Holt-Winters' Method

Holt-Winters' method is an exponential smoothing approach to handling seasonal data, where seasonal adjustments are made to the trend. To handle the seasonality it adds a third parameter to the model. Per NIST, it is sometimes referred to as triple exponential smoothing.

Holt-Winters', as well as any exponential (or other) smoothing model, works well with deterministic effects, such as seasonality that can be attributed to known events throughout the course of a year. For stochastic time series it may not be as effective, and we should consider an ARMA / ARIMA model.

The Holt-Winters' method often incorporates a linear trend, but can also accommodate non-linear trends.

• Exponential trend

• Damped trend

Two basic approaches include additive and multiplicative seasonal models.

Additive seasonal model

The additive seasonal model is used when the amplitude of the seasonal variation does not change steadily over time.

y_t = (B₀ + B₁*t) + S_t + e_t (alternatively: = L_t + S_t + e_t)

Image source

Multiplicative seasonal model

The multiplicative seasonal model is used when the amplitude of the seasonal variation does change steadily over time.

y_t = (B₀ + B₁*t) * S_t + e_t (alternatively: = L_t * S_t + e_t)

Image source

• This resource and this source show a slightly different model with a multiplicative error term: L_t * S_t * e_t

• They also suggest the following: "To fit a multiplicative model, take logarithms of the data, then analyse the log data as before."