What are the level, trend, and seasonal components for Winters' method?

Winters' method employs a level component, a trend component, and a seasonal component at each period. It uses three weights, or smoothing parameters, to update the components at each period. Initial values for the level and trend components are obtained from a linear regression on time. Initial values for the seasonal component are obtained from a indicator variable regression using detrended data. The following equations are the Winters' method smoothing equations.

Smoothing equation for the additive model

  • Lt= α (Yt - St-p) + (1 - α) [Lt-1 + Tt-1]
  • Tt = γ [Lt - Lt-1] + (1 - γ)Tt-1
  • St = δ (Yt - Lt) + (1 - δ) St-p
  • = Lt-1 + Tt-1 + St-p

Smoothing equation for the multiplicative model

  • Lt= α (Yt / St-p) + (1 - α) [Lt-1 + Tt-1]
  • Tt = γ [Lt - Lt-1] + (1 - γ)Tt-1
  • St = δ (Yt / Lt) + (1 - δ) St-p
  • = (Lt-1 + Tt-1) St-p

Notation

TermDescription
LtThe level at time t
αThe weight for the level
TtThe trend at time t
γThe weight for the trend
StThe seasonal component at time t
δThe weight for the seasonal component
pThe seasonal period
YtThe data value at time t
The fitted value, or one-period-ahead forecast, at time t
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