aTSA (version 3.1.2)

# expsmooth: Simple Exponential Smoothing

## Description

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

## Usage

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

## Arguments

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.
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.

## Value

• A list with class "es" containing the following components:
• estimatethe smoothed values.
• predthe predicted values when lead > 0.
• accuratethe accurate measurements.

## Details

Simple exponential smoothing is a weighted average between the most recent observation and the most recent forecasting, with weights $\alpha$ and $1 - \alpha$, respectively. To be precise, the smoothing equation of single exponential smoothing (constant model, trend = 1) is given by $$level[t] = \alpha *x[t] + (1 - \alpha)*level[t-1],$$ 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.

Winters, Holt, MA
x <- rnorm(100)
expsmooth(x,trend = 3) # trend = 3: a quadratic model