State space models for exponential smoothing

Forecasting in R

Rob J. Hyndman

Professor of Statistics at Monash University

Innovations state space models

  • Each exponential smoothing method can be written as an "innovations state space model"

ch3_vid4_slides.004.png

Forecasting in R

Innovations state space models

  • Each exponential smoothing method can be written as an "innovations state space model"

ch3_vid4_slides.005.png

Forecasting in R

Innovations state space models

  • Each exponential smoothing method can be written as an "innovations state space model"

ch3_vid4_slides.006.png

Forecasting in R

Innovations state space models

  • Each exponential smoothing method can be written as an "innovations state space model"

ch3_vid4_slides.007.png

Forecasting in R

Innovations state space models

  • Each exponential smoothing method can be written as an "innovations state space model"

ch3_vid4_slides.008.png

  • ETS models: Error, Trend, Seasonal
Forecasting in R

ETS models

  • Parameters: estimated using the "likelihood", the probability of the data arising from the specified model
  • For models with additive errors, this is equivalent to minimizing SSE
  • Choose the best model by minimizing a corrected version of Akaike's Information Criterion ($AIC_c$)
Forecasting in R

Example: Australian air traffic

ets(ausair)

ETS(M,A,N)
Call:
 ets(y = ausair)
  Smoothing parameters:
    alpha = 0.9999
    beta  = 0.0186
  Initial states:
    l = 6.5249
    b = 0.7562
  sigma:  0.0763
     AIC     AICc      BIC
234.5273 236.0273 243.6705
Forecasting in R

Example: Australian air traffic

ausair %>% ets() %>% forecast() %>% autoplot()

Forecasts from ETS(M,A,N)

Forecasting in R

Example: Monthly cortecosteroid drug sales

ets(h02)

ETS(M,Ad,M)
Call:
 ets(y = h02)
  Smoothing parameters:
    alpha = 0.1953
    beta  = 1e-04 
    gamma = 1e-04
    phi   = 0.9798
  Initial states:
    l = 0.3945
    b = 0.0085
    s=0.874 0.8197 0.7644 0.7693 0.6941 1.2838
           1.326 1.1765 1.1621 1.0955 1.0422 0.9924
  sigma:  0.0676
       AIC       AICc        BIC
-122.90601 -119.20871  -63.17985
Forecasting in R

Example: Monthly cortecosteroid drug sales

h02 %>% ets() %>% forecast() %>% autoplot()

Forecats from ETS(M,Ad,M)

Forecasting in R

Let's practice!

Forecasting in R

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