Time Series Analysis using In a Nutshell

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1 1 Time Series Analysis using In a Nushell dr. JJM J.J.M. Rijpkema Eindhoven Universiy of Technology, dep. Mahemaics & Compuer Science P.O.Box 513, 5600 MB Eindhoven, NL 2012 [email protected] Sochasic Processes:, X,,, X 1 2 X Sochasic Process Individual sochass, ha migh be dependen Daa Collecion Model Fi Model Idenificaion Parameer Esimaion Model Validaion laborforce - daa 14 x Realizaion 7 Model Use year 2 Overview Inroducion & Preliminaries Exploraory Daa Analysis Time Sequence Plo (Parial) Auocorrelaion Funcion Exponenial Smoohing Mehods Simple Exponenial Smoohing Hol and Hol-Winers Models ARMA Models The Bare Essenials Model Fiing More Time Series Modeling 2IS55 TSA wih R 1

2 3 Inroducion & Preliminaries Sochasic Processes: X, X,,, X 1 2 Sochasic Process Individual sochass, ha migh be dependen Daa Collecion Model Fi Model Idenificaion Parameer Esimaion Model Validaion laborforce - daa 14 x Realizaion 7 Model Use year Relevan Models? 2IS55 Resricion Exploraory Daa Analysis? 4 References: hp://a-lile-book-of-r-for-ime-series.readhedocs.org/ Coghlan, Avril, A Lile Book of R for Time Series, Cowperwai, Paul S.P. e al., Inroducory Time Series wih R, chapers 1& 2 2IS55 TSA wih R 2

3 5 Sofware: hp:// projec org/ Meaphor: Download and Insall : CRAN mirror: hp://cran-mirror.cs.uu.nl/ 2IS55 TSA wih R 3

4 Sar -Console: Alernaive: R-sudio hp://rsudio.org/ 2IS55 TSA wih R 4

5 Insall package forecas (& series ): Neherlands (Urech) Load package forecas & series : > library(forecas) 2IS55 TSA wih R 5

6 11 Exploraory Daa Analysis Example: Age of Deah of 42 Successive Kings of England > load("d:/.../kings.rdaa") General Properies: > class(kings) [1] "numeric" > kings [1] [23] Conversion o ime series objec: > kings.s <- s(kings) > kings.s Time Series: Sar = 1 End = 42 Frequency = 1 [1] [23] Time Sequence Plo: > plo(kings.s) kings.s Time Exploraory Daa Analysis Main Properies: Trend? Cyclic Variaion? Irregular Variaion? Sudden Changes? Ouliers? Missing Values? 2IS55 TSA wih R 6

7 13 Auocorrelaion Funcion: Are successive observaions relaed? Scaerplo &S Sample Correlaion Coefficien i of f{( {(x,x -1 )} > lag.plo(kings.s,do.lines=f) r x x x1 x x x 1 2 Approximaion! r=0.4 kings.s k lag 1 Auocorrelaion Coefficien a ime-lag 1 14 Generalisaion: r k x x xk x x 2 x > lag.plo(kings.s,lags=12,do.lines=f) kings.s kings.s kings.s lag 1 lag 2 lag 3 kings.s lag 4 kings.s lag 5 kings.s lag kings.s kings.s kings.s lag 7 lag 8 lag 9 kings.s lag 10 kings.s lag kings.s lag IS55 TSA wih R 7

8 15 Auocorrelaion Funcion: > acf(kings.s)$acf [,1] [1,] > acf(kings.s) [2,] [3,] [4,] [5,] [6,] [7,] [8,] [9,] [10,] [11,] [12,] [13,] [14,] [15,] [16,] [17,] Parial Auocorrelaion: Direc Pahway x -k Indirec Pahway x x -k-1 x -k-2 x -2 x -1 Auocorrelaion Funcion: Measure for he overall correlaion beween {x -k } and {x } Boh hrough he direc and he indirec pahway! Parial Auocorrelaion Funcion: Measure for he direc correlaion beween {x -k } and {x } Indirec pahway correced for! 2IS55 TSA wih R 8

9 17 (Parial) Auocorrelaion Funcion: > pacf(kings.s) Parial > acf(kings.s) EDA-Summary: > sdisplay(kings.s) kings.s P IS55 TSA wih R 9

10 19 Example: Birhs per Monh in New York Ciy > load("d:/.../birhs.rdaa") > birhs.s <- s(birhs,sar=c(1946,1),frequency=12) > plo(birhs.s) birhs.s Time Exploraory Daa Analysis Main Properies: Trend? Seasonal Variaion? Cyclic Variaion? Irregular Variaion? Sudden Changes? Ouliers? Missing Values? 20 EDA-Summary: sdisplay(birhs.s) birhs.s P IS55 TSA wih R 10

11 21 Example: Monhly Sales for Ausralian Souvenir Shop > load("d:/.../souvenir.rdaa") > souvenir.s <- s(souvenir,sar=c(1987,1),frequency=12) > plo(souvenir.s) souvenir.s 4e+04 8e+04 Exploraory Daa Analysis Main Properies: Trend? Seasonal Variaion? 0e Time Cyclic Variaion? Irregular Variaion? Sudden Changes? Ouliers? Missing Values? 22 EDA-Summary: sdisplay(souvenir.s) souvenir.s 1e P e IS55 TSA wih R 11

12 23 Time Sequence Plo afer Log-Trafo: > souvenir.log <- log(souvenir.s) > plo(souvenir.log) souvenir.log Exploraory Daa Analysis Main Properies: Trend? Seasonal Variaion? Cyclic Variaion? Irregular Variaion? Sudden Changes? Ouliers? Missing Values? 24 EDA-Summary: > sdisplay(souvenir.log.s) souvenir.log P IS55 TSA wih R 12

13 25 Example: Annual Rainfall in London > load("d:/.../rain.rdaa") > rain.s <- s(rain,sar=1813) > plo(rain.s) rain.s Exploraory Daa Analysis Main Properies: Trend? Time Cyclic Variaion? Irregular Variaion? Sudden Changes? Ouliers? Missing Values? 26 EDA-Summary: > sdisplay(rain.s) rain.s P IS55 TSA wih R 13

14 27 Example: Annual Diameer of Women s Skirs > load("d:/.../skirs.rdaa") > skirs.s <- s(skirs,sar=1866) > plo(skirs.s) skirs.s Time Exploraory Daa Analysis Main Properies: Trend? Cyclic Variaion? Irregular Variaion? Sudden Changes? Ouliers? Missing Values? 28 EDA-Summary: > sdisplay(skirs.s) skirs.s P IS55 TSA wih R 14

15 29 Example: Volcanic Dus Veil Index > load("d:/.../volcano.rdaa") > volcano.s <- s(volcano, sar=1500) > plo(volcano.s) volcano.s Exploraory Daa Analysis Main Properies: Trend? Time Cyclic Variaion? Irregular Variaion? Sudden Changes? Ouliers? Missing Values? 30 EDA-Summary: > sdisplay(volcano.s) volcano.s P IS55 TSA wih R 15

16 31 Types of Variaion: Trend Long erm changes in mean Can be esimaed & modeled Can be correced for Seasonal Variaion Periodic variaions over ime Can be esimaed & modeled Can be correced for Saionary Time Series No sysemaic change in mean and variance No sricly periodic variaions Oher Cyclic Variaion Business cycle (~7 year) Oher Irregular Flucuaions Random or wih srucure? 32 Simple Exponenial Smoohing Series wih NO Trend and No Seasonaliy Example: Annual Rainfall in London rain.s Saionary Series No rend (??) No Seasonaliy Basic Idea: one-sep ahead predicion xˆ 1 c x c x c x T T T T Geomeric Weighs: c 1 i i Remark: mainly used for shor-erm forecass! 2IS55 TSA wih R 16

17 33 Inerpreaion: xˆ T 1 xt 1 xt 1 1 xt2 1 1 ˆ 1 xˆt x T xt 1 Weighed average of pas and presen ˆ xˆ 1 x x 1 T1 T T 1 α =0 pas α =1 presen Overview: Simple Exponenial Smoohing x 1 Lˆ 1 xˆ 1 Lˆ ˆ 1 Lˆ x L x Lˆ Remark: Simple Exponenial Smoohing allows for updaes of level esimaes Example: fixed parameer = 0.2 Adequae iniial value for he level esimae needed ˆL x 1 1 > rain.ses1 <- HolWiners(rain.s,alpha=0.2,bea=FALSE,gamma=FALSE) Available Informaion: Smoohing Parameers: > names(rain.ses1) [1] "fied" "x" "alpha" "bea "gamma [6] "coefficiens" "seasonal" "SSE" "call" > rain.ses1 Hol-Winers exponenial smoohing wihou rend and wihou seasonal componen. Smoohing parameers: alpha: 0.2 bea : FALSE gamma: FALSE 34 Final Componen Esimae: Coefficiens: [,1] a IS55 TSA wih R 17

18 35 Running Componen Esimaes: > rain.ses1$fied xha level xha level xˆ 1 L ˆ 1 1 > plo(rain.ses1$fied) α = In-Sample Predicion vs. Realizaion: > plo(rain.ses1) Observed / Fied Running SSE T s0 1 > rain.ses1$sse [1] x xˆ Remark: In-sample comparison uses he same daa for fiing and validaion. Independen hold-ou sample for validaion would be preferred! 2IS55 TSA wih R 18

19 37 Goodness-of-Fi Measures: Bias: Mean Error n 1 ME x ˆ x n 1 The closer o 0, he Mean Percenage Error 1 n x ˆ x MPE n 1 x Variabiliy: Mean Absolue Error n 1 MAE x ˆ x n 1 Mean Absolue Percenage Error MAPE n 1 x xˆ x 100 n > accuracy(rain.ses1) Roo Mean Squared Error n 1 RMSE x ˆ x n 1 In-Sample Accuracy: 2 Mean Absolue Scaled Error 1 ˆ n x x MASE n q 1 ME RMSE MAE MPE MAPE MASE Hol-Winers Forecass: > rain.ses1.fore <- forecas.holwiners(rain.ses1,h=10) > rain.ses1.fore Poin Forecas Lo 80 Hi 80 Lo 95 Hi > plo.forecas(rain.ses1.fore) Forecass from HolWiners IS55 TSA wih R 19

20 39 Diagnosics: In-Sample Forecas Errors Consan over ime? > sdisplay(rain.ses1.fore$residuals) X Non-zero Auocorrelaions? k 2 2 i 2 BL NN 2 N # p i 1 N i r P > Box.es(rain.ses1.fore$residuals,lag=20,ype="Ljung-Box") Box-Ljung es daa: rain.ses1.fore$residuals X-squared = , df = 20, p-value = Example: opimal esimaed Principle: Opimal value for he parameer o be deermined from running 1-sep ahead predicion: > rain.ses2 <- HolWiners(laborforce.s,bea=FALSE,gamma=FALSE) Smoohing Parameers: T 0 > rain.ses2 1 2 min x x ˆ 1 (or relaed, eg. AIC or BIC) Smoohing parameers: alpha: bea : FALSE gamma: FALSE 1 1 ˆ 1 xˆt xt xt 1 More emphasis on he pas! 40 Final Componen Esimae: Coefficiens: [,1] [,1] a IS55 TSA wih R 20

21 41 Running Componen Esimaes: > rain.ses2$fied xha level xha xˆ 1 L ˆ 1 1 > plo(rain.ses2$fied) level opimal α 42 In-Sample Predicion vs. Realizaion: > plo(rain.ses2) Observed / Fied Running SSE T s0 1 > rain.ses2$sse [1] x xˆ In-Sample Accuracy: > accuracy(rain.ses2) ME RMSE MAE MPE MAPE MASE IS55 TSA wih R 21

22 43 Hol-Winers Forecass: > rain.ses2.fore <- forecas.holwiners(rain.ses2,h=10) > rain.ses2.fore Poin Forecas Lo 80 Hi 80 Lo 95 Hi > plo.forecas(rain.ses2.fore) Forecass from HolWiners Diagnosics: In-Sample Forecas Errors Consan over ime? > sdisplay(rain.ses2.fore$residuals) X Non-zero Auocorrelaions? k 2 2 i 2 BL NN 2 N # p i 1 N i r P > Box.es(rain.ses2.fore$residuals,lag=20,ype="Ljung-Box") Box-Ljung es daa: rain.ses2.fore$residuals X-squared = , df = 20, p-value = IS55 TSA wih R 22

23 45 Hol s Exponenial Smoohing Series wih Trend bu No Seasonaliy Example: Annual Diameer of Women s Skirs skirs.s No Seasonaliy (annual daa!) Time Principle: x T 1 Lˆ 1 Tˆ 1 Hol s Exponenial Smoohing xˆ 1 Lˆ Tˆ T ˆ T L ˆ ˆ x L T T x Lˆ Tˆ 1 T ˆ L ˆ L ˆ T ˆ Remarks: Hol s ES allows for updaes of level and rend esimaions Two parameer version of Exponenial Smoohing Special: Brown s Exponenial Smoohing Boh parameers are equal: = β Similar o ARIMA(0,2,1) model (o be discussed) Pracical: Adequae iniial values for he level and rend esimaes needed ˆ ˆL x T ˆ x x Opimal value for he parameers and o be deermined from running 1-sep ahead predicion:, T min x x ˆ 1 (or relaed, eg. AIC or BIC) 2IS55 TSA wih R 23

24 47 Example: opimal and esimaed > skirs.hes <- HolWiners(x=skirs.s,gamma=FALSE) Smoohing Parameers: > skirs.hes Smoohing parameers: alpha: bea : 1 gamma: FALSE T Lˆ ˆ ˆ x L T T ˆ L ˆ L ˆ T ˆ Main emphasis on recen values! Final Componen Esimae: Coefficiens: [,1] a b Running Componen Esimaes: > skirs.hes$fied xha level rend xha xˆ 1 Lˆ Tˆ T > plo(skirs.hes$fied) opimal α and leve el rend IS55 TSA wih R 24

25 49 In-Sample Predicion vs. Realizaion: > plo(skirs.hes) Observed / Fied Running SSE T s0 1 > skirs.hes$sse [1] x xˆ In-Sample Accuracy: > accuracy(skirs.hes) ME RMSE MAE MPE MAPE MASE Hol-Winers Forecass: > skirs.hes.fore <- forecas.holwiners(skirs.hes,h=19) > skirs.hes.fore Poin Forecas Lo 80 Hi 80 Lo 95 Hi > plo.forecas(skirs.hes.fore) ?? IS55 TSA wih R 25

26 51 Diagnosics: In-Sample Forecas Errors Consan over ime? > sdisplay(skirs.hes.fore$residuals) X Non-zero Auocorrelaions? k 2 2 i 2 BL NN 2 N # p i 1 N i r P > Box.es(skirs.hes.fore$residuals,lag=20,ype="Ljung-Box") Box-Ljung es daa: skirs.fore$residuals X-squared = , df = 20, p-value = Hol-Winers Exponenial Smoohing (Addiive Seasonaliy) Example: Birhs per Monhs in New York Ciy 52 birhs.s Trend? Addiive Seasonaliy? Time Pi Principle: i x TS 1 Lˆ 1 Tˆ 1 Iˆ 1 Addiive Hol-Winer s ES xˆ 1 Lˆ Tˆ Iˆ TS s TS TS 1 Lˆ x Iˆ Lˆ Tˆ s T ˆ L ˆ L ˆ T ˆ Iˆ x Lˆ Iˆ s TS x Lˆ Tˆ Iˆ 2IS55 TSA wih R 26

27 53 Remarks: Hol-Winer s ES allows for updaes of level, rend and seasonaliy esimaions Three parameer version of Exponenial Smoohing Similar o SARIMA (0,1,1)x(0,1,1) s model (o be discussed) Special: Simple Seasonal ES No rend (T =0), only seasonaliy! Simple Addiive Seasonal ES x S 1 Lˆ 1 ˆ 1 xˆ 1 Lˆ Iˆ S 1 1 s Lˆ x Iˆ 1 Lˆ S 1 s S x Lˆ Iˆ I TS 1 Iˆ x Lˆ Iˆ s Pracical: Adequae iniial values for he level, rend and seasonaliy esimaes needed Opimal values for he parameers, and o be deermined from:,, T min x x ˆ 1 (or relaed, eg. AIC or BIC) Example: opimal and esimaed birhs.hw <- HolWiners(birhs.s,seasonal="addiive") 54 Smoohing Parameers: birhs.hw Smoohing parameers: alpha: bea : gamma: T TS 1 Lˆ ˆ ˆ x L T T ˆ L ˆ L ˆ T ˆ Iˆ x Lˆ Iˆ s Emphasis for rend on he pas! Final Componen Esimaes: Coefficiens: [,1] a b s s s s s s s s s s s s IS55 TSA wih R 27

28 l Running Componen Esimaes: > birhs.hw$fied xˆ 1 Lˆ Tˆ Iˆ TS s 55 xha level rend season Jan Feb Mar Apr > plo(birhs.hw$fied) May Jun Jul Aug Sep Oc Nov Dec Jan Feb opimal α, and xha level rend season In-Sample Predicion vs. Realizaion: > plo(birhs.hw) 30 Observed / Fied Running SSE T s0 1 > birhs.hw$sse [1] x xˆ In-Sample Accuracy: > accuracy(birhs.hw) ME RMSE MAE MPE MAPE MASE IS55 TSA wih R 28

29 57 Hol-Winers Forecass: > birhs.hw.fore <- forecas.holwiners(birhs.hw,h=48) > birhs.hw.fore Poin Forecas Lo 80 Hi 80 Lo 95 Hi 95 Jan Feb Mar Apr May > plo.forecas(birhs.hw.fore) Jun Jul Aug Sep Oc Nov Dec Jan Feb Diagnosics: In-Sample Forecas Errors Consan over ime? > sdisplay(birhs.hw.fore$residuals) X Non-zero Auocorrelaions? k 2 2 i 2 BL NN 2 N # p i 1 N i r P > Box.es(birhs.hw.fore$residuals,lag=20,ype="Ljung-Box") daa: birhs.hw.fore$residuals X-squared = , df = 35, p-value = IS55 TSA wih R 29

30 59 ARMA Models: he Bare Essenials Sochasic Processes:, X,,, X 1 2 X Sochasic Process Individual sochass, ha migh be dependen Model Fi Model Idenificaion Parameer esimaion Model Validaion laborforce - daa x Realizaion: year Purely Random Processes Moving Average Processes: MA(q) Auoregressive Processes: AR(p) ARMA(p,q) & ARIMA(p,d,q) Processes 60 Purely Random Process: (whie noise) Z Z Z,,,, 1 2 Z ~ 0, 2 N Z and muually independen Basic Building Block Simulaion: > s.sim <- arima.sim(lis(order=c(0,0,0)),n=100) > plo(s.sim) im s.si Exploraory Daa Analysis Main Properies: Trend? Seasonal Variaion? Cyclic Variaion? Irregular Variaion? Time 2IS55 TSA wih R 30

31 61 EDA-Summary: > sdisplay(s.sim) s.sim P Moving Average Process: MA(q) X Z Z Z 1 1 q q Inerpreaion: Presen Value = Moving Average of Pas Disurbances (=shock) Process is mainly influenced by random evens from he pas: Economics?? MA-models are ofen used o model ime series which show shor erm dependencies beween successive observaions Operaor Noaion: B X X 1 backshif operaor B X B Z q wih: B 1B B q 1 q q 2IS55 TSA wih R 31

32 63 Example: MA(2) Process X Z 0.3Z 0.4Z 1 2 Operaor Noaion: X B Z q wih: B 10.3 B q 0.4 B 2 Simulaion: 2 3 > s.sim <- arima.sim(lis(order= + c(0,0,2), ma=c(-0.3,-0.4)),n=100) > plo(s.sim) s.sim EDA-Summary: > sdisplay(s.sim) s.sim MA-specific P IS55 TSA wih R 32

33 65 Auoregressive Process: AR(p) X X X Z 1 1 p p Inerpreaion: Presen Value = Moving Average of Pas Values + Disurbance Process is mainly influenced by pas values of he process! AR-models are ofen used o model ime series which show longer erm dependencies beween successive observaions Operaor Noaion: B X X 1 backshif operaor B B X Z p wih: B 1B B p 1 p p 66 Example: AR(2) Process X 0.3X 0.4X Z 1 2 Operaor Noaion: B X Z p wih: B 10.3 B p 0.4 B 2 Simulaion: > s.sim <- arima.sim(lis(order= + c(2,0,0), ar=c(0.3,0.4)),n=100) > plo(s.sim) s.sim Time 2IS55 TSA wih R 33

34 67 EDA-Summary: > sdisplay(s.sim) s.sim AR-specific P Auoregressive Moving Average Process: ARMA(p,q) X X X Z Z Z p p q q Inerpreaion: Process is influenced boh by levels and by disurbances from he pas! Operaor Noaion: BX B Z p q backshif operaor B B 1B B p 1 p p B 1B B q 1 q q 2IS55 TSA wih R 34

35 69 Example: ARMA(1,1) Process X 0.6X Z 0.4Z 1 1 Operaor Noaion: BX B Z p q 1 0.6B 1 B B q 10.4 B Simulaion: > s.sim <- arima.sim(lis(order= + c(1,0,1),ar=c(0.6),ma=c(0.4)),n=100) > plo(s.sim) s.sim EDA-Summary: > sdisplay(s.sim) s.sim MA-specific? AR-specific? P IS55 TSA wih R 35

36 ARMA Model Fiing saionary series wihou seasonaliy 71 Sochasic Processes: X, X,,, X 1 2 Sochasic Process Individual sochass, ha migh be dependen Model Fi Model Idenificaion Parameer esimaion Model Validaion laborforce - daa 14 x Realizaion: year Theoreical Finger Prin Auocorrelaion Parial Auocorrelaion EDA - Finger Prin Auocorrelaion Parial Auocorrelaion 72 Example: Volcanic Dus Veil Index > plo(volcano.s); sdisplay(volcano.s) volcano.s Saionary? MA(3)? P AR(2)? ARMA(1,1)? IS55 TSA wih R 36

37 73 Conjecure 1: MA(3)-model? > volcano.arma.0.3 <- Arima(volcano.s,order=c(0,0,3)) > volcano.arma.0.3 Series: volcano.s ARIMA(0,0,3) wih non-zero mean Coefficiens: ma1 ma2 ma3 inercep X X s.e X Z 0.74Z 10.45Z2 0.19Z3 sigma^2 esimaed as 4852: log likelihood= AIC= AICc= BIC= Model Forecass: > volcano.arma.0.3.fore <- + forecas(volcano.arma.0.3,h=19) > plo.forecas(volcano.arma.0.3.fore) In-Sample Accuracy: > accuracy(volcano.arma.0.3) ME RMSE MAE MPE MAPE MASE Inf Inf In-Sample Diagnosics: > sdiag(volcano.arma.0.3) arma Consan over ime? 0 8 Sandardized Residuals Time Non-zero Auocorrelaions? of Residuals X k 2 2 i 2 BL NN 2 N # p i 1 N i r p value 0.0 p values for Ljung-Box saisic lag 2IS55 TSA wih R 37

38 75 Conjecure 2: AR(2)-model? > volcano.arma.2.0 <- Arima(volcano.s,order=c(2,0,0)) > volcano.arma.2.0 Series: volcano.s ARIMA(2,0,0) wih non-zero mean Coefficiens: ar1 ar2 inercep X X s.e X 0.75X X2 Z sigma^2 esimaed as 4870: log likelihood= AIC= AICc= BIC= Model Forecass: > volcano.arma.2.0.fore <- + forecas(volcano.arma.2.0,h=19) > plo.forecas(volcano.arma.2.0.fore) In-Sample Accuracy: > accuracy(volcano.arma.2.0) ME RMSE MAE MPE MAPE MASE Inf Inf In-Sample Diagnosics: > sdiag(volcano.arma.2.0) arma Consan over ime? 0 8 Sandardized Residuals Time Non-zero Auocorrelaions? of Residuals X k 2 2 i 2 BL NN 2 N # p i 1 N i r p value 0.0 p values for Ljung-Box saisic lag 2IS55 TSA wih R 38

39 77 Conjecure 3: ARMA(1,1)-model? > volcano.arma.1.1 <- Arima(volcano.s,order=c(1,0,1)) > volcano.arma.1.1 Series: volcano.s ARIMA(1,0,1) wih non-zero mean Coefficiens: ar1 ma1 inercep X X s.e X 0.3X 1 0.4X 2 0.4X3 Z sigma^2 esimaed as 4883: log likelihood= AIC= AICc= BIC= Model Forecass: > volcano.arma.1.1.fore <- + forecas(volcano.arma.1.1,h=19) > plo.forecas(volcano.arma.1.1.fore) In-Sample Accuracy: > accuracy(volcano.arma.1.1) ME RMSE MAE MPE MAPE MASE Inf Inf In-Sample Diagnosics: > sdiag(volcano.arma.1.1) arma 1) Consan over ime? 0 8 Sandardized Residuals Time Non-zero Auocorrelaions? of Residuals X k 2 2 i 2 BL NN 2 N # p i 1 N i r p value 0.0 p values for Ljung-Box saisic lag 2IS55 TSA wih R 39

40 year year year year year 79 ARIMA Model Fiing non-saionary series skirs.s Saionary? Level / Variaion? kings.s Time Time Remedy: Remove dominan non-saionariy hrough finie differencing: X X X X 1 1 B If successful hen fi ARMA(p,q) on X ARIMA(p,1,q) on X 80 ARMA(p,q) model on W : laborforce - AR(aic) noise BW B Z p q 14 x 104 laborforce - derend piecewise W d X laborforce - daa ARIMA(p,d,q)-model on X : 14 x 104 laborforce - AR(aic) noise d B 1B X B Z p q laborforce - daa 2IS55 TSA wih R 40

41 81 Example: Annual Diameer of Women s Skirs rs.s ski > plo(skirs.s) > skirs.d1 <- diff(skirs.s,differences=1) > plo(skirs.d1) skirs.dif skirs.dif > skirs.d2 <- diff(skirs.s,differences=2) > plo(skirs.d2) EDA-Summary: > sdisplay(skirs.s) P P > sdisplay(skirs.d1) > sdisplay(skirs.d2) P IS55 TSA wih R 41

42 83 2 Model: W X X X X 2X X > sdisplay(skirs.d2) MA(?) P AR(?) ARMA(p,q) on W ARIMA(p,2,q) on X 84 Conjecure: ARIMA(1,2,1)-model? > skirs.arima <- Arima(skirs.s,order=c(1,2,1)) > skirs.arima Series: skirs.s ARIMA(1,2,1) Coefficiens: ar1 ma s.e W X W 0.31W Z 0.01Z 1 1 sigma^2 esimaed as 388.7: log likelihood= AIC= AICc= BIC= Model Forecass: > skirs.arima fore <- + forecas(skirs.arima.1.2.1,h=19) > plo(skirs.arima fore) IS55 TSA wih R 42

43 year year year year 85 SARIMA Model Fiing birhs.s Time Saionary? Level / Variaion? Seasonaliy? Addiive / Muliplicaive? souvenir.s 0e+00 4e+04 8e Time Remedy: Remove dominan non-saionariy hrough finie differencing: X X X X 1 1 B Remove dominan seasonaliy hrough seasonal differencing: X X X 1B X s S s 86 d D d s Combined adjusmen: 1B 1B ARMA(p,q) for W : W X X s D laborforce - AR(aic) noise BW B Z p q laborforce - aperiodic Generalisaion o accoun for changes in seasonaliy: laborforce - AR(aic) noise s B B W s B B p P Z q Q laborforce - aperiodic 2IS55 TSA wih R 43

44 year year year year year 87 Equivalen: laborforce - AR(aic) noise s B B W s B B p P Z q Q 14 x 104 laborforce - derend piecewise W X d D s laborforce - daa SARIMA (p,d,q)x(p,d,q) s -model: laborforce - AR(aic) noise B B s B B s d D X p P s Z q Q laborforce - daa 14 x Example: Birhs per Monh in New York Ciy > plo(birhs.s); sdisplay(birhs.s) birhs.s P > birhs.d1.sd1<-diff(diff(birhs.s,differences=1),lag=12,differences=1) > plo(birhs.d1.sd1); sdisplay(birhs.d1.sd1) birhs.d1.sd P IS55 TSA wih R 44

45 89 Model: 12 1B1B 12 W X X sdisplay(birhs.d1.sd1,lag.max=40) MA(?), MA s (?) AR(?), AR s (?) P ARMA(p,q) on W SARIMA(p,1,q)x(P,1,Q) 12 on X 90 Conjecure: SARIMA(3,1,3)x(1,1,0) 12 -model? > birhs.sarima <-arima(birhs.s,order=c(3,1,3), + seasonal=lis(order=c(1,1,0),period=12)) > birhs.sarima Series: birhs.s ARIMA(3,1,3)(1,1,0)[12] Coefficiens: ar1 ar2 ar3 ma1 ma2 ma3 sar s.e sigma^2 esimaed as : log likelihood= AIC= AICc=373.5 BIC= Model Forecass: > birhs.sarima fore<- + forecas(birhs.sarima ,h=12) > plo(birhs.sarima fore) IS55 TSA wih R 45

46 91 More Time Series Modeling. Paul S. Cowperwaih e al., Inroducory Time Series wih R, ISBN Journal of Saisical Forecasing, July 2008, Volume 27, Issue 3 hp:// 2IS55 TSA wih R 46

47 93 Rob. J. Hyndman e al., l Rober M. Shumway e al., l Forecasing wih Exponenial Smoohing, ISBN Time Series Analysis and is Applicaions wih R examples, ISBN Jonahan D. Cryer e al., Time Series Analysis wih Applicaions in R, ISBN Suggesions for Improvemens of TSA R are welcomed: [email protected] 2IS55 TSA wih R 47