HoltWinters {ts} | R Documentation |
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 = "additive", start.periods = 3, l.start = NULL, b.start = NULL, s.start = NULL)
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]) |
The additive Holt-Winters prediction function (for time series with period length p) is
Yhat[t+h] = a[t] + h * b[t] + s[t + 1 + (h - 1) mod p],
where a[t], b[t] and s[t] are given by
a[t] = α (Y[t] - s[t-p]) + (1-α) (a[t-1] + b[t-1])
b[t] = β (a[t] - a[t-1]) + (1-β) 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
Yhat[t+h] = (a[t] + h * b[t]) * s[t + 1 + (h - 1) mod p],
where a[t], b[t] and s[t] are given by
a[t] = α (Y[t] / s[t-p]) + (1-α) (a[t-1] + b[t-1])
b[t] = β (a[t] - a[t-1]) + (1-β) b[t-1]
s[t] = gamma (Y[t] / a[t]) + (1-gamma) s[t-p]
The function tries to find the optimal values of α and/or β and/or gamma by minimizing the squared one-step prediction error if they are omitted.
For seasonal models, 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.). 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:
fitted |
The filtered time series |
x |
The original series |
alpha |
alpha used for filtering |
beta |
beta used for filtering |
coefficients |
A vector with named components a, b, s1, ..., sp
containing the estimated values for the level, trend and seasonal
components |
seasonal |
The specified seasonal -parameter |
SSE |
The final sum of squared errors achieved in optimizing |
call |
The call used |
David Meyer david.meyer@ci.tuwien.ac.at
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, 324342.
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)