Seasonal and Related Models Vladas Pipiras STOR UNC CH February 2022 What is this all about Consider the following time series and its analysis ts0 scan airpass txt ts log ts0 par mfrow c 1 2 plot ts ts0 plot ts ts n length ts tt seq 1 n tt2 tt 2 cos12 cos 2 pi tt 12 sin12 sin 2 pi tt 12 cos6 cos 2 pi tt 6 sin6 sin 2 pi tt 6 cos4 cos 2 pi tt 4 sin4 sin 2 pi tt 4 rg lm ts tt tt2 cos12 sin12 cos6 sin6 cos4 sin4 summary rg Call lm formula ts tt tt2 cos12 sin12 cos6 sin6 cos4 Residuals sin4 1 Timets0020406080100120140100300500Timets0204060801001201405 05 56 06 5 1Q Min Max 0 128827 3Q 0 038911 Median 0 173540 0 035102 0 002494 Coefficients Intercept 4 736e 00 1 319e 02 tt 2 148e 05 tt2 1 415e 01 cos12 4 927e 02 sin12 2 275e 02 cos6 7 872e 02 sin6 2 731e 02 cos4 8 706e 03 sin4 Signif codes 0 0 001 0 01 0 05 0 1 1 Residual standard error 0 05584 on 135 degrees of freedom Multiple R squared 0 9849 Adjusted R squared F statistic 1100 on 8 and 135 DF Estimate Std Error t value Pr t 2e 16 1 417e 02 334 223 29 247 4 508e 04 2e 16 7 132 5 45e 11 3 011e 06 6 582e 03 21 499 2e 16 7 471 8 93e 12 6 594e 03 3 457 0 000732 6 582e 03 6 584e 03 2e 16 11 957 6 582e 03 4 149 5 87e 05 6 582e 03 0 984 p value 2 2e 16 1 323 0 188144 ts2 residuals rg par mfrow c 1 2 plot ts ts lines fitted values rg col red plot ts ts2 par mfrow c 1 2 acf ts2 lag max 30 pacf ts2 lag max 30 2 Timets0204060801001201405 05 56 06 5Timets2020406080100120140 0 15 0 050 05 ts20 diff ts lag 12 ts2a diff ts20 par mfrow c 1 2 plot ts ts20 plot ts ts2a par mfrow c 1 2 acf ts2a lag max 30 pacf ts2a lag max 30 3 051015202530 0 40 00 40 8LagACFSeries ts2051015202530 0 3 0 10 10 3LagPartial ACFSeries ts2Timets200204060801001200 00 10 20 3Timets2a020406080100120 0 15 0 050 050 15051015202530 0 40 00 40 8LagACFSeries ts2a051015202530 0 3 0 10 1LagPartial ACFSeries ts2a Note that the resulting series exhibit strong partial correlations at some lags which are multiples of the period s That is b s b 2s b s b 2s are large Data Theory goals of the topic Introduce seasonal models that can capture the above behavior Discuss a few related issues and also another model with seasonal periodic features Seasonal models Motivating seasonal models Consider time series data for r years and 12 months Year Month 1 2 r 1 Y1 Y13 2 Y2 Y14 12 Y12 Y24 Y12r Y12 r 1 1 Y12 r 1 2 There are j 1 12 column series Yj 12t t 0 r 1 Assume the same AR model for each j 1 12 Yj 12t 1Yj 12 t 1 P Yj 12 t P Uj 12t with Uj 12t WN 0 2 This is the same as B12 Yt Ut with z 1 1z P zP Terminology between year model B Example P 1 1 0 7 To incorporate dependence between months take another AR model for Ut with z 1 1z pzp and Zt WN 0 2 Z Combining the two models What one obtains is known as a seasonal ARIMA model De nition d D non negative integers Xt is a seasonal ARIMA p d 0 P D 0 s time series with period s if the di erenced time series Yt I B d I Bs DXt is a causal ARMA time series de ned by B Ut Zt B B12 Yt Zt B Bs Yt Zt with z 1 1z pzp z 1 1z P zP and Zt WN 0 2 Z MA parts can be added similarly B Note the multiplicative nature of SARIMA models Example SARIMA 1 0 0 1 0 0 12 with 1 0 7 1 0 6 library astsa set seed 123 ts e1 sarima sim ar 0 7 sar 0 6 S 12 n 400 ts e1 as numeric ts e1 4 par mfrow c 1 3 plot ts ts e1 acf ts e1 lag max 30 pacf ts e1 lag max 30 set seed 1 ts e2 arima sim list order c 13 0 0 ar c 7 rep 0 10 6 7 6 n 400 par mfrow c 1 1 plot ts ts e2 par mfrow c 1 2 acf ts e2 lag max 30 pacf ts e2 lag max 30 library forecast 5 Timets e10100200300400 4 20240510152025300 00 20 40 60 81 0LagACFSeries ts e1051015202530 0 20 00 20 40 6LagPartial ACFSeries ts e1Timets e20100200300400 4 202 ar acf ARMAacf ar c 7 rep 0 10 6 7 6 lag max 30 pacf F ar pacf ARMAacf ar c 7 rep 0 10 6 7 6 lag max 30 pacf T par mfrow c 1 2 plot ar acf type h plot ar pacf type h H Homework 5 will ask you to explore similarly another SARIMA model A number of R functions used earlier model selection tting forecasting etc allow for seasonal models library forecast set seed 123 ts e1 sarima sim ar 0 7 sar 0 6 S 12 n 400 auto mod auto arima ts e1 max p 2 max q 2 max d 0 start p 0 start q 0 max P 2 max Q 2 max D 0 start P 0 start Q 0 allowdrift FALSE ic aic auto mod Series ts e1 ARIMA 1 0 0 1 0 0 12 with zero mean Coefficients sar1 ar1 6 0510152025300 00 40 8LagACFSeries ts e2051015202530 0 20 20 40 6LagPartial ACFSeries ts e20510152025300 20 40 60 81 0Indexar acf051015202530 0 40 00 4Indexar pacf log likelihood 560 87 BIC 1139 71 0 6061 0 5298 s e 0 0396 0 0426 sigma 2 estimated as 0 9612 AIC 1127 73 AICc 1127 79 sarima model Arima ts e1 order c 1 0 0 seasonal c 1 0 0 method ML include mean TRUE sarima model Series ts e1 ARIMA 1 0 0 1 0 0 12 with non zero mean Coefficients ar1 0 6057 s e 0 0396 sigma 2 estimated as 0 9634 AIC 1129 62 h 12 ts e1 forecast forecast auto mod h plot ts e1 forecast sar1 0 5295 0 0426 mean 0 0851 0 2541 AICc 1129 72 BIC 1145 59 log likelihood 560 81 Back to the time series motivating this topic ts0 scan airpass txt ts1 log ts0 ts ts data ts1 frequency 12 auto ts auto arima ts max p 2 max q 2 max d 1 start p 0 start q 0 max P 2 max Q 2 max D 1 start P 0 start Q 0 allowdrift FALSE ic aic auto ts Series ts 7 Forecasts from …