Exponential smoothing and non-negative data Muhammad Akram Rob J - PowerPoint PPT Presentation

Exponential smoothing and non-negative data 1 Exponential smoothing and non-negative data Muhammad Akram Rob J Hyndman J Keith Ord Business & Economic Forecasting Unit

Exponential smoothing and non-negative data 2 Outline Exponential smoothing models 1 Problems with some of the models 2 A new model for positive data 3 Conclusions 4

Exponential smoothing and non-negative data Exponential smoothing models 3 Outline Exponential smoothing models 1 Problems with some of the models 2 A new model for positive data 3 Conclusions 4

Exponential smoothing and non-negative data Exponential smoothing models 4 Problem Most forecasting methods in business are based on exponential smoothing.

Exponential smoothing and non-negative data Exponential smoothing models 4 Problem Most forecasting methods in business are based on exponential smoothing. Most time series in business are inherently non-negative.

Exponential smoothing and non-negative data Exponential smoothing models 4 Problem Most forecasting methods in business are based on exponential smoothing. Most time series in business are inherently non-negative. But exponential smoothing models for non-negative data don’t always work!

Exponential smoothing and non-negative data Exponential smoothing models 4 Problem Most forecasting methods in business are based on exponential smoothing. Most time series in business are inherently non-negative. But exponential smoothing models for non-negative Forecasts from ETS(A,A,N) 30 data don’t always work! 20 They can produce 1 negative forecasts 10 0 0 5 10 15 20 25 30 35 Time

Exponential smoothing and non-negative data Exponential smoothing models 4 Problem Most forecasting methods in business are based on exponential smoothing. Most time series in business are inherently non-negative. But exponential smoothing models for non-negative Forecasts from ETS(A,M,N) 10 data don’t always work! They can produce 5 1 negative forecasts They can produce infinite 2 0 forecast variance −5 0 5 10 15 20 25 30 Time

Exponential smoothing and non-negative data Exponential smoothing models 4 Problem Most forecasting methods in business are based on exponential smoothing. Most time series in business are inherently non-negative. But exponential smoothing models for non-negative Simulation from ETS(M,N,N) data don’t always work! 40 They can produce 1 30 negative forecasts y 20 They can produce infinite 2 10 forecast variance They can converge almost 3 0 0 500 1000 1500 2000 surely to zero.

Exponential smoothing and non-negative data Exponential smoothing models 5 Taxonomy of models 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 Ad (Additive damped) Ad,N Ad,A Ad,M M (Multiplicative) M,N M,A M,M Md (Multiplicative damped) Md,N Md,A Md,M

Exponential smoothing and non-negative data Exponential smoothing models 5 Taxonomy of models 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 Ad (Additive damped) Ad,N Ad,A Ad,M M (Multiplicative) M,N M,A M,M Md (Multiplicative damped) Md,N Md,A Md,M General notation ETS ( Error,Trend,Seasonal )

Exponential smoothing and non-negative data Exponential smoothing models 5 Taxonomy of models 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 Ad (Additive damped) Ad,N Ad,A Ad,M M (Multiplicative) M,N M,A M,M Md (Multiplicative damped) Md,N Md,A Md,M General notation ETS ( Error,Trend,Seasonal ) E xponen T ial S moothing

Exponential smoothing and non-negative data Exponential smoothing models 5 Taxonomy of models 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 Ad (Additive damped) Ad,N Ad,A Ad,M M (Multiplicative) M,N M,A M,M Md (Multiplicative damped) Md,N Md,A Md,M General notation ETS ( Error,Trend,Seasonal ) E xponen T ial S moothing ETS(A,N,N) : Simple exponential smoothing with additive errors

Exponential smoothing and non-negative data Exponential smoothing models 5 Taxonomy of models 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 Ad (Additive damped) Ad,N Ad,A Ad,M M (Multiplicative) M,N M,A M,M Md (Multiplicative damped) Md,N Md,A Md,M General notation ETS ( Error,Trend,Seasonal ) E xponen T ial S moothing ETS(A,A,N) : Holt’s linear method with additive errors

Exponential smoothing and non-negative data Exponential smoothing models 5 Taxonomy of models 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 Ad (Additive damped) Ad,N Ad,A Ad,M M (Multiplicative) M,N M,A M,M Md (Multiplicative damped) Md,N Md,A Md,M General notation ETS ( Error,Trend,Seasonal ) E xponen T ial S moothing ETS(A,A,A) : Additive Holt-Winters’ method with additive errors

Exponential smoothing and non-negative data Exponential smoothing models 5 Taxonomy of models 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 Ad (Additive damped) Ad,N Ad,A Ad,M M (Multiplicative) M,N M,A M,M Md (Multiplicative damped) Md,N Md,A Md,M General notation ETS ( Error,Trend,Seasonal ) E xponen T ial S moothing ETS(M,A,M) : Multiplicative Holt-Winters’ method with multiplicative errors

Exponential smoothing and non-negative data Exponential smoothing models 5 Taxonomy of models 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 Ad (Additive damped) Ad,N Ad,A Ad,M M (Multiplicative) M,N M,A M,M Md (Multiplicative damped) Md,N Md,A Md,M General notation ETS ( Error,Trend,Seasonal ) E xponen T ial S moothing ETS(A,A d ,N) : Damped trend method with addi- tive errors

Exponential smoothing and non-negative data Exponential smoothing models 5 Taxonomy of models 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 Ad (Additive damped) Ad,N Ad,A Ad,M M (Multiplicative) M,N M,A M,M Md (Multiplicative damped) Md,N Md,A Md,M General notation ETS ( Error,Trend,Seasonal ) E xponen T ial S moothing There are 30 separate models in the ETS framework

Exponential smoothing and non-negative data Exponential smoothing models 6 Innovations state space model No trend or seasonality and multiplicative errors Example: ETS(M,N,N) y t = ℓ t − 1 ( 1 + ε t ) ℓ t = α y t + ( 1 − α ) ℓ t − 1 0 ≤ α ≤ 1 ε t is white noise with mean zero.

Exponential smoothing and non-negative data Exponential smoothing models 6 Innovations state space model No trend or seasonality and multiplicative errors Example: ETS(M,N,N) y t = ℓ t − 1 ( 1 + ε t ) ℓ t = ℓ t − 1 ( 1 + αε t ) 0 ≤ α ≤ 1 ε t is white noise with mean zero.

Exponential smoothing and non-negative data Exponential smoothing models 6 Innovations state space model No trend or seasonality and multiplicative errors Example: ETS(M,N,N) y t = ℓ t − 1 ( 1 + ε t ) ℓ t = ℓ t − 1 ( 1 + αε t ) 0 ≤ α ≤ 1 ε t is white noise with mean zero. All exponential smoothing models can be written using analogous state space equations.

1 1฀3 Exponential smoothing and non-negative data Exponential smoothing models 7 New book! State space modeling Springer Series in Statistics framework Prediction intervals Rob J.Hyndman · Anne B.Koehler Model selection J.Keith Ord · Ralph D.Snyder Maximum likelihood Forecasting estimation with Exponential All the important research Smoothing results in one place with consistent notation The State Space Approach Many new results 375 pages but only US$39.95.

1 1฀3 Exponential smoothing and non-negative data Exponential smoothing models 7 New book! State space modeling Springer Series in Statistics framework Prediction intervals Rob J.Hyndman · Anne B.Koehler Model selection J.Keith Ord · Ralph D.Snyder Maximum likelihood Forecasting estimation with Exponential All the important research Smoothing results in one place with consistent notation The State Space Approach Many new results 375 pages but only US$39.95. www.exponentialsmoothing.net

Exponential smoothing and non-negative data Problems with some of the models 8 Outline Exponential smoothing models 1 Problems with some of the models 2 A new model for positive data 3 Conclusions 4

Exponential smoothing and non-negative data Problems with some of the models 9 Negative forecasts Forecasts from ETS(A,A,N) 30 20 10 0 0 5 10 15 20 25 30 35 Time Could solve by taking logs or some other Box-Cox transformation. However, this limits models to be additive in the transformed space.