| HoltWinters {stats} | 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 = 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())
Arguments
x |
An object of class |
alpha |
|
beta |
|
gamma |
|
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
( |
optim.start |
Vector with named components |
optim.control |
Optional list with additional control parameters
passed to |
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 - 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].
Value
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 |
Author(s)
David Meyer David.Meyer@wu.ac.at
References
Holt CC (1957). “Forecasting Seasonals and Trends by Exponentially Weighted Moving Averages.” ONR Research Memorandum 52, Carnegie Institute of Technology. Reprinted in Holt (2004).
Holt CC (2004). “Forecasting Seasonals and Trends by Exponentially Weighted Moving Averages.” International Journal of Forecasting, 20(1), 5–10. doi:10.1016/j.ijforecast.2003.09.015.
Winters PR (1960). “Forecasting Sales by Exponentially Weighted Moving Averages.” Management Science, 6(3), 324–342. doi:10.1287/mnsc.6.3.324.
See Also
Examples
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)