aTSA (version 3.1.2)

# Holt: Holt's Two-parameter Exponential Smoothing

## Description

Performs Holt's two-parameter exponential smoothing for linear trend or damped trend.

## Usage

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

## Arguments

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.
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.

## Value

• A list with class "Holt" containing the following components:
• estimatethe estimate values.
• alphathe smoothing parameter used for level.
• betathe smoothing parameter used for trend.
• phithe smoothing parameter used for damped trend.
• predthe predicted values, only available for lead > 0.
• accuratethe accurate measurements.

## Details

Holt's two parameter is used to forecast a time series with trend, but wihtout seasonal pattern. For the additive model (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).

## References

R. J. Hyndman and G. Athanasopoulos, "Forecasting: principles and practice," 2013. [Online]. Available: http://otexts.org/fpp/.

HoltWinters, expsmooth, Winters

## Examples

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")