# Holt: Holt's Two-parameter Exponential Smoothing In aTSA: Alternative Time Series Analysis

## Description

Performs Holt's two-parameter exponential smoothing for linear trend or damped trend.

## Usage

 ```1 2``` ```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 `0.2`. `beta` the parameter for the trend smoothing. The default is `0.1057`. `lead` the number of steps ahead for which prediction is required. The default is `0`. `damped` a logical value indicating a damped trend. See details. The default is `FALSE`. `phi` a smoothing parameter for damped trend. The default is `0.98`, only valid for `damped = TRUE`. `plot` a logical value indicating to print the plot of original data v.s smoothed data. The default is `TRUE`.

## 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-step-ahead forecast is given by hat{x}[t+h|t] = level[t] + h*b[t], where

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

b[t] = β*(level[t] - level[t-1]) + (1-β)*b[t-1],

in which b[t] is the trend component. For the multiplicative (`type = "multiplicative"`) model, the h-step-ahead forecast is given by hat{x}[t+h|t] = level[t] + h*b[t], where

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

b[t] = β*(level[t] / level[t-1]) + (1-β)*b[t-1].

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 two-parameter smoothing. If `damped = TRUE`, the additive model becomes

hat{x}[t+h|t] = level[t] + (φ + φ^{2} + ... + φ^{h})*b[t],

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

b[t] = β*(level[t] - level[t-1]) + (1-β)*φ*b[t-1].

The multiplicative model becomes

hat{x}[t+h|t] = level[t] *b[t]^(φ + φ^{2} + ... + φ^{h}),

level[t] = α *x[t] + (1-α)*(b[t-1]^{φ} * level[t-1]),

b[t] = β*(level[t] / level[t-1]) + (1-β)*b[t-1]^{φ}.

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 `lead` > 0. `accurate` the accurate measurements.

## Note

Missing values are removed before analysis.

Debin Qiu

## References

R. J. Hyndman and G. Athanasopoulos, "Forecasting: principles and practice," 2013. [Online]. Available: http://otexts.org/fpp/.

`HoltWinters`, `expsmooth`, `Winters`

## Examples

 ```1 2 3 4 5 6 7 8 9``` ```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") ```

### Example output

```Attaching package: 'aTSA'

The following object is masked from 'package:graphics':

identify
```

aTSA documentation built on May 1, 2019, 8:47 p.m.