| HoltWinters {stats} | 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 = 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())
x |
An object of class |
alpha |
alpha parameter of Holt-Winters Filter. |
beta |
beta parameter of Holt-Winters Filter. If set to
|
gamma |
gamma parameter used for the seasonal component.
If set to |
seasonal |
Character string to select an |
start.periods |
Start periods used in the autodetection of start values. Must be at least 2. |
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]) |
optim.start |
Vector with named components |
optim.control |
Optional list with additional control parameters
passed to |
The additive Holt-Winters prediction function (for time series with period length p) is
Yhat[t+h] = a[t] + h * b[t] + s[t - p + 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] = γ (Y[t] - a[t]) + (1-γ) 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 - p + 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] = γ (Y[t] / a[t]) + (1-γ) 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 α and/or
β and/or γ 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:
fitted |
A 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). |
x |
The original series |
alpha |
alpha used for filtering |
beta |
beta used for filtering |
gamma |
gamma used for filtering |
coefficients |
A vector with named components |
seasonal |
The specified |
SSE |
The final sum of squared errors achieved in optimizing |
call |
The call used |
David Meyer David.Meyer@wu.ac.at
C. C. Holt (1957) Forecasting seasonals and trends by exponentially weighted moving averages, ONR Research Memorandum, Carnegie Institute of Technology 52. (reprint at http://dx.doi.org/10.1016/j.ijforecast.2003.09.015).
P. R. Winters (1960). Forecasting sales by exponentially weighted moving averages. Management Science, 6, 324–342. doi: 10.1287/mnsc.6.3.324.
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)