```
Holt(x, type = c("additive", "multiplicative"), alpha = 0.2,
beta = 0.1057, lead = 0, damped = FALSE, phi = 0.98, plot = TRUE)
```

x

a numeric vector or univariate time series.

type

the type of interaction between the level and the linear trend. See
details.

alpha

the parameter for the level smoothing. The default is

`0.2`

.beta

the parameter for the trend smoothing. The default is

`0.1057`

.lead

the number of steps ahead for which prediction is required.
The default is

`0`

.damped

a logical value indicating a damped trend. See details. The default is

`FALSE`

.phi

a smoothing parameter for damped trend. The default is

`0.98`

, only valid
for `damped = TRUE`

.plot

a logical value indicating to print the plot of original data v.s smoothed
data. The default is

`TRUE`

.- A list with class "
`Holt`

" containing the following components: estimate the estimate values. alpha the smoothing parameter used for level. beta the smoothing parameter used for trend. phi the smoothing parameter used for damped trend. pred the predicted values, only available for `lead`

> 0.accurate the accurate measurements.

`type = "additive"`

), the
$h$-step-ahead forecast is given by $hat{x}[t+h|t] = level[t] + h*b[t]$,
where
$$level[t] = \alpha *x[t] + (1-\alpha)*(b[t-1] + level[t-1]),$$
$$b[t] = \beta*(level[t] - level[t-1]) + (1-\beta)*b[t-1],$$
in which $b[t]$ is the trend component.
For the multiplicative (`type = "multiplicative"`

) model, the
$h$-step-ahead forecast is given by $hat{x}[t+h|t] = level[t] + h*b[t]$,
where
$$level[t] = \alpha *x[t] + (1-\alpha)*(b[t-1] * level[t-1]),$$
$$b[t] = \beta*(level[t] / level[t-1]) + (1-\beta)*b[t-1].$$Compared with the Holt's linear trend that displays a constant increasing or
decreasing, the damped trend generated by exponential smoothing method shows a
exponential growth or decline, which is a situation between simple exponential
smoothing (with 0 increasing or decreasing rate) and Holt's two-parameter smoothing.
If `damped = TRUE`

, the additive model becomes
$$hat{x}[t+h|t] = level[t] + (\phi + \phi^{2} + ... + \phi^{h})*b[t],$$
$$level[t] = \alpha *x[t] + (1-\alpha)*(\phi*b[t-1] + level[t-1]),$$
$$b[t] = \beta*(level[t] - level[t-1]) + (1-\beta)*\phi*b[t-1].$$
The multiplicative model becomes
$$hat{x}[t+h|t] = level[t] *b[t]^(\phi + \phi^{2} + ... + \phi^{h}),$$
$$level[t] = \alpha *x[t] + (1-\alpha)*(b[t-1]^{\phi} * level[t-1]),$$
$$b[t] = \beta*(level[t] / level[t-1]) + (1-\beta)*b[t-1]^{\phi}.$$
See Chapter 7.4 for more details in R. J. Hyndman and G. Athanasopoulos (2013).

`HoltWinters`

, `expsmooth`

, `Winters`

x <- (1:100)/100 y <- 2 + 1.2*x + rnorm(100) ho0 <- Holt(y) # with additive interaction ho1 <- Holt(y,damped = TRUE) # with damped trend # multiplicative model for AirPassengers data, # although seasonal pattern exists. ho2 <- Holt(AirPassengers,type = "multiplicative")