About the Dataset:

US Department of Tourism maintains statistics on visitors to America. They have very details datasets. We are going to focus on just a general statistic that keeps track of monthly visitors. See details at http://travel.trade.gov/research/monthly/arrivals/index.asp

• Why is necessary to forecast this data?

  • Travel and tourism is the largest services export industry for the United States.Estimates of future demand at destination level are very important in managing and planning tourism development and the necessary investment. It is required to predict travel demand for every major global market and particular traveller segments. These forecasting models will give insights in the economic fundamentals of origin markets along with the changing dynamics of the tourism industry and traveller preferences.

Import Data:

library(readr)

## Warning: package 'readr' was built under R version 3.3.3

library(lattice)
library(TTR)
library(fpp)

## Loading required package: forecast

## Loading required package: zoo

## 
## Attaching package: 'zoo'

## The following objects are masked from 'package:base':
## 
##     as.Date, as.Date.numeric

## Loading required package: timeDate

## This is forecast 7.3

## Loading required package: fma

## Loading required package: tseries

## Loading required package: expsmooth

## Loading required package: lmtest

US_tourism <- read_csv("D:/BF/FINAL/Final_Travel.csv")

## Parsed with column specification:
## cols(
##   Year = col_integer(),
##   Month = col_character(),
##   Value = col_integer()
## )

travel <- US_tourism$Value
plot(travel)

US_tourism_TS<- ts(travel,start=c(1999,1),frequency = 12)
# Subset of the data to remove values I do not want in my analysis: As we can see from the plot, there is a steep downfall in the tourism in the end of the year 2001 which may be beacause of the WTC 9/11 attack which happened in September 2011
# Therefore not considering this irregular component from my analysis
# Taking the data from January 2002 till August 2016
US_tourism_TSW <- window(US_tourism_TS, c(2002,1), c(2016,8), frequency = 12)

Plot and Inference

• . Show a time series plot.

plot(US_tourism_TSW, ylab="US_Visitors", type="o", pch =20)

• Please summaries your observations of the times series plot

  • We can see from the time series plot that the USA Tourist Visitors frequency shows an upward trend with strong seasonal variation
  • There is no evidence of any cyclic behaviour here
  • The year on year trend clearly shows that the visitors though increasing linearly, some dips are seen during the start and end months of each year
  • We see a small dip in the year 2009
  • These seasonal fluctuations seem to increase as level of the time series increases

Central Tendency

• What are the min, max, mean, median, 1st and 3rd Quartile values of the times series?

summary(US_tourism_TSW)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## 1197000 1735000 2056000 2206000 2569000 4075000

hist(US_tourism_TSW)

#the default percentile values
quantile(US_tourism_TSW)

##      0%     25%     50%     75%    100% 
## 1197397 1735330 2055772 2569408 4075384

• Show the box plot

cycle(US_tourism_TSW)

##      Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
## 2002   1   2   3   4   5   6   7   8   9  10  11  12
## 2003   1   2   3   4   5   6   7   8   9  10  11  12
## 2004   1   2   3   4   5   6   7   8   9  10  11  12
## 2005   1   2   3   4   5   6   7   8   9  10  11  12
## 2006   1   2   3   4   5   6   7   8   9  10  11  12
## 2007   1   2   3   4   5   6   7   8   9  10  11  12
## 2008   1   2   3   4   5   6   7   8   9  10  11  12
## 2009   1   2   3   4   5   6   7   8   9  10  11  12
## 2010   1   2   3   4   5   6   7   8   9  10  11  12
## 2011   1   2   3   4   5   6   7   8   9  10  11  12
## 2012   1   2   3   4   5   6   7   8   9  10  11  12
## 2013   1   2   3   4   5   6   7   8   9  10  11  12
## 2014   1   2   3   4   5   6   7   8   9  10  11  12
## 2015   1   2   3   4   5   6   7   8   9  10  11  12
## 2016   1   2   3   4   5   6   7   8

This will print the cycle across the years.

cycle(US_tourism_TSW)

##      Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
## 2002   1   2   3   4   5   6   7   8   9  10  11  12
## 2003   1   2   3   4   5   6   7   8   9  10  11  12
## 2004   1   2   3   4   5   6   7   8   9  10  11  12
## 2005   1   2   3   4   5   6   7   8   9  10  11  12
## 2006   1   2   3   4   5   6   7   8   9  10  11  12
## 2007   1   2   3   4   5   6   7   8   9  10  11  12
## 2008   1   2   3   4   5   6   7   8   9  10  11  12
## 2009   1   2   3   4   5   6   7   8   9  10  11  12
## 2010   1   2   3   4   5   6   7   8   9  10  11  12
## 2011   1   2   3   4   5   6   7   8   9  10  11  12
## 2012   1   2   3   4   5   6   7   8   9  10  11  12
## 2013   1   2   3   4   5   6   7   8   9  10  11  12
## 2014   1   2   3   4   5   6   7   8   9  10  11  12
## 2015   1   2   3   4   5   6   7   8   9  10  11  12
## 2016   1   2   3   4   5   6   7   8

This will aggregate the cycles and display a year on year trend.

boxplot(US_tourism_TSW~cycle(US_tourism_TSW))

Box plot across the months will give us a sense on seasonal effect

a = boxplot(US_tourism_TSW, las =2, col = "royalblue2")

text(a$stats[3],"Median", pos = 4, offset = 8.5)
text(max(a$stats),"Max", pos = 4, offset = 8.5)
text(min(a$stats),"Min", pos = 4, offset = 8.5)
text(a$stats[2], "1st Quartile", pos = 4, offset = 8.5)
text(a$stats[4], "3rd Quartile", pos = 4, offset = 8.5)
text(a$out[2], "Outlier", pos = 4, offset = 8.5)

• Can you summarize your observation about the time series from the summary stats and box plot?

• Summary of the time series summary stats and boxplots:

  • The data is found to be a little right skewed as per the histogram of the time series values above
  • Summary of the time series gives the minimum and the maximum value of the dataset. Also, we see the mean is significantly greater than the median and is justified for a distribution that is skewed to the right (that is, bunched up toward the left and with a “tail” stretching toward the right)
  • Boxplots show the data accordingly and goes with the above summary.
  • Quantiles are the 25th and 75th percentiles for the jobless claims in the sample; i.e. they are the first and third quartiles. In short, 25% of the claims are below 254500 and 75% are above 254500, 75% are below 341400 and 25% are above 341400, and the other 50% are in between.
  • Boxplots show the data accordingly and goes with the above summary.
  • The bottom and top of the box are always the 25th and 75th percentile, and the band near the middle of the box is always the median.
  • The ends of the whiskers represent the minimum and maximum values that do not exceed a certain distance from the middle 50% of the data.
  • A single point denotes the outliers in the data which represent a big value. We do see such a values.

Decomposition

• Plot the decomposition of the time series.

STL_US_tourism_TSW <- stl(US_tourism_TSW, s.window ="periodic", robust = TRUE)
plot(STL_US_tourism_TSW)

• Is the times series seasonal?

  • Yes, it is Seasonal, as we can clearly see a seasonal variation with an upward trend in the plot.

• Is the decomposition additive or multiplicative?

  • The seasonal variation looks to be about the same magnitude across time. Hence, Additive; the magnitude of the seasonal pattern in the data does not depend on the magnitude of the data.
  • In other words, the magnitude of the seasonal pattern does not change as the series goes up or down.

• If seasonal, what are the values of the seasonal monthly indices?

decomp_US_tourism_TSW <- decompose(US_tourism_TSW, type = "additive")
decomp_US_tourism_TSW$seasonal

##             Jan        Feb        Mar        Apr        May        Jun
## 2002 -414382.74 -522276.94 -173650.06  -48804.42   30903.31   76409.67
## 2003 -414382.74 -522276.94 -173650.06  -48804.42   30903.31   76409.67
## 2004 -414382.74 -522276.94 -173650.06  -48804.42   30903.31   76409.67
## 2005 -414382.74 -522276.94 -173650.06  -48804.42   30903.31   76409.67
## 2006 -414382.74 -522276.94 -173650.06  -48804.42   30903.31   76409.67
## 2007 -414382.74 -522276.94 -173650.06  -48804.42   30903.31   76409.67
## 2008 -414382.74 -522276.94 -173650.06  -48804.42   30903.31   76409.67
## 2009 -414382.74 -522276.94 -173650.06  -48804.42   30903.31   76409.67
## 2010 -414382.74 -522276.94 -173650.06  -48804.42   30903.31   76409.67
## 2011 -414382.74 -522276.94 -173650.06  -48804.42   30903.31   76409.67
## 2012 -414382.74 -522276.94 -173650.06  -48804.42   30903.31   76409.67
## 2013 -414382.74 -522276.94 -173650.06  -48804.42   30903.31   76409.67
## 2014 -414382.74 -522276.94 -173650.06  -48804.42   30903.31   76409.67
## 2015 -414382.74 -522276.94 -173650.06  -48804.42   30903.31   76409.67
## 2016 -414382.74 -522276.94 -173650.06  -48804.42   30903.31   76409.67
##             Jul        Aug        Sep        Oct        Nov        Dec
## 2002  503879.35  470083.42  171812.04  130449.21 -269600.93   45178.10
## 2003  503879.35  470083.42  171812.04  130449.21 -269600.93   45178.10
## 2004  503879.35  470083.42  171812.04  130449.21 -269600.93   45178.10
## 2005  503879.35  470083.42  171812.04  130449.21 -269600.93   45178.10
## 2006  503879.35  470083.42  171812.04  130449.21 -269600.93   45178.10
## 2007  503879.35  470083.42  171812.04  130449.21 -269600.93   45178.10
## 2008  503879.35  470083.42  171812.04  130449.21 -269600.93   45178.10
## 2009  503879.35  470083.42  171812.04  130449.21 -269600.93   45178.10
## 2010  503879.35  470083.42  171812.04  130449.21 -269600.93   45178.10
## 2011  503879.35  470083.42  171812.04  130449.21 -269600.93   45178.10
## 2012  503879.35  470083.42  171812.04  130449.21 -269600.93   45178.10
## 2013  503879.35  470083.42  171812.04  130449.21 -269600.93   45178.10
## 2014  503879.35  470083.42  171812.04  130449.21 -269600.93   45178.10
## 2015  503879.35  470083.42  171812.04  130449.21 -269600.93   45178.10
## 2016  503879.35  470083.42

# the values of the seasonal monthly indices:
decomp_US_tourism_TSW$figure

##  [1] -414382.74 -522276.94 -173650.06  -48804.42   30903.31   76409.67
##  [7]  503879.35  470083.42  171812.04  130449.21 -269600.93   45178.10

#the seasonal effects values for all the months. We can see that the seasonal effect values are repeated each year (month-wise) in the $seasonal object from January onwards.
plot(decomp_US_tourism_TSW$figure, main = "Seasonal Indices")

• For which month is the value of time series high and for which month is it low?

#month for which the value of time series is high: 
US_tourism[which.max(US_tourism_TS),]

## # A tibble: 1 × 3
##    Year Month   Value
##   <int> <chr>   <int>
## 1  2015  July 4075384

#month for which the value of time series is low: 
US_tourism[which.min(US_tourism_TS),]

## # A tibble: 1 × 3
##    Year    Month   Value
##   <int>    <chr>   <int>
## 1  2001 November 1118797

• Can you think of the reason behind the value being high in those months and low in those months?

  • Reasons for a steep downfall in year-2002 : 9/11 crisis.
  • Reasons for a low count of visitors during start and end of each year: Snow Weather
  • Reasons for a high count of visitors each mid-year : Good Pleasant Weather

• Show the plot for time series adjusted for seasonality. Overlay this with the line for actual time series?

SA_US_tourism_TSW <- seasadj(STL_US_tourism_TSW)
plot(SA_US_tourism_TSW, main = "Time Series Adjusted for Seasonality")

plot(US_tourism_TSW)
lines(SA_US_tourism_TSW, col = "red")

• • Yes, the seasonality has big fluctuations to the actual value of time series

Naïve Method

• Output

snaive_forecast <- snaive(US_tourism_TSW)
plot(snaive_forecast)

naive_forecast <- naive(US_tourism_TSW)
plot(naive_forecast)
lines(snaive_forecast$mean,col="black")

• Perform Residual Analysis for this technique.

  1 Do a plot of residuals. What does the plot indicate?

res <- residuals(naive_forecast)
fit <- fitted(naive_forecast)
act <- naive_forecast$x
plot(res, main="Residuals from the Naïve method Forecasts", 
ylab="", xlab="Time")

• • The time plot of the residuals shows that the variation of the residuals stays much the same across the historical data, so the residual variance can be treated as constant
  • The remaining residuals show that the model has captured the patterns in the data quite well, also there is a decent amount of autocorrelation in the residuals (seen from the significant spikes in the ACF plot below)

2 Do a Histogram plot of residuals. What does the plot indicate?

hist(res, xlab = "Residuals", main="Histogram of Residuals from the Naïve method Forecasts")

• • The histogram plot of the residuals suggests that the residuals may not follow a normal distribution
  3 Do a plot of fitted values vs. residuals. What does the plot indicate?

xyplot(res~fit, pch=16, cex = 1.5, abline=0)

• • A plot of the residuals against the fitted values should show no pattern. We do not see any pattern here.
  • If a pattern is observed, there may be “heteroscedasticity” in the errors. That is, the variance of the residuals may not be constant.
  4 Do a plot of actual values vs. residuals. What does the plot indicate?

xyplot(res~act, pch = 16, cex = 1.5, abline = 0)

• • This plot shows no systematic patterns and the variation in the residuals does not seem to change with the size of the actual values
  5 Do an ACF plot of the residuals? What does this plot indicate?

Acf(res, main="ACF of residuals from the Naive method Forecasts")

• • The above figure shows an autocorrelation plot for the USA Tourist Visitors from 1999-2017 for Naive Forecast.
  • The plot shows the value of an autocorrelation function (Acf) on the vertical axis that ranges from -1 to 1
  • The horizontal axis of an autocorrelation plot shows the size of the lag between the elements of the time series.
  • On the graph, there is a vertical line (a “spike”) corresponding to each lag. The height of each spike shows the value of the autocorrelation function for the lag.
  • In this scenario, the spikes show a seasonal variation in terms of statistical significance for lags upto 24, i.e, some exceed the significance bounds - lags 1,6,12,18,24.
  • This goes very well in explanation with the seasonality variation that we have already discussed before (time series plot)
  • If a spike is significantly away from zero, that is evidence of autocorrelation. A spike that’s close to zero is evidence against autocorrelation.
  • I can see that all the Plots above and ACF, both show a seasonal pattern throughout the lags; i.e. the data here is exhibiting a trend that is going up with increase in period.
• Print the 5 measures of accuracy for this forecasting technique

accuracy_naive <- accuracy(naive_forecast)
accuracy_snaive <- accuracy(snaive_forecast)
accuracy_naive

##                    ME   RMSE      MAE        MPE     MAPE     MASE
## Training set 15081.85 301462 240545.6 -0.2643395 11.06543 1.397038
##                    ACF1
## Training set -0.2024705

accuracy_snaive

##                    ME     RMSE      MAE      MPE     MAPE MASE      ACF1
## Training set 115998.2 205853.8 172182.5 4.595344 7.741379    1 0.5732204

• Forecast
  • Time series value for next year. Show table and plot

naive_forecast <- naive(US_tourism_TSW)
naive_forecast

##          Point Forecast   Lo 80   Hi 80   Lo 95   Hi 95
## Sep 2016        3836721 3450382 4223060 3245866 4427576
## Oct 2016        3836721 3450382 4223060 3245866 4427576
## Nov 2016        3836721 3450382 4223060 3245866 4427576
## Dec 2016        3836721 3450382 4223060 3245866 4427576
## Jan 2017        3836721 3450382 4223060 3245866 4427576
## Feb 2017        3836721 3450382 4223060 3245866 4427576
## Mar 2017        3836721 3450382 4223060 3245866 4427576
## Apr 2017        3836721 3450382 4223060 3245866 4427576
## May 2017        3836721 3450382 4223060 3245866 4427576
## Jun 2017        3836721 3450382 4223060 3245866 4427576

plot(naive_forecast)

snaive_forecast <- snaive(US_tourism_TSW,12)
snaive_forecast

##          Point Forecast   Lo 80   Hi 80   Lo 95   Hi 95
## Sep 2016        3551833 3288021 3815645 3148367 3955299
## Oct 2016        3500510 3236698 3764322 3097044 3903976
## Nov 2016        2822990 2559178 3086802 2419524 3226456
## Dec 2016        3145903 2882091 3409715 2742437 3549369
## Jan 2017        2645349 2381537 2909161 2241883 3048815
## Feb 2017        2405849 2142037 2669661 2002383 2809315
## Mar 2017        2897042 2633230 3160854 2493576 3300508
## Apr 2017        2895487 2631675 3159299 2492021 3298953
## May 2017        3213869 2950057 3477681 2810403 3617335
## Jun 2017        3293795 3029983 3557607 2890329 3697261
## Jul 2017        3931057 3667245 4194869 3527591 4334523
## Aug 2017        3836721 3572909 4100533 3433255 4240187

plot(snaive_forecast)

• Summarize this forecasting technique

summary(naive_forecast)

## 
## Forecast method: Naive method
## 
## Model Information:
## $drift
## [1] 0
## 
## $drift.se
## [1] 0
## 
## $sd
## [1] 301948.4
## 
## $call
## naive(y = US_tourism_TSW)
## 
## 
## Error measures:
##                    ME   RMSE      MAE        MPE     MAPE     MASE
## Training set 15081.85 301462 240545.6 -0.2643395 11.06543 1.397038
##                    ACF1
## Training set -0.2024705
## 
## Forecasts:
##          Point Forecast   Lo 80   Hi 80   Lo 95   Hi 95
## Sep 2016        3836721 3450382 4223060 3245866 4427576
## Oct 2016        3836721 3450382 4223060 3245866 4427576
## Nov 2016        3836721 3450382 4223060 3245866 4427576
## Dec 2016        3836721 3450382 4223060 3245866 4427576
## Jan 2017        3836721 3450382 4223060 3245866 4427576
## Feb 2017        3836721 3450382 4223060 3245866 4427576
## Mar 2017        3836721 3450382 4223060 3245866 4427576
## Apr 2017        3836721 3450382 4223060 3245866 4427576
## May 2017        3836721 3450382 4223060 3245866 4427576
## Jun 2017        3836721 3450382 4223060 3245866 4427576

summary(snaive_forecast)

## 
## Forecast method: Seasonal naive method
## 
## Model Information:
## $drift
## [1] 0
## 
## $drift.se
## [1] 0
## 
## $sd
## [1] 170580.3
## 
## $call
## snaive(y = US_tourism_TSW, h = 12)
## 
## 
## Error measures:
##                    ME     RMSE      MAE      MPE     MAPE MASE      ACF1
## Training set 115998.2 205853.8 172182.5 4.595344 7.741379    1 0.5732204
## 
## Forecasts:
##          Point Forecast   Lo 80   Hi 80   Lo 95   Hi 95
## Sep 2016        3551833 3288021 3815645 3148367 3955299
## Oct 2016        3500510 3236698 3764322 3097044 3903976
## Nov 2016        2822990 2559178 3086802 2419524 3226456
## Dec 2016        3145903 2882091 3409715 2742437 3549369
## Jan 2017        2645349 2381537 2909161 2241883 3048815
## Feb 2017        2405849 2142037 2669661 2002383 2809315
## Mar 2017        2897042 2633230 3160854 2493576 3300508
## Apr 2017        2895487 2631675 3159299 2492021 3298953
## May 2017        3213869 2950057 3477681 2810403 3617335
## Jun 2017        3293795 3029983 3557607 2890329 3697261
## Jul 2017        3931057 3667245 4194869 3527591 4334523
## Aug 2017        3836721 3572909 4100533 3433255 4240187

• • From the accuracy measures that we saw for the both Naive Forecasts:
  • It is obvious that the accuracy values are way too high in terms of the RSME and not suitable to use for such a data
  • However, comparing both the Naive Methods, Seasonal Naïve method seems better for this data, although it can still be improved
  • In one year, i.e by the end of year 2017, it predicts the value of time series for USA Tourist Visitors will be 3836721 - Aug 2017

Simple Moving Averages

• Plot the graph for time series

plot(US_tourism_TSW)

• Show the Simple Moving average of order 3 on the plot above in Red

ma3_forecast <- ma(US_tourism_TSW,order=3)
plot(US_tourism_TSW)
lines(ma3_forecast,col="Red", lwd = 2)

• Show the Simple Moving average of order 6 on the plot above in Blue

ma3_forecast <- ma(US_tourism_TSW,order=3)
plot(US_tourism_TSW)
lines(ma3_forecast,col="Red", lwd = 2)
ma6_forecast <- ma(US_tourism_TSW,order=6)
lines(ma6_forecast,col="Blue", lwd = 2)

• Show the Simple Moving average of order 12 on the plot above in Green

ma3_forecast <- ma(US_tourism_TSW,order=3)
plot(US_tourism_TSW)
lines(ma3_forecast,col="Red", lwd = 2)
ma6_forecast <- ma(US_tourism_TSW,order=6)
lines(ma6_forecast,col="Blue", lwd = 2)
ma12_forecast <- ma(US_tourism_TSW,order=12)
lines(ma12_forecast,col="Green", lwd = 2)

• Show the forecast of next 12 months using one of the simple average order that you feel works best for time series

ma1_forecast <- ma(US_tourism_TSW,order=1)
plot(US_tourism_TSW)
lines(ma1_forecast,col="Purple", lwd = 2)

MA_forecast <- forecast(ma1_forecast,h=12)
plot(MA_forecast)

• • The above plot shows that as we increase the order, Simple Moving average does not fit properly with the actual data - it fluctuates all the more. Therefore taking the least order :- One

• What are your observations of the plot as the moving average order goes up?

  • The order of the moving average is inversely proportional to the residuals which is seen in the above 4 plots, as the order goes on increasing the residual factor also increases for this data.
  • Therefore, for better prediction, the order of the simple moving average should be small in this case.

Simple Smoothing

• Perform a simple smoothing forecast for next 12 months for the time series

forecast_ets<-ets(US_tourism_TSW)
forecast_ets

## ETS(M,N,M) 
## 
## Call:
##  ets(y = US_tourism_TSW) 
## 
##   Smoothing parameters:
##     alpha = 0.4147 
##     gamma = 1e-04 
## 
##   Initial states:
##     l = 1615326.4368 
##     s=1.0256 0.8825 1.0596 1.0823 1.2087 1.2262
##            1.0286 1.0079 0.972 0.9245 0.7721 0.81
## 
##   sigma:  0.0469
## 
##      AIC     AICc      BIC 
## 4988.083 4991.083 5035.640

ets_forecast <- forecast.ets(forecast_ets, h=12)
ets_forecast

##          Point Forecast   Lo 80   Hi 80   Lo 95   Hi 95
## Sep 2016        3437539 3231023 3644055 3121700 3753378
## Oct 2016        3365511 3146587 3584435 3030695 3700327
## Nov 2016        2803153 2607859 2998448 2504476 3101831
## Dec 2016        3257644 3016566 3498721 2888948 3626340
## Jan 2017        2572956 2372011 2773901 2265637 2880275
## Feb 2017        2452432 2251354 2653511 2144909 2759955
## Mar 2017        2936402 2684727 3188077 2551498 3321306
## Apr 2017        3087290 2811679 3362901 2665780 3508800
## May 2017        3201511 2904728 3498294 2747620 3655401
## Jun 2017        3267131 2953460 3580802 2787413 3746850
## Jul 2017        3894858 3508463 4281252 3303918 4485797
## Aug 2017        3839291 3446509 4232072 3238584 4439998

plot(ets_forecast)

• 1 What is the value of alpha? What does that value signify?

  The value of alpha = 0.4147

  • The estimated value of alpha is not that low, telling us that the estimate of the current value of the level is based upon both recent and past observations in the time series; i.e recent and old values in the time series are both given weightage

  2 What is the value of initial state?

forecast_ets$initstate

##            l           s1           s2           s3           s4 
## 1.615326e+06 1.025606e+00 8.825179e-01 1.059568e+00 1.082254e+00 
##           s5           s6           s7           s8           s9 
## 1.208728e+00 1.226218e+00 1.028589e+00 1.007929e+00 9.719781e-01 
##          s10          s11          s12 
## 9.244650e-01 7.720981e-01 8.100486e-01

• 3 What is the value of sigma? What does the sigma signify?

  • The value of sigma : 0.0469

  • It signifies the standard deviation of the residuals. Standard deviation is the extent of deviation for a group as a whole

• Perform Residual Analysis for this technique.

  1 Do a plot of residuals. What does the plot indicate?

plot(forecast_ets$residuals, main="Residuals from Simple Smoothing method Forecasts", 
ylab="", xlab="Time")

• • The time plot of the residuals shows that the variation of the residuals stays much the same across the historical data, so the residual variance can be treated as constant.
  • There is a downward spike in the residuals which suggests there might be something unusual happening in that year.
  • The remaining residuals show that the model has captured the patterns in the data quite well, also there is a decent amount of autocorrelation in the residuals (seen from the significant spikes in the ACF plot below)

2 Do a Histogram plot of residuals. What does the plot indicate?

hist(forecast_ets$residuals, xlab = "Residuals", main="Histogram of Residuals from Simple Smoothing method Forecasts")

• • The histogram plot shows that the residuals is roughly centered around zero .It is slighly skewed to the left as compared to the standard normal curve.
  3 Do a plot of fitted values vs. residuals. What does the plot indicate?

xyplot(forecast_ets$residuals ~ forecast_ets$fitted, pch=16, cex = 1.5, abline=0)

• • If a pattern is observed, there may be “heteroscedasticity” in the errors. That is, the variance of the residuals may not be constant.
  • However, this plot of the residuals against the fitted values should shows no pattern, there is tight correlation
  4 Do a plot of actual values vs. residuals. What does the plot indicate?

xyplot(forecast_ets$residuals ~ forecast_ets$x, pch = 16, cex = 1.5, abline = 0)

• • This plot shows no systematic patterns, scattered across, less correlation and the variation in the residuals does not seem to change with the size of the actual values.
  5 Do an ACF plot of the residuals? What does this plot indicate?

acf(forecast_ets$residuals, main = "ACF of Residuals of Simple Smoothing")

• • On the graph, there is a vertical line (a “spike”) corresponding to each lag. The height of each spike shows the value of the autocorrelation function for the lag.
  • In this scenario, the spikes show a seasonal variation in terms of statistical significance for lag of 0.0 which is positively above the significance bound
  • I can see that the Plots above and ACF, both show a seasonal pattern throughout the lags; i.e. the data here is exhibiting a trend that is going up with increase in period.
  • This goes very well in explanation with the seasonality variation that we have already discussed before (time series plots)
• Print the 5 measures of accuracy for this forecasting technique

accuracy_ets <- accuracy(ets_forecast)
accuracy_ets

##                    ME     RMSE      MAE      MPE     MAPE      MASE
## Training set 21543.29 96382.89 72295.69 0.756152 3.456318 0.4198782
##                    ACF1
## Training set 0.02090026

• Forecast
  • Time series value for next year. Show table and plot

ets_forecast

##          Point Forecast   Lo 80   Hi 80   Lo 95   Hi 95
## Sep 2016        3437539 3231023 3644055 3121700 3753378
## Oct 2016        3365511 3146587 3584435 3030695 3700327
## Nov 2016        2803153 2607859 2998448 2504476 3101831
## Dec 2016        3257644 3016566 3498721 2888948 3626340
## Jan 2017        2572956 2372011 2773901 2265637 2880275
## Feb 2017        2452432 2251354 2653511 2144909 2759955
## Mar 2017        2936402 2684727 3188077 2551498 3321306
## Apr 2017        3087290 2811679 3362901 2665780 3508800
## May 2017        3201511 2904728 3498294 2747620 3655401
## Jun 2017        3267131 2953460 3580802 2787413 3746850
## Jul 2017        3894858 3508463 4281252 3303918 4485797
## Aug 2017        3839291 3446509 4232072 3238584 4439998

plot(ets_forecast)

• Summarize this forecasting technique

summary(ets_forecast)

## 
## Forecast method: ETS(M,N,M)
## 
## Model Information:
## ETS(M,N,M) 
## 
## Call:
##  ets(y = US_tourism_TSW) 
## 
##   Smoothing parameters:
##     alpha = 0.4147 
##     gamma = 1e-04 
## 
##   Initial states:
##     l = 1615326.4368 
##     s=1.0256 0.8825 1.0596 1.0823 1.2087 1.2262
##            1.0286 1.0079 0.972 0.9245 0.7721 0.81
## 
##   sigma:  0.0469
## 
##      AIC     AICc      BIC 
## 4988.083 4991.083 5035.640 
## 
## Error measures:
##                    ME     RMSE      MAE      MPE     MAPE      MASE
## Training set 21543.29 96382.89 72295.69 0.756152 3.456318 0.4198782
##                    ACF1
## Training set 0.02090026
## 
## Forecasts:
##          Point Forecast   Lo 80   Hi 80   Lo 95   Hi 95
## Sep 2016        3437539 3231023 3644055 3121700 3753378
## Oct 2016        3365511 3146587 3584435 3030695 3700327
## Nov 2016        2803153 2607859 2998448 2504476 3101831
## Dec 2016        3257644 3016566 3498721 2888948 3626340
## Jan 2017        2572956 2372011 2773901 2265637 2880275
## Feb 2017        2452432 2251354 2653511 2144909 2759955
## Mar 2017        2936402 2684727 3188077 2551498 3321306
## Apr 2017        3087290 2811679 3362901 2665780 3508800
## May 2017        3201511 2904728 3498294 2747620 3655401
## Jun 2017        3267131 2953460 3580802 2787413 3746850
## Jul 2017        3894858 3508463 4281252 3303918 4485797
## Aug 2017        3839291 3446509 4232072 3238584 4439998

• • From the accuracy measures that we saw for the Simple Smoothing Method, we know:
  • It is evident that the accuracy values are much decent than the previous methods in terms of the RMSE
  • MPE is close to 0, So it says that our forecast is unbiased.
  • In one year, i.e by the end of year 2017, it predicts the value of time series will be 3839291
  • Therefore, we can say that the method explains all the trends in the dataset

Holt-Winters

• Perform a simple smoothing forecast for next 12 months for the time series

HW_US_tourism_TSW <- HoltWinters(US_tourism_TSW)

HW_US_tourism_TSW

## Holt-Winters exponential smoothing with trend and additive seasonal component.
## 
## Call:
## HoltWinters(x = US_tourism_TSW)
## 
## Smoothing parameters:
##  alpha: 0.3272698
##  beta : 0.02480278
##  gamma: 0.7684698
## 
## Coefficients:
##            [,1]
## a   3009108.503
## b      9420.709
## s1   339813.279
## s2   249777.147
## s3  -387797.749
## s4     1381.222
## s5  -465808.327
## s6  -668332.884
## s7  -149157.471
## s8   -15436.641
## s9   313995.641
## s10  314927.905
## s11  929179.097
## s12  824051.837

plot(HW_US_tourism_TSW, lwd=2)

• • We see from the plot that the Holt-Winters exponential method is successful in predicting the seasonal peaks, which occurs roughly around each mid-year, although they tend to lag-lead the observed values a little bit

To make forecasts for future times not included in the original time series, we use the “forecast.HoltWinters()” function in the “forecast” package

HW_forecast_US_tourism_TSW <- forecast.HoltWinters(HW_US_tourism_TSW, h= 12)
plot(HW_forecast_US_tourism_TSW)

• • The forecasts are shown as a blue line, and the dark grey and light grey shaded areas show 80% and 95% prediction intervals, respectively

  1 What is the value of alpha? What does that value signify?

  The value of alpha = 0.3272698

  • The estimated value of alpha is low, telling us that the estimate of the current value of the level is based mostly upon not very recent observations in the time series; i.e older values in the time series are weighted more heavily

  2 What is the value of beta? What does that value signify?

  The value of beta =0.02480278v

  • The value of beta is low, telling us that the estimate of the current value of the level is based mostly upon not very recent observations in the time series

  3 What is the value of gamma? What does that value signify?

  The value of gamma = 0.7684698

  • The valus of gamma is very high meaning that recent values in the time series are weighted more heavily.

  4 What is the value of initial state for the level, trend and seasonality? What do these values signify?

HW_US_tourism_TSW$coefficients

##           a           b          s1          s2          s3          s4 
## 3009108.503    9420.709  339813.279  249777.147 -387797.749    1381.222 
##          s5          s6          s7          s8          s9         s10 
## -465808.327 -668332.884 -149157.471  -15436.641  313995.641  314927.905 
##         s11         s12 
##  929179.097  824051.837

• • These values : a,b,s1 contain the initial estimated values for the level, trend and seasonal components respectively.

  5 What is the value of sigma? What does the sigma signify?

  The value of sigma is:

sd(complete.cases(HW_forecast_US_tourism_TSW$residuals))

## [1] 0.2527768

• • As we know,“sigma” (Standard deviation of residuals), so I have computed the standard deviation using the “sd()” function

• Perform Residual Analysis for this technique.

  1 Do a plot of residuals. What does the plot indicate?

resHW <- residuals(HW_forecast_US_tourism_TSW)
fitHW <- fitted(HW_forecast_US_tourism_TSW)
actHW <- HW_forecast_US_tourism_TSW$x
plot(resHW, main="Residuals from the Holt-Winter method Forecasts", 
ylab="", xlab="Time")

• • The time plot of the residuals shows that the variation of the residuals stays much the same across the historical data, so the residual variance can be treated as constant
  • There is a downward spike in the residuals (2002) which suggests there might be something unusual happening in that year. (We already know about this spike : discussed previously)

2 Do a Histogram plot of residuals. What does the plot indicate?

hist(resHW, xlab = "Residuals", main="Histogram of Residuals from the Holt-Winter method Forecasts")

• • The histogram plot of the residuals suggests that the residuals can be considered to follow a normal distribution
  3 Do a plot of fitted values vs. residuals. What does the plot indicate?

xyplot(resHW ~ fitHW, pch=16, cex = 1.5, abline=0)

• • The scatterplot relation is quite random. There is no strong correlation between the residuals and fitted values
  • The residuals roughly form a “horizontal band” around the 0 line. This suggests that the variances of the error terms are equal.
  4 Do a plot of actual values vs. residuals. What does the plot indicate?

xyplot(resHW ~ actHW, pch = 16, cex=1.5, abline = 0)

• • Though the scatterplot relation is quite random, there is a little correlation between the residuals and actual values
  • The residuals roughly form a “horizontal band” around the 0 line. This suggests that the variances of the error terms are equal. Some outlier seen.
  5 Do an ACF plot of the residuals? What does this plot indicate?

Acf(resHW, main = "ACF of Residuals of Holt-Winter Forecasts")

• • I can see that the Plots of residuals above and ACF, both show a seasonal pattern throughout the lags; i.e. the data here is exhibiting a trend that is going up with increase in period.
  • In this scenario, the spikes show a seasonal variation in terms of the lags.
  • This goes very well in explanation with the seasonality variation that we have already discussed before (time series plots)
• Print the 5 measures of accuracy for this forecasting technique

accuracy_HW <- accuracy(HW_forecast_US_tourism_TSW)
accuracy_HW

##                    ME     RMSE      MAE       MPE     MAPE      MASE
## Training set 15175.03 123247.5 95446.26 0.4461066 4.438431 0.5543318
##                 ACF1
## Training set 0.20383

• Forecast
  • Time series value for next year. Show table and plot

HW_forecast_US_tourism_TSW

##          Point Forecast   Lo 80   Hi 80   Lo 95   Hi 95
## Sep 2016        3358342 3201116 3515569 3117886 3598799
## Oct 2016        3277727 3111894 3443560 3024107 3531347
## Nov 2016        2649573 2475167 2823979 2382841 2916304
## Dec 2016        3048173 2865214 3231131 2768361 3327984
## Jan 2017        2590404 2398902 2781906 2297527 2883281
## Feb 2017        2397300 2197255 2597345 2091357 2703242
## Mar 2017        2925896 2717300 3134492 2606876 3244916
## Apr 2017        3069038 2851876 3286199 2736917 3401158
## May 2017        3407891 3182143 3633638 3062640 3753141
## Jun 2017        3418243 3183886 3652601 3059825 3776662
## Jul 2017        4041915 3798919 4284912 3670284 4413547
## Aug 2017        3946209 3694541 4197877 3561316 4331102

plot(HW_forecast_US_tourism_TSW)

• Summarize this forecasting technique

summary(HW_forecast_US_tourism_TSW)

## 
## Forecast method: HoltWinters
## 
## Model Information:
## Holt-Winters exponential smoothing with trend and additive seasonal component.
## 
## Call:
## HoltWinters(x = US_tourism_TSW)
## 
## Smoothing parameters:
##  alpha: 0.3272698
##  beta : 0.02480278
##  gamma: 0.7684698
## 
## Coefficients:
##            [,1]
## a   3009108.503
## b      9420.709
## s1   339813.279
## s2   249777.147
## s3  -387797.749
## s4     1381.222
## s5  -465808.327
## s6  -668332.884
## s7  -149157.471
## s8   -15436.641
## s9   313995.641
## s10  314927.905
## s11  929179.097
## s12  824051.837
## 
## Error measures:
##                    ME     RMSE      MAE       MPE     MAPE      MASE
## Training set 15175.03 123247.5 95446.26 0.4461066 4.438431 0.5543318
##                 ACF1
## Training set 0.20383
## 
## Forecasts:
##          Point Forecast   Lo 80   Hi 80   Lo 95   Hi 95
## Sep 2016        3358342 3201116 3515569 3117886 3598799
## Oct 2016        3277727 3111894 3443560 3024107 3531347
## Nov 2016        2649573 2475167 2823979 2382841 2916304
## Dec 2016        3048173 2865214 3231131 2768361 3327984
## Jan 2017        2590404 2398902 2781906 2297527 2883281
## Feb 2017        2397300 2197255 2597345 2091357 2703242
## Mar 2017        2925896 2717300 3134492 2606876 3244916
## Apr 2017        3069038 2851876 3286199 2736917 3401158
## May 2017        3407891 3182143 3633638 3062640 3753141
## Jun 2017        3418243 3183886 3652601 3059825 3776662
## Jul 2017        4041915 3798919 4284912 3670284 4413547
## Aug 2017        3946209 3694541 4197877 3561316 4331102

• • From the accuracy measures that we saw for the Holt-Winter Method, we know:
  • It is seen from the table that the accuracy values are decent enough
  • MPE is close to 0, So our forecast is unbiased.
  • In one year, i.e by the end of year 2017, it predicts the value of time series will be 3947060
  • Therefore, we can say that the method explain all the trends in the dataset

ARIMA or Box-Jenkins

• Is Time Series data Stationary? How did you verify? Please post the output from one of the test.

adf.test(US_tourism_TSW)

## Warning in adf.test(US_tourism_TSW): p-value smaller than printed p-value

## 
##  Augmented Dickey-Fuller Test
## 
## data:  US_tourism_TSW
## Dickey-Fuller = -11.032, Lag order = 5, p-value = 0.01
## alternative hypothesis: stationary

• • Since the p value is less thn 0.05, The time series is not stationary.
• How many differences are needed to make it stationary?

nsdiffs(US_tourism_TSW)

## [1] 1

• • 1 difference is needed to make the dataset stationary

• Is Seasonality component needed?

  • Yes, the sesonality component is needed

• Plot the Time Series chart of the differenced series.

tsdisplay(diff(US_tourism_TSW,12))

+ The residual plot obtained is not stationary; Hence doing the differetiation one more time

tsdisplay(diff(diff(US_tourism_TSW,12)))

• Plot the Acf and Pacf plot of the differenced series

US_tourism_TSW_diff <- diff(diff(US_tourism_TSW,12))
Acf(US_tourism_TSW_diff, lag.max = 36)

Pacf(US_tourism_TSW_diff, lag.max = 36)

• Based of the ACF and PACF, which are the possible ARIMA model possible?

  • Based on the PCF plot the spikes are decreasing exponentially having two significant spikes
  • on the seasonal lags i.e. 12,24,36 we can see two significant lags.
  • Based on the ACF plot there is one spike at the beginning, which is neither exponentially increasing or decreasing
  • on the seasonal lags i.e. 12,24,36 we can see two significant lags.
  • Therefore, the possible arima models are (0,1,2)(2,1,2)[12]

• Show the AIC, BIC and Sigma^2 for the possible models?

arima_fit <- Arima(US_tourism_TSW, order=c(0,1,2), seasonal=c(2,1,2))
arima_fit

## Series: US_tourism_TSW 
## ARIMA(0,1,2)(2,1,2)[12]                    
## 
## Coefficients:
##           ma1     ma2     sar1     sar2    sma1    sma2
##       -0.5552  0.2355  -1.1425  -0.6561  0.6170  0.0465
## s.e.   0.0920  0.1152   0.1184   0.1433  0.1671  0.2162
## 
## sigma^2 estimated as 1.027e+10:  log likelihood=-2112.67
## AIC=4239.34   AICc=4240.06   BIC=4261

• • AIC=4239.34
  • BIC=4261
  • Sigma^2 = 1.027e+10

tsdisplay(residuals(arima_fit))

• • But the above plot shows that there is some pattern in the ACF and the PCF plot
  • Therefore the above model is not good.

auto.arima(US_tourism_TSW, stepwise = FALSE, approximation = FALSE)

## Series: US_tourism_TSW 
## ARIMA(0,1,2)(2,1,1)[12]                    
## 
## Coefficients:
##           ma1     ma2     sar1     sar2    sma1
##       -0.5586  0.2425  -1.1258  -0.6276  0.5921
## s.e.   0.0908  0.1114   0.0910   0.0624  0.1163
## 
## sigma^2 estimated as 1.022e+10:  log likelihood=-2112.69
## AIC=4237.39   AICc=4237.93   BIC=4255.95

• Based on the above AIC, BIC and Sigma^2 values, which model will you select?

  • Based on the above AIC, BIC and Sigma^2 I would select model (0,1,2)(2,1,1)

arima_fit1 <- Arima(US_tourism_TSW, order=c(0,1,2), seasonal=c(2,1,1))
arima_fit1    

## Series: US_tourism_TSW 
## ARIMA(0,1,2)(2,1,1)[12]                    
## 
## Coefficients:
##           ma1     ma2     sar1     sar2    sma1
##       -0.5586  0.2425  -1.1258  -0.6276  0.5921
## s.e.   0.0908  0.1114   0.0910   0.0624  0.1163
## 
## sigma^2 estimated as 1.022e+10:  log likelihood=-2112.69
## AIC=4237.39   AICc=4237.93   BIC=4255.95

• What is the final formula for ARIMA with the coefficients?

arima_fit2 <- auto.arima(US_tourism_TSW, stepwise = FALSE, approximation = FALSE)
arima_fit2

## Series: US_tourism_TSW 
## ARIMA(0,1,2)(2,1,1)[12]                    
## 
## Coefficients:
##           ma1     ma2     sar1     sar2    sma1
##       -0.5586  0.2425  -1.1258  -0.6276  0.5921
## s.e.   0.0908  0.1114   0.0910   0.0624  0.1163
## 
## sigma^2 estimated as 1.022e+10:  log likelihood=-2112.69
## AIC=4237.39   AICc=4237.93   BIC=4255.95

• Perform Residual Analysis for this technique.

  1 Do a plot of residuals. What does the plot indicate?

res <- residuals(arima_fit2)
fit <- fitted(arima_fit2)
act <- arima_fit2$x
plot(res, main="Residuals of the Holt-Winter Forecasts", xlab="Time")

• • The time plot of the residuals shows that the variation of the residuals stays much the same across the historical data, so the residual variance can be treated as constant

2 Do a Histogram plot of residuals. What does the plot indicate?

hist(res, xlab = "Residuals", main="Histogram of Residuals from the Arima Forecasts")

• • The histogram plot of the residuals suggests that the residuals follows a beautiful normal distribution
  3 Do a plot of fitted values vs. residuals. What does the plot indicate?

xyplot(res ~ fit, pch=16, cex = 1.5, abline=0)

• • The scatterplot relation is quite random. There is no strong correlation between the residuals and fitted values
  4 Do a plot of actual values vs. residuals. What does the plot indicate?

xyplot(res ~ act, pch = 16, cex=1.5, abline = 0)

• • Though the scatterplot relation is quite random, there is a little correlation between the residuals and actual values
  • The residuals roughly form a “horizontal band” around the 0 line. This suggests that the variances of the error terms are equal. Some outlier seen.
  5 Do an ACF plot of the residuals? What does this plot indicate?

Acf(res, main = "ACF of Residuals of Arima Forecasts")

• • The above ACF plot shows that there is not much pattern left
  • There are significant lags at 6 and 7
• Print the 5 measures of accuracy for this forecasting technique

accuracy_arima <- accuracy(arima_fit2)
accuracy_arima

##                    ME     RMSE      MAE        MPE     MAPE      MASE
## Training set 430.4661 95778.21 69782.74 -0.1387604 3.166028 0.4052835
##                    ACF1
## Training set 0.02986358

• Forecast
  • Next one year. Show table and plot

arima_forecast <- forecast(arima_fit2,h=12)
plot(arima_forecast)

+ Next two years. Show table and plot

arima_forecast <- forecast(arima_fit2,h=24)
plot(arima_forecast)

• Summarize this forecasting technique

summary(arima_forecast)

## 
## Forecast method: ARIMA(0,1,2)(2,1,1)[12]
## 
## Model Information:
## Series: US_tourism_TSW 
## ARIMA(0,1,2)(2,1,1)[12]                    
## 
## Coefficients:
##           ma1     ma2     sar1     sar2    sma1
##       -0.5586  0.2425  -1.1258  -0.6276  0.5921
## s.e.   0.0908  0.1114   0.0910   0.0624  0.1163
## 
## sigma^2 estimated as 1.022e+10:  log likelihood=-2112.69
## AIC=4237.39   AICc=4237.93   BIC=4255.95
## 
## Error measures:
##                    ME     RMSE      MAE        MPE     MAPE      MASE
## Training set 430.4661 95778.21 69782.74 -0.1387604 3.166028 0.4052835
##                    ACF1
## Training set 0.02986358
## 
## Forecasts:
##          Point Forecast   Lo 80   Hi 80   Lo 95   Hi 95
## Sep 2016        3424479 3294931 3554027 3226353 3622606
## Oct 2016        3387667 3246061 3529274 3171099 3604236
## Nov 2016        2759190 2592154 2926226 2503731 3014649
## Dec 2016        3164014 2974939 3353090 2874848 3453180
## Jan 2017        2717708 2508907 2926510 2398374 3037042
## Feb 2017        2426245 2199427 2653064 2079356 2773134
## Mar 2017        2888452 2644946 3131958 2516041 3260862
## Apr 2017        3186597 2927476 3445718 2790305 3582888
## May 2017        3390784 3116937 3664631 2971971 3809597
## Jun 2017        3372468 3084647 3660289 2932284 3812652
## Jul 2017        4048860 3747713 4350006 3588296 4509424
## Aug 2017        4012055 3698148 4325963 3531975 4492135
## Sep 2017        3561230 3213754 3908706 3029811 4092649
## Oct 2017        3489157 3123064 3855250 2929267 4049048
## Nov 2017        2864522 2476065 3252979 2270428 3458616
## Dec 2017        3260740 2851137 3670342 2634307 3887172
## Jan 2018        2746188 2316480 3175896 2089006 3403370
## Feb 2018        2548432 2099517 2997346 1861876 3234987
## Mar 2018        3076444 2609112 3543776 2361721 3791167
## Apr 2018        3179619 2694568 3664670 2437797 3921440
## May 2018        3571821 3069676 4073966 2803857 4339785
## Jun 2018        3515563 2996887 4034239 2722317 4308809
## Jul 2018        4235398 3700702 4770093 3417651 5053144
## Aug 2018        4115946 3565696 4666196 3274411 4957480

• • Arima Forecasts give very good results in terms of accuracy
  • By the end of year 2017, it predicts the value of time series will be 4115946
  • Therefore, we can say that the method explain all the trends in the dataset
  • According to the histogram plot the residuals is normally distributed
  • Therefore, we can say that the model explain all the trends in the dataset
  • The amount of error in model is consistent across the full range of your observed data

Accuracy Measures

• Show a table of all the forecast method above with their accuracy measures

final_accuracy <- rbind(accuracy_naive,accuracy_snaive,accuracy_ets,accuracy_HW, accuracy_arima)

rownames(final_accuracy) <- c("Naive Method", "Seasonal Naive","ETS", "Holt-Winter", "Arima")

final_accuracy

##                         ME      RMSE       MAE        MPE      MAPE
## Naive Method    15081.8514 301461.96 240545.59 -0.2643395 11.065429
## Seasonal Naive 115998.1585 205853.83 172182.52  4.5953436  7.741379
## ETS             21543.2855  96382.89  72295.69  0.7561520  3.456318
## Holt-Winter     15175.0336 123247.45  95446.26  0.4461066  4.438431
## Arima             430.4661  95778.21  69782.74 -0.1387604  3.166028
##                     MASE        ACF1
## Naive Method   1.3970383 -0.20247055
## Seasonal Naive 1.0000000  0.57322037
## ETS            0.4198782  0.02090026
## Holt-Winter    0.5543318  0.20382996
## Arima          0.4052835  0.02986358

• Separately define each forecast method and why it is useful.

  1 Naive Forecast:
  • Simple model that assumes very recent data provides best predictors of the future.
  • It is also known as “no-change” forecast.
  • Side Effects of Naive forecast is the Random fluctuations are tracked as faithfully as fundamental changes.
  • It is used only for comparison with the forecasts generated by the better techniques.
  2 Simple Moving Average:
  • It is the unweighted mean of the previous n data.
  • It is used when recent observations are better than all observation.
  • As new data comes in , newest value is added and oldest value is dropped.
  • Equal weights are assigned to each observation. It does not handle trend or seasonality well, although it does better than simple average method.
  • When the time series is stationary, the moving average can be very effective as the observations are nearby across time.
  3 Simple Smoothing:
  • When forecaster believes more-recent observations are likely to contain more information, this is the technique to use.
  • It is more appropriate for data with no predictable upward or downward trend.
  • The main aim is to estimate the current level. The level estimate is then used to forecast future values.
  • This method continually revises estimate in light of more recent experiences. Thus, most recent observation receives most weight, the one after little lower and so forth.
    
  • Best for short term forecasting. It assumes extreme fluctuations represent randomness in a series of historical observation.
  4 Holt Winters:
  • It is also Known as Triple Exponential Smoothing i.e. one for the level, one for trend, and one for seasonality.
  • There is additive method and multiplicative method.
  • It is Builds on Holts method.
  • It is used when forecast data points in a series, provided that the series is “seasonal”, i.e. repetitive over some period.
  5 Arima:
  • ARIMA is fitted to time series data either to better understand the data or to predict future points in the series.
  • ARIMA models are applied in some cases where data show evidence of non-stationarity, where an initial differencing step can be applied one or more times to eliminate the non-stationarity.
  • The AR part of ARIMA indicates that the evolving variable of interest is regressed on its own lagged values.
  • The MA part indicates that the regression error is actually a linear combination of error terms whose values occurred contemporaneously and at various times in the past.
  • The I indicates that the data values have been replaced with the difference between their values and the previous values.

• Show the best and worst forecast method for each of the accuracy measures.

Best:

+ ME:   Mean Error : 430.4661 - lowest -> Arima Method
+ RMSE: Root Mean Squared Error: 95778.21 -> (Penalizes large errors) lowest -> Arima Method
+ MAE:  Mean Absolute Error: 69782.74 -> lowest -> Arima Method
+ MPE:  Mean Percentage Error: -0.1387604 -> closest to zero -> Arima Method
+ MAPE: Mean Absolute Percentage Error: 3.166028 -> lowest -> Arima Method
+ MASE: Mean Absolute Scaled Error: 0.4052835 -> lowest -> Arima Method
+ ACF1: Autocorrelation of errors at lag 1: 0.02986358 -> lowest -> Arima Method

Worse:

+ ME:   Mean Error : 115998.1585 -> Seasonal Naive
+ RMSE: Root Mean Squared Error: 301461.96  -> Naive
+ MAE:  Mean Absolute Error: 240545.59 -> Naive
+ MPE:  Mean Percentage Error: 4.5953436  -> Seasonal Naive
+ MAPE: Mean Absolute Percentage Error: 11.065429 -> Naive Method
+ MASE: Mean Absolute Scaled Error: 1.3970383 -> Naive Method
+ ACF1: Autocorrelation of errors at lag 1: 0.57322037 -> Seasonal Naive Method

Based on all the accuracy measures that we see above, Forecasts performed by the ARIMA METHOD seems to be the best forecast method for the given dataset.

Conclusion

• Summarize your analysis of time series value over the time period.

  • From the time series value over the time-period we can say that the seasonality plays a major role as compared to the trend.

• Based on your analysis and forecast above, do you think the value of the time series will increase, decrease or stay flat over the next year? How about next 2 years?

#Naive Method
snaive_forecast <- snaive(US_tourism_TSW, 24)
plot(snaive_forecast)

naive_forecast <- naive(US_tourism_TSW, 24)
plot(naive_forecast)
lines(snaive_forecast$mean,col="black")

#SimpleSmoothing
ma1_forecast <- ma(US_tourism_TSW,order=1)
forecast_ma <- forecast(ma1_forecast,h=24)
plot(forecast_ma)

#Holt-Winter
HW_forecast <- HoltWinters(US_tourism_TSW)
HW_forecast_US <- forecast.HoltWinters(HW_forecast, h= 24)
plot(HW_forecast_US)

#Arima
arima_forecast <- forecast(arima_fit2,h=24)
plot(arima_forecast)

arima_forecast

##          Point Forecast   Lo 80   Hi 80   Lo 95   Hi 95
## Sep 2016        3424479 3294931 3554027 3226353 3622606
## Oct 2016        3387667 3246061 3529274 3171099 3604236
## Nov 2016        2759190 2592154 2926226 2503731 3014649
## Dec 2016        3164014 2974939 3353090 2874848 3453180
## Jan 2017        2717708 2508907 2926510 2398374 3037042
## Feb 2017        2426245 2199427 2653064 2079356 2773134
## Mar 2017        2888452 2644946 3131958 2516041 3260862
## Apr 2017        3186597 2927476 3445718 2790305 3582888
## May 2017        3390784 3116937 3664631 2971971 3809597
## Jun 2017        3372468 3084647 3660289 2932284 3812652
## Jul 2017        4048860 3747713 4350006 3588296 4509424
## Aug 2017        4012055 3698148 4325963 3531975 4492135
## Sep 2017        3561230 3213754 3908706 3029811 4092649
## Oct 2017        3489157 3123064 3855250 2929267 4049048
## Nov 2017        2864522 2476065 3252979 2270428 3458616
## Dec 2017        3260740 2851137 3670342 2634307 3887172
## Jan 2018        2746188 2316480 3175896 2089006 3403370
## Feb 2018        2548432 2099517 2997346 1861876 3234987
## Mar 2018        3076444 2609112 3543776 2361721 3791167
## Apr 2018        3179619 2694568 3664670 2437797 3921440
## May 2018        3571821 3069676 4073966 2803857 4339785
## Jun 2018        3515563 2996887 4034239 2722317 4308809
## Jul 2018        4235398 3700702 4770093 3417651 5053144
## Aug 2018        4115946 3565696 4666196 3274411 4957480

• According to the analysis done above, we can see that by the end of 2 years, the USA tourist visitors is forecasted to be 4115946 around Aug 2018.