Question

Paul has just completed an analysis into the sales of a computer game over the past 3 years and the result is shown in the table below (sales are in S 000s): Period Sales Period Sales Sales Year 2007 Year 2008 Year 2009 Period 1 40 38 38 a) b) c) Plot the series. From the raw data, calculate the centred moving averages and plot this on the graph. Comment on both series of data. Use the additive decomposition model to obtain seasonal differences for each period. Adjusted the differences if needed. Obtain rough forecasts for 2010, d)

Answer 1

Answer:

Given that:

a) Plot of the series

plot of the series of sales year 2007,2008,and 2009 given below

Fig-A)

Time Series Plot of sales sales 2 ¢ ¢ 18 ¢ 10 il 14 16 Index

b) From the raw data,calculate the central moving average and plot this on graph.Comment on both series of data

The calculate the moving average of first two observation of sales and then next two observation so on corresponding all observation respectively. Simillarly for centered moving average calculated from moving average column respectively

C15 C13 Period C14 sales C16 Moving Average Centered moving Average 35 40 w na u 38.75 41.25 46.25 55.00 55.00 43.25 38.50 42.75 48.50 56.50 56.75 45.75 39.50 42.50 48.25 57.00 38.75 41.25 46.25 55.00 55.00 43.25 38.50 42.75 48.50 56.50 56.75 45.75 39.50 42.50 48.25 57.00

Centered Moving Averag Trend Variable Actual Fits - Forecasts + Moving Average Length 2 sales Accuracy Measures MAPE 16.983 MAD 7.9375 MSD 87.1406 30 2 4 6 8 12 14 16 18 10 Index

Fig-B)

Centered moving Average 2 4 6 8 10 12 14 16 18

comment:

The difference between Fig-A) and Fig-B) is that in Fig B) elliminate the short time flucation and also reduce the effecto of extreme values.

The question c) and d) solve in R- software given below solution in attach images

c) Use the additive decomposition model to obtain seasonal differences for each period.Adjusted the differences if needed

The first seasonal difference is taken for sales to stationary the data i.e all means and variance are constant at point

### Import data in R > data<-read.csv(file.choose ,header=T) > str(data) 'data.frame': 18 obs. of 3 variables: $ year : int 2007 2007 2007 2007 2007 2007 2008 2008 2008 2008 ... $ Period: int 1 2 3 4 5 6 1 2 3 4 ... $ sales : int 35 40 40 45 55 65 35 38 43 47 ... > library(tseries) Registered 53 method overwritten by 'quantmod': method from as.zoo.data. frame zoo 'tseries' version: 0.10-47 tseries' is a package for time series analysis and computational finance. See 'library (help="tseries") for details. Warning message: package 'tseries' was built under R version 3.6.3 > diff <- diff(data$sales, differences = 1) > diff [1] 5 0 5 10 10 -30 3 5 4 10 8 -25 [13] -2 4 6 7 15 > plot(diff) > library(forecast) Warning message: package forecast' was built under R version 3.6.3 > d<-(ts (data$sales)) U U

>da-(ts (datassales) > d Time Series: Start = 1 End = 18 Frequency = 1 [1] 35 40 40 45 55 65 35 38 43 47 57 65 40 38 42 48 55 [18] 70 > #decompose(d, type = "multiplicative") > mymodel<-auto.arima(d) > mymodel Series: d ARIMA(0,0,1) with non-zero mean Coefficients: mal 0.4540 s.e. 0.1913 mean 47.8821 3.2845 sigma12 estimated as 106.9: log likelihood=-66.64 AIC=139.28 AICC=141 BIC=141.95 > accuracy (mymodel) ME RMSE MAE MPE Training set 0.2141529 9.746569 7.732818 -3.5709 MAPE MASE ACF1 Training set 16.3817 0.8822678 0.02228293 > auto.arima(d, ic="aic", trace = TRUE) 50

> auto.arima(d, ic="aic", trace = TRUE) ARIMA(2,0,2) with non-zero mean : Inf ARIMA(0,0,0) with non-zero mean : 140.2278 ARIMA(1,0,0) with non-zero mean : 140.3279 ARIMA(0,0,1) with non-zero mean : 139.2815 ARIMA(0,0,0) with zero mean : 193.0703 ARIMA(1,0,1) with non-zero mean : 141.2443 ARIMA(0,0,2) with non-zero mean : 141. 1812 ARIMA(1,0,2) with non-zero mean : Inf ARIMA(0,0,1) with zero mean 178.0718 Best model: ARIMA(0,0,1) with non-zero mean Series: d ARIMA(0,0,1) with non-zero mean Coefficients: mal mean 0.4540 47.8821 s.e. 0.1913 3.2845 sigma 2 estimated as 106.9: log likelihood=-66.64 AIC=139.28 AICC=141 BIC=141.95

d) Obtain rough forecasts for 2010

The forecast for 2010 is given below

> #plot. ts (mymodel $residuals) > myforcat<-forecast (mymodel, level = c(95), h=6) > myforcat Point Forecast LO 95 Hi 95 56.64806 36.38635 76.90977 47.88209 25.63015 70.13404 47.88209 25.63015 70.13404 47.88209 25.63015 70.13404 47.88209 25.63015 70.13404 47.88209 25.63015 70.13404 19 20