688738 -51.428443 51.42844 1.0033014
## ACF1 Theil's U
## Training set 0.3944743 NA
## Test set -0.3078260 4.77675
### Additive Seasonal Model with quadratic trend
train.lm.qtns <- tslm(train.ts ~ trend + I(trend^2) + season)
train.lm.qtns.pred <- forecast(train.lm.qtns, h = nValid, level = 0)
plot(train.lm.qtns.pred, xlab = "Monthly", ylab = "Mobile App Visits", bty = "l", col = 'black',flty = 2)
lines(train.lm.qtns.pred$fitted, lwd = 2, col = "blue")
lines(valid.ts)
summary(train.lm.qtns)
##
## Call:
## tslm(formula = train.ts ~ trend + I(trend^2) + season)
##
## Residuals:
## Min 1Q Median 3Q Max
## -47428101 -18669535 -2525287 17621512 88444246
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -40472852 25346295 -1.597 0.1246
## trend 11273206 2331960 4.834 7.88e-05 ***
## I(trend^2) -115282 60933 -1.892 0.0717 .
## season2 -6955548 28619756 -0.243 0.8102
## season3 -11770706 28652883 -0.411 0.6852
## season4 13570361 28700162 0.473 0.6410
## season5 40319931 28756876 1.402 0.1748
## season6 -9203547 28819877 -0.319 0.7525
## season7 -13267339 28887583 -0.459 0.6505
## season8 -13839265 28959959 -0.478 0.6375
## season9 -32958969 29038507 -1.135 0.2686
## season10 31991600 29126234 1.098 0.2839
## season11 -33094518 29227633 -1.132 0.2697
## season12 -21157149 29348633 -0.721 0.4786
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 35040000 on 22 degrees of freedom
## Multiple R-squared: 0.8817, Adjusted R-squared: 0.8119
## F-statistic: 12.62 on 13 and 22 DF, p-value: 2.51e-07
accuracy(train.lm.qtns.pred, valid.ts)
## ME RMSE MAE MPE MAPE MASE
## Training set 1.396984e-09 27389176 21047477 52.22086 98.03125 0.2464380
## Test set -5.248747e+07 54690985 52487470 -31.61438 31.61438 0.6145586
## ACF1 Theil's U
## Training set 0.3007832 NA
## Test set -0.5454901 2.956618
### multiplicative Seasonal Model with trend
train.lm.tnms <- tslm(train.ts ~ trend*season)
train.lm.tnms.pred <- forecast(train.lm.tnms, h = nValid, level = 0)
plot(train.lm.tnms.pred, xlab = "Monthly", ylab = "Mobile App Visits", bty = "l", col = 'black',flty = 2)
lines(train.lm.tnms.pred$fitted, lwd = 2, col = "blue")
lines(valid.ts)
summary(train.lm.tnms)
##
## Call:
## tslm(formula = train.ts ~ trend * season)
##
## Residuals:
## Min 1Q Median 3Q Max
## -36361012 -17129455 -6095840 10491863 45600892
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -11816678 33173232 -0.356 0.72787
## trend 6718902 2037815 3.297 0.00637 **
## season2 -4994039 48094172 -0.104 0.91901
## season3 3205569 49330289 0.065 0.94926
## season4 -25795207 50618286 -0.510 0.61957
## season5 -57787891 51954304 -1.112 0.28780
## season6 -7884998 53334734 -0.148 0.88492
## season7 2739833 54756219 0.050 0.96092
## season8 2043426 56215643 0.036 0.97160
## season9 520296 57710128 0.009 0.99295
## season10 -9526150 59237020 -0.161 0.87492
## season11 24591634 60793879 0.405 0.69296
## season12 27724724 62378460 0.444 0.66462
## trend:season2 -37130 2881906 -0.013 0.98993
## trend:season3 -821565 2881906 -0.285 0.78045
## trend:season4 2687433 2881906 0.933 0.36945
## trend:season5 6028893 2881906 2.092 0.05837 .
## trend:season6 199124 2881906 0.069 0.94605
## trend:season7 -569238 2881906 -0.198 0.84673
## trend:season8 -531637 2881906 -0.184 0.85672
## trend:season9 -1352457 2881906 -0.469 0.64727
## trend:season10 2099664 2881906 0.729 0.48024
## trend:season11 -2332378 2881906 -0.809 0.43409
## trend:season12 -1904349 2881906 -0.661 0.52123
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 34580000 on 12 degrees of freedom
## Multiple R-squared: 0.9372, Adjusted R-squared: 0.8167
## F-statistic: 7.781 on 23 and 12 DF, p-value: 0.0003454
accuracy(train.lm.tnms.pred, valid.ts)
## ME RMSE MAE MPE MAPE
## Training set -2.328306e-09 19966432 17031787 6.777368 43.20576
## Test set -8.972845e+07 102561537 89728449 -54.025813 54.02581
## MASE ACF1 Theil's U
## Training set 0.1994196 0.6728281 NA
## Test set 1.0506011 -0.2950032 5.656306
### on the basis of MAPE, we will use model polynomial model
Visits.lm.tns <- tslm(appVisits.ts ~ trend + I(trend^2))
Visits.lm.tns.pred <- forecast(Visits.lm.tns, h = 4, level = 0)
plot(Visits.lm.tns.pred, xlab = "Monthly", ylab = "Mobile App Visits", bty = "l", col = 'black',flty = 2)
lines(Visits.lm.tns.pred$fitted, lwd = 2, col = "blue")
summary(Visits.lm.tns.pred)
##
## Forecast method: Linear regression model
##
## Model Information:
##
## Call:
## tslm(formula = appVisits.ts ~ trend + I(trend^2))
##
## Coefficients:
## (Intercept) trend I(trend^2)
## -51912812 13128194 -180670
##
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 3.633113e-09 34519251 23538175 57.9854 95.79685 0.3033527
## ACF1
## Training set 0.190379
##
## Forecasts:
## Point Forecast Lo 0 Hi 0
## May 2017 182636375 182636375 182636375
## Jun 2017 180768934 180768934 180768934
## Jul 2017 178540153 178540153 178540153
## Aug 2017 175950031 175950031 175950031
names(Visits.lm.tns.pred)
## [1] "model" "mean" "lower" "upper" "level"
## [6] "x" "method" "newdata" "residuals" "fitted"
round(Visits.lm.tns.pred$mean,0)
## May Jun Jul Aug
## 2017 182636375 180768934 178540153 175950031
accuracy(Visits.lm.tns.pred)
## ME RMSE MAE MPE MAPE MASE
## Training set 3.633113e-09 34519251 23538175 57.9854 95.79685 0.3033527
## ACF1
## Training set 0.190379
# Calcualting Residuals of training period
train.lm.qt <- tslm(train.ts ~ trend + I(trend^2))
train.lm.qt.pred <- forecast(train.lm.qt, h = nValid, level = 0)
res <- residuals(train.lm.qt)
#par(mfrow=c(1,2))
plot(res, ylab="Residuals",xlab="Year")
Acf(res, lag.max = 12, main="ACF of residuals")
hist(res, ylab = "Frequency", xlab = "Forecast Error", bty = "l", main ="")
## ACF plot shows that errors are random
################# Now use Smoothing Methods ###########################
ma.trailing <- rollmean(train.ts, k = 12, align = "right")
last.ma <- tail(ma.trailing, 1)
ma.trailing.pred <- ts(rep(last.ma, nValid), start = c (2014,nTrain+1), end = c(2014,totalRecords), freq = 12)
plot(train.ts, xlab = "Monthly", ylab = "Mobile App Visits", bty = "l", col = 'black')
lines(ma.trailing, lwd = 2, col = "blue")
lines(ma.trailing.pred, lwd = 2, col = "blue", lty = 2)
lines(valid.ts)
summary(ma.trailing.pred)
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 189700000 189700000 189700000 189700000 189700000 189700000
accuracy(ma.trailing.pred, valid.ts)
## ME RMSE MAE MPE MAPE ACF1
## Test set -20955158 23372636 20955158 -12.84802 12.84802 -0.6362683
## Theil's U
## Test set 1.3043
#Simple exponential smoothing
#The result is a series with no monthly seasonality.
diff.ts <- diff(appVisits.ts, lag = 12)
diff.ts
## Jan Feb Mar Apr May Jun Jul
## 2015 56724935 60150103 83956121 87228567 98432021 106403283 125464690
## 2016 104528716 100212420 57579980 138523480 207515056 59629345 22127243
## 2017 18226967 -9650587 26569989 -68405483
## Aug Sep Oct Nov Dec
## 2015 127835211 92853175 169556022 121039625 91088680
## 2016 20659144 35941520 42089555 -15763050 24460594
## 2017
totalRecords <- length(appVisits.ts)
nValid <- 4
nTrain <- totalRecords - nValid
train.ts <- window(diff.ts, end = c(2014,nTrain))
train.ts
## Jan Feb Mar Apr May Jun Jul
## 2015 56724935 60150103 83956121 87228567 98432021 106403283 125464690
## 2016 104528716 100212420 57579980 138523480 207515056 59629345 22127243
## Aug Sep Oct Nov Dec
## 2015 127835211 92853175 169556022 121039625 91088680
## 2016 20659144 35941520 42089555 -15763050 24460594
valid.ts <- window(diff.ts, start = c (2014,nTrain+1), end = c(2014,totalRecords))
valid.ts
## Jan Feb Mar Apr
## 2017 18226967 -9650587 26569989 -68405483
#The ets function chooses the optimal ?? and initial level value by maximizing something called the likelihood over
#the training period. In the case of simple exponential smoothing model, maximizing the likelihood
#is equivalent to minimizing the RMSE in the training period
#(the RMSE in the training period is equal to ??, the standard deviation of the training residuals.)
ses <- ets(train.ts, model = "ANN", alpha = 0.2)
ses.pred <- forecast(ses, h = nValid, level = 0)
ses.pred
## Point Forecast Lo 0 Hi 0
## Jan 2017 44146557 44146557 44146557
## Feb 2017 44146557 44146557 44146557
## Mar 2017 44146557 44146557 44146557
## Apr 2017 44146557 44146557 44146557
plot(ses.pred, ylab = "Visits (Difference)", xlab = "Time", bty = "l", xaxt = "n", main ="", flty = 2)
lines(ses.pred$fitted, lwd = 2, col = "blue")
lines(valid.ts)
accuracy(ses.pred,valid.ts)
## ME RMSE MAE MPE MAPE MASE
## Training set -8597834 46992556 37747688 -26.44339 89.54415 0.4925129
## Test set -52461335 64309516 52461335 128.40736 232.58565 0.6844892
## ACF1 Theil's U
## Training set 0.3676872 NA
## Test set -0.3936157 1.013092
## Holt's Winter model for trend
totalRecords <- length(appVisits.ts)
nValid <- 4
nTrain <- totalRecords - nValid
train.ts <- window(appVisits.ts,start = c (2014,1), end = c(2014,nTrain))
train.ts
## Jan Feb Mar Apr May Jun Jul
## 2014 2869521 3229880 4684880 8562608 12314911 14010825 16747894
## 2015 59594456 63379983 88641001 95791175 110746932 120414108 142212584
## 2016 164123172 163592403 146220981 234314655 318261988 180043453 164339827
## Aug Sep Oct Nov Dec
## 2014 21862188 27516353 45598418 38226274 62578002
## 2015 149697399 120369528 215154440 159265899 153666682
## 2016 170356543 156311048 257243995 143502849 178127276
valid.ts <- window(appVisits.ts, start = c (2014,nTrain+1), end = c(2014,totalRecords))
valid.ts
## Jan Feb Mar Apr
## 2017 182350139 153941816 172790970 165909172
ses.opt <- ets(train.ts, model = "MAN", restrict = FALSE)
ses.opt.pred <- forecast(ses.opt, h = nValid, level = 0)
accuracy(ses.pred, valid.ts)
## ME RMSE MAE MPE MAPE MASE
## Training set -8597834 46992556 37747688 -26.44339 89.54415 0.4925129
## Test set 124601468 125030745 124601468 73.73870 73.73870 1.6257375
## ACF1 Theil's U
## Training set 0.3676872 NA
## Test set -0.6362683 6.174715
accuracy(ses.opt.pred, valid.ts)
## ME RMSE MAE MPE MAPE MASE
## Training set 4505258 40662195 25982820 11.058455 25.761235 0.30422435
## Test set -3214459 10310554 7996170 -2.272052 4.921869 0.09362455
## ACF1 Theil's U
## Training set -0.0901213 NA
## Test set -0.6501759 0.5206197
ses.opt
## ETS(M,Ad,N)
##
## Call:
## ets(y = train.ts, model = "MAN", restrict = FALSE)
##
## Smoothing parameters:
## alpha = 0.4558
## beta = 0.1247
## phi = 0.8
##
## Initial states:
## l = -6483708.473
## b = 4040627.5949
##
## sigma: 0.4331
##
## AIC AICc BIC
## 1372.740 1375.636 1382.241
plot(ses.opt.pred, ylab = "Visits (Differenced)", xlab = "Time", bty = "l", xaxt = "n", main ="", flty = 2)
lines(ses.opt.pred$fitted, lwd = 2, col = "blue")
lines(valid.ts)
## Using MMA model we will forecast next 4 months of visits
appVisits <- ets(appVisits.ts, model = "MAN", restrict = FALSE)
appVisits.pred <- forecast(appVisits, h = 4, level = 0)
appVisits.pred
## Point Forecast Lo 0 Hi 0
## May 2017 164937047 164937047 164937047
## Jun 2017 163971799 163971799 163971799
## Jul 2017 163199601 163199601 163199601
## Aug 2017 162581842 162581842 162581842
appVisits.pred$mean
## May Jun Jul Aug
## 2017 164937047 163971799 163199601 162581842
accuracy(appVisits.pred)
## ME RMSE MAE MPE MAPE MASE
## Training set 4065329 38895955 24370173 9.847372 23.75835 0.3140753
## ACF1
## Training set -0.1122878
appVisits
## ETS(M,Ad,N)
##
## Call:
## ets(y = appVisits.ts, model = "MAN", restrict = FALSE)
##
## Smoothing parameters:
## alpha = 0.4922
## beta = 0.1082
## phi = 0.8
##
## Initial states:
## l = -6483708.6573
## b = 4040628.5953
##
## sigma: 0.412
##
## AIC AICc BIC
## 1532.177 1534.723 1542.311