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)

Value

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.

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.
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.

Author

Debin Qiu

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/.

Examples

Run this code

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")

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