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 `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 if
`gamma`

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 (\(s_1[0] \ldots s_p[0]\))

optim.start

Vector with named components `alpha`

,
`beta`

, and `gamma`

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.

An object of class `"HoltWinters"`

, a list with components:

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).

The original series

alpha used for filtering

beta used for filtering

gamma used for filtering

A vector with named components `a, b, s1, ..., sp`

containing the estimated values for the level, trend and seasonal
components

The specified `seasonal`

parameter

The final sum of squared errors achieved in optimizing

The call used

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]`

.

C. C. Holt (1957)
Forecasting seasonals and trends by exponentially weighted moving averages,
*ONR Research Memorandum, Carnegie Institute of Technology* **52**.
(reprint at https://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.
10.1287/mnsc.6.3.324.

```
# NOT RUN {
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)
# }
```

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