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 |

`beta` |
the smoothing parameter for linear component. The default is |

`gamma` |
the smoothing parameter for quadratic component. The default is |

`lead` |
the number of steps ahead for which prediction is required.
The default is |

`plot` |
a logical value indicating to print the plot of original data v.s smoothed
data. The default is |

### 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.

### Value

A list with class `"es"`

containing the following components:

`estimate` |
the smoothed values. |

`pred` |
the predicted values when |

`accurate` |
the accurate measurements. |

### Note

Missing values are removed before the analysis.

### Author(s)

Debin Qiu

### See Also

### 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
```

*aTSA*version 3.1.2.1 Index]