Holt {aTSA}  R Documentation 
Holt's Twoparameter Exponential Smoothing
Description
Performs Holt's twoparameter 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 
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 
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
stepahead forecast is given by hat{x}[t+ht] = level[t] + h*b[t]
,
where
level[t] = \alpha *x[t] + (1\alpha)*(b[t1] + level[t1]),
b[t] = \beta*(level[t]  level[t1]) + (1\beta)*b[t1],
in which b[t]
is the trend component.
For the multiplicative (type = "multiplicative"
) model, the
h
stepahead forecast is given by hat{x}[t+ht] = level[t] + h*b[t]
,
where
level[t] = \alpha *x[t] + (1\alpha)*(b[t1] * level[t1]),
b[t] = \beta*(level[t] / level[t1]) + (1\beta)*b[t1].
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 twoparameter smoothing.
If damped = TRUE
, the additive model becomes
hat{x}[t+ht] = level[t] + (\phi + \phi^{2} + ... + \phi^{h})*b[t],
level[t] = \alpha *x[t] + (1\alpha)*(\phi*b[t1] + level[t1]),
b[t] = \beta*(level[t]  level[t1]) + (1\beta)*\phi*b[t1].
The multiplicative model becomes
hat{x}[t+ht] = level[t] *b[t]^(\phi + \phi^{2} + ... + \phi^{h}),
level[t] = \alpha *x[t] + (1\alpha)*(b[t1]^{\phi} * level[t1]),
b[t] = \beta*(level[t] / level[t1]) + (1\beta)*b[t1]^{\phi}.
See Chapter 7.4 for more details in R. J. Hyndman and G. Athanasopoulos (2013).
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 
accurate 
the accurate measurements. 
Note
Missing values are removed before analysis.
Author(s)
Debin Qiu
References
R. J. Hyndman and G. Athanasopoulos, "Forecasting: principles and practice," 2013. [Online]. Available: http://otexts.org/fpp/.
See Also
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")