1. Would you consider neural networks for this task? Explain why.

Even though neural network outputs are difficult to explain, I would consider using a neural network because the outputs can produce forecasts at different time horizons, which would work for what is being requested of the wine sales. The short term wine sale forecasts are easy to produce via a neural network.

2. Use neural networks to forecast fortified wine sales, as follows: - Partition the data using the period until December 1993 as the training period. - Run a neural network using R’s nnetar with 11 nonseasonal lags (i.e., p = 11). Leave all other arguments at their default.

setwd("C:/Users/larms.LA-INSP5559/Documents/R/win-library/3.3/17_0429assignment8")
WineSales<- read.csv("awines2.csv", stringsAsFactors = FALSE)
head(WineSales)
##      Month Fortified
## 1 1/1/1980      2585
## 2 2/1/1980      3368
## 3 3/1/1980      3210
## 4 4/1/1980      3111
## 5 5/1/1980      3756
## 6 6/1/1980      4216
str(WineSales)
## 'data.frame':    180 obs. of  2 variables:
##  $ Month    : chr  "1/1/1980" "2/1/1980" "3/1/1980" "4/1/1980" ...
##  $ Fortified: int  2585 3368 3210 3111 3756 4216 5225 4426 3932 3816 ...
tail(WineSales, 20)
##         Month Fortified
## 161  5/1/1993      2329
## 162  6/1/1993      2660
## 163  7/1/1993      2923
## 164  8/1/1993      2626
## 165  9/1/1993      2132
## 166 10/1/1993      1772
## 167 11/1/1993      2526
## 168 12/1/1993      2755
## 169  1/1/1994      1154
## 170  2/1/1994      1568
## 171  3/1/1994      1965
## 172  4/1/1994      2659
## 173  5/1/1994      2354
## 174  6/1/1994      2592
## 175  7/1/1994      2714
## 176  8/1/1994      2294
## 177  9/1/1994      2416
## 178 10/1/1994      2016
## 179 11/1/1994      2799
## 180 12/1/1994      2467
library(forecast)

winesalests<- ts(WineSales$Fortified, start = c(1980, 1), frequency = 12)

#Partition the data
winevalidationlen<- 12
winetraininglen<- length(winesalests)-winevalidationlen
winesalestrain<- window(winesalests, end = c(1980, winetraininglen))
winesalesvalid<- window(winesalests, start = c(1980, winetraininglen + 1))
#plot the series
plot(winesalests, xlab = "Year", ylab = "Wine Sales (Thousands)", main = "Fortified Wine Sales")

#run a neural network
set.seed(8373493)
wineNN<- nnetar(winesalestrain, p=11)
wineNN
## Series: winesalestrain 
## Model:  NNAR(11,1,6)[12] 
## Call:   nnetar(y = winesalestrain, p = 11)
## 
## Average of 20 networks, each of which is
## a 12-6-1 network with 85 weights
## options were - linear output units 
## 
## sigma^2 estimated as 5734
summary(wineNN$model[[1]])
## a 12-6-1 network with 85 weights
## options were - linear output units 
##   b->h1  i1->h1  i2->h1  i3->h1  i4->h1  i5->h1  i6->h1  i7->h1  i8->h1 
##   -0.14    1.60   -1.88    0.65    4.43   -2.46   -0.67    0.05   -3.61 
##  i9->h1 i10->h1 i11->h1 i12->h1 
##   -2.41    1.52    2.98    1.65 
##   b->h2  i1->h2  i2->h2  i3->h2  i4->h2  i5->h2  i6->h2  i7->h2  i8->h2 
##    0.44   -0.83    1.07    0.71    0.73    0.00    0.66    0.38   -1.07 
##  i9->h2 i10->h2 i11->h2 i12->h2 
##    0.09    0.00    0.09   -1.09 
##   b->h3  i1->h3  i2->h3  i3->h3  i4->h3  i5->h3  i6->h3  i7->h3  i8->h3 
##    1.21    3.92   -0.71    1.73   -2.67    1.49    0.36   -0.03    3.92 
##  i9->h3 i10->h3 i11->h3 i12->h3 
##    0.30   -3.39    1.96   -5.85 
##   b->h4  i1->h4  i2->h4  i3->h4  i4->h4  i5->h4  i6->h4  i7->h4  i8->h4 
##   -2.17    3.83    1.14    0.28   -1.49   -0.99    3.75    2.98   -4.88 
##  i9->h4 i10->h4 i11->h4 i12->h4 
##   -0.52   -1.34    0.72   -1.59 
##   b->h5  i1->h5  i2->h5  i3->h5  i4->h5  i5->h5  i6->h5  i7->h5  i8->h5 
##   -0.89    1.48   -1.48   -0.52   -1.33    0.44   -0.22   -0.45    0.73 
##  i9->h5 i10->h5 i11->h5 i12->h5 
##   -0.91    0.29    0.24   -0.83 
##   b->h6  i1->h6  i2->h6  i3->h6  i4->h6  i5->h6  i6->h6  i7->h6  i8->h6 
##   -0.65   -1.81   -0.12    0.84    0.43   -0.82   -2.19   -1.92    3.76 
##  i9->h6 i10->h6 i11->h6 i12->h6 
##   -0.03   -0.85    0.71   -0.97 
##  b->o h1->o h2->o h3->o h4->o h5->o h6->o 
##  1.98  1.59 -2.75  1.39 -1.27 -1.82 -2.04

(a) Create a time plot for the actual and forecasted series over the training period. Create also a time plot of the forecast errors for the training period. Interpret what you see in the plots.

set.seed(8373493)
winepred<- forecast(wineNN, h = winevalidationlen)
winepred
##           Jan      Feb      Mar      Apr      May      Jun      Jul
## 1994 1319.548 1407.169 1855.154 2049.701 2132.175 2376.346 2758.660
##           Aug      Sep      Oct      Nov      Dec
## 1994 2581.042 2139.324 1798.984 2306.188 2608.601
yrange = range(winesalests)
plot(c(1980, 1994), yrange, type = "n", xlab = "Year", ylab = "Wine Sales", main = "Actual & Forecasted Wine Sales", bty = "l", xaxt = "n", yaxt = "n")
lines(winesalestrain, col = "blue", lwd = 2)
lines(wineNN$fitted, col ="red", lwd = 1)
axis(1, at = seq(1980, 1994, 1))
axis(2, at = seq(1000, 6000, 500), labels = format(seq(1000,6000, 500)))

plot(wineNN$residuals, main = "Residuals  of NN Model", col = "purple")
abline(h = 0)

There aren’t many errors or deviations from the actuals to the NN model. It seems like it has overfit the model.

(b) Use the neural network to forecast sales for each month in the validation period (January 1994 to December 1994).

winepred
##           Jan      Feb      Mar      Apr      May      Jun      Jul
## 1994 1319.548 1407.169 1855.154 2049.701 2132.175 2376.346 2758.660
##           Aug      Sep      Oct      Nov      Dec
## 1994 2581.042 2139.324 1798.984 2306.188 2608.601
accuracy(winepred)
##                       ME     RMSE      MAE        MPE     MAPE      MASE
## Training set -0.09094804 75.71998 57.53445 -0.2129987 2.097739 0.2067678
##                      ACF1
## Training set -0.009408954
yrange = range(winesalests)
plot(c(1980, 1994), yrange, type = "n", xlab = "Year", ylab = "Wine Sales", main = "Actual & Forecasted Wine Sales", bty = "l", xaxt = "n", yaxt = "n")
lines(winesalestrain, col = "blue", lwd = 2)
lines(winepred$fitted, col ="red", lwd = 1)
axis(1, at = seq(1980, 1994, 1))
axis(2, at = seq(1000, 6000, 500), labels = format(seq(1000,6000, 500)))

3. Compare your neural network to an exponential smoothing model used to forecast fortified wine sales.

(a) Use R’s ets function to automatically select and fit an exponential smoothing model to the training period until December 1993. Which model did ets fit?

winesalesETS<- ets(winesalestrain, model = "ZZZ", restrict = FALSE)
winesalesETS
## ETS(M,A,M) 
## 
## Call:
##  ets(y = winesalestrain, model = "ZZZ", restrict = FALSE) 
## 
##   Smoothing parameters:
##     alpha = 0.0555 
##     beta  = 9e-04 
##     gamma = 1e-04 
## 
##   Initial states:
##     l = 4040.0811 
##     b = -6.7983 
##     s=1.1316 1.0399 0.8877 0.9505 1.2722 1.3862
##            1.1463 1.1097 0.9345 0.8513 0.6996 0.5903
## 
##   sigma:  0.0859
## 
##      AIC     AICc      BIC 
## 2755.038 2759.118 2808.145
accuracy(winesalesETS)
##                     ME     RMSE      MAE       MPE     MAPE      MASE
## Training set -25.32466 287.8687 224.6507 -1.317643 7.229271 0.8073515
##                    ACF1
## Training set 0.05168201

ETS fit the Holt-Winter’s model.

(b) Use this exponential smoothing model to forecast sales for each month in 1994.

winesalesETSpred<- forecast(winesalesETS, winevalidationlen)
winesalesETSpred
##          Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## Jan 1994       1289.829 1147.913 1431.745 1072.788 1506.871
## Feb 1994       1521.475 1353.802 1689.148 1265.041 1777.909
## Mar 1994       1842.645 1639.237 2046.054 1531.559 2153.732
## Apr 1994       2013.011 1790.409 2235.614 1672.571 2353.452
## May 1994       2379.117 2115.554 2642.679 1976.033 2782.201
## Jun 1994       2445.906 2174.435 2717.376 2030.728 2861.083
## Jul 1994       2943.532 2616.195 3270.870 2442.913 3444.151
## Aug 1994       2688.471 2388.895 2988.047 2230.309 3146.633
## Sep 1994       1998.782 1775.592 2221.971 1657.443 2340.120
## Oct 1994       1857.773 1649.880 2065.666 1539.829 2175.717
## Nov 1994       2165.635 1922.749 2408.521 1794.173 2537.097
## Dec 1994       2344.995 2081.384 2608.606 1941.836 2748.153

(c) How does the neural network compare to the exponential smoothing model in terms of predictive performance in the training period? In the validation period?

accuracy(winepred, winesalesvalid)
##                        ME      RMSE       MAE        MPE      MAPE
## Training set  -0.09094804  75.71998  57.53445 -0.2129987  2.097739
## Test set     138.75901697 289.14325 245.23423  5.1737224 10.881000
##                   MASE         ACF1 Theil's U
## Training set 0.2067678 -0.009408954        NA
## Test set     0.8813246 -0.040116767 0.6427012
accuracy(winesalesETSpred, winesalesvalid)
##                     ME     RMSE      MAE       MPE      MAPE      MASE
## Training set -25.32466 287.8687 224.6507 -1.317643  7.229271 0.8073515
## Test set     125.56906 328.9246 256.3940  4.443793 10.858860 0.9214307
##                     ACF1 Theil's U
## Training set  0.05168201        NA
## Test set     -0.01105575 0.7140459

After reviewing the MAPE values, it looks like the NN model forecasted much better than the ETS model considering the distance between the MAPE values in the training and test sets.