12/7/2011. Procedures to be Covered. Time Series Analysis Using Statgraphics Centurion. Time Series Analysis. Example #1 U.S.

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1 Tme Seres Analyss Usng Sagraphcs Cenuron Nel W. Polhemus, CTO, SaPon Technologes, Inc. Procedures o be Covered Descrpve Mehods (me sequence plos, auocorrelaon funcons, perodograms) Smoohng Seasonal Decomposon Forecasng User-specfed models Auomac model selecon Copyrgh 20 by SaPon Technologes, Inc. Web se: 2 Tme Seres Analyss Analyss of sequenally ordered daa. Sandard noaon: Y, =,2,,n Man complcaon for analyss s ha he observaons are usually no ndependen. Insead, hey are auocorrelaed. Example # U.S. Unemploymen Rae Defned as he percenage of he cvlan labor force ha are acvely lookng for a job bu can fnd one. (Noe: people who have gven up lookng for jobs or are no avalable for work are no consdered par of he labor force.) Source: November 20: Unadjused rae: 8.2% Seasonally adjused rae: 8.6% 3 4 Tme Sequence Plo Tme Seres Componens Classc me seres model consss of 4 componens:. Trend (T) long-erm changes n he underlyng process mean. 2. Cycle (C) cyclcal varaons around he mean wh no fxed perod. 3. Seasonaly (S) cyclcal varaons around he mean wh a fxed perod. 4. Randomness (R) random flucuaons. 5 6

2 Basc Models Mulplcave model Smoohng Sagraphcs has 2 ypes of smoohers: Y = T * C * S * R Addve model Lnear smoohers weghed movng averages ha esmae he underlyng rend-cycle by averagng ou he oher componens. Y = T + C + S + R Nonlnear smoohers varaons on runnng medans ha reduce he mpac of solaed oulers. Noe: akng logarhms urns a mulplcave model no an addve model. 7 8 Lnear Smoohers These smoohers are bascally weghed movng averages of he form S m jm W Y j j Movng Average wh Span of 2 Sagraphcs offers smple movng averages, exponenally weghed movng averages, and movng averages developed by Spencer and Henderson. Excep for he EWMA, he movng averages are cenered around me. 9 0 Adjusmens Varous adjusmens may be appled o he me seres daa before s smoohed. Nonlnear Smoohers These smoohers were developed by John Tukey and are based on runnng medans of eher 3 or 5 observaons. A ypcal smooher s he 3RSSH smooher: 3 smooh he daa wh a runnng medan of 3. R repea he runnng medan agan and agan unl nohng changes. S spl he mesas and dales and repea he 3R operaon. S spl he mesas and dales and repea he 3R operaon. H apply a smple 3-erm MA as proposed by Hannng. 2 2

3 Smoohng wh 3RSSH Smoohng wh 3RSSH and MA(2) 3 4 Seasonal Adjusmen Sep # Smooh daa o esmae rend-cycle Removes he seasonal componen from a me seres, leavng he oher componens behnd. For a mulplcave model: Y adj Y S TC S R S T C R 5 6 Sep #2 Dvde daa by smooh and hen average each season o esmae seasonaly Sep #3 Dvde daa by seasonal ndex o remove seasonaly 7 8 3

4 Seasonal Subseres Plo Shows Changes on a Season by Season Bass Forecasng Used o predc fuure values of he me seres. Types of models: Auoprojecve models forecass he fuure based solely on he pas behavor of he me seres self. Inpu/oupu models ncludes nformaon from oher predcor varables, usually leadng ndcaors Mehods Sagraphcs Procedures User-Specfed Model fs a parcular model seleced by he user. Auomac Model Selecon fs many models and auomacally pcks he bes accordng o some opmzaon creron Auomac Forecasng Daa Inpu Auomac Forecasng Analyss Opons

5 Akake Informaon Creron (AIC) Resuls Unadjused Rae Mnmzes he roo mean squared error RMSE plus a penaly based on he number of esmaed model parameers c. 2c AIC 2lnRMSE n Can be hough of as a radeoff beween model precson and model complexy (whch can lead o bas) Resuls Adjused Rae Example #2 Closng prces of Apple Inc. common sock Daa conss of closng prces on each radng day beween Jan. 4, 200 and Dec. 2, 20. Source: Low durng perod: $92.05 on Feb. 4, 200 Hgh durng perod: $ on Oc. 8, 20 Laes close: $ on Dec. 2, Tme Sequence Plo AAPL close Auomac Forecasng AAPL close

6 Random Walk Model The random walk model mples ha he dfferences beween consecuve closng prces are random. Sascal Tme Seres Models A me seres may be hough of conssng of a deermnsc componen and a sochasc componen. Y Y The pas hsory has no effec on he behavor of he me seres n he fuure. Thus he bes forecas for omorrow s closng prce s oday s closng prce. Y 3 32 Deermnsc Componen The deermnsc componen may be: a consan mean: m a rend over me such as: a + b* a cyclcal effec such as: a*sn(w*+f) a funcon of one or more npu varables: m*x -k +b a combnaon of he above Sochasc Componen The sochasc componen may ofen be represened as a lnear combnaon of random shocks a -k a he presen and n he pas: k 0 k a k ARIMA Models Common model forms nclude: Auoregressve AR(p) models: Y m fy m f2 Y 2 m a Movng average MA(q) models: Y m a a a 2 2 Mxed ARMA(p,q) models: Y m fy m a a Models nvolvng dfferencng : ARIMA(p,d,q) Y Y fy Y 2 a a Auocorrelaon Funcon Plos he lnear correlaon beween observaons separaed by varous lags: r corr( Y, Y k k ) nk Y Y Y k Y n 2 Y Y Auoregressve models show exponenal decay, whle movng average models show spkes a low lags. Falure o decay s a sgn of nonsaonary

7 ACF for Adjused Unemploymen Rae ACF for Frs Dfferences Paral Auocorrelaon Funcon Used o deermne he order of auoregressve model needed o represen he me seres adequaely. Auoregressve models show spkes a low lags, whle movng average models show exponenal decay. PACF for Frs Dfferences Unemploymen Rae Model Auomac forecasng seleced an ARIMA(,,) model: Y Y 2 a 0. a Y Y ˆ 0.6 a Perodogram Performs an analyss of varance across he Fourer frequences. q Y a0 a sn2f b cos2 f n 2q I N 2 2 f a b 2 f n

8 Perodogram for Frs Dfferences Crosscorrelaon Funcon Plos he lnear correlaon beween wo me seres separaed by varous lags: r xy ( k) corr( Y, X k ) Plo of Resduals from Random Walk Model for AAPL Crosscorrelaons wh for Frs Dfferences of BUD X-Y Plo wh LOWESS Smooh Example #3 Inpu/Oupu Models Oupu me seres: Anheser Busch sock prce (BUD) Inpu me seres X: Apple Inc sock prce (AAPL) Showed ha here appears o be a small negave correlaon beween changes n AAPL oday and changes n BUD omorrow. We wll buld a model for BUD ha ncludes AAPL as a lagged regressor n an ARIMA model

9 Sep : Forecas Inpu Tme Seres and Add Forecass o Daa Shee Sep 2: Add Regressor o Forecasng Model Fed Model Forecasng Resuls wh Inpu Varable BUD BUD AAPL AAPL The effec on he forecass s very slgh snce AAPL dd no change much on he las day of he observed daa. More Informaon Go o Or send e-mal o nfo@sagraphcs.com 53 9