Forecasting. Summary. Sample StatFolio: tsforecast.sgp. STATGRAPHICS Centurion Rev. 9/16/2013

Transcription

1 STATGRAPHICS Cenurion Rev. 9/16/2013 Forecasing Summary... 1 Daa Inpu... 3 Analysis Opions... 5 Forecasing Models... 9 Analysis Summary Time Sequence Plo Forecas Table Forecas Plo Model Comparisons Residual Plos Residual Auocorrelaions Residual Auocorrelaion Funcion Residual Parial Auocorrelaions Residual Parial Auocorrelaion Funcion Residual Periodogram Table Residual Periodogram Plo Residual Inegraed Periodogram Tess For Randomness Residual Crosscorrelaions Residual Crosscorrelaion Plo Save Resuls Calculaions Summary The Forecasing procedure is designed o forecas fuure values of ime series daa. A ime series consiss of a se of sequenial numeric daa aken a equally spaced inervals, usually over a period of ime or space. The models provided o forecas fuure values include a moving average, a random walk, various ypes of exponenial smoohers, rend models, and parameric ARIMA models. Saisics are calculaed o compare he fi of up o 5 models a any one ime. This procedure is designed for users who wish o selec heir own model. The Auomaic Forecasing procedure ries various models and auomaically selecs he model ha is bes according o a specified goodness-of-fi crieria. Sample SaFolio: sforecas.sgp 2013 by SaPoin Technologies, Inc. Forecasing - 1

2 STATGRAPHICS Cenurion Rev. 9/16/2013 Sample Daa: The file golden gae.sgd conains monhly raffic volumes on he Golden Gae Bridge in San Francisco for a period of n = 168 monhs from January, 1968 hrough December, The able below shows a parial lis of he daa from ha file: Monh Traffic 1/ / / / / / / / / / / / / The daa were obained from a publicaion of he Golden Gae Bridge. As an exercise, he daa for he las wo years (1980 and 1981) will no be used o esimae he forecasing model, bu will be used insead o calculae validaion saisics by SaPoin Technologies, Inc. Forecasing - 2

3 STATGRAPHICS Cenurion Rev. 9/16/2013 Daa Inpu The daa inpu dialog box requess he name of he column conaining he ime series daa: Daa: numeric column conaining n equally spaced numeric observaions. Time indices: ime, dae or oher index associaed wih each observaion. Each value in his column mus be unique and arranged in ascending order. Sampling Inerval: If ime indices are no provided, his defines he inerval beween successive observaions. For example, he daa from he Golden Gae Bridge were colleced once every monh, beginning in January, Seasonaliy: he lengh of seasonaliy s, if any. The daa is seasonal if here is a paern ha repeas a a fixed period. For example, monhly daa such as raffic on he Golden Gae Bridge have a seasonaliy of s = 12. Hourly daa ha repea every day 2013 by SaPoin Technologies, Inc. Forecasing - 3

4 STATGRAPHICS Cenurion Rev. 9/16/2013 have a seasonaliy of s = 24. If no enry is made, he daa is assumed o be nonseasonal (s = 1). Trading Days Adjusmen: a numeric variable wih n observaions used o normalize he original observaions, such as he number of working days in a monh. The observaions in he Daa column will be divided by hese values before being ploed or analyzed. There mus be enough enries in his column o cover boh he observed daa and he number of periods for which forecass are requesed. Selec: subse selecion. Number of Forecass: number of periods following he end of he daa for which forecass are desired. Wihhold for Validaion: number of periods m a he end of he series o wihhold for validaion purposes. The daa in hose periods will no be used o esimae he forecasing model. However, saisics will be calculaed describing how well he esimaed model is able o forecas hose observaions. In he curren example, he raffic daa is monhly beginning in January, 1968, and has a seasonaliy of s = 12. m = 24 observaions a he end of he series will be wihheld for validaion purpose, while forecass will be generaed for he nex 36 monhs by SaPoin Technologies, Inc. Forecasing - 4

5 STATGRAPHICS Cenurion Rev. 9/16/2013 Analysis Opions The Forecasing procedure is conrolled by he Analysis Opions dialog box: Model: he model o which he oher seings on he dialog box apply. Up o five forecasing models may be considered a he same ime, labeled A, B, C, D, and E. Mah: Before fiing a model, he daa may be ransformed using any of he indicaed operaions. Wih he excepion of he Box-Cox ransformaion, he selecions are selfexplanaory. The Box-Cox ransformaion is used when necessary o make he daa more Gaussian. For a deailed discussion, see he documenaion for he Box-Cox Transformaions procedure. Seasonal: seasonally adjus he daa using he indicaed mehod before fiing he model. Seasonal adjusmens are designed o remove any seasonal componen from he daa. The mehods used are discussed in he documenaion for he Seasonal Decomposiion procedure. Inflaion: adjuss he daa for inflaion using he specified inflaion rae before fiing he modelif applied a he beginning of he period, he adjusmen is 2013 by SaPoin Technologies, Inc. Forecasing - 5

6 STATGRAPHICS Cenurion Rev. 9/16/2013 y y ( 0 1) 1 (1) where 0 is he index of he firs observaion. If applied a he middle of he period, he adjusmen is y y (1 0.5) ) ( 0 (2) Noe: Transformaions are applied o he daa before he forecasing model is fi. If more han one ransformaion is requesed, hey are applied in he following order: 1. rading day adjusmen 2. inflaion adjusmen 3. mah adjusmen 4. seasonal adjusmen Afer he forecass are generaed, inverse ransformaions are applied o he forecass in reverse order. Type: he ype of forecasing model o be fi. For an explanaion of he differen ypes of models, see he discussion below by SaPoin Technologies, Inc. Forecasing - 6

7 STATGRAPHICS Cenurion Rev. 9/16/2013 Parameers and Terms: opions for differen forecasing models. o Alpha, bea, and gamma: parameers for he Exponenial Smoohing models. Each parameer mus be greaer han 0 and less han 1. The lower he value of a parameer, he greaer he amoun of smoohing ha is performed. o Order: he number of erms in he Moving Average model. o AR, MA, SAR, and SMA: he order of he various componens of he ARIMA models, referred o as p, q, P, and Q respecively in he discussion below. o Opimize: wheher opimal values of he parameers should be found. If checked, he parameer values specified are used as saring values for he search procedures. If no checked, he values enered will be used in he model. o Consan: wheher a consan erm should be included when fiing a Random Walk or ARIMA model. Differencing: he order of seasonal and nonseasonal differencing o be applied when fiing he ARIMA models, referred o as d and D in he discussion below. Esimaion Buon: displays a dialog box ha conrols he nonlinear esimaion procedure used when opimizing he exponenial smoohing and ARIMA models. Sopping Crierion 1: The algorihm is assumed o have converged when he relaive change in he residuals sums of squares from one ieraion o he nex is less han his value. Sopping Crierion 2: The algorihm is assumed o have converged when he relaive change in all parameer esimaes from one ieraion o he nex is less han his value by SaPoin Technologies, Inc. Forecasing - 7

8 STATGRAPHICS Cenurion Rev. 9/16/2013 Maximum Ieraions: Esimaion sops if convergence is no achieved wihin his many ieraions. Backforecasing: Uses a mehod called backforecasing o forecas values prior o ime = 1. These values are used o generae he iniial values which are needed o generae forecass for small values of. For deails, see Box, Jenkins and Reinsel (1994). Regression Buon: adds addiional independen variables o he forecasing model when esimaing a rend or ARIMA model. Typically, such variables are lagged values of leading indicaors. Variables: values of X variables o include in he model. If you wish o include a column named X bu lag he daa by 3 rows so ha he model includes a erm involving X -3, ener LAG(X,3) insead of jus X. Noe: Whichever leer is seleced in he Model field when he dialog box is closed is aken o be he primary model. This is he model used when generaing all of he ables and plos (excep for he Model Comparisons pane, which compares hem all) by SaPoin Technologies, Inc. Forecasing - 8

9 STATGRAPHICS Cenurion Rev. 9/16/2013 Forecasing Models Each of he forecasing models akes a differen approach for forecasing fuure values. In he discussions below, he following noaion will be used: Y = observed value a ime, = 1, 2,, n n = sample size (number of observaions used o fi he model) F (k) = forecas for ime +k made a ime e = one period ahead forecasing errors calculaed from e = Y - F -1 (1) (3) Given ha m observaions a he end of he ime series have been wihheld for validaion purposes, wo imporan validaion saisics are: RMSE = roo mean squared error over he validaion period, given by RMSE m i1 e m 2 ni (4) MAPE = mean absolue percenage error over he validaion period, given by m eni / Y i i MAPE m % (5) The RMSE esimaes he sandard deviaion of he one-ahead forecas errors. The MAPE esimaes he average percenage one-ahead forecasing error. Small values of RMSE and MAPE are desirable by SaPoin Technologies, Inc. Forecasing - 9

10 Traffic STATGRAPHICS Cenurion Rev. 9/16/2013 Random Walk Model The random walk model is very simple. Wihou a consan, i uses he curren value of he ime series o forecas all fuure values, i.e., F (k) = Y for all k 1 (6) This model is ofen used for daa ha does no have a fixed mean and for which he hisory of he process is irrelevan given is curren posiion. The ime series is hus equally likely o go up or down a any poin in ime. If a consan is included, hen he forecas is given by F ( k) Y kˆ (7) where ˆ esimaes he average change from one period o he nex. The forecas funcion for such a model is a sraigh line wih slope equal o ˆ. For he sample daa, he random walk model could be used if a consan is included and he ime series is firs seasonally adjused. The resuls are shown below: Model Consan Seasonal adj. Validaion RMSE Validaion MAPE Random walk Yes Muliplicaive % Time Sequence Plo for Traffic Random walk wih drif acual forecas 95.0% limis /68 1/72 1/76 1/80 1/84 1/88 The plo shows: 1. Observed daa: shown using poin symbols. 2. One-ahead forecass: shown as a solid line passing hrough he daa. 3. Forecass for fuure values: exension of he forecass pas he end of he daa % predicion limis: he red bounds around he forecass. Noe he wide predicion limis, ypical of random walk models by SaPoin Technologies, Inc. Forecasing - 10

11 Traffic STATGRAPHICS Cenurion Rev. 9/16/2013 Trend Models The Mean, Linear Trend, Quadraic Trend, Exponenial Trend, and S-Curve models all fi various ypes of regression models o he daa, using ime as he independen variable. The models are fi by leas squares, resuling in esimaes of up o 3 coefficiens: a, b, and c. Forecass from he models are as follows: Mean model: F ( k) Y where Y is he average of he daa up o and including ime. (8) Linear rend: ( k) aˆ bˆ( k) (9) F Quadraic rend: F ( k) ˆ ˆ( ˆ( k Exponenial rend: F ( k) expa ˆ bˆ( k) S-Curve: ( k) expa ˆ bˆ /( k) F 2 a b k) c ) (10) (11) (12) Since hey weigh all daa equally, regression models are ofen no he bes mehods for forecasing ime series daa. For he sample daa, he bes-fiing rend model is he Quadraic Trend, fi afer seasonally adjusing he daa. Model Seasonal adj. Validaion RMSE Validaion MAPE Quadraic rend Muliplicaive % Time Sequence Plo for Traffic Quadraic rend = ^ acual forecas 95.0% limis /68 1/72 1/76 1/80 1/84 1/ by SaPoin Technologies, Inc. Forecasing - 11

12 Traffic STATGRAPHICS Cenurion Rev. 9/16/2013 Moving Average The Moving Average model uses he average of he mos recen c observaions o forecas fuure values. The forecass are given by: c 1 Y i i 0 F ( k) c for all k 1 (13) Such a model can rack a series ha moves up and down, bu ends o lag behind he acual series. Experimening wih various orders of moving averages, i was found ha an average of c = 2 observaions gave he bes fi for he raffic daa during he validaion period. Model Order Seasonal adj. Validaion RMSE Validaion MAPE Moving average 2 Muliplicaive % Time Sequence Plo for Traffic Simple moving average of 2 erms acual forecas 95.0% limis /68 1/72 1/76 1/80 1/84 1/88 Noe ha he forecas funcion has no rend, which is couner-inuiive given he observed behavior. However, he one-monh-ahead forecass appear o be very good by SaPoin Technologies, Inc. Forecasing - 12

13 STATGRAPHICS Cenurion Rev. 9/16/2013 Exponenial Smoohing The Simple Exp. Smoohing, Brown s Linear Exp. Smoohing, and Quadraic Exp. Smoohing models esimae rends similar o he Mean, Linear Trend, and Quadraic Trend models, respecively. However, hey do so by weighing recen observaions more heavily han observaions ha are furher in he pas. To generae he forecass, up o hree passes of an exponenial smooher are made: S Y (14) ( 1 ) S 1 S S ( 1 ) S (15) 1 S S ( 1 ) S (16) 1 The iniial values a ime = 0 are deermined by backforecasing (unless suppressed using he Esimaion buon on he Analysis Opions dialog box), which firs smoohes he ime series backwards and hen uses he backforecass o iniialize he forward smoohing. The forecass are hen generaed from: Simple smoohing: F ( k) S (17) Linear smoohing: F k) 2S S k S S Quadraic smoohing: ( (18) 1 3S 3S S k 2 2(1 ) F (k)= 2 2 k S 2S S 2 2(1 ) (6 5 ) S (10 8 ) S (4 3 ) S (19) The Quadraic Exp. Smooher gives he bes resuls of he hree procedures during he validaion period when forecasing one period ahead. However, exrapolaion of a quadraic rend ino he fuure is always problemaic, as can be seen in he plo below by SaPoin Technologies, Inc. Forecasing - 13

14 Traffic STATGRAPHICS Cenurion Rev. 9/16/2013 Model Quadraic exp. smoohing Alpha (opimized) Seasonal adj. Validaion RMSE Validaion MAPE Muliplicaive % Time Sequence Plo for Traffic Brown's quadraic exp. smoohing wih alpha = /68 1/72 1/76 1/80 1/84 1/88 acual forecas 95.0% limis This example illusraes several imporan facs: 1. I is imporan o look a he resuls whenever a forecasing model is fi o be sure ha he resuls make sense. 2. Models ha are good for shor-erm forecasing may no behave well a forecasing values far ino he fuure. 3. Models involving polynomials of order 2 or higher can behave erraically by SaPoin Technologies, Inc. Forecasing - 14

15 Traffic STATGRAPHICS Cenurion Rev. 9/16/2013 Hol s Linear Exponenial Smoohing Hol s Linear Exp. Smoohing is similar o Brown s Linear Exp. Smoohing in ha i generaes forecass ha follow a linear rend. However, Hol s procedure uses wo smoohing consans, and, one o esimae he level of he series a ime and a second o esimae he slope. The procedure is as follows: 1. Smooh he daa o esimae he level using S ( 1) S 1 T 1 Y (20) 2. Smooh he firs smooh o esimae he slope using T S S 1 ( 1 ) T 1 (21) 3. Calculae he forecass using F ( k) S kt (22) The following shows he resuls of opimizing Hol s smooher afer seasonally adjusing he daa: Model Hol s linear exponenial smoohing Alpha (opimized) Bea (opimized) Seasonal adj. Validaion RMSE Validaion MAPE Muliplicaive % Time Sequence Plo for Traffic Hol's linear exp. smoohing wih alpha = and bea = acual forecas % limis /68 1/72 1/76 1/80 1/84 1/88 The resuls look quie reasonable by SaPoin Technologies, Inc. Forecasing - 15

16 STATGRAPHICS Cenurion Rev. 9/16/2013 Winer s Exponenial Smoohing All of he forecasing mehods described above handle he seasonaliy by firs seasonally adjusing he daa, hen applying he forecasing model, and hen puing back he seasonaliy. Winer s Exp. Smoohing procedure handles he seasonaliy direcly by esimaing seasonaliy a he same ime ha i esimaes he level and rend. I exends Hol s procedure by adding an addiional parameer o use in a hird smooher. The procedure is as follows: 1. Esimae he seasonaliy by smoohing he raio of he daa o he esimaed level a ime using: I Y Is (23) S ( 1 ) where s is he lengh of seasonaliy. 2. Esimae he level of he series by smoohing he daa divided by he esimaed seasonaliy using S Y ( 1) S 1 T 1 (24) I s 3. Esimae he slope of he series using T S S 1 ( 1 ) T 1 (25) 4. Calculae he forecass using S kt I sm F ( k) (26) The following shows he resuls of opimizing Winer s smooher: Model Winer s seasonal exp. smoohing Alpha Bea Gamma Validaion Validaion (opimized) (opimized) (opimized) RMSE MAPE % 2013 by SaPoin Technologies, Inc. Forecasing - 16

17 Traffic STATGRAPHICS Cenurion Rev. 9/16/2013 Time Sequence Plo for Traffic Winer's exp. smoohing wih alpha = , bea = , gamma = acual forecas % limis /68 1/72 1/76 1/80 1/84 1/88 Is performance on he raffic daa is no as good as wih some of he oher mehods. Also, he parameers are hard o esimae numerically and may vary quie a bi depending on he saring values of he search procedure. ARIMA Models The final choice of forecasing models, he ARIMA models, are he mos general and include many of he oher models are special cases. ARIMA models (shor for AuoRegressive, Inegraed, Moving Average ), express he observaion a ime as a linear funcion of previous observaions, a curren error erm, and a linear combinaion of previous error erms. The general form of he model is mos easily expressed in erms of he backwards operaor B, which operaes on he ime index of a daa value such ha B j Y = Y -j. Using his operaor, he model akes he form 2 p s 2s Ps d s D 1 B B... B 1 B B... B (1 B) (1 B ) Z 2 q s 2s Qs 1 B B... B 1 B B B a... (27) where Z Y (28) and a is a random error or shock o he sysem a ime, usually assumed o be random observaions from a normal disribuion wih mean 0 and sandard deviaion a. For a saionary series, represens he process mean. Oherwise, i is relaed o he slope of he forecas funcion. is someimes assumed o equal 0. The above model is ofen referred o as an ARIMA(p,d,q)x(P,D,Q)s model. I consiss of several erms: 2013 by SaPoin Technologies, Inc. Forecasing - 17

18 STATGRAPHICS Cenurion Rev. 9/16/ A nonseasonal auoregressive erm of order p. 2. Nonseasonal differencing of order d. 3. A nonseasonal moving average erm of order q. 4. A seasonal auoregressive erm of order P 5. Seasonal differencing of order D. 6. A seasonal moving average erm of order Q. While he general model looks formidable, he mos commonly used models are relaively simple special cases. These include: AR(1) auoregressive of order 1 The observaion a ime is expressed as a mean plus a muliple of he deviaion from he mean a he previous ime period plus a random shock: Y Y a 1 1 (29) AR(2) auoregressive of order 2 The observaion a ime is expressed as a mean plus muliples of he deviaions from he mean a he 2 previous ime periods plus a random shock: Y Y 1 2 Y a 1 2 (30) MA(1) moving average of order 1 The observaion a ime is expressed as a mean plus a random shock a he curren ime period plus a muliple of he random shock a he previous ime period: Y a a (31) 1 1 MA(2) moving average of order 2 The observaion a ime is expressed as a mean plus a random shock a he curren ime period plus muliples of he random shocks a he 2 previous ime periods: Y a a a (32) ARMA(1,1) mixed model wih 2 firs order erms The observaion a ime is expressed as a mean plus a muliple of he deviaion from he mean a he previous ime period plus a random shock a he curren ime period plus a muliple of he random shock a he previous ime period: Y Y 1 a 1a 1 (33) by SaPoin Technologies, Inc. Forecasing - 18

19 STATGRAPHICS Cenurion Rev. 9/16/2013 ARIMA(0,1,1) moving average of order 1 applied o he firs differences The difference beween he curren period and he previous period is expressed as a random shock a he curren ime period plus a muliple of he random shock a he previous ime period: Y Y a a (34) I can be shown ha his model is equivalen o he Simple Exponenial Smoohing model. ARIMA(0,2,2) moving average of order 2 applied o he second differences The difference of he differences is expressed as a random shock a he curren ime period plus muliples of he random shocks a he 2 previous ime periods: Y Y 1 Y 1 Y 2 a 1a1 2a2 (35) This model is equivalen o he Hol s Linear Exponenial Smoohing model. ARIMA(0,1,1)x(0,1,1)s seasonal and nonseasonal MA erms of order 1 The observaion a ime is expressed as a combinaion of he observaion one season ago plus he difference beween he observaion las period and is counerpar one season ago plus muliple of he shocks o hi he sysem his period, las period, and wo periods one season ago: Y Y s Y Y a a a a (36) 1 s s 1 1 s1 Many economic ime series wih a seasonal componen can be well represened by his model. I also does very well on he Golden Gae Bridge raffic daa: 2013 by SaPoin Technologies, Inc. Forecasing - 19

20 Traffic STATGRAPHICS Cenurion Rev. 9/16/2013 Model MA(1) SMA(1) Validaion Validaion RMSE MAPE ARIMA % Time Sequence Plo for Traffic ARIMA(0,1,1)x(0,1,1)12 acual forecas 95.0% limis /68 1/72 1/76 1/80 1/84 1/88 Inuiively, he model expresses he difference in he raffic his monh compared o he same monh las year as being equal o he difference observed las monh, plus a combinaion of he noise observed las monh, las year, and 13 monhs ago. The classic reference for consrucing ARIMA models is Box, Jenkins and Reinsel (1994) by SaPoin Technologies, Inc. Forecasing - 20

21 STATGRAPHICS Cenurion Rev. 9/16/2013 Analysis Summary The resuls of fiing a forecasing model are displayed in he Analysis Summary. As an example, he able below shows he resuls of fiing he ARIMA(0,1,1)x(0,1,1)12 model o he Golden Gae Bridge raffic daa: Forecasing - Traffic Daa variable: Traffic (Golden Gae Bridge Traffic Volume) Number of observaions = 168 Sar index = 1/68 Sampling inerval = 1.0 monh(s) Lengh of seasonaliy = 12 Forecas Summary Nonseasonal differencing of order: 1 Seasonal differencing of order: 1 Forecas model seleced: ARIMA(0,1,1)x(0,1,1)12 Number of forecass generaed: 36 Number of periods wihheld for validaion: 24 Esimaion Validaion Saisic Period Period RMSE MAE MAPE ME MPE ARIMA Model Summary Parameer Esimae Snd. Error P-value MA(1) SMA(1) Backforecasing: yes Esimaed whie noise variance = wih 153 degrees of freedom Esimaed whie noise sandard deviaion = Number of ieraions: 6 There are several imporan secions in he oupu: Daa Summary: he op secion summarizes he inpu daa and he lengh of seasonaliy s, if any. Forecas Summary: indicaes any ransformaions ha were made o he daa, as well as he ype of model ha was fi. The number of periods m ha were wihheld for validaion purposes is also shown. Table of Saisics: shows saisics calculaed from he one-ahead forecas errors during boh he esimaion and validaion periods. In addiion o he roo mean squared error (RMSE) and mean absolue percenage error (MAPE) described earlier, he program also displays he mean absolue error (MAE), he mean error (ME), and he mean percenage error (MPE). Ideally, RMSE, MAE, and MAPE will be small, since hey measure he variabiliy of he forecas errors. ME and MPE should be close o zero if he forecass are no biased by SaPoin Technologies, Inc. Forecasing - 21

22 STATGRAPHICS Cenurion Rev. 9/16/2013 ARIMA Model Summary displays saisics for he coefficiens of he fied ARIMA model. A similar able is displayed when rend models are fi using leas squares regression. Of ineres are: o Esimae: he esimaed coefficien. o Snd. error: he sandard error of he coefficien. o : he value of a saisic calculaed by dividing he esimaed coefficien by is sandard error. o P-value: wo-sided P-value calculaed from Suden s disribuion wih he degrees of freedom indicaed below he able. Small P-values (less han 0.05 if operaing a he 5% significance level) correspond o saisically significan coefficiens. If any P-values are greaer han 0.05, consideraion should be given o reducing he complexiy of he model. o Esimaed whie noise sandard deviaion: esimae of he sandard deviaion of he noise ˆ ha is unaccouned for by he model. a o Number of ieraions: he number of ieraions used by he nonlinear esimaion procedure. In he example, he fied ARIMA model has wo parameers, boh of which are saisically significan. The ME and MPE are close o zero in boh he esimaion and he validaion periods, indicaing lile bias in he one-monh ahead forecass. Examining he RMSE, MAE, and MAPE, he model appears o do no worse (and possibly beer) during he validaion period han in he esimaion period by SaPoin Technologies, Inc. Forecasing - 22

23 Traffic STATGRAPHICS Cenurion Rev. 9/16/2013 Time Sequence Plo The Time Sequence Plo displays he daa, he forecass, and he forecas limis: Time Sequence Plo for Traffic ARIMA(0,1,1)x(0,1,1)12 acual forecas 95.0% limis /68 1/72 1/76 1/80 1/84 1/88 The plo shows: 1. The observed daa Y, including any replacemens for missing values, shown as poin symbols. 2. The one-sep ahead forecass F (1), displayed as a solid line hrough he poins. These are creaed using he fied model, forecasing each ime period +1 using only he informaion available a ime. The one-ahead forecas errors e are observable as he verical disance beween he observaions and he solid line. 3. Forecass for fuure values F n+m (k) made a ime = n+m, he las ime a which observed daa is available. These are shown by he exension of he solid forecas line beyond he las observaion. 4. Probabiliy limis for he forecass a he 100(1-)% confidence level, calculaed assuming ha he noise in he sysem follows a normal disribuion. The limis are given by ( k) z / 2 Vˆ( k) (37) F n m where V ˆ( k ) equals he esimaed variance of he forecas k periods pas he end of he daa. The formula for he variance depends on he model used, as oulined in he Calculaions secion. I should be noed ha he limis are only valid if several assumpions hold, including: a. The proper model has been seleced. b. The seleced model was valid for all of he hisorical daa by SaPoin Technologies, Inc. Forecasing - 23

24 STATGRAPHICS Cenurion Rev. 9/16/2013 c. The seleced model coninues o be valid in he fuure. d. The shocks o he sysem follow a normal disribuion. e. The model has been esimaed from a long enough ime series ha he model esimaion error is small compared o he variabiliy of he error erm (excep for models esimaed by linear regression which include he model esimaion error). In pracice, he limis should be regarded as no more han an approximaion of how far he ime series may sray from he forecased values in he fuure. The forecased paern for he Golden Gae Bridge raffic shows a coninued upward rend wih a srong seasonal oscillaion. Alhough he forecas limis may seem quie wide, hey mus allow for he possibiliy of dramaic evens such as were observed wice in he pas. Pane Opions Confidence Level: he percenage o use for he probabiliy limis. Forecas Table The Forecas Table displays he forecass for boh he hisorical and fuure periods. A porion of he oupu is shown below: Forecas Table for Traffic Model: ARIMA(0,1,1)x(0,1,1)12 V = wihheld for validaion Period Daa Forecas Residual 7/ V 8/ V 9/ V 10/ V 11/ V 12/ V Lower 95.0% Upper 95.0% Period Forecas Limi Limi 1/ / / / / / by SaPoin Technologies, Inc. Forecasing - 24

25 Traffic STATGRAPHICS Cenurion Rev. 9/16/2013 The op secion of he oupu shows: Period: he ime period corresponding o each hisorical observaion. Daa: he observed daa value Y, including any replacemens for missing values. Forecas: he forecas for ime using all of he informaion available a ime -1. Residual: he one-ahead forecas error e, calculaed by subracing he forecas from he observed daa value. V: indicaes ha he corresponding observaion was no used o fi he model bu insead was included in he validaion se. The boom secion of he oupu shows: Period: he ime corresponding o periods beyond he end of he observed daa. Forecas: he forecas F n+m (k) for ime period using all of he available daa. Limis: he probabiliy limis for he forecass. For example, he forecased raffic on he Golden Gae Bridge in June of 1982, made a he end of 1981, was The 95% limis ranged from 96.5 o Forecas Plo The Forecas Plo shows he las several observaions, he forecass, and he forecas limis: Forecas Plo for Traffic ARIMA(0,1,1)x(0,1,1)12 acual forecas 99.0% limis /81 12/82 12/83 12/84 12/ by SaPoin Technologies, Inc. Forecasing - 25

26 STATGRAPHICS Cenurion Rev. 9/16/2013 I is similar o he Time Sequence Plo, excep ha i gives a closer view of he forecass. Model Comparisons The Model Comparisons pane displays saisics ha compare each of he models seleced on he Analysis Opions dialog box. Model Comparison Daa variable: Traffic Number of observaions = 168 Sar index = 1/68 Sampling inerval = 1.0 monh(s) Lengh of seasonaliy = 12 Number of periods wihheld for validaion: 24 Models (A) ARIMA(0,1,1)x(0,1,1)12 (B) Winer's exp. smoohing wih alpha = , bea = , gamma = Esimaion Period Model RMSE MAE MAPE ME MPE (A) (B) Model RMSE RUNS RUNM AUTO MEAN VAR (A) OK OK OK OK *** (B) OK OK OK OK *** Validaion Period Model RMSE MAE MAPE ME MPE (A) (B) Key: RMSE = Roo Mean Squared Error RUNS = Tes for excessive runs up and down RUNM = Tes for excessive runs above and below median AUTO = Box-Pierce es for excessive auocorrelaion MEAN = Tes for difference in mean 1s half o 2nd half VAR = Tes for difference in variance 1s half o 2nd half OK = no significan (p >= 0.05) * = marginally significan (0.01 < p <= 0.05) ** = significan (0.001 < p <= 0.01) *** = highly significan (p <= 0.001) The ables labeled Esimaion Period and Validaion Period display saisics calculaed from he one-ahead forecas errors e in heir respecive periods: RMSE: he roo mean squared error. MAE: he mean absolue error. MAPE: he mean absolue percenage error. ME: he mean error. MPE: he mean percenage error by SaPoin Technologies, Inc. Forecasing - 26

27 Residual STATGRAPHICS Cenurion Rev. 9/16/2013 Beer models have smaller RMSE, MAE, and MAPE values, which measure he variance of he forecasing errors. ME and MPE are measures of bias and should be close o 0. For he esimaion period only, several ess are applied o he forecas errors o deermine wheher he model has accouned for all of he srucure in he daa. These ess are designed o deermine wheher he residuals form a random ime series ( whie noise ) and are described in he Time Series Descripive Mehods documenaion. Included are: RUNS: a es based on he number of runs up and down. RUNM: a es based on he number of runs above and below he median. AUTO: a chi-squared es based on he firs k residual auocorrelaions, where k is se by Pane Opions in he able displaying he residual auocorrelaions. MEAN: a -es comparing he mean residuals in he firs and second halves of he daa. VAR: an F-es comparing he variance of he residuals in he wo halves. If he enry for a paricular es is OK, hen he es is no saisically significan a he 5% significance level and he assumpion of random residuals is no rejeced. Oherwise, he number of sars (*) indicaes he significance level a which he assumpion of random residuals would be rejeced. Boh of he models fi o he raffic daa pass all of he ess excep ha comparing he wo variances. The laer es is highly significan. As will be seen when he residual plos are examined, his failure is due o he presence of hree large residuals during he second half of he esimaion period. Residual Plos The Residual Plos display he one-ahead forecas errors e in several ways. The defaul plo displays he residuals in sequenial order: 8 Residual Plo for adjused Traffic ARIMA(0,1,1)x(0,1,1) /68 1/71 1/74 1/77 1/80 1/ by SaPoin Technologies, Inc. Forecasing - 27

28 percenage STATGRAPHICS Cenurion Rev. 9/16/2013 Noice he hree large spikes occurring in March and April of 1974 and May of Traffic in hose monhs changed much more han normal. Using Pane Opions, a normal probabiliy plo of he residuals can be displayed insead: Residual Plo for adjused Traffic ARIMA(0,1,1)x(0,1,1) Residual If he residuals come from a normal disribuion, hey should fall close o he line. The plo above shows some curvaure away from he line in he ails, plus 3 ouliers. Pane Opions Three differen plos made be displayed: 1. Time Sequence Plo a plo of he residuals versus ime. 2. Probabiliy Plo (Horz.) a probabiliy plo wih he percenages displayed on he horizonal axis. 3. Probabiliy Plo (Ver.) a probabiliy plo wih he percenages displayed on he verical axis (as shown above) by SaPoin Technologies, Inc. Forecasing - 28

29 STATGRAPHICS Cenurion Rev. 9/16/2013 Residual Auocorrelaions I is also useful o examine he auocorrelaions of he residuals. The residual auocorrelaion a lag k measures he srengh of he correlaion beween residuals k ime periods apar. The residual lag k auocorrelaion is calculaed from r k nk 1 e ee e n e e 1 k 2 (38) If a model describes all of he dynamic srucure in a ime series, hen he residuals should be random and all of heir auocorrelaions should be insignifican. The Residual Auocorrelaions pane displays he residual auocorrelaions ogeher wih large lag sandard errors and probabiliy limis: Esimaed Auocorrelaions for residuals Daa variable: Traffic Model: ARIMA(0,1,1)x(0,1,1)12 Lower 95.0% Upper 95.0% Lag Auocorrelaion Snd. Error Prob. Limi Prob. Limi Any auocorrelaions ha fall ouside he probabiliy limis are saisically significan a he indicaed level. The SaAdvisor highlighs any such auocorrelaions in red by SaPoin Technologies, Inc. Forecasing - 29

30 Auocorrelaions STATGRAPHICS Cenurion Rev. 9/16/2013 Pane Opions Number of lags: maximum lag k a which o calculae he auocorrelaion. Confidence level: value of 100(1-)% used o calculae he probabiliy limis. Residual Auocorrelaion Funcion The Residual Auocorrelaion Funcion plo displays he residual auocorrelaions and probabiliy limis: Residual Auocorrelaions for adjused Traffic 1 ARIMA(0,1,1)x(0,1,1) lag Bars exending beyond he upper or lower limi correspond o saisically significan auocorrelaions. For he raffic daa, he only esimae ha is close o a probabiliy limi is he esimae a k = 2. In fac, a sligh reducion in he RMSE during he esimaion period can be achieved by increasing he order of he nonseasonal MA erm from 1 o 2. However, he performance of he model during he validaion period is worse hen wih he curren model, so he simpler model has been seleced by SaPoin Technologies, Inc. Forecasing - 30

31 STATGRAPHICS Cenurion Rev. 9/16/2013 Residual Parial Auocorrelaions If he model fis well, he residual parial auocorrelaions should also be insignifican. The Residual Parial Auocorrelaions pane displays he residual parial auocorrelaions ogeher wih large lag sandard errors and probabiliy limis: Esimaed Parial Auocorrelaions for residuals Daa variable: Traffic Model: ARIMA(0,1,1)x(0,1,1)12 Parial Lower 95.0% Upper 95.0% Lag Auocorrelaion Snd. Error Prob. Limi Prob. Limi The SaAdvisor highlighs any significan parial auocorrelaions in red. Pane Opions Number of lags: maximum lag k a which o calculae he parial auocorrelaion. Confidence level: value of 100(1-)% used o calculae he probabiliy limis by SaPoin Technologies, Inc. Forecasing - 31

32 Parial Auocorrelaions STATGRAPHICS Cenurion Rev. 9/16/2013 Residual Parial Auocorrelaion Funcion The Residual Parial Auocorrelaion Funcion plos he residual parial auocorrelaions and probabiliy limis: Residual Parial Auocorrelaions for adjused Traffic ARIMA(0,1,1)x(0,1,1) lag All of he coefficiens should be wihin he limis, as in he plo above. Residual Periodogram Table I is also useful o examine he residuals in he frequency domain, by considering how much variabiliy exiss a differen frequencies. As described in he Time Series Descripive Mehods documenaion, he periodogram plos he power a each of he Fourier frequencies. If he residuals are random, here should approximaely equal power a all frequencies, which is why a random ime series is ofen called whie noise. The Residual Periodogram pane displays he following able: Periodogram for residuals Daa variable: Traffic Model: ARIMA(0,1,1)x(0,1,1)12 Cumulaive Inegraed i Frequency Period Ordinae Sum Periodogram E E E by SaPoin Technologies, Inc. Forecasing - 32

33 STATGRAPHICS Cenurion Rev. 9/16/2013 The able includes: Frequency: he i-h Fourier frequency f i = i/n. Period: he period associaed wih he Fourier frequency, given by 1/ f i. This is he number of observaions in a complee cycle a ha frequency. Ordinae: he periodogram ordinae I(f i ). Cumulaive Sum: he sum of he periodogram ordinaes a all frequencies up o and including he i-h. Inegraed Periodogram: he cumulaive sum divided by he sum of he periodogram ordinaes a all of he Fourier frequencies. This column represens he proporion of he power in he ime series a or below he i-h frequency. Unlike he periodogram for he original raffic series, here is no large spike a a frequency of once every 12 monhs. Pane Opions Remove mean: check o subrac he mean from he ime series before calculaing he periodogram. Taper: percen of he daa a each end of he ime series o which a daa aper will be applied before he periodogram is calculaed. Following Bloomfield (2000), STATGRAPHICS uses a cosine aper ha downweighs observaions close o i = 1 and i = n. This is useful for correcing bias if he periodogram ordinaes are o be smoohed in order o creae an esimae of he underlying specral densiy funcion by SaPoin Technologies, Inc. Forecasing - 33

34 Ordinae STATGRAPHICS Cenurion Rev. 9/16/2013 Residual Periodogram Plo The Residual Periodogram plos he periodogram ordinaes of he residuals: 40 Residual Periodogram for adjused Traffic ARIMA(0,1,1)x(0,1,1) frequency If he residuals are random, here should be no noiceable spikes. Allowing for some naural skewness in he disribuion of he ordinaes, he above plo shows no large peaks. Pane Opions Remove mean: check o subrac he mean from he ime series before calculaing he periodogram. Poins: if checked, poin symbols will be displayed. Lines: if checked, he ordinaes will be conneced by a line. Taper: percen of he daa a each end of he ime series o which a daa aper will be applied before he periodogram is calculaed by SaPoin Technologies, Inc. Forecasing - 34

35 Ordinae STATGRAPHICS Cenurion Rev. 9/16/2013 Residual Inegraed Periodogram The Residual Inegraed Periodogram displays he cumulaive sums of he residual periodogram ordinaes, divided by he sum of he ordinaes over all of he Fourier frequencies: Periodogram for Residuals frequency A diagonal line is included on he plo, ogeher wih 95% and 99% Kolmogorov- Smirnov bounds. If he residuals are random, he inegraed periodogram should fall wihin hose bounds 95% and 99% of he ime. For he raffic daa, he residuals do appear o be whie noise. Tess For Randomness The Tess for Randomness pane displays he resuls of addiional ess run o deermine wheher or no he residuals are purely random: Tess for Randomness of residuals Daa variable: Traffic Model: ARIMA(0,1,1)x(0,1,1)12 (1) Runs above and below median Median = Number of runs above and below median = 79 Expeced number of runs = 78.0 Large sample es saisic z = P-value = (2) Runs up and down Number of runs up and down = 99 Expeced number of runs = Large sample es saisic z = P-value = (3) Box-Pierce Tes Tes based on firs 24 auocorrelaions Large sample es saisic = P-value = Three ess are performed: 2013 by SaPoin Technologies, Inc. Forecasing - 35

36 STATGRAPHICS Cenurion Rev. 9/16/ Runs above and below median: couns he number of imes he series goes above or below is median. This number is compared o he expeced value for a random ime series. Small P-values (less han 0.05 if operaing a he 5% significance level) indicae ha he residuals are no purely random. 2. Runs up and down: couns he number of imes he series goes up or down. This number is compared o he expeced value for a random ime series. Small P- values indicae ha he ime series is no purely random. 3. Box-Pierce Tes: consrucs a es saisic based on he firs k residual auocorrelaions by calculaing: Q n k 2 r i i1 (39) This saisic is compared o a chi-squared disribuion wih k degrees of freedom. As wih he oher wo ess, small P-values indicae ha he residuals are no purely random. Since he P-values for all hree ess are well above 0.05, here is no reason o doub ha he residuals are whie noise. Pane Opions Number of Lags: number of lags k o include in he Box-Pierce es. Residual Crosscorrelaions The Residual Crosscorrelaions pane displays crosscorrelaions beween he residuals and a second series, specified using Pane Opions. The crosscorrelaion beween one ime series Y a ime and a second ime series X a ime -k is denoed by c xy (k). A ypical use of crosscorrelaions is in idenifying leadings indicaors or an inpu-oupu relaionship. For example, Box, Jenkins and Reinsel (1994) presen daa from he inpu and oupu of a gas furnace a 9 second inervals, conained in he file furnace.sgd. The daa consis of: 1. Oupu series Y: % Co2 in oule gas 2013 by SaPoin Technologies, Inc. Forecasing - 36

37 STATGRAPHICS Cenurion Rev. 9/16/ Inpu series X: inpu gas rae in cubic fee per minue The oupu ime series is well described by an ARIMA(3,1,0) model. The able below shows he residual auocorrelaions for he oupu model residuals and he similarly differenced inpu ime series: Esimaed Crosscorrelaions for residuals wih DIFF(Inpu) Daa variable: Oupu Model: ARIMA(3,1,0) Lag Crosscorrelaion Some large negaive correlaions are noiceable, peaking a k = 3. This suggess ha changes in he inpu gas rae are correlaed wih he residuals from he fied oupu model and could herefore be used o improve he forecass. Pane Opions Second Time Series: he observaions for he X ime series. Noe he use of he DIFF operaor o calculae he firs differences of he Inpu column by SaPoin Technologies, Inc. Forecasing - 37

38 Crosscorrelaions STATGRAPHICS Cenurion Rev. 9/16/2013 Number of Lags: maximum lag k (boh posiive and negaive) a which o calculae he crosscorrelaions Residual Crosscorrelaion Plo The Residual Crosscorrelaion Plo displays he esimaed crosscorrelaions: Esimaed Crosscorrelaions for Residuals wih DIFF(Inpu) 1 ARIMA(3,1,0) lag Noe he large negaive correlaion peaking a lag 3. This implies ha he changes in he inpu are correlaed wih he residuals from he oupu model. They could hus be used o help forecas he oupu values by SaPoin Technologies, Inc. Forecasing - 38

39 STATGRAPHICS Cenurion Rev. 9/16/2013 Save Resuls The following resuls can be saved o he daashee: 1. Daa he original observaions, ogeher wih any inerpolaed replacemens for missing values. 2. Adjused daa ime series daa afer any adjusmens have been made. 3. Forecass forecased values wihin and beyond he sampling period. 4. Upper forecas limis upper probabiliy limis for he forecass. 5. Lower forecas limis lower probabiliy limis for he forecass. 6. Residuals one-sep ahead forecas errors. 7. Auocorrelaions residual auocorrelaions. 8. Parial auocorrelaions residual parial auocorrelaions. 9. Crosscorrelaions crosscorrelaions beween he residuals and a second ime series. 10. Residual periodogram ordinaes calculaed periodogram ordinaes for he residuals. 11. Fourier frequencies Fourier frequencies corresponding o he residual periodogram ordinaes by SaPoin Technologies, Inc. Forecasing - 39

40 STATGRAPHICS Cenurion Rev. 9/16/2013 Calculaions Error Saisics validaion period RMSE = roo mean squared error RMSE m i1 e m 2 ni (40) MAPE = mean absolue percenage error m eni / Y i i MAPE m % (41) MAE = mean absolue error m eni i MAE 1 (42) m ME = mean error m eni i ME 1 (43) m MPE = mean percenage error m eni i Y MPE ni % (44) m 2013 by SaPoin Technologies, Inc. Forecasing - 40

41 STATGRAPHICS Cenurion Rev. 9/16/2013 Variance funcion for forecass Random walk model V ˆ ( k) kˆ (45) a Mean model V ˆ 1 ( k) ˆ a 1 n (46) Moving average model V ˆ 1 ( k) ˆ a 1 c (47) Simple Exponenial Smoohing The variance funcion is deermined from he equivalen ARIMA(0,1,1) model. ˆ ˆ 2 V( k) a 1 ( k 1) (48) Brown s Linear Exponenial Smoohing The variance funcion is deermined from he equivalen ARIMA(0,2,2) model. 2 2 k( k 1)(2k 1) 1 Vˆ( k) ˆ 1 ( 1) ( 1) a k 0 01k k (49) 6 where 0 = (2-) and 1 = 2 Brown s Quadraic Exponenial Smoohing The variance funcion is deermined from he equivalen ARIMA(0,3,3) model. Hol s Linear Exponenial Smoohing The variance funcion is deermined from he equivalen ARIMA(0,2,2) model. 2 2 k( k 1)(2k 1) 1 Vˆ( k) ˆ 1 ( 1) ( 1) a k 0 01k k (50) by SaPoin Technologies, Inc. Forecasing - 41

42 STATGRAPHICS Cenurion Rev. 9/16/2013 where 0 = and 1 = Winer's Exponenial Smoohing The variance funcion is deermined from he equivalen ARIMA(0,1,s+1)(0,1,0)s model by SaPoin Technologies, Inc. Forecasing - 42

43 STATGRAPHICS Cenurion Rev. 9/16/2013 Trend models Forecas limis are calculaed from regression formulas for predicing a new observaion a ime = n + m + k, including use of Suden s disribuion wih he appropriae number of degrees of freedom. ARIMA Models Calculaed following he mehods of Box, Jenkins and Reinsel (1994), which involves finding he funcion o express he observaion a ime in erms of curren and previous shocks by SaPoin Technologies, Inc. Forecasing - 43