Predictive Analytics : QM901.1x Prof U Dinesh Kumar, IIMB. All Rights Reserved, Indian Institute of Management Bangalore

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1 Predicive Analyics : QM901.1x All Righs Reserved, Indian Insiue of Managemen Bangalore

2 Predicive Analyics : QM901.1x Those who have knowledge don predic. Those who predic don have knowledge. - Lao Tzu All Righs Reserved, Indian Insiue of Managemen Bangalore

3 I hink here is a world marke for may be 5 compuers - Thomas Wason, Chairman of IBM 1943 Predicive Analyics : QM901.1x Compuers in fuure weigh no more han 1.5 ons - Popular Mechanics, K ough o be enough for everybody - Bill Gaes, 1981??? All Righs Reserved, Indian Insiue of Managemen Bangalore

4 Forecasing Predicive Analyics : QM901.1x Forecasing is a process of esimaion of an unknown even/parameer such as demand for a produc. Forecasing is commonly used o refer ime series daa. Time series is a sequence of daa poins measured a successive ime inervals. All Righs Reserved, Indian Insiue of Managemen Bangalore

5 Corporae Sraegy Predicive Analyics : QM901.1x Business Forecasing Produc and Marke Planning Produc and Marke Sraegy Aggregae Forecasing Aggregae Producion Planning Resource Planning - Medium o Long Range Planning Iem Forecasing Maser Producion Planning Producion Capaciy - Shor Range Planning Demand forecasing a SKU Level Maerials Requiremen Planning Capaciy Requiremen Planning All Righs Reserved, Indian Insiue of Managemen Bangalore

6 Forecasing mehods Predicive Analyics : QM901.1x Qualiaive Techniques. Exper opinion (or Asrologers) Quaniaive Techniques. Time series echniques such as exponenial smoohing Casual Models. Uses informaion abou relaionship beween sysem elemens (e.g regression). All Righs Reserved, Indian Insiue of Managemen Bangalore

7 Predicive Analyics : QM901.1x All Righs Reserved, Indian Insiue of Managemen Bangalore

8 Why Time Series Analysis? Predicive Analyics : QM901.1x Time series analysis helps o idenify and explain: Any sysemic variaion in he series of daa which is due o seasonaliy. Cyclical paern ha repea. Trends in he daa. Growh raes in he rends. All Righs Reserved, Indian Insiue of Managemen Bangalore

9 Time Series Componens Predicive Analyics : QM901.1x Trend Cyclical Seasonal Irregular All Righs Reserved, Indian Insiue of Managemen Bangalore

10 Trend Componen Predicive Analyics : QM901.1x Persisen upward or downward paern Due o consumer behaviour, populaion, economy, echnology ec. All Righs Reserved, Indian Insiue of Managemen Bangalore

11 Cyclical Componen Predicive Analyics : QM901.1x Repeaing up and down movemens. Due o ineracion of facors influencing economy such as recession. Usually 2-10 years duraion. All Righs Reserved, Indian Insiue of Managemen Bangalore

12 Seasonal Componen Predicive Analyics : QM901.1x Regular paern of up and down movemens. Due o weaher, cusoms, fesivals ec. Occurs wihin one year. All Righs Reserved, Indian Insiue of Managemen Bangalore

13 Seasonal Vs Cyclical Predicive Analyics : QM901.1x When a cyclical paern in he daa has a period of less han one year, i is referred as seasonal variaion. When he cyclical paern has a period of more han one year we refer o i as cyclical variaion. All Righs Reserved, Indian Insiue of Managemen Bangalore

14 Irregular Componen Predicive Analyics : QM901.1x Erraic flucuaions Due o random variaion or unforeseen evens Whie Noise All Righs Reserved, Indian Insiue of Managemen Bangalore

15 Demand Demand Demand Demand More han one year gap Predicive Analyics : QM901.1x Random movemen Time (a) Trend Time (b) Cycle Less han one year gap Time (c) Seasonal paern Time (d) Trend wih seasonal paern All Righs Reserved, Indian Insiue of Managemen Bangalore

16 Predicive Analyics : QM901.1x All Righs Reserved, Indian Insiue of Managemen Bangalore

17 Predicive Analyics : QM901.1x Time Series Daa Addiive and Muliplicaive Models 1. Addiive Forecasing Model Y Seasonaliy Cyclical T S C R Trend Random 2. Muliplicaive Forecasing Model Y Seasonaliy Cyclical T S C R Trend Random All Righs Reserved, Indian Insiue of Managemen Bangalore

18 Time Series Daa Decomposiion Predicive Analyics : QM901.1x Muliplicaive Forecasing Model Y Seasonali y T S Trend S Y T Y / T is called deseasonalized daa All Righs Reserved, Indian Insiue of Managemen Bangalore

19 Time Series Techniques Predicive Analyics : QM901.1x Moving Average. Exponenial Smoohing. Auo-regression Models (AR Models). ARIMA (Auo-regressive Inegraed Moving Average) Models. All Righs Reserved, Indian Insiue of Managemen Bangalore

20 Predicive Analyics : QM901.1x Moving Average Mehod All Righs Reserved, Indian Insiue of Managemen Bangalore

21 Moving Average (Rolling Average) Predicive Analyics : QM901.1x Simple moving average. Used mainly o capure rend and smooh shor erm flucuaions. Mos recen daa are given equal weighs. Weighed moving average Uses unequal weighs for daa All Righs Reserved, Indian Insiue of Managemen Bangalore

22 Daa Predicive Analyics : QM901.1x Demand for coninenal breakfas a he Die Anoher Day Hospial. Daily daa beween 1 Ocober January 2015 (115 days) All Righs Reserved, Indian Insiue of Managemen Bangalore

23 Simple moving average Predicive Analyics : QM901.1x The forecas for period +1 (F +1 ) is given by he average of he n mos recen daa. F 1 1 n Y in1 i F 1 Forecas for period 1 Y i Daa corresponding o ime period i All Righs Reserved, Indian Insiue of Managemen Bangalore

24 Simple moving average Predicive Analyics : QM901.1x The forecas for period +1 (F +1 ) is given by he average of he n mos recen daa. F 1 1 n Y in1 i F 1 Forecas for period 1 Y i Daa corresponding o ime period i All Righs Reserved, Indian Insiue of Managemen Bangalore

25 Demand for Coninenal Breakfas a DAD Hospial Moving Average Predicive Analyics : QM901.1x Acual Demand Forecas F Yi i71 All Righs Reserved, Indian Insiue of Managemen Bangalore

26 Measures of aggregae error Predicive Analyics : QM901.1x Mean absolue error MAE MAE 1 n n E 1 Mean absolue percenage error MAPE Mean squared error MSE 1 MAPE n 1 MSE n n 1 E Y n 2 E 1 Roo mean squared error RMSE RMSE 1 n n 2 E 1 E = F - Y All Righs Reserved, Indian Insiue of Managemen Bangalore

27 DAD Forecasing MAPE and RMSE Predicive Analyics : QM901.1x Mean absolue percenage error MAPE Roo mean squared error RMSE or 10.68% All Righs Reserved, Indian Insiue of Managemen Bangalore

28 Predicive Analyics : QM901.1x All Righs Reserved, Indian Insiue of Managemen Bangalore

29 Exponenial Smoohing Predicive Analyics : QM901.1x Form of Weighed Moving Average Weighs decline exponenially. Larges weigh is given o he presen observaion, less weigh o immediaely preceding observaion and so on. Requires smoohing consan () Ranges from 0 o 1 All Righs Reserved, Indian Insiue of Managemen Bangalore

30 Predicive Analyics : QM901.1x Exponenial Smoohing Nex forecas = (presen acual value) + (1-) presen forecas All Righs Reserved, Indian Insiue of Managemen Bangalore

31 Simple Exponenial Smoohing Equaions Predicive Analyics : QM901.1x Smoohing Equaions F 1 * Y (1 ) * F F 1 Y 1 F +1 is he forecased value a ime +1 All Righs Reserved, Indian Insiue of Managemen Bangalore

32 Simple Exponenial Smoohing Equaions Predicive Analyics : QM901.1x Smoohing Equaions F Y 1 ( 1) Y 2 (1 ) Y All Righs Reserved, Indian Insiue of Managemen Bangalore

33 Exponenial Smoohing Forecas Predicive Analyics : QM901.1x Demand Exponenial Smoohing Forecas F ) * * Y (1 F All Righs Reserved, Indian Insiue of Managemen Bangalore

34 DAD Forecasing MAPE and RMSE Exponenial Smoohing Predicive Analyics : QM901.1x Mean absolue percenage error MAPE Roo mean squared error RMSE or 9.06% = All Righs Reserved, Indian Insiue of Managemen Bangalore

35 Predicive Analyics : QM901.1x Choice of For smooh daa, ry a high value of a, forecas responsive o mos curren daa. For noisy daa ry low a forecas more sable less responsive All Righs Reserved, Indian Insiue of Managemen Bangalore

36 Predicive Analyics : QM901.1x All Righs Reserved, Indian Insiue of Managemen Bangalore Opimal ) (1 / n F Y F n F Y Min

37 Predicive Analyics : QM901.1x All Righs Reserved, Indian Insiue of Managemen Bangalore

38 Double exponenial Smoohing Hol s model Predicive Analyics : QM901.1x Simple exponenial smoohing may produce consisenly biased forecass in he presence of a rend. Hol's mehod (double exponenial smoohing) is appropriae when demand has a rend bu no seasonaliy. Sysemaic componen of demand = Level + Trend All Righs Reserved, Indian Insiue of Managemen Bangalore

39 Hol s mehod Predicive Analyics : QM901.1x Hol s mehod can be used o forecas when here is a linear rend presen in he daa. The mehod requires separae smoohing consans for slope and inercep. All Righs Reserved, Indian Insiue of Managemen Bangalore

40 Predicive Analyics : QM901.1x All Righs Reserved, Indian Insiue of Managemen Bangalore Hol s Mehod Hol s Equaions Forecas Equaion ) (1 ) ( ) ( ) ( ) (1 ) ( T L L T ii T L Y L i T L F 1 Equaion for inercep or level Equaion for Slope (Trend)

41 Predicive Analyics : QM901.1x All Righs Reserved, Indian Insiue of Managemen Bangalore Iniial values of L and T L 1 is in general se o Y 1. T 1 can be se o any one of he following values (or use regression o ge iniial values): 1) ) /( ( 3 / ) ( ) ( ) ( ) ( n Y Y T Y Y Y Y Y Y T Y Y T n

42 60 Predicive Analyics : QM901.1x Demand Forecas Double exponenial smoohing = ; = 0.05 All Righs Reserved, Indian Insiue of Managemen Bangalore

43 DAD Forecasing MAPE and RMSE Double Exponenial Smoohing Predicive Analyics : QM901.1x Mean absolue percenage error MAPE Roo mean squared error RMSE or 9.3% = ; = 0.05 All Righs Reserved, Indian Insiue of Managemen Bangalore

44 Predicive Analyics : QM901.1x All Righs Reserved, Indian Insiue of Managemen Bangalore

45 Forecasing Accuracy Predicive Analyics : QM901.1x The forecas error is he difference beween he forecas value and he acual value for he corresponding period. E E Y F Y F Forecas error a period Acual value a ime Forecas for ime period period All Righs Reserved, Indian Insiue of Managemen Bangalore

46 Measures of aggregae error Predicive Analyics : QM901.1x Mean absolue error MAE MAE 1 n n E 1 Mean absolue percenage error MAPE Mean squared error MSE 1 MAPE n 1 MSE n n 1 E Y n 2 E 1 Roo mean squared error RMSE RMSE 1 n n 2 E 1 All Righs Reserved, Indian Insiue of Managemen Bangalore

47 Predicive Analyics : QM901.1x Forecasing Power of a Model All Righs Reserved, Indian Insiue of Managemen Bangalore

48 Predicive Analyics : QM901.1x All Righs Reserved, Indian Insiue of Managemen Bangalore Theil s coefficien (U Saisic) n n Y Y F Y U ) ( ) ( The value U is he relaive forecasing power of he mehod agains naïve echnique. If U < 1, he echnique is beer han naïve forecas If U > 1, he echnique is no beer han he naïve forecas.

49 Theil s coefficien for DAD Hospial Predicive Analyics : QM901.1x Mehod U Moving Average wih 7 periods Exponenial Smoohing Double exponenial Smoohing All Righs Reserved, Indian Insiue of Managemen Bangalore

50 Predicive Analyics : QM901.1x All Righs Reserved, Indian Insiue of Managemen Bangalore Theil s Coefficien n n F U2, 1 n n n Y Y Y Y Y F Y F Y U U1 is bounded beween 0 and 1, wih values closure o zero indicaing greaer accuracy. If U2 = 1, here is no difference beween naïve forecas and he forecasing echnique If U2 < 1, he echnique is beer han naïve forecas If U2 > 1, he echnique is no beer han he naïve forecas.

51 Predicive Analyics : QM901.1x All Righs Reserved, Indian Insiue of Managemen Bangalore

52 Seasonal Effec Predicive Analyics : QM901.1x Seasonal effec is defined as he repeiive and predicable paern of daa behaviour in a ime-series around he rend line. In seasonal effec he ime period mus be less han one year, such as, days, weeks, monhs or quarers. All Righs Reserved, Indian Insiue of Managemen Bangalore

53 Seasonal effec Predicive Analyics : QM901.1x Idenificaion of seasonal effec provides beer undersanding of he ime series daa. Seasonal effec can be eliminaed from he ime-series. This process is called deseasonalizaion or seasonal adjusing. All Righs Reserved, Indian Insiue of Managemen Bangalore

54 Seasonal Adjusing Predicive Analyics : QM901.1x Seasonal adjusmen in muliplicaive model TS Y Seasonal effec 100 T T Seasonal adjusmen in addiive model Y S T S S T All Righs Reserved, Indian Insiue of Managemen Bangalore

55 Seasonal Index Predicive Analyics : QM901.1x Mehod of simple averages Raio-o-moving average mehod All Righs Reserved, Indian Insiue of Managemen Bangalore

56 Mehod of simple averages Predicive Analyics : QM901.1x Average he unadjused daa by period ( for example daily or monhly). Calculae he average of daily (or monhly) averages. Seasonal index for day i (or monh i) is he raio of daily average of day i (or monh i) o he average of daily (or monhly) averages imes 100. All Righs Reserved, Indian Insiue of Managemen Bangalore

57 Example: DAD Example - Demand for Coninenal Breakfas 5 Ocober 1 November 2014 Daa Predicive Analyics : QM901.1x DAY Week (5-11 Ocober) Week 2 (12-18 Ocober) Week 3 (19-25 OCT) Week 4 (26 OCT - 1 NOV) Sunday Monday Tuesday Wednesday Thursday Friday Saurday All Righs Reserved, Indian Insiue of Managemen Bangalore

58 Seasonaliy Index Predicive Analyics : QM901.1x DAY Week1 (5-11 Ocober) Week 2 (12-18 Ocober) Week 3 (19-25 OCT) Week 4 (26 Oc-1 Nov) Daily Average Seasonaliy Index Sunday % Monday % Tuesday % Wednesday % Thursday % Friday % Saurday % Average of daily averages Toal = Number of seasons x 100 All Righs Reserved, Indian Insiue of Managemen Bangalore

59 Deseasonalized Daa Predicive Analyics : QM901.1x Dae Demand Seasonal Index De-seasonalized Daa 10/5/ % /6/ % /7/ % /8/ % /9/ % /10/ % /11/ % All Righs Reserved, Indian Insiue of Managemen Bangalore

60 Trend Predicive Analyics : QM901.1x Trend is calculaed using regression on deseasonalized daa. Deseasonalized daa is obained by dividing he acual daa wih is seasonaliy index. All Righs Reserved, Indian Insiue of Managemen Bangalore

61 Predicive Analyics : QM901.1x SUMMARY OUTPUT Regression Saisics Muliple R R Square Adjused R Square Sandard Error Observaions 28 ANOVA df SS MS F Significance F Regression Residual Toal Coefficiens Sandard Error Sa P-value Lower 95% Upper 95% Inercep E Day All Righs Reserved, Indian Insiue of Managemen Bangalore

62 Forecas Predicive Analyics : QM901.1x F = T x S F 29 = T 29 x S 29 F 29 = ( x 29) x = Y 29 = 46 All Righs Reserved, Indian Insiue of Managemen Bangalore

63 Predicive Analyics : QM901.1x Forecasing using mehod of averages in he presence of seasonaliy All Righs Reserved, Indian Insiue of Managemen Bangalore

64 Predicive Analyics : QM901.1x All Righs Reserved, Indian Insiue of Managemen Bangalore

65 Predicive Analyics : QM901.1x All Righs Reserved, Indian Insiue of Managemen Bangalore Auo-correlaion Auo correlaion is a correlaion of a variable observed a wo ime poins (e.g. Y and Y -1 or Y and Y -3 ). Auo-correlaion of lag k, k, is given by: observaions number of oal n ) ( n n k k k Y Y Y Y Y Y

66 Auo-correlaion Funcion (ACF) Predicive Analyics : QM901.1x A k-period plo of auocorrelaions is called auocorrelaion funcion (ACF) or a correlogram. All Righs Reserved, Indian Insiue of Managemen Bangalore

67 Auo-Correlaion Predicive Analyics : QM901.1x Auo-correlaion of lag k is auo-correlaion beween Y and Y +k. To es wheher he auocorrelaion a lag k is significanly differen from 0, he following hypohesis es is used: H 0 : k = 0 H A : k 0 For any k, rejec H 0 if k > 1.96/ n. Where n is he number of observaions. All Righs Reserved, Indian Insiue of Managemen Bangalore

68 ACF for Demand for Coninenal Breakfas Predicive Analyics : QM901.1x All Righs Reserved, Indian Insiue of Managemen Bangalore

69 Parial Auo-correlaion Predicive Analyics : QM901.1x Parial auo correlaion of lag k is an auo correlaion beween Y and Y -k wih linear dependence beween he inermedia values (Y -k+1, Y -k+2,, Y -1 ) removed. All Righs Reserved, Indian Insiue of Managemen Bangalore

70 Predicive Analyics : QM901.1x Parial Auo-correlaion Funcion A k-period plo of parial auocorrelaions is called parial auocorrelaion funcion (PACF). All Righs Reserved, Indian Insiue of Managemen Bangalore

71 Parial Auo-Correlaion Predicive Analyics : QM901.1x Parial auo-correlaion of lag k is auo-correlaion beween Y and Y +k afer he removal of linear dependence of Y +1 o Y +k-1. To es wheher he parial auocorrelaion a lag k is significanly differen from 0, he following hypohesis es is used: H 0 : pk = 0 H A : pk 0 For any k, rejec H 0 if pk > 1.96/ n. Where n is he number of observaions. All Righs Reserved, Indian Insiue of Managemen Bangalore

72 PACF Demand for Coninenal Breakfas a DAD hospial Predicive Analyics : QM901.1x All Righs Reserved, Indian Insiue of Managemen Bangalore

73 Saionariy Predicive Analyics : QM901.1x A ime series is saionary, if: Mean () is consan over ime. Variance () is consan over ime. The covariance beween wo ime periods (Y ) and (Y +k ) depends only on he lag k no on he ime. We assume ha he ime series is saionary before applying forecasing models All Righs Reserved, Indian Insiue of Managemen Bangalore

74 ACF Plo of non-saionary and saionary process Predicive Analyics : QM901.1x Non-saionary Saionary All Righs Reserved, Indian Insiue of Managemen Bangalore

75 Whie Noise Predicive Analyics : QM901.1x Whie noise is a daa uncorrelaed across ime ha follow normal disribuion wih mean 0 and consan sandard deviaion. In forecasing we assume ha he residuals are whie Noise. All Righs Reserved, Indian Insiue of Managemen Bangalore

76 Predicive Analyics : QM901.1x Residual Whie Noise All Righs Reserved, Indian Insiue of Managemen Bangalore

77 Predicive Analyics : QM901.1x All Righs Reserved, Indian Insiue of Managemen Bangalore

78 Auo-Regression Predicive Analyics : QM901.1x Auo-regression is a regression of Y on iself observed a differen ime poins. Tha is, we use Y as he response variable and Y -1, Y -2 ec. as explanaory variables. All Righs Reserved, Indian Insiue of Managemen Bangalore

79 Predicive Analyics : QM901.1x All Righs Reserved, Indian Insiue of Managemen Bangalore AR(1) Parameer Esimaion n n n n Y Y Y Y Y Y Y OLS Esimae

80 AR(1) Process Predicive Analyics : QM901.1x Y = Y -1 + Y = (Y ) + Y = X Y X 0 1 i0 i i All Righs Reserved, Indian Insiue of Managemen Bangalore

81 Auo-regressive process (AR(p)) Predicive Analyics : QM901.1x Assume {Y } is purely random wih mean zero and consan sandard deviaion (Whie Noise). Then he auoregressive process of order p or AR(p) process is Y 0 Y 1 1 2Y 2... Y p p AR(p) process models each fuure observaion as a funcion p previous observaions. All Righs Reserved, Indian Insiue of Managemen Bangalore

82 Predicive Analyics : QM901.1x Model Idenificaion in AR(p) Process All Righs Reserved, Indian Insiue of Managemen Bangalore

83 Pure AR Model Idenificaion Predicive Analyics : QM901.1x Model ACF PACF AR(1) Exponenial Decay: Posiive side if ø1 > 0 and alernaing in sign saring on negaive side if ø1 < 0. Spike a lag 1, hen cus off o zero. Spike posiive if ø1 > 0 and negaive side if ø1 < 0. AR(p) Exponenial decay: paern depends on signs of ø1, ø2, ec Spikes a lags 1 o p, hen cus of o zero. Y 0 Y 1 1 2Y 2... Y p p All Righs Reserved, Indian Insiue of Managemen Bangalore

84 Parial Auo-Correlaion Predicive Analyics : QM901.1x Parial auo-correlaion of lag k is auo-correlaion beween Y and Y +k afer he removal of linear dependence of Y +1 o Y +k-1. To es wheher he auocorrelaion a lag k is significanly differen from 0, he following hypohesis es is used: H 0 : k = 0 H A : k 0 For any k, rejec H0 if k > 1.96/ n. Where n is he number of observaions. All Righs Reserved, Indian Insiue of Managemen Bangalore

85 PACF Funcion DAD Coninenal Breakfas Predicive Analyics : QM901.1x Firs wo PAC are differen from zero. All Righs Reserved, Indian Insiue of Managemen Bangalore

86 ACF Funcion - DAD Coninenal Breakfas Predicive Analyics : QM901.1x All Righs Reserved, Indian Insiue of Managemen Bangalore

87 Acual Vs Fi- Coninenal Breakfas Daa Predicive Analyics : QM901.1x All Righs Reserved, Indian Insiue of Managemen Bangalore

88 AR(2) Model Predicive Analyics : QM901.1x MAPE = 8.8% All Righs Reserved, Indian Insiue of Managemen Bangalore

89 Residual Whie Noise Predicive Analyics : QM901.1x All Righs Reserved, Indian Insiue of Managemen Bangalore

90 Predicive Analyics : QM901.1x (Y ) = (Y ) (Y ) Y = (X - ) All Righs Reserved, Indian Insiue of Managemen Bangalore

91 Predicive Analyics : QM901.1x All Righs Reserved, Indian Insiue of Managemen Bangalore

92 Moving Average Process Predicive Analyics : QM901.1x A moving average process is a ime series regression model in which he value a ime, Y, is a linear funcion of pas errors. Firs order moving average, MA(1), is given by: Y = All Righs Reserved, Indian Insiue of Managemen Bangalore

93 Moving Average Process MA(q) Predicive Analyics : QM901.1x {Y} is a moving average process of order q (wrien MA(q)) if for some consans 0, 1,... q We have., Y q q MA(q) models each fuure observaion as a funcion of q previous errors All Righs Reserved, Indian Insiue of Managemen Bangalore

94 Pure MA Model Idenificaion Predicive Analyics : QM901.1x Model ACF PACF MA(1) Spike a lag 1 hen cus of o zero. Spike posiive if 1 > 0 and negaive side if 1 < 0. Exponenial decay. On negaive side if 1> 0 on posiive side if 1< 0. MA(q) Spikes a lags 1 o q, hen cus off o zero. Exponenial decay or sine wave. Exac paern depends on signs of 1, 2 ec. All Righs Reserved, Indian Insiue of Managemen Bangalore

95 ACF Funcion - DAD Coninenal Breakfas Predicive Analyics : QM901.1x All Righs Reserved, Indian Insiue of Managemen Bangalore

96 SPSS Oupu Predicive Analyics : QM901.1x All Righs Reserved, Indian Insiue of Managemen Bangalore

97 Acual Vs Fied Value Predicive Analyics : QM901.1x All Righs Reserved, Indian Insiue of Managemen Bangalore

98 Residual Plo Predicive Analyics : QM901.1x All Righs Reserved, Indian Insiue of Managemen Bangalore

99 Predicive Analyics : QM901.1x All Righs Reserved, Indian Insiue of Managemen Bangalore

100 Predicive Analyics : QM901.1x All Righs Reserved, Indian Insiue of Managemen Bangalore ARMA(p,q) Model q q p p Y Y Y Y AR(p) Model MA(q) Model

101 ARMA(p,q) Model Idenificaion Predicive Analyics : QM901.1x ARMA(p,q) models are no easy o idenify. We usually sar wih pure AR and MA process. The following hump rule may be used. Process ACF PACF ARMA(p,q) Tails off afer q lags Tails off o zero afer p lags The final ARMA model may be seleced based on parameers such as RMSE, MAPE, AIC and BIC. All Righs Reserved, Indian Insiue of Managemen Bangalore

102 Predicive Analyics : QM901.1x ACF PACF All Righs Reserved, Indian Insiue of Managemen Bangalore

103 ARMA(2,1) Model Oupu Predicive Analyics : QM901.1x All Righs Reserved, Indian Insiue of Managemen Bangalore

104 AR(2) Model Predicive Analyics : QM901.1x MAPE = 8.8% All Righs Reserved, Indian Insiue of Managemen Bangalore

105 Residual Whie Noise Predicive Analyics : QM901.1x All Righs Reserved, Indian Insiue of Managemen Bangalore

106 Forecasing Model Evaluaion Predicive Analyics : QM901.1x Akaike s informaion crieria AIC = -2LL + 2m Where m is he number of variables esimaed in he model Bayesian Informaion Crieria BIC = -2LL + m ln(n) AIC and BIC can be inerpreed as disance from rue model Where m is he number of variables esimaed in he model and n is he number of observaions All Righs Reserved, Indian Insiue of Managemen Bangalore

107 Final Model Selecion Predicive Analyics : QM901.1x Model AR(2) MA(1) ARMA(2,1) BIC AR(2) has he leas BIC All Righs Reserved, Indian Insiue of Managemen Bangalore

108 Predicive Analyics : QM901.1x All Righs Reserved, Indian Insiue of Managemen Bangalore

109 ARIMA Predicive Analyics : QM901.1x ARIMA has he following hree componens: Auo-regressive componen: Funcion of pas values of he ime series. Inegraion Componen: Differencing he ime series o make i a saionary process. Moving Average Componen: Funcion of pas error values. All Righs Reserved, Indian Insiue of Managemen Bangalore

110 Inegraion (d) Predicive Analyics : QM901.1x Used when he process is non-saionary. Insead of observed values, differences beween observed values are considered. When d=0, he observaions are modelled direcly. If d = 1, he differences beween consecuive observaions are modelled. If d = 2, he differences of he differences are modelled. All Righs Reserved, Indian Insiue of Managemen Bangalore

111 ARIMA (p, d, q) Predicive Analyics : QM901.1x The q and p values are idenified using auo-correlaion funcion (ACF) and Parial auo-correlaion funcion (PACF) respecively. The value d idenifies he level of differencing. Usually p+q <= 4 and d <= 2. All Righs Reserved, Indian Insiue of Managemen Bangalore

112 Differencing Predicive Analyics : QM901.1x Differencing is a process of saionary process. making a non-saionary process ino In differencing, we creae a new process X, where X = Y Y -1. All Righs Reserved, Indian Insiue of Managemen Bangalore

113 Predicive Analyics : QM901.1x All Righs Reserved, Indian Insiue of Managemen Bangalore ARIMA(p,1,q) Process q q p p X X X X Where X = Y Y -1

114 Ljung-Box Tes Predicive Analyics : QM901.1x Ljung-Box is a es on he auocorrelaions of residuals. The es saisic is: Q m n( n 2) m k 1 2 rk n k n = number of observaions in he ime series. k = paricular ime lag checked m = he number of ime lags o be esed r k = sample auocorrelaion funcion of he k h residual erm. H0: The model does no exhibi lack of fi HA: The model exhibis lack of fi Q saisic is approximae chi-square disribuion wih m p q degrees of freedom if ARMA orders are correcly specified. All Righs Reserved, Indian Insiue of Managemen Bangalore

115 AR(2) Ljung-Box Tes Predicive Analyics : QM901.1x P-value is 0.740, hus he model doesn show lack of fi All Righs Reserved, Indian Insiue of Managemen Bangalore

116 Box-Jenkins Mehodology Predicive Analyics : QM901.1x Idenificaion: Idenify he ARIMA model using ACF & PACF plos. This would give he values of p, q and d. Esimaion: Esimae he model parameers (using maximum likelihood) Diagnosics: Check he residual for any issue such as no providing Whie Noise. All Righs Reserved, Indian Insiue of Managemen Bangalore