Holt-Winters Filtering

Computes Holt-Winters Filtering of a given time series. Unknown parameters are determined by minimizing the squared prediction error.

HoltWinters(x, alpha = NULL, beta = NULL, gamma = NULL,
            seasonal = c("additive", "multiplicative"),
            start.periods = 2, l.start = NULL, b.start = NULL,
            s.start = NULL,
            optim.start = c(alpha = 0.3, beta = 0.1, gamma = 0.1),
            optim.control = list())
An object of class ts
$alpha$ parameter of Holt-Winters Filter.
$beta$ parameter of Holt-Winters Filter. If set to FALSE, the function will do exponential smoothing.
$gamma$ parameter used for the seasonal component. If set to FALSE, an non-seasonal model is fitted.
Character string to select an "additive" (the default) or "multiplicative" seasonal model. The first few characters are sufficient. (Only takes effect if gamma is non-zero).
Start periods used in the autodetection of start values. Must be at least 2.
Start value for level (a[0]).
Start value for trend (b[0]).
Vector of start values for the seasonal component ($s_1[0] \ldots s_p[0]$)
Vector with named components alpha, beta, and gamma containing the starting values for the optimizer. Only the values needed must be specified. Ignored in the one-parameter case.
Optional list with additional control parameters passed to optim if this is used. Ignored in the one-parameter case.

The additive Holt-Winters prediction function (for time series with period length p) is $$\hat Y[t+h] = a[t] + h b[t] + s[t - p + 1 + (h - 1) \bmod p],$$ where $a[t]$, $b[t]$ and $s[t]$ are given by $$a[t] = \alpha (Y[t] - s[t-p]) + (1-\alpha) (a[t-1] + b[t-1])$$ $$b[t] = \beta (a[t] -a[t-1]) + (1-\beta) b[t-1]$$ $$s[t] = \gamma (Y[t] - a[t]) + (1-\gamma) s[t-p]$$

The multiplicative Holt-Winters prediction function (for time series with period length p) is $$\hat Y[t+h] = (a[t] + h b[t]) \times s[t - p + 1 + (h - 1) \bmod p].$$ where $a[t]$, $b[t]$ and $s[t]$ are given by $$a[t] = \alpha (Y[t] / s[t-p]) + (1-\alpha) (a[t-1] + b[t-1])$$ $$b[t] = \beta (a[t] - a[t-1]) + (1-\beta) b[t-1]$$ $$s[t] = \gamma (Y[t] / a[t]) + (1-\gamma) s[t-p]$$ The data in x are required to be non-zero for a multiplicative model, but it makes most sense if they are all positive.

The function tries to find the optimal values of $\alpha$ and/or $\beta$ and/or $\gamma$ by minimizing the squared one-step prediction error if they are NULL (the default). optimize will be used for the single-parameter case, and optim otherwise.

For seasonal models, start values for a, b and s are inferred by performing a simple decomposition in trend and seasonal component using moving averages (see function decompose) on the start.periods first periods (a simple linear regression on the trend component is used for starting level and trend). For level/trend-models (no seasonal component), start values for a and b are x[2] and x[2] - x[1], respectively. For level-only models (ordinary exponential smoothing), the start value for a is x[1].


  • An object of class "HoltWinters", a list with components:
  • fittedA multiple time series with one column for the filtered series as well as for the level, trend and seasonal components, estimated contemporaneously (that is at time t and not at the end of the series).
  • xThe original series
  • alphaalpha used for filtering
  • betabeta used for filtering
  • gammagamma used for filtering
  • coefficientsA vector with named components a, b, s1, ..., sp containing the estimated values for the level, trend and seasonal components
  • seasonalThe specified seasonal parameter
  • SSEThe final sum of squared errors achieved in optimizing
  • callThe call used


C. C. Holt (1957) Forecasting trends and seasonals by exponentially weighted moving averages, ONR Research Memorandum, Carnegie Institute of Technology 52.

P. R. Winters (1960) Forecasting sales by exponentially weighted moving averages, Management Science 6, 324--342.

See Also

predict.HoltWinters, optim.

  • HoltWinters
  • print.HoltWinters
  • residuals.HoltWinters
library(stats) <testonly>od <- options(digits = 5)</testonly> require(graphics) ## Seasonal Holt-Winters (m <- HoltWinters(co2)) plot(m) plot(fitted(m)) (m <- HoltWinters(AirPassengers, seasonal = "mult")) plot(m) ## Non-Seasonal Holt-Winters x <- uspop + rnorm(uspop, sd = 5) m <- HoltWinters(x, gamma = FALSE) plot(m) ## Exponential Smoothing m2 <- HoltWinters(x, gamma = FALSE, beta = FALSE) lines(fitted(m2)[,1], col = 3) <testonly>options(od)</testonly>
Documentation reproduced from package stats, version 3.3, License: Part of R 3.3

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