Answer to Question #199412 in Statistics and Probability for ROJER

Question

The given table shows the rainfall of Gujarat Region. Forecast the rainfall using

Exponential Smoothing. Use Alpha =0.2, 0.5 and 0.8. Data is available from 1997 to

2016, use this series for the calculation and forecast the rainfall for the year 2017. To

know, what extent the prediction is correct, actual rainfall for 2017 (1024.4millimeters) is provided Based on MSE and MAD, find out which alpha values

among the three suggestions are relatively near to actual value?

SUBDIVISION YEAR ANNUAL (in MM)

Gujarat Region 1997 1068.9

Gujarat Region 1998 1070

Gujarat Region 1999 568.4

Gujarat Region 2000 550.6

Gujarat Region 2001 849

Gujarat Region 2002 637.2

Gujarat Region 2003 1160.3

Gujarat Region 2004 1005.8

Gujarat Region 2005 1316.4

Gujarat Region 2006 1478

Gujarat Region 2007 1178.9

Gujarat Region 2008 911.1

Gujarat Region 2009 641.6

Gujarat Region 2010 1088.7

Gujarat Region 2011 890.5

Gujarat Region 2012 714

Gujarat Region 2013 1118.6

Gujarat Region 2014 705.7

Gujarat Region 2015 622.9

Gujarat Region 2016 764.9

Answer 1

Compute the exponential forecasts with the smoothening constant 0.2:

The general formula to obtain forecast value using simple exponential smoothing is given below:

$F_{t+1}=αA_t+(1-α)F_t$

$F_{t+1}$ is the forecast value for future observation

F_t is the forecast value for preceding observation

A_t is the actual value for preceding observation

α is the smoothing constant

Find the forecast values:

The forecast values using simple exponential smoothing are obtained as given below:

$F_1=1068.9 \\ F_2 =αA_1+(1-α)F_1 \\ = (0.2 \times 1068.9)+(0.8 \times 1068.9) \\ = 1068.9 \\ F_3 = αA_2+(1-α)F_2 \\ = (0.2 \times 1070) + (0.8 \times 1068.9) \\ = 1069.12$

Similarly, the forecast values for the remaining years using simple exponential smoothing are given below:

Compute the value of MSE and MAD:

The required calculations are obtained in the table given below:

The mean square error is obtained as given below:

$MSE = \frac{\sum (Error)^2}{Total \;number \;of \;considered \;forecast \;values} \\ = \frac{1555814}{20} \\ = 77790.7$

The mean absolute error is obtained as given below:

$MAD = \frac{\sum |Error|}{Total\; number \;of\; forecast \;values} \\ = \frac{4738.474}{20} \\ = 236.9237$

Compute the exponential forecasts with the smoothening constant 0.5:

The forecast values using simple exponential smoothing are obtained as given below:

$F_1=1068.9 \\ F_2=αA_1+(1-α)F_1 \\ = (0.5 \times 1068.9)+(0.5 \times 1068.9) \\ = 1068.9 \\ F_3 = αA_2+(1-α)F_2 \\ = (0.5 \times 1070) + (0.5 \times 1068.9) \\ = 1069.45$

Similarly, the forecast values for the remaining years using simple exponential smoothing are given below:

Compute the value of MSE and MAD:

The required calculations are obtained in the table given below:

The mean square error is obtained as given below:

$MSE = \frac{\sum (Error)^2}{Total \;number \;of \;considered \;forecast \;values} \\ = \frac{1523621}{20} \\ = 76181.05$

The mean absolute error is obtained as given below:

$MAD = \frac{\sum |Error|}{Total\; number \;of\; forecast \;values} \\ = \frac{4799.774}{20} \\ = 239.9887$

Compute the exponential forecasts with the smoothening constant 0.8:

The forecast values using simple exponential smoothing are obtained as given below:

$F_1=1068.9 \\ F_2=αA_1+(1-α)F_1 \\ = (0.8 \times 1068.9)+(0.2 \times 1068.9) \\ = 1068.9 \\ F_3 = αA_2+(1-α)F_2 \\ = (0.8 \times 1070) + (0.2 \times 1068.9) \\ = 1069.78$

Similarly, the forecast values for the remaining years using simple exponential smoothing are given below:

Compute the value of MSE and MAD:

The required calculations are obtained in the table given below:

The mean square error is obtained as given below:

$MSE = \frac{\sum (Error)^2}{Total \;number \;of \;considered \;forecast \;values} \\ = \frac{1589780}{20} \\ = 79489$

The mean absolute error is obtained as given below:

$MAD = \frac{\sum |Error|}{Total\; number \;of\; forecast \;values} \\ = \frac{4680.155}{20} \\ = 234.00775$

Based on MSE α=0.5 is better and based on MAD α=0.8 is better.