time series paper

1

Analysis of Data Using Triple Exponential Smoothing and Commentary Treating Mbeya

As Supply Region For Beans And Kinondoni As The Market Region For Beans.

By: MAWDO GIBBA

2

Original Data Set for Kidondoni

Year Month Kinondoni Units 100KG

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

2004 48545 41875 44308 43600 36885 37885 37708 40583 48462 50154 53167 60444

2005 61364 58955 59318 46500 53500 56583 48654 49455 50682 48583 57333 55563

2006 61611 72818 75000 80000 68208 63333 57318 56731 59583 60625 60750 62500

2007 66591 68500 0 0 68227 67936 64469 67479 63925 78714 86119 88300

2008 95800 105056 104500 89463 86417 92929 89164 100750 100500 105000 109250 121875

2009 115273 107773 102808 103227 103917 97577 107404 104500 105417 102292 108125 110682

2010 116583 111479 114231 111786 111923 112538 103042 103462 98792 97538 100417 101250

2011 100000 110550 64423 109500 127692 123050 128542 133222 127269 141625 151538 150750

2012 140308 139808 136615 138682 130385 136538 137500 148300 145833 141731 153462 156818

2013 160417 168125 162292 154773 151923 150833 150179 144583 149167 146591 152500 150682

2014 156458 157727 167708 164500 167500 172500 156250 145208 150000 150000 150000 0

Original Data Set for Mbeya

Year Month Mbeya Units 100 KG

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

2004 41667 34500 30893 29250 26731 28115 32292 31958 34423 37385 44792 45944

2005 45375 43614 37000 32500 40000 40909 42500 43000 39818 43023 41389 47313

2006 48250 54000 50455 48438 52208 52729 46295 44750 46545 50406 56500 61375

2007 60333 52875 48462 48864 49583 49300 49538 51731 53455 56679 71654 87042

2008 94231 102222 109500 98750 92045 78962 79091 82650 91250 95500 97208 102250

2009 106250 103167 102154 95273 97500 87500 100365 98846 102500 105542 101958 101273

2010 104458 101250 94712 90786 89481 89808 85000 85923 90833 89792 94583 96986

2011 100938 116111 135208 124500 123292 120500 115682 121136 121000 143125 149154 155300

2012 160615 164000 139077 110682 114077 122654 112692 129500 133333 135417 149808 151458

2013 160792 149292 127500 113864 110000 104167 110179 112708 110000 108864 118654 129545

2014 132708 135000 135000 135100 134654 132500 125000 115000 116538 118571 123125 123462

3

Introduction

The objective of the assignment is to analyze the data and give comment on the analysis of the

data. Several methods were used to approach the data analysis and all approaches cannot be

presented. As per the requirement of the assignment question, we resorted in reporting the

HoltWinters exponential smoothing to be specific, the triple exponential smoothing.

The HoltWinters seasonal method otherwise the triple exponential smoothing comprises the

forecast equation and three smoothing equations one for the level, one for trend, and one for the

seasonal component, with smoothing parameters α, β and γ. We use m to denote the period of the

seasonality, m=12, for this case, because we are dealing with monthly data.

There are two variations to this method that differ in the nature of the seasonal component. The

additive method is preferred when the seasonal variations are roughly constant through the

series, while the multiplicative method is preferred when the seasonal variations are changing

proportional to the level of the series. With the additive method, the seasonal component is

expressed in absolute terms in the scale of the observed series, and in the level equation the

series is seasonally adjusted by subtracting the seasonal component. Within each year the

seasonal component will add up to approximately zero. With the multiplicative method, the

seasonal component is expressed in relative terms (percentages) and the series is seasonally

adjusted by dividing through by the seasonal component. Within each year, the seasonal

component will sum up to approximately m.

The objectives of using this method is analyse the data for both Mbeya and Kinondoni include

the following:

Estimate the HoltWinter Model using either the seasonal multiplicative or seasonal

additive methods whichever of the two methods that best fits or describes the data.

Calculate the Level, Trend and Seasonality in order to generate the fitted values or

predicted values of the model used for both Mbeya and Kinondoni.

Do a forecast for the whole period of year 2015 for both Mbeya and Kinondoni.

Do residual analysis and forecast measurement accuracy analysis to test the fit and

accuracy of the fitted time series model using the techniques of MAD, MSE, MAPE,

THEIL’s U and error scatter plot.

4

The chart below shows the plot of the Supply prices of beans for Mbeya and as can be seen from

the seasonal variation from the plot, there is an increasing variation of supply price of bean over

the period. For this kind of seasonal variation, the multiplicative method of the HoltWinter triple

exponential smoothing is preferred, that is when the seasonal variations are changing

proportional to the level of the series. Therefore we used the seasonal multiplicative approach of

triple exponential smoothing to analyse the supply price of beans for Mbeya.

The table below shows the smoothing parameters that were used in fitting the models. An alpha

level of 0.2, beta level of 0.1 and gamma level of 0.1 were used to smoothen the multiplicative

seasonal variation by using the HoltWinters triple exponential smoothing for Supply price of

bean in Mbeya. Since the selection of the smoothing parameters is objective, thus we selected the

parameters that will give the best smoothing to the model.

Smoothing Parameters

Alpha 0.2 Beta 0.1 Gamma 0.1

Time

Supp

ly

2 4 6 8 10 12

4000

080

000

1200

00

5

The table below shows the coefficients generated from the model, “a” which is the first or initial

smoothed level value and “b” which is the first smoothed trend value and seasonal indices “S”.

Since there are twelve seasons in the given data, so we expect to have seasonal indices from

to . The trend and level are initialized at period S. These initialization values are estimated as:

( )

(

)

Coefficients

a 1.23059E-05

b -0.0458759

S1 1.152692

S2 1.123397

S3 1.005243

S4 0.88383

S5 0.9451797

S6 0.938037

S7 0.9013298

S8 0.8973297

S9 0.9244399

S10 0.9754336

S11 1.087385

S12 1.129305

The table below gives the fitted or predicted values of the HoltWinters Triple Exponential

smoothing for the supply price of beans for Mbeya.

The whole of year 2004 is without fitted or predicted values that is from January 2004 through

December 2004. Fitted values begin from period two that is from January 2005 down to the

December 2014. For the Winter’s multiplicative method, the level, trend and seasonality are

generated as:

( )( )

( ) ( )

6

( )

( )

Fitted Values

period Month Level Trend Season

2005 Jan 40058.46 34271.17 663.889423 1.1466548

2005 Feb 39469.93 35862.38 756.620803 1.0778538

2005 Mar 34363.64 37387.95 833.515721 0.8990665

2005 Apr 31003.04 38807.93 892.162411 0.7809312

2005 May 39331.74 40083.47 930.500201 0.958984

2005 Jun 41373.23 41153.34 944.437012 0.9827889

2005 Jul 39337.16 42003.3 934.989904 0.9161322

2005 Aug 39857.94 43628.77 1004.037687 0.8930189

2005 Sep 43859.48 45336.5 1074.407021 0.9450253

2005 Oct 47260.87 45555.59 988.875264 1.0153917

2005 Nov 55662.1 45709.74 905.402713 1.1940776

2005 Dec 53423.46 44224.5 666.337831 1.1900751

Month Level Trend Season

2006 Jan 51470 43863.93 563.64748 1.1585146

2006 Feb 48228.41 43871.69 508.059077 1.086721

2006 Mar 41657.92 45441.96 614.279369 0.9045012

2006 Apr 38263.25 48001.41 48001.41 0.7839189

2006 May 50390.34 51406.08 1068.384095 0.9602831

2006 Jun 52982.86 52853.03 1106.240877 0.9819047

2006 Jul 50714.21 53907.56 1101.070027 0.9219318

2006 Aug 49470.54 54049.95 1005.201584 0.8985633

2006 Sep 51496.56 54004.46 900.132983 0.9379281

2006 Oct 55079.06 53848.75 794.547849 1.0079747

2006 Nov 63574.17 53716.08 701.826096 1.1682583

2006 Dec 63411.8 53206.84 580.719909 1.1789307

7


2007 Jan 62229.11 53442.03 546.166482 1.152643

2007 Feb 59420.78 53659.19 513.266245 1.0968818

2007 Mar 49058.32 52978.93 393.913683 0.9191626

2007 Apr 42885.99 53243.09 380.938379 0.7997532

2007 May 53592.31 55118.99 530.434716 0.9630343

2007 Jun 54243.12 54816.79 447.170595 0.981528

2007 Jul 49983.25 54256.73 346.447617 0.9153909

2007 Aug 48896.06 54505.9 336.719567 0.8915705

2007 Sep 51999.3 55478.56 400.313902 0.9305718

2007 Oct 56680.81 56191.73 431.599978 1.0010151

2007 Nov 66047.57 56622.97 431.563822 1.1576218

2007 Dec 68849.72 58023.15 528.424932 1.1758817


2008 Jan 71844.71 61645.81 837.848602 1.1498161

2008 Feb 73486.23 66377.55 1227.237617 1.0869975

2008 Mar 68546.69 72891.97 1755.955908 0.9182666

2008 Apr 69699.21 83567.62 2647.925858 0.8084297

2008 May 92625.78 93402.51 3366.622505 0.9571831

2008 Jun 97426.16 96647.78 3354.487217 0.9742395

2008 Jul 90730.27 96211.79 2975.439512 0.9147374

2008 Aug 88995.64 96642.4 2720.956218 0.8956585

2008 Sep 93754.66 97946.38 2579.258568 0.9326443

2008 Oct 102617.9 99988.53 2525.547575 1.0010125

2008 Nov 120585.1 101091.94 2383.334083 1.1653517

2008 Dec 121682.8 99463.25 1982.131764 1.1994904

8


2009 Jan 117518.8 97948.16 1658.114675 1.1767966

2009 Feb 111198.9 97948.16 1466.597666 1.1185354

2009 Mar 95078.44 97978.61 1322.982676 0.9574715

2009 Apr 85206.43 100779.56 1470.7794 0.833312

2009 May 101772.8 104666.37 1712.383315 0.9567024

2009 Jun 102705 105485.52 1623.0596 0.9588866

2009 Jul 95255.81 103937.2 1305.921391 0.9051024

2009 Aug 95985.17 106372.1 1418.818997 0.8904755

2009 Sep 102292.8 108433.45 1483.072885 0.9306403

2009 Oct 110933.7 109961.07 1487.526778 0.9953797

2009 Nov 128120.5 110365.25 1379.192864 1.1465491

2009 Dec 127957.9 107180.75 922.823472 1.1836601


2010 Jan 121507.4 103594.7 471.935752 1.1675927

2010 Feb 112672.3 101146.19 179.891379 1.1119773

2010 Mar 95582.76 99271.67 -25.549884 0.9630882

2010 Apr 83277.83 99065.29 -43.632623 0.8410062

2010 May 96244.45 100807.18 134.919506 0.9534619

2010 Jun 94260.31 99523.39 -6.951818 0.9471834

2010 Jul 89508.68 98576.32 -100.963413 0.908945

2010 Aug 86833.57 97483.29 -200.170245 0.8925862

2010 Sep 90155.04 97079.09 -220.573175 0.9307911

2010 Oct 95972.64 97004.19 -206.005851 0.9914715

2010 Nov 107315.8 95551.42 -330.681903 1.1270214

2010 Dec 107471.4 92961.19 -556.636939 1.163053

9


2011 Jan 103713.4 90601.47 -736.944953 1.1541077

2011 Feb 97704.02 89383.58 -785.040032 1.1027724

2011 Mar 88044.4 91936.85 -451.209049 0.962385

2011 Apr 86234.52 101287.04 528.930849 0.8469646

2011 May 106448.4 110851.88 1432.521777 0.9480252

2011 Jun 110988.1 115837.8 1787.861676 0.9435701

2011 Jul 110106.1 119641.82 1989.477855 0.9052449

2011 Aug 111458 122863.21 2112.668578 0.8918358

2011 Sep 120587.5 127146.24 2329.704862 0.9313502

2011 Oct 130095.6 129564.53 2338.564126 0.9862968

2011 Nov 153065.9 134545.18 2602.772616 1.1160638

2011 Dec 160353.9 136446.94 2532.671356 1.1537945


2012 Jan 161859.1 138103.56 2445.066251 1.1516237

2012 Feb 159713.9 140332.56 2423.459561 1.1187895

2012 Mar 145969.2 143522.22 2500.079271 0.9996364

2012 Apr 128568.1 144643.36 2362.184954 0.8745802

2012 May 139024.2 142915.32 1953.162765 0.9596577

2012 Jun 134037.6 139669.3 1433.244931 0.9499303

2012 Jul 127151.1 138705.83 1193.572769 0.9088755

2012 Aug 123548.3 136717.64 875.396308 0.8979252

2012 Sep 130356.4 138918.7 1007.962946 0.9316049

2012 Oct 140794 140565.69 1071.866233 0.994044

2012 Nov 157620.1 140555.72 963.682559 1.1137703

2012 Dec 162203.2 140116.58 823.400369 1.1508669

10


2013 Jan 160793.5 139072.67 636.668776 1.1509145

2013 Feb 157352.6 139709.08 636.642716 1.1211785

2013 Mar 138818.6 138907.84 492.854597 0.9958245

2013 Apr 118785.6 137127.48 265.533046 0.864568

2013 May 128954.1 136254.5 151.681846 0.9453685

2013 Jun 124662.8 132396.29 -249.306777 0.9433647

2013 Jul 114458.8 127801.73 -683.832417 0.9004148

2013 Aug 113019.1 126167.26 -778.896059 0.9013527

2013 Sep 116227 125319.33 -785.799931 0.933299

2013 Oct 121177.4 123199.12 -919.24098 0.9909836

2013 Nov 131594.2 119794.8 -1167.74864 1.1093099

2013 Dec 131516.4 116294.04 -1401.04978 1.1446859


2014 Jan 130183.4 114548.55 -1435.493801 1.1509136

2014 Feb 125230.9 113551.77 -1391.621809 1.1165363

2014 Mar 111478.7 113910.05 -1216.632018 0.9892212

2014 Apr 100564.6 117448.93 -741.080478 0.8616784

2014 May 116537.9 124723.69 60.503161 0.9339155

2014 Jun 120143.5 128663.8 448.46345 0.930535

2014 Jul 118929.3 131768.05 714.042382 0.897701

2014 Aug 121370.9 133834.59 849.292067 0.9011541

2014 Sep 124499.6 133269.93 707.897217 0.9292555

2014 Oct 130511.4 132264.28 536.542137 0.9827606

2014 Nov 143784.2 130370.85 293.544642 1.1004082

2014 Dec 145003.2 126909.57 -81.937301 1.1433091

The table below gives the forecasted values for the whole of year 2015 that is from January 2015

to December 2015. The forecasted value of supply price of beans at the beginning of the year

2015 is 141320.8 with confidence level of one percent error, as this forecasted value must lie

between its corresponding upper and lower bound. The same applies to the rest of the other

months down to December 2015 with forecasted value of 100803.96 as it also lies between its

corresponding upper and lower bounds of one percent error.

11

Forecast For Supply For The Whole Of Year 2015

Confidence Bound

period Month Forecasted Values

Upper Bound Lower Bound

2015 Jan 141320.8 148543.9 134097.75

2015 Feb 137213.9 146311.4 128116.32

2015 Mar 122321.1 132670.7 111971.46

2015 Apr 107141.7 118422 95861.46

2015 May 114145.2 127922.7 100367.71

2015 Jun 112852.3 128417.8 97286.78

2015 Jul 108022.7 124925.8 91119.55

2015 Aug 107131.6 125875.1 88388.11

2015 Sep 109944.2 131180.8 88707.54

2015 Oct 115561.4 139964 91158.78

2015 Nov 128325.7 157652.2 98999.12

2015 Dec 132754.7 164705.4 100803.96

The chart below depicts the smoothened seasonal multiplicative forecast for the supply price of

beans for Mbeya. The line with dots is the smoothened line.

Time

Supp

ly

2 4 6 8 10 12

4000

080

000

1200

00

12

RESIDUAL ANALYSIS

Time Period Month Supply(Y) Residual/Errors ( ) ABSOLUTE(Error) ( )

( )

1 2005 Jan 45375 5316.54388 28265638.83 5316.54388 0.117169011

2 2005 Feb 43614 4144.074014 17173349.43 4144.074014 0.095017059 -1761 2058890625

3 2005 Mar 37000 2636.363709 6950413.606 2636.363709 0.071253073 -6614 1902180996

4 2005 Apr 32500 1496.958855 2240885.814 1496.958855 0.046060272 -4500 1369000000

5 2005 May 40000 668.25893 446569.9975 668.25893 0.016706473 7500 1056250000

6 2005 Jun 40909 -464.225635 215505.4402 464.225635 0.011347763 909 1600000000

7 2005 Jul 42500 3162.844947 10003588.16 3162.844947 0.074419881 1591 1673546281

8 2005 Aug 43000 3142.057288 9872524.001 3142.057288 0.0730711 500 1806250000

9 2005 Sep 39818 -4041.48365 16333590.07 4041.483647 0.101498911 -3182 1849000000

10 2005 Oct 43023 -4237.86687 17959515.62 4237.866871 0.098502356 3205 1585473124

11 2005 Nov 41389 -14273.1014 203721423.7 14273.1014 0.344852531 -1634 1850978529

12 2005 Dec 47313 -6110.46121 37337736.15 6110.461206 0.129149731 5924 1713049321

13 2006 Jan 48250 -3219.99891 10368392.97 3219.998909 0.066735729 937 2238519969

14 2006 Feb 54000 5771.591052 33311263.27 5771.591052 0.106881316 5750 2328062500

15 2006 Mar 50455 8797.08092 77388632.71 8797.08092 0.174354988 -3545 2916000000

16 2006 Apr 48438 10174.75325 103525603.7 10174.75325 0.210057254 -2017 2545707025

17 2006 May 52208 1817.661356 3303892.805 1817.661356 0.034815763 3770 2346239844

18 2006 Jun 52729 -253.864126 64446.99447 253.864126 0.004814507 521 2725675264

19 2006 Jul 46295 -4419.20824 19529401.47 4419.20824 0.095457571 -6434 2780347441

20 2006 Aug 44750 -4720.53961 22283494.16 4720.539605 0.105486919 -1545 2143227025

21 2006 Sep 46545 -4951.563 24517976.17 4951.563003 0.106382275 1795 2002562500

22 2006 Oct 50406 -4673.05914 21837481.72 4673.059139 0.092708391 3861 2166437025

23 2006 Nov 56500 -7074.16519 50043813.12 7074.165189 0.125206464 6094 2540764836

24 2006 Dec 61375 -2036.80472 4148573.451 2036.804716 0.033186228 4875 3192250000

25 2007 Jan 60333 -1896.11137 3595238.308 1896.111365 0.031427434 -1042 3766890625

26 2007 Feb 52875 -6545.7827 42847271.12 6545.782697 0.123797309 -7458 3640070889

27 2007 Mar 48462 -596.320672 355598.3439 596.320672 0.012304913 -4413 2795765625

28 2007 Apr 48864 5978.008841 35736589.7 5978.008841 0.122339736 402 2348565444

13

29 2007 May 49583 -4009.31035 16074569.48 4009.31035 0.080860584 719 2387690496

30 2007 Jun 49300 -4943.12106 24434445.8 4943.121059 0.100266147 -283 2458473889

31 2007 Jul 49538 -445.248384 198246.1235 445.248384 0.008988017 238 2430490000

32 2007 Aug 51731 2834.941689 8036894.38 2834.941689 0.054801602 2193 2454013444

33 2007 Sep 53455 1455.697014 2119053.797 1455.697014 0.027232196 1724 2676096361

34 2007 Oct 56679 -1.809652 3.274840361 1.809652 3.19281E-05 3224 2857437025

35 2007 Nov 71654 5606.4266 31432019.22 5606.4266 0.078243037 14975 3212509041

36 2007 Dec 87042 18192.28101 330959088.4 18192.28101 0.209005779 15388 5134295716

37 2008 Jan 94231 22386.28741 501145864.2 22386.28741 0.237568183 7189 7576309764

38 2008 Feb 102222 28735.77179 825744580.3 28735.77179 0.281111422 7991 8879481361

39 2008 Mar 109500 40953.30872 1677173495 40953.30872 0.374002819 7278 10449337284

40 2008 Apr 98750 29050.78701 843948226 29050.78701 0.294185185 -10750 11990250000

41 2008 May 92045 -580.784623 337310.7783 580.784623 0.00630979 -6705 9751562500

42 2008 Jun 78962 -18464.1621 340925282.8 18464.16212 0.233836049 -13083 8472282025

43 2008 Jul 79091 -11639.2688 135472579 11639.26883 0.147163 129 6234997444

44 2008 Aug 82650 -6345.63523 40267086.41 6345.635225 0.076777196 3559 6255386281

45 2008 Sep 91250 -2504.66249 6273334.164 2504.662485 0.027448356 8600 6831022500

46 2008 Oct 95500 -7117.87427 50664134.18 7117.874274 0.074532715 4250 8326562500

47 2008 Nov 97208 -23377.0905 546488361.3 23377.09052 0.240485253 1708 9120250000

48 2008 Dec 102250 -19432.7702 377632556 19432.77016 0.190051542 5042 9449395264

49 2009 Jan 106250 -11268.8285 126986496.1 11268.82851 0.106059562 4000 10455062500

50 2009 Feb 103167 -8031.92238 64511777.09 8031.922378 0.0778536 -3083 11289062500

51 2009 Mar 102154 7075.557626 50063515.72 7075.557626 0.069263638 -1013 10643429889

52 2009 Apr 95273 10066.57169 101335865.6 10066.57169 0.105660278 -6881 10435439716

53 2009 May 97500 -4272.81053 18256909.79 4272.810526 0.043823698 2227 9076944529

54 2009 Jun 87500 -15204.9783 231191364.8 15204.97829 0.17377118 -10000 9506250000

55 2009 Jul 100365 5109.194962 26103873.16 5109.194962 0.050906142 12865 7656250000

56 2009 Aug 98846 2860.825815 8184324.344 2860.825815 0.028942252 -1519 10073133225

57 2009 Sep 102500 207.248607 42951.9851 207.248607 0.002021938 3654 9770531716

58 2009 Oct 105542 -5391.66906 29070095.28 5391.669063 0.051085531 3042 10506250000

59 2009 Nov 101958 -26162.4962 684476207.4 26162.4962 0.25660072 -3584 11139113764

14

60 2009 Dec 101273 -26684.8904 712083376.4 26684.89041 0.263494618 -685 10395433764

61 2010 Jan 104458 -17049.4441 290683545.8 17049.44415 0.163218175 3185 10256220529

62 2010 Feb 101250 -11422.3009 130468956.9 11422.30086 0.112812848 -3208 10911473764

63 2010 Mar 94712 -870.763578 758229.2088 870.763578 0.009193804 -6538 10251562500

64 2010 Apr 90786 7508.172213 56372649.98 7508.172213 0.082701873 -3926 8970362944

65 2010 May 89481 -6763.445 45744188.31 6763.445003 0.075585264 -1305 8242097796

66 2010 Jun 89808 -4452.31087 19823072.05 4452.310866 0.049575883 327 8006849361

67 2010 Jul 85000 -4508.67749 20328172.66 4508.677485 0.053043265 -4808 8065476864

68 2010 Aug 85923 -910.568699 829135.3556 910.568699 0.010597497 923 7225000000

69 2010 Sep 90833 677.95677 459625.382 677.95677 0.007463772 4910 7382761929

70 2010 Oct 89792 -6180.63749 38200279.72 6180.637485 0.06883283 -1041 8250633889

71 2010 Nov 94583 -12732.8076 162124390.4 12732.80764 0.134620467 4791 8062603264

72 2010 Dec 96986 -10485.3885 109943371.6 10485.38848 0.108112392 2403 8945943889

73 2011 Jan 100938 -2775.3451 7702540.419 2775.345099 0.027495543 3952 9406284196

74 2011 Feb 116111 18406.97962 338816898.7 18406.97962 0.158529163 15173 10188479844

75 2011 Mar 135208 47163.59617 2224404804 47163.59617 0.348822527 19097 13481764321

76 2011 Apr 124500 38265.47759 1464246775 38265.47759 0.307353234 -10708 18281203264

77 2011 May 123292 16843.5592 283705486.4 16843.5592 0.136615183 -1208 15500250000

78 2011 Jun 120500 9511.949563 90477184.49 9511.949563 0.078937341 -2792 15200917264

79 2011 Jul 115682 5575.888645 31090534.18 5575.888645 0.04820014 -4818 14520250000

80 2011 Aug 121136 9678.036755 93664395.43 9678.036755 0.079893977 5454 13382325124

81 2011 Sep 121000 412.553865 170200.6915 412.553865 0.003409536 -136 14673930496

82 2011 Oct 143125 13029.39908 169765240.5 13029.39908 0.091035103 22125 14641000000

83 2011 Nov 149154 -3911.87406 15302758.63 3911.874056 0.026227081 6029 20484765625

84 2011 Dec 155300 -5053.91437 25542050.44 5053.914368 0.032542913 6146 22246915716

85 2012 Jan 160615 -1244.13884 1547881.456 1244.138841 0.007746094 5315 24118090000

86 2012 Feb 164000 4286.0662 18370363.47 4286.0662 0.02613455 3385 25797178225

87 2012 Mar 139077 -6892.20911 47502546.4 6892.209109 0.049556786 -24923 26896000000

88 2012 Apr 110682 -17886.1354 319913838.5 17886.13537 0.161599315 -28395 19342411929

89 2012 May 114077 -24947.1584 622360712.5 24947.15841 0.218687013 3395 12250505124

90 2012 Jun 122654 -11383.593 129586190.3 11383.59303 0.092810614 8577 13013561929

15

91 2012 Jul 112692 -14459.1398 209066724.7 14459.13983 0.128306711 -9962 15044003716

92 2012 Aug 129500 5951.746403 35423285.25 5951.746403 0.045959432 16808 12699486864

93 2012 Sep 133333 2976.630901 8860331.521 2976.630901 0.022324788 3833 16770250000

94 2012 Oct 135417 -5376.96661 28911769.88 5376.966606 0.039706733 2084 17777688889

95 2012 Nov 149808 -7812.10662 61029009.86 7812.106621 0.05214746 14391 18337763889

96 2012 Dec 151458 -10745.1603 115458470 10745.16031 0.070944818 1650 22442436864

97 2013 Jan 160792 -1.499669 2.24900711 1.499669 9.32676E-06 9334 22939525764

98 2013 Feb 149292 -8060.60765 64973395.69 8060.60765 0.053992228 -11500 25854067264

99 2013 Mar 127500 -11318.6179 128111111.4 11318.61791 0.088773474 -21792 22288101264

100 2013 Apr 113864 -4921.60544 24222200.13 4921.605442 0.043223542 -13636 16256250000

101 2013 May 110000 -18954.0998 359257898.5 18954.09978 0.172309998 -3864 12965010496

102 2013 Jun 104167 -20495.8085 420078166 20495.8085 0.196759132 -5833 12100000000

103 2013 Jul 110179 -4279.83546 18316991.56 4279.835459 0.038844385 6012 10850763889

104 2013 Aug 112708 -311.141184 96808.83638 311.141184 0.002760595 2529 12139412041

105 2013 Sep 110000 -6227.02011 38775779.39 6227.020105 0.056609274 -2708 12703093264

106 2013 Oct 108864 -12313.3507 151618606.1 12313.35072 0.113107646 -1136 12100000000

107 2013 Nov 118654 -12940.164 167447843.1 12940.16395 0.109057966 9790 11851370496

108 2013 Dec 129545 -1971.3787 3886333.987 1971.378702 0.015217714 10891 14078771716

109 2014 Jan 132708 2524.643648 6373825.549 2524.643648 0.01902405 3163 16781907025

110 2014 Feb 135000 9769.12237 95435751.88 9769.12237 0.072363869 2292 17611413264

111 2014 Mar 135000 23521.28282 553250745.5 23521.28282 0.174231725 0 18225000000

112 2014 Apr 135100 34535.36482 1192691423 34535.36482 0.255628163 100 18225000000

113 2014 May 134654 18116.10655 328193316.5 18116.10655 0.134538198 -446 18252010000

114 2014 Jun 132500 12356.52445 152683696.5 12356.52445 0.093256788 -2154 18131699716

115 2014 Jul 125000 6070.689121 36853266.4 6070.689121 0.048565513 -7500 17556250000

116 2014 Aug 115000 -6370.92726 40588714.14 6370.927259 0.055399367 -10000 15625000000

117 2014 Sep 116538 -7961.63239 63387590.3 7961.632389 0.068317908 1538 13225000000

118 2014 Oct 118571 -11940.4186 142573596.6 11940.41861 0.10070269 2033 13581105444

119 2014 Nov 123125 -20659.1712 426801353.1 20659.17116 0.167790223 4554 14059082041

120 2014 Dec 123462 -21541.1813 464022491.3 21541.18129 0.174476206 337 15159765625

16

Measurements of forecast accuracy were determined using the techniques of MAD, MAPE, MSE, THEIL’s U and residual plot. As

the Measurement of forecast accuracy table shows large values for MAD and MSE, the MAPE is about 10 percent which is acceptable

and the Theil’s U, which is slightly larger than zero. Since the Theil’s U is different than one and it is fairly close to zero then we can

conclude that the multiplicative triple exponential smoothing forecast produced are better than the naïve forecast.

Measurement of Forecast Accuracy

MAD 9486.243991

MAPE 9.985910844

MSE 172610906.8

Theil’s U 0.131246765

17

-40000

-30000

-20000

-10000

0

10000

20000

30000

40000

50000

60000

0 20 40 60 80 100 120 140

Res

idu

als

Time

Residual Analysis

Residual

18

ANALYSIS FOR KINONDONI

The plot shows the demand prices of beans (100Kg) for Kinondoni. As can be seen from the

plot, there is an increasing seasonal variation in the demand price for beans, but the increase in

the demand price for beans looks stable as indicated by the oscillations. In this situation it is

appropriate to use the additive seasonal HoltWinters triple exponential smoothing. Also due to

the presence of zero demand prices for beans in some months in the data set, the additive

seasonal method will do better than the multiplicative seasonal method of triple exponential

smoothing. The plotted data has almost shown constant variability. Hence the choice of additive

seasonal triple exponential smoothing. The demand price for beans for Kinondoni is analyzed

using the seasonal additive triple exponential smoothing.

In this regard also the smoothing parameters as shown in the table below are an alpha level of

0.2; beta level of 0.1 and gamma level of 0.1, were use in smoothing the forecast. Since the

choice of these smoothing parameters is subjective, we choose the parameters that give the best

smoothing forecast.

SMOOTHING PARAMETERS

Alpha 0.2

Beta 0.1

Gamma 0.1

Time

Dem

and

2 4 6 8 10 12

050

000

1000

00

19

The table below gives the initialization values and initial seasonal indices for the seasonal

additive triple exponential smoothing model for demand price for beans for Kidondoni. The

value “a” which is the initial level value, “b” which is the initial trend value and to are

the initial seasonal indices, as shown in the table. These initial seasonal indices are computed as:

…………….

COEFFICIENTS

a 124873.5843

b -3254.0222

S1 6258.4307

S2 6081.4272

S3 -488.5466

S4 -5026.8528

S5 130.885

S6 1076.7557

S7 -5911.6547

S8 -5914.4704

S9 -3857.1757

S10 -2569.8874

S11 1451.726

S12 -7447.197

The table below shows the fitted or predicted values of the seasonal additive triple exponential

smoothing model of demand price for beans for Kidondoni. There are no predicted or fitted

values for the year 2004 that is from January 2004 down to December 2004. The predicted

values commenced from January 2005 to December 2014. The various components and the

predicted values are computed using;

( ) ( )( )

( ) ( )

( ) ( )

20

Fitted Values


2005 Jan 54430.42 45988.98 800.118 7641.326

2005 Feb 53521.18 48175.81 938.7896 4406.576

2005 Mar 55556.24 50201.36 1047.466 4307.41

2005 Apr 44586.25 52001.18 1122.701 -8537.63

2005 May 53021.85 53506.63 1160.976 -1645.76

2005 Jun 57400.81 54763.24 1170.539 1467.035

2005 Jul 47758.52 55770.22 1154.183 -9165.88

2005 Aug 50738.91 57103.5 1172.093 -7536.67

2005 Sep 58170.46 58018.81 1146.414 -994.757

2005 Oct 58615.16 57667.53 996.6451 -49.0069

2005 Nov 59604.61 56657.74 796.0018 796.0018

2005 Dec 65706.48 56999.42 750.5696 7956.493


2006 Jan 64465 55721.29 547.7 8196.013

2006 Feb 61030.09 55698.19 490.6199 4841.282

2006 Mar 63881.12 58546.39 726.378 4608.351

2006 Apr 54060.77 61496.55 948.7556 -8384.53

2006 May 67493.18 67633.15 1467.54 -1607.51

2006 Jun 72127.1 69243.65 1481.837 1401.61

2006 Jul 61178.38 68966.67 1305.955 -9094.24

2006 Aug 63089.91 69500.55 1228.747 -7639.39

2006 Sep 68965.25 69457.51 1101.569 -1593.83

2006 Oct 68744.98 68682.63 913.9239 -851.58

2006 Nov 70693.22 67972.56 751.5244 1969.139

2006 Dec 74433.11 66735.44 552.6599 7145.015

21


2007 Jan 73183.17 64901.48 313.9976 7967.692

2007 Feb 69863.51 63897.04 182.1542 5784.315

2007 Mar 69459.24 63806.49 154.884 5497.861

2007 Apr 42525.84 50069.53 -1234.3 -6309.39

2007 May 36694.92 40330.06 -2084.82 -1550.32

2007 Jun 43795.57 44551.66 -1454.18 698.0819

2007 Jul 37551.13 47925.57 -971.367 -9403.07

2007 Aug 43756.67 52337.78 -433.01 -8148.1

2007 Sep 54346.26 56649.23 41.4368 -2344.41

2007 Oct 57338.25 58606.42 233.0117 -1501.18

2007 Nov 64948.79 63114.58 660.5266 1173.681

2007 Dec 75283.45 68009.15 1083.931 6190.365


2008 Jan 80480.97 71696.39 1344.262 7440.319

2008 Feb 83430.34 76104.46 1650.642 5675.234

2008 Mar 84104.51 82080.23 2083.156 -58.8782

2008 Apr 81022.09 88242.49 2491.066 -9711.46

2008 May 96053.86 92421.73 2659.884 972.2462

2008 Jun 98250.71 93154.25 2467.146 2629.317

2008 Jul 89668.12 94557.05 2360.712 -7249.64

2008 Aug 92917.26 96816.94 2350.63 -6250.31

2008 Sep 101663.3 100734.1 2507.285 -1578.11

2008 Oct 105701.7 103008.7 2484.019 208.8816

2008 Nov 110689.7 105352.4 2469.986 2867.298

2008 Dec 117207.4 107534.5 2441.192 7231.69

22


2009 Jan 122109.6 110909.2 2534.545 8665.841

2009 Feb 121879.5 112076.4 2397.813 7405.287

2009 Mar 115341.4 111652.9 2115.682 1572.761

2009 Apr 104090.8 111261.9 1865.015 -9036.19

2009 May 115003.2 112954.2 1847.74 201.2971

2009 Jun 116414.3 112584.7 1626.015 2203.58

2009 Jul 104402.6 110443.3 1249.269 -7289.97

2009 Aug 107978.4 112292.8 1309.298 -5623.69

2009 Sep 112475 112906.4 1239.73 -1671.18

2009 Oct 113985.9 112734.6 1098.57 152.75

2009 Nov 115111.2 111494.4 864.6926 2752.121

2009 Dec 119291.9 110961.8 724.9693 7605.101


2010 Jan 118636.5 109964.8 552.7716 8118.915

2010 Feb 116895.4 110106.9 511.7018 6276.765

2010 Mar 110508.8 109535.3 403.3749 570.0908

2010 Apr 102055.7 110683.1 477.8193 -9105.29

2010 May 113093.8 113107 672.4261 -685.602

2010 Jun 114890.9 113545.3 649.0093 696.5969

2010 Jul 107275.8 113723.7 601.9516 -7049.86

2010 Aug 108094.2 113478.9 517.2756 -5901.97

2010 Sep 111258.6 113069.7 424.6314 -2235.82

2010 Oct 110393.6 111001.1 175.3004 -782.76

2010 Nov 110716.7 108605.2 -81.8115 2193.227

2010 Dec 113092 106463.5 -287.805 6916.31

23


2011 Jan 111237.3 103807.3 -524.645 7954.635

2011 Feb 106129.3 101035.2 -749.39 5843.458

2011 Mar 101376.8 101170 -660.975 867.8686

2011 Apr 83391.29 93118.2 -1400.05 -8326.86

2011 May 95282.75 96939.89 -877.878 -779.269

2011 Jun 102822.5 102543.9 -229.693 508.3664

2011 Jul 99145.96 106359.7 174.8563 -7388.56

2011 Aug 106904 112413.7 762.7771 -6272.54

2011 Sep 116496.1 118440.1 1289.138 -3233.14

2011 Oct 121577.2 121883.8 1504.596 -1811.21

2011 Nov 130672.8 127398 1905.551 1369.255

2011 Dec 141768.4 133476.6 2322.855 5968.95


2012 Jan 147153.9 137595.8 2502.488 7055.653

2012 Feb 147291.8 138729.1 2365.57 6197.117

2012 Mar 139725.3 139597.9 2215.895 -2088.44

2012 Apr 137107.2 141191.7 2153.688 -6238.17

2012 May 147659 143660.4 2185.183 1813.471

2012 Jun 146357 142390.7 1839.703 2126.563

2012 Jul 138873.1 142266.6 1643.323 -5036.88

2012 Aug 141084.1 143635.3 1615.862 -4167.1

2012 Sep 146083.3 146694.4 1760.18 -2371.31

2012 Oct 149952.3 148404.5 1755.174 -207.385

2012 Nov 153144.7 148515.4 1590.748 3038.472

2012 Dec 158454.2 150169.7 1597.095 6687.48

24


2013 Jan 159511.9 151439.5 1564.371 6507.981

2013 Feb 160365.8 153184.9 1582.474 5598.417

2013 Mar 155719.6 156319.2 1737.658 -2337.27

2013 Apr 155128.3 159371.4 1869.106 -6112.18

2013 May 163463 161169.4 1862.001 431.5504

2013 Jun 163695.7 160723.4 1631.202 1341.043

2013 Jul 156009.3 159782.1 1373.948 -5146.72

2013 Aug 157657.5 159990 1257.342 -3589.83

2013 Sep 157236.9 158632.4 995.8526 -2391.33

2013 Oct 157983.6 158014.3 834.4538 -865.09

2013 Nov 160240.7 156570.2 606.601 3063.86

2013 Dec 162637 155628.7 451.7876 6556.582


2014 Jan 160482.5 153689.5 212.6868 6580.393

2014 Feb 159448.6 153097.2 132.1961 6219.154

2014 Mar 151171.4 152885.1 97.76448 -1811.47

2014 Apr 150578.1 156290.2 428.4964 -6140.61

2014 May 159718.4 159503.1 706.9347 -491.646

2014 Jun 162940.9 161766.3 862.5674 312.0305

2014 Jul 159981.3 164540.7 1053.749 -5613.15

2014 Aug 161191.5 164848.2 979.1224 -4635.79

2014 Sep 160253.2 162630.6 659.4516 -3036.92

2014 Oct 159917.3 161239.4 454.3887 -1776.5

2014 Nov 162411 159710.4 256.0422 2444.607

2014 Dec 163092.2 157484.2 7.821917 5600.179

25

The table below shows the forecasted values for demand price for beans in Kidondoni for the

entire period of year 2015. That is from January 2015 to December 2015. The forecasted demand

price for beans is all presented in the table, with confidence level of one percent error. This

means that the forecasted demand values for beans for the whole of year 2015 must fall within

the specified upper and lower bounds for each month.

Forecast Values for the whole of Year 2015

CONFIDENCE BOUNDS

period Month Forecast Values

Upper Bound Lower Bound

2015 Jan 127877.99 168578.2 87177.77

2015 Feb 124446.97 166120.5 82773.43

2015 Mar 114622.97 157426 71819.95

2015 Apr 106830.64 150922.3 62738.94

2015 May 108734.36 154275 63193.73

2015 Jun 106426.21 153575.3 59277.14

2015 Jul 96183.77 145098.6 47268.94

2015 Aug 92926.94 143761.5 42092.37

2015 Sep 91730.21 144634.2 38826.18

2015 Oct 89763.48 144881.9 34645.09

2015 Nov 90531.07 148003.5 33058.66

2015 Dec 78378.12 138338.8 18417.42

The chart below shows the smoothened seasonal additive triple exponential forecast for demand

price for beans for Kidondoni. The red line depicts the smoothened line for the demand price for

beans.

Time

Dem

and

2 4 6 8 10 12

050

000

1000

00

26

RESIDUAL ANALYSIS

Time Period Month Supply(Y) Residual/Errors ( ) ABSOLUTE(Error) ( )

( )

1 2005 Jan 61364 6933.5796 48074526.07 6933.5796 0.112990998

2 2005 Feb 58955 5433.8241 29526444.35 5433.8241 0.092169012 -2409 3765540496

3 2005 Mar 59318 3761.7598 14150836.79 3761.7598 0.063416835 363 3475692025

4 2005 Apr 46500 1913.7483 3662432.556 1913.7483 0.041155877 -12818 3518625124

5 2005 May 53500 478.1474 228624.9361 478.1474 0.008937335 7000 2162250000

6 2005 Jun 56583 -817.813 668818.103 817.813 0.014453334 3083 2862250000

7 2005 Jul 48654 895.4834 801890.5197 895.4834 0.018405134 -7929 3201635889

8 2005 Aug 49455 -1283.9142 1648435.673 1283.9142 0.025961262 801 2367211716

9 2005 Sep 50682 -7488.4624 56077069.12 7488.4624 0.147753885 1227 2445797025

10 2005 Oct 48583 -10032.165 100644334.6 10032.165 0.206495379 -2099 2568665124

11 2005 Nov 57333 -2271.6087 5160206.086 2271.6087 0.039621312 8750 2360307889

12 2005 Dec 55563 -10143.4816 102890219 10143.4816 0.182558206 -1770 3287072889

13 2006 Jan 61611 -2854.0049 8145343.969 2854.0049 0.046322976 6048 3087246969

14 2006 Feb 72818 11787.9066 138954742 11787.9066 0.161881768 11207 3795915321

15 2006 Mar 75000 11118.8791 123629472.4 11118.8791 0.148251721 2182 5302461124

16 2006 Apr 80000 25939.2303 672843668.6 25939.2303 0.324240379 5000 5625000000

17 2006 May 68208 714.8171 510963.4865 714.8171 0.01047996 -11792 6400000000

18 2006 Jun 63333 -8794.0977 77336154.36 8794.0977 0.138854905 -4875 4652331264

19 2006 Jul 57318 -3860.3797 14902531.43 3860.3797 0.067350216 -6015 4011068889

20 2006 Aug 56731 -6358.9073 40435702.05 6358.9073 0.112088757 -587 3285353124

21 2006 Sep 59583 -9382.2475 88026568.15 9382.2475 0.157465175 2852 3218406361

22 2006 Oct 60625 -8119.9757 65934005.37 8119.9757 0.133937744 1042 3550133889

23 2006 Nov 60750 -9943.2244 98867711.47 9943.2244 0.163674476 125 3675390625

24 2006 Dec 62500 -11933.1146 142399224.1 11933.1146 0.190929834 1750 3690562500

25 2007 Jan 66591 -6592.1671 43456667.07 6592.1671 0.098994866 4091 3906250000

26 2007 Feb 68500 -1363.5104 1859160.611 1363.5104 0.019905261 1909 4434361281

27 2007 Mar 0 -69459.2384 4824585799 69459.2384 UNDEFINED -68500 4692250000

28 2007 Apr 0 -42525.8355 1808446685 42525.8355 UNDEFINED 0 0

27

29 2007 May 68227 31532.0752 994271766.4 31532.0752 0.462164175 68227 0

30 2007 Jun 67936 24140.4344 582760573 24140.4344 0.355340827 -291 4654923529

31 2007 Jul 64469 26917.8703 724571741.5 26917.8703 0.417531997 -3467 4615300096

32 2007 Aug 67479 23722.3318 562749026 23722.3318 0.351551324 3010 4156251961

33 2007 Sep 63925 9578.743 91752317.46 9578.743 0.149843457 -3554 4553415441

34 2007 Oct 78714 21375.7472 456922568.4 21375.7472 0.271562202 14789 4086405625

35 2007 Nov 86119 21170.2116 448177859.2 21170.2116 0.245825098 7405 6195893796

36 2007 Dec 88300 13016.5545 169430691.1 13016.5545 0.147412848 2181 7416482161

37 2008 Jan 95800 15319.028 234672618.9 15319.028 0.159906347 7500 7796890000

38 2008 Feb 105056 21625.6649 467669382.4 21625.6649 0.205848927 9256 9177640000

39 2008 Mar 104500 20395.4884 415975947.1 20395.4884 0.195172138 -556 11036763136

40 2008 Apr 89463 8440.9074 71248917.74 8440.9074 0.09435082 -15037 10920250000

41 2008 May 86417 -9636.8645 92869157.39 9636.8645 0.111515842 -3046 8003628369

42 2008 Jun 92929 -5321.7084 28320580.29 5321.7084 0.05726639 6512 7467897889

43 2008 Jul 89164 -504.1183 254135.2604 504.1183 0.005653832 -3765 8635799041

44 2008 Aug 100750 7832.7442 61351881.7 7832.7442 0.077744359 11586 7950218896

45 2008 Sep 100500 -1163.2879 1353238.738 1163.2879 0.011575004 -250 10150562500

46 2008 Oct 105000 -701.6452 492305.9867 701.6452 0.006682335 4500 10100250000

47 2008 Nov 109250 -1439.719 2072790.799 1439.719 0.013178206 4250 11025000000

48 2008 Dec 121875 4667.6417 21786879.04 4667.6417 0.038298599 12625 11935562500

49 2009 Jan 115273 -6836.5827 46738863.01 6836.5827 0.059307754 -6602 14853515625

50 2009 Feb 107773 -14106.525 198994047.6 14106.525 0.130891086 -7500 13287864529

51 2009 Mar 102808 -12533.3761 157085516.5 12533.3761 0.121910514 -4965 11615019529

52 2009 Apr 103227 -863.767 746093.4303 863.767 0.008367646 419 10569484864

53 2009 May 103917 -11086.2381 122904675.2 11086.2381 0.106683585 690 10655813529

54 2009 Jun 97577 -18837.2881 354843423 18837.2881 0.193050494 -6340 10798742889

55 2009 Jul 107404 3001.4539 9008725.514 3001.4539 0.027945457 9827 9521270929

56 2009 Aug 104500 -3478.4152 12099372.3 3478.4152 0.03328627 -2904 11535619216

57 2009 Sep 105417 -7057.9779 49815052.04 7057.9779 0.066952938 917 10920250000

58 2009 Oct 102292 -11693.8798 136746824.8 11693.8798 0.114318615 -3125 11112743889

59 2009 Nov 108125 -6986.1673 48806533.54 6986.1673 0.064611952 5833 10463653264

28

60 2009 Dec 110682 -8609.8834 74130092.16 8609.8834 0.077789373 2557 11691015625

61 2010 Jan 116583 -2053.4919 4216828.983 2053.4919 0.017613991 5901 12250505124

62 2010 Feb 111479 -5416.3459 29336802.91 5416.3459 0.048586244 -5104 13591595889

63 2010 Mar 114231 3722.2229 13854943.32 3722.2229 0.03258505 2752 12427567441

64 2010 Apr 111786 9730.339 94679497.05 9730.339 0.087044344 -2445 13048721361

65 2010 May 111923 -1170.8422 1370871.457 1170.8422 0.01046114 137 12496109796

66 2010 Jun 112538 -2352.8819 5536053.235 2352.8819 0.020907444 615 12526757929

67 2010 Jul 103042 -4233.803 17925087.84 4233.803 0.041088129 -9496 12664801444

68 2010 Aug 103462 -4632.2087 21457357.44 4632.2087 0.044772078 420 10617653764

69 2010 Sep 98792 -12466.5493 155414851.4 12466.5493 0.126189867 -4670 10704385444

70 2010 Oct 97538 -12855.5949 165266320.2 12855.5949 0.131800887 -1254 9759859264

71 2010 Nov 100417 -10299.6523 106082837.5 10299.6523 0.102568811 2879 9513661444

72 2010 Dec 101250 -11842.0003 140232971.1 11842.0003 0.116958028 833 10083573889

73 2011 Jan 100000 -11237.2805 126276473 11237.2805 0.112372805 -1250 10251562500

74 2011 Feb 110550 4420.7435 19542973.09 4420.7435 0.039988634 10550 10000000000

75 2011 Mar 64423 -36953.841 1365586365 36953.841 0.573612545 -46127 12221302500

76 2011 Apr 109500 26108.71 681664737.9 26108.71 0.238435708 45077 4150322929

77 2011 May 127692 32409.2532 1050359693 32409.2532 0.253808016 18192 11990250000

78 2011 Jun 123050 20227.4597 409150125.9 20227.4597 0.164384069 -4642 16305246864

79 2011 Jul 128542 29396.0392 864127120.6 29396.0392 0.228688205 5492 15141302500

80 2011 Aug 133222 26318.0359 692639013.6 26318.0359 0.197550224 4680 16523045764

81 2011 Sep 127269 10772.8872 116055098.6 10772.8872 0.084646593 -5953 17748101284

82 2011 Oct 141625 20047.7827 401913591.2 20047.7827 0.141555394 14356 16197398361

83 2011 Nov 151538 20865.2117 435357059.3 20865.2117 0.137689634 9913 20057640625

84 2011 Dec 150750 8981.6187 80669474.47 8981.6187 0.05957956 -788 22963765444

85 2012 Jan 140308 -6845.8953 46866282.46 6845.8953 0.04879191 -10442 22725562500

86 2012 Feb 139808 -7483.7504 56006520.05 7483.7504 0.053528771 -500 19686334864

87 2012 Mar 136615 -3110.3396 9674212.427 3110.3396 0.02276719 -3193 19546276864

88 2012 Apr 138682 1574.7669 2479890.789 1574.7669 0.011355236 2067 18663658225

89 2012 May 130385 -17274.0062 298391290.2 17274.0062 0.132484612 -8297 19232697124

90 2012 Jun 136538 -9819.0006 96412772.78 9819.0006 0.07191405 6153 17000248225

29

91 2012 Jul 137500 -1373.0824 1885355.277 1373.0824 0.009986054 962 18642625444

92 2012 Aug 148300 7215.8944 52069131.99 7215.8944 0.048657413 10800 18906250000

93 2012 Sep 145833 -250.2559 62628.01548 250.2559 0.001716044 -2467 21992890000

94 2012 Oct 141731 -8221.3023 67589811.51 8221.3023 0.05800638 -4102 21267263889

95 2012 Nov 153462 317.3522 100712.4188 317.3522 0.002067953 11731 20087676361

96 2012 Dec 156818 -1636.2214 2677220.47 1636.2214 0.010433888 3356 23550585444

97 2013 Jan 160417 905.1505 819297.4277 905.1505 0.005642485 3599 24591885124

98 2013 Feb 168125 7759.2106 60205349.14 7759.2106 0.046151439 7708 25733613889

99 2013 Mar 162292 6572.393 43196349.75 6572.393 0.040497332 -5833 28266015625

100 2013 Apr 154773 -355.2736 126219.3309 355.2736 0.002295449 -7519 26338693264

101 2013 May 151923 -11539.9539 133170536 11539.9539 0.075959229 -2850 23954681529

102 2013 Jun 150833 -12862.6574 165447955.4 12862.6574 0.085277475 -1090 23080597929

103 2013 Jul 150179 -5830.3064 33992472.72 5830.3064 0.038822381 -654 22750593889

104 2013 Aug 144583 -13074.4835 170942118.8 13074.4835 0.090428913 -5596 22553732041

105 2013 Sep 149167 -8069.9392 65123918.69 8069.9392 0.05410003 4584 20904243889

106 2013 Oct 146591 -11392.6446 129792351 11392.6446 0.077717217 -2576 22250793889

107 2013 Nov 152500 -7740.6666 59917919.41 7740.6666 0.05075847 5909 21488921281

108 2013 Dec 150682 -11955.0427 142923046 11955.0427 0.079339554 -1818 23256250000

109 2014 Jan 156458 -4024.532 16196857.82 4024.532 0.025722763 5776 22705065124

110 2014 Feb 157727 -1721.5823 2963845.616 1721.5823 0.01091495 1269 24479105764

111 2014 Mar 167708 16536.5979 273459070.1 16536.5979 0.098603513 9981 24877806529

112 2014 Apr 164500 13921.9133 193819669.9 13921.9133 0.084631692 -3208 28125973264

113 2014 May 167500 7781.636 60553858.84 7781.636 0.046457528 3000 27060250000

114 2014 Jun 172500 9559.065 91375723.67 9559.065 0.05541487 5000 28056250000

115 2014 Jul 156250 -3731.3169 13922725.81 3731.3169 0.023880428 -16250 29756250000

116 2014 Aug 145208 -15983.5378 255473480.6 15983.5378 0.110073397 -11042 24414062500

117 2014 Sep 150000 -10253.1452 105126986.5 10253.1452 0.068354301 4792 21085363264

118 2014 Oct 150000 -9917.3279 98353392.68 9917.3279 0.066115519 0 22500000000

119 2014 Nov 150000 -12411.0126 154033233.8 12411.0126 0.082740084 0 22500000000

120 2014 Dec 0 -163092.204 26599066908 163092.2037 UNDEFINED -150000 22500000000

30

Measurement of Forecast Accuracy

MSE 430478559.1

MAD 11803.53433

MAPE CAN NOT BE COMPUTED

Theil U 0.184349811

For the measurement of forecast accuracy of the seasonal additive triple exponential smoothing, we made use of the techniques of

MAD, MSE, MAPE, THEIL’S U and the residual plot to check for the best fit of the model. From the measurement of forecast

accuracy table, MSE and MAD are extremely large; MAPE is undefined and could not be computed as a result of the presences of

zero demand prices for some months in Kidondoni. Basing our argument or conclusion on the THEIL’S U statistics which is barely

close to zero; we can then conclude that the forecast using the seasonal additive triple exponential smoothing is better than the naïve

forecast.

31

From the residual plot, the errors are not random and still visible pattern exist. This problem

could be attributable to the existence of zero demand prices for some months in the data set.

Presence of outliers in a data set can distort the analysis of a data.

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