expsmooth {aTSA} R Documentation

## 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`. `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`.

### Details

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 = x. For example, hat{x}[1|0] = level, hat{x}[2|1] = level,..., 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.

### Value

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

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

### Note

Missing values are removed before the analysis.

Debin Qiu

### See Also

`Winters`, `Holt`, `MA`

### Examples

```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
```

[Package aTSA version 3.1.2 Index]