Rob J Hyndman Forecasting: Principles and Practice 9. State space models Forecasting:PrinciplesandPractice 1 Outline 1 Recall ETS models 2 Simple structural models 3 Linear Gaussian state space models 4 Kalman filter 5 ARIMA models in state space form 6 Kalman smoothing 7 Time varying parameter models Forecasting:PrinciplesandPractice RecallETSmodels 2 Exponential smoothing methods Seasonal Component Trend N A M Component (None) (Additive) (Multiplicative) N (None) N,N N,A N,M A (Additive) A,N A,A A,M A (Additivedamped) A ,N A ,A A ,M d d d d M (Multiplicative) M,N M,A M,M M (Multiplicativedamped) M ,N M ,A M ,M d d d d General notation E T S : ExponenTial Smoothing Examples: A,N,N: Simpleexponentialsmoothingwithadditiveerrors A,A,N: Holt’slinearmethodwithadditiveerrors M,A,M: MultiplicativeHolt-Winters’methodwithmultiplicativeerrors Forecasting:PrinciplesandPractice RecallETSmodels 3 Exponential smoothing methods Seasonal Component Trend N A M Component (None) (Additive) (Multiplicative) N (None) N,N N,A N,M A (Additive) A,N A,A A,M A (Additivedamped) A ,N A ,A A ,M d d d d M (Multiplicative) M,N M,A M,M M (Multiplicativedamped) M ,N M ,A M ,M d d d d General notation E T S : ExponenTial Smoothing Examples: A,N,N: Simpleexponentialsmoothingwithadditiveerrors A,A,N: Holt’slinearmethodwithadditiveerrors M,A,M: MultiplicativeHolt-Winters’methodwithmultiplicativeerrors Forecasting:PrinciplesandPractice RecallETSmodels 3 Exponential smoothing methods Seasonal Component Trend N A M Component (None) (Additive) (Multiplicative) N (None) N,N N,A N,M A (Additive) A,N A,A A,M A (Additivedamped) A ,N A ,A A ,M d d d d M (Multiplicative) M,N M,A M,M M (Multiplicativedamped) M ,N M ,A M ,M d d d d General notation E T S : ExponenTial Smoothing ↑ Trend Examples: A,N,N: Simpleexponentialsmoothingwithadditiveerrors A,A,N: Holt’slinearmethodwithadditiveerrors M,A,M: MultiplicativeHolt-Winters’methodwithmultiplicativeerrors Forecasting:PrinciplesandPractice RecallETSmodels 3 Exponential smoothing methods Seasonal Component Trend N A M Component (None) (Additive) (Multiplicative) N (None) N,N N,A N,M A (Additive) A,N A,A A,M A (Additivedamped) A ,N A ,A A ,M d d d d M (Multiplicative) M,N M,A M,M M (Multiplicativedamped) M ,N M ,A M ,M d d d d General notation E T S : ExponenTial Smoothing ↑ (cid:45) Trend Seasonal Examples: A,N,N: Simpleexponentialsmoothingwithadditiveerrors A,A,N: Holt’slinearmethodwithadditiveerrors M,A,M: MultiplicativeHolt-Winters’methodwithmultiplicativeerrors Forecasting:PrinciplesandPractice RecallETSmodels 3 Exponential smoothing methods Seasonal Component Trend N A M Component (None) (Additive) (Multiplicative) N (None) N,N N,A N,M A (Additive) A,N A,A A,M A (Additivedamped) A ,N A ,A A ,M d d d d M (Multiplicative) M,N M,A M,M M (Multiplicativedamped) M ,N M ,A M ,M d d d d General notation E T S : ExponenTial Smoothing (cid:37) ↑ (cid:45) Error Trend Seasonal Examples: A,N,N: Simpleexponentialsmoothingwithadditiveerrors A,A,N: Holt’slinearmethodwithadditiveerrors M,A,M: MultiplicativeHolt-Winters’methodwithmultiplicativeerrors Forecasting:PrinciplesandPractice RecallETSmodels 3 Exponential smoothing methods Seasonal Component Trend N A M Component (None) (Additive) (Multiplicative) N (None) N,N N,A N,M A (Additive) A,N A,A A,M A (Additivedamped) A ,N A ,A A ,M d d d d M (Multiplicative) M,N M,A M,M M (Multiplicativedamped) M ,N M ,A M ,M d d d d General notation E T S : ExponenTial Smoothing (cid:37) ↑ (cid:45) Error Trend Seasonal Examples: A,N,N: Simpleexponentialsmoothingwithadditiveerrors A,A,N: Holt’slinearmethodwithadditiveerrors M,A,M: MultiplicativeHolt-Winters’methodwithmultiplicativeerrors Forecasting:PrinciplesandPractice RecallETSmodels 3 Exponential smoothing methods Innovations state space models Seasonal Component (cid:229) Trend N A M All ETS models can be written in innovations Component (None) (Additive) (Multiplicative) state space form. N (None) N,N N,A N,M A (Additive) A,N A,A A,M (cid:229) Additive and multiplicative versions give the A (Additivedamped) A ,N A ,A A ,M d d d d same point forecasts but different prediction M (Multiplicative) M,N M,A M,M intervals. M (Multiplicativedamped) M ,N M ,A M ,M d d d d General notation E T S : ExponenTial Smoothing (cid:37) ↑ (cid:45) Error Trend Seasonal Examples: A,N,N: Simpleexponentialsmoothingwithadditiveerrors A,A,N: Holt’slinearmethodwithadditiveerrors M,A,M: MultiplicativeHolt-Winters’methodwithmultiplicativeerrors Forecasting:PrinciplesandPractice RecallETSmodels 3 Innovations state space models Let x = ((cid:96) ,b ,s ,s ,...,s ) and ε ∼iid N(0,σ2). t t t t t−1 t−m+1 t y = h(x )+k(x )ε t t−1 t−1 t (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) µ e t t x = f(x ) + g(x )ε t t−1 t−1 t Additive errors: k(x) = 1. y = µ + ε . t t t Multiplicative errors: k(x ) = µ . y = µ (1 + ε ). t−1 t t t t ε = (y − µ )/µ is relative error. t t t t Forecasting:PrinciplesandPractice RecallETSmodels 4