expsmooth: Simple Exponential Smoothing
In deman007/aTSA: Alternative Time Series Analysis

Description Usage Arguments Details Value Note Author(s) See Also Examples

Performs a simple exponential smoothing for univariate time series with no trend or seasonal pattern.

1 2	expsmooth(x, trend = 1, alpha = 0.2, beta = 0.10557, gamma = 0.07168, lead = 0, plot = TRUE)

`x`	a numeric vector or univariate time series.
`trend`	the type of trend. See details.
`alpha`	the smoothing parameter for constant component. The default is `0.2`.
`beta`	the smoothing parameter for linear component. The default is `0.10557`.
`gamma`	the smoothing parameter for quadratic component. The default is `0.07168`.
`lead`	the number of steps ahead for which prediction is required. The default is `0`.
`plot`	a logical value indicating to print the plot of original data v.s smoothed data. The default is `TRUE`.

Simple exponential smoothing is a weighted average between the most recent observation and the most recent forecasting, with weights α and 1 - α, respectively. To be precise, the smoothing equation of single exponential smoothing (constant model, trend = 1) is given by

level[t] = α *x[t] + (1 - α)*level[t-1],

and the forecasting equation is

hat{x}[t+1|t] = level[t],

for t = 1,...,n. The initial value level[0] = x[1]. For example, hat{x}[1|0] = level[0], hat{x}[2|1] = level[1],..., etc.

Let x1[t] be the smoothed values of single exponential smoothing. The double exponential smoothing (trend = 2, a linear model) is to apply a single exponential smoothing again to the smoothed sequence x1[t], with a new smoothing parameter beta. Similarly, we denote the smoothed values of double exponential smoothing to be x2[t]. The triple exponential smoothing (trend = 3, a quadratic model) is to apply the single exponential smoothing to the smoothed sequence x2[t] with a new smoothing parameter gamma. The default smoothing parameters (weights) alpha, beta, gamma are taken from the equation 1 - 0.8^{1/trend} respectively, which is similar to the FORECAST procedure in SAS.

A list with class "es" containing the following components:

`estimate`	the smoothed values.
`pred`	the predicted values when `lead` > 0.
`accurate`	the accurate measurements.

Missing values are removed before the analysis.

Debin Qiu

Winters, Holt, MA

x <- rnorm(100)
es <- expsmooth(x) # trend = 1: a constant model
plot(x,type = "l")
lines(es$estimate,col = 2)
expsmooth(x,trend = 2) # trend = 2: a linear model
expsmooth(x,trend = 3) # trend = 3: a quadratic model