HoltWinters {ts}

R Documentation

Holt-Winters Filtering

Description

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

Usage

HoltWinters(x, alpha = NULL, beta = NULL, gamma = NULL,
          seasonal = "additive", start.periods = 3,
          l.start = NULL, b.start = NULL, s.start = NULL)

Arguments

x	An object of class ts
alpha	alpha parameter of Holt-Winters Filter
beta	beta parameter of Holt-Winters Filter. If set to 0, the function will do exponential smoothing.
gamma	gamma parameter used for the seasonal component. If set to 0, an non-seasonal model is fitted.
seasonal	Selects an "additive" or "multiplicative" seasonal model. (Only takes effect if gamma is non-zero).
start.periods	Start periods used in the autodetection of start values. Must be at least 3.
l.start	Start value for level (a[0]).
b.start	Start value for trend (b[0]).
s.start	Vector of start values for the seasonal component (s_1[0]...s_p[0])

Details

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 + 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 + 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 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 omitted.

Start values for a, b and s are detected 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.)

Value

An object of class "HoltWinters", a list with components:

fitted	The filtered time series
x	The original series
alpha	alpha used for filtering
beta	beta used for filtering

(gamma)gamma used for filtering \itemcoefficientsA vector with named components a, b, s1, ..., sp containing the estimated values for the level, trend and seasonal components \itemseasonalThe specified seasonal-parameter \itemSSEThe final sum of squared errors achieved in optimizing \itemcallThe call used

Author(s)

David Meyer david.meyer@ci.tuwien.ac.at

References

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

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

See Also

predict.HoltWinters

Examples

library(ts)
data(co2)


(m <- HoltWinters(co2))
plot(m)

data(AirPassengers)
(m <- HoltWinters(AirPassengers, seasonal = "mult"))
plot(m)


data(uspop)
x <- uspop + rnorm(uspop, sd = 5)
m <- HoltWinters(x, gamma = 0)
plot(m)


m2 <- HoltWinters(x, gamma = 0, beta = 0)
lines(fitted(m2), col = 3)