Holt's Two-parameter Exponential Smoothing
Performs Holt's two-parameter exponential smoothing for linear trend or damped trend.
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 |
beta |
the parameter for the trend smoothing. The default is |
lead |
the number of steps ahead for which prediction is required.
The default is |
damped |
a logical value indicating a damped trend. See details. The default is
|
phi |
a smoothing parameter for damped trend. The default is |
plot |
a logical value indicating to print the plot of original data v.s smoothed
data. The default is |
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] = α *x[t] + (1-α)*(b[t-1] + level[t-1]),
b[t] = β*(level[t] - level[t-1]) + (1-β)*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] = α *x[t] + (1-α)*(b[t-1] * level[t-1]),
b[t] = β*(level[t] / level[t-1]) + (1-β)*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] + (φ + φ^{2} + ... + φ^{h})*b[t],
level[t] = α *x[t] + (1-α)*(φ*b[t-1] + level[t-1]),
b[t] = β*(level[t] - level[t-1]) + (1-β)*φ*b[t-1].
The multiplicative model becomes
hat{x}[t+h|t] = level[t] *b[t]^(φ + φ^{2} + ... + φ^{h}),
level[t] = α *x[t] + (1-α)*(b[t-1]^{φ} * level[t-1]),
b[t] = β*(level[t] / level[t-1]) + (1-β)*b[t-1]^{φ}.
See Chapter 7.4 for more details in R. J. Hyndman and G. Athanasopoulos (2013).
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 |
accurate |
the accurate measurements. |
Missing values are removed before analysis.
Debin Qiu
R. J. Hyndman and G. Athanasopoulos, "Forecasting: principles and practice," 2013. [Online]. Available: http://otexts.org/fpp/.
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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