Forecasting: principles and practice. Rob J Hyndman 3.2 Forecasting with multiple seasonality
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1 Forecasting: principles and practice Rob J Hyndman 3.2 Forecasting with multiple seasonality 1
2 Outline 1 Time series with complex seasonality 2 STL with multiple seasonal periods 3 Dynamic harmonic regression 4 TBATS model 5 Lab session 21 2
3 Examples autoplot(gasoline) + xlab("year") + ylab("thousands of barrels per day") + ggtitle("weekly US finished motor gasoline products") Weekly US finished motor gasoline products Thousands of barrels per day
4 Examples 5 minute call volume at North American bank 400 Call volume Weeks 400 Call volume Weeks 4
5 Examples Turkish daily electricity demand 25 Electricity Demand (GW) Year 5
6 Outline 1 Time series with complex seasonality 2 STL with multiple seasonal periods 3 Dynamic harmonic regression 4 TBATS model 5 Lab session 21 6
7 STL with multiple seasonal periods calls %>% mstl() %>% autoplot() + xlab("week") Week Data Trend Seasonal169 Seasonal845 Remainder 7
8 . STL with multiple seasonal periods calls %>% stlf() %>% autoplot() + xlab("week") Forecasts from STL + ETS(M,N,N) level Week 8
9 . STL with multiple seasonal periods calls %>% stlf() %>% autoplot(include=5*169) + xlab("week") Forecasts from STL + ETS(M,N,N) level Week 9
10 Outline 1 Time series with complex seasonality 2 STL with multiple seasonal periods 3 Dynamic harmonic regression 4 TBATS model 5 Lab session 21 10
11 Dynamic harmonic regression fit <- auto.arima(calls, seasonal=false, lambda=0, xreg=fourier(calls, K=c(10,10))) fit %>% forecast(xreg=fourier(calls, K=c(10,10), h=2*169)) %>% autoplot(include=5*169) + ylab("call volume") + xlab("weeks") 400 Forecasts from Regression with ARIMA(3,1,4) errors Call volume level Weeks 11
12 Outline 1 Time series with complex seasonality 2 STL with multiple seasonal periods 3 Dynamic harmonic regression 4 TBATS model 5 Lab session 21 12
13 TBATS model TBATS Trigonometric terms for seasonality Box-Cox transformations for heterogeneity ARMA errors for short-term dynamics Trend (possibly damped) Seasonal (including multiple and non-integer periods) 13
14 TBATS model y observation at time t (y ω t 1)/ω if ω 0; log y t if ω = 0. l t 1 + φb t 1 + M i=1 l l t 1 + φb t 1 + αd t t m i + d t b (1 φ)b + φb t 1 + βd t d p i=1 k i φ i d t i + j,t q θ j ε t j + ε t j, s(i) j,t 1 cos λ(i) j + s (i) j,t 1 sin λ(i) j + γ 1 (i) d t j, s(i) j,t 1 sin λ(i) j + s (i) j,t 1 cos λ(i) j + γ (i) 2 d t 14
15 TBATS model y observation at time t (y ω t 1)/ω if ω 0; log y t if ω = 0. l t 1 + φb t 1 + M i=1 l l t 1 + φb t 1 + αd t t m i + d t Box-Cox transformation b (1 φ)b + φb t 1 + βd t d p i=1 k i φ i d t i + j,t q θ j ε t j + ε t j, s(i) j,t 1 cos λ(i) j + s (i) j,t 1 sin λ(i) j + γ 1 (i) d t j, s(i) j,t 1 sin λ(i) j + s (i) j,t 1 cos λ(i) j + γ (i) 2 d t 14
16 TBATS model y observation at time t (y ω t 1)/ω if ω 0; log y t if ω = 0. l t 1 + φb t 1 + M i=1 l l t 1 + φb t 1 + αd t t m i + d t Box-Cox transformation M seasonal periods b (1 φ)b + φb t 1 + βd t d p i=1 k i φ i d t i + j,t q θ j ε t j + ε t j, s(i) j,t 1 cos λ(i) j + s (i) j,t 1 sin λ(i) j + γ 1 (i) d t j, s(i) j,t 1 sin λ(i) j + s (i) j,t 1 cos λ(i) j + γ (i) 2 d t 14
17 TBATS model y observation at time t (y ω t 1)/ω if ω 0; log y t if ω = 0. l t 1 + φb t 1 + M i=1 l l t 1 + φb t 1 + αd t t m i + d t b (1 φ)b + φb t 1 + βd t d p i=1 k i φ i d t i + j,t q θ j ε t j + ε t j, Box-Cox transformation M seasonal periods global and local trend s(i) j,t 1 cos λ(i) j + s (i) j,t 1 sin λ(i) j + γ 1 (i) d t j, s(i) j,t 1 sin λ(i) j + s (i) j,t 1 cos λ(i) j + γ (i) 2 d t 14
18 TBATS model y observation at time t (y ω t 1)/ω if ω 0; log y t if ω = 0. l t 1 + φb t 1 + M i=1 l l t 1 + φb t 1 + αd t t m i + d t b (1 φ)b + φb t 1 + βd t d p i=1 k i φ i d t i + j,t q θ j ε t j + ε t j, Box-Cox transformation M seasonal periods global and local trend ARMA error s(i) j,t 1 cos λ(i) j + s (i) j,t 1 sin λ(i) j + γ 1 (i) d t j, s(i) j,t 1 sin λ(i) j + s (i) j,t 1 cos λ(i) j + γ (i) 2 d t 14
19 TBATS model y observation at time t (y ω t 1)/ω if ω 0; log y t if ω = 0. l t 1 + φb t 1 + M i=1 l l t 1 + φb t 1 + αd t t m i + d t b (1 φ)b + φb t 1 + βd t d p i=1 k i φ i d t i + j,t q θ j ε t j + ε t j, s(i) j,t 1 Box-Cox transformation M seasonal periods global and local trend ARMA error Fourier-like seasonal terms + s (i) j,t 1 sin λ(i) j + γ 1 (i) d t cos λ(i) j j, s(i) j,t 1 sin λ(i) j + s (i) j,t 1 cos λ(i) j + γ (i) 2 d t 14
20 TBATS model y observation at time t (yt ω 1)/ω if ω 0; log TBATS y t if ω = 0. l t 1 + φb t 1 + M i=1 l l t 1 + φb t 1 + αd t t m i + d t b (1 φ)b + φb t 1 + βd t d p q φ i d t i + i=1seasonal k i Trigonometric Box-Cox ARMA Trend j,t θ j ε t j + ε t j, s(i) j,t 1 Box-Cox transformation M seasonal periods global and local trend ARMA error Fourier-like seasonal terms + s (i) j,t 1 sin λ(i) j + γ 1 (i) d t cos λ(i) j j, s(i) j,t 1 sin λ(i) j + s (i) j,t 1 cos λ(i) j + γ (i) 2 d t 14
21 . Complex seasonality gasoline %>% tbats() %>% forecast() %>% autoplot() Forecasts from TBATS(1, {0,0},, {<52.18,12>}) 10 9 level Time 15
22 . Complex seasonality calls %>% tbats() %>% forecast() %>% autoplot() Forecasts from TBATS(0.555, {0,0},, {<169,6>, <845,4>}) level Time 16
23 . Complex seasonality telec %>% tbats() %>% forecast() %>% autoplot() Forecasts from TBATS(0.005, {4,2},, {<7,3>, <354.37,7>, <365.25,3>}) level Time 17
24 TBATS model TBATS Trigonometric terms for seasonality Box-Cox transformations for heterogeneity ARMA errors for short-term dynamics Trend (possibly damped) Seasonal (including multiple and non-integer periods) Handles non-integer seasonality, multiple seasonal periods. Entirely automated Prediction intervals often too wide Very slow on long series 18
25 Outline 1 Time series with complex seasonality 2 STL with multiple seasonal periods 3 Dynamic harmonic regression 4 TBATS model 5 Lab session 21 19
26 Lab Session 21 20
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