HoltWinters
Holt-Winters Filtering
Computes Holt-Winters Filtering of a given time series. Unknown parameters are determined by minimizing the squared prediction error.
- Keywords
- ts
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
ts
- alpha
- $alpha$ parameter of Holt-Winters Filter.
- beta
- $beta$ parameter of Holt-Winters Filter. If set to
FALSE
, the function will do exponential smoothing. - gamma
- $gamma$ parameter used for the seasonal component.
If set to
FALSE
, an non-seasonal model is fitted. - seasonal
- Character string to select an
"additive"
(the default) or"multiplicative"
seasonal model. The first few characters are sufficient. (Only takes effect ifgamma
is non-zero). - 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 ($s_1[0] \dots s_p[0]$)
- optim.start
- Vector with named components
alpha
,beta
, andgamma
containing the starting values for the optimizer. Only the values needed must be specified. Ignored in the one-parameter case. - optim.control
- Optional list with additional control parameters
passed to
optim
if this is used. Ignored in the one-parameter case.
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
- 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
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
"HoltWinters"
, a list with components:
References
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
Examples
library(stats)
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)