aTSA (version 3.1.2)

# Winters: Winters Three-parameter Smoothing

## Description

Performs Winters three-parameter smoothing for a univariate time series with seasonal pattern.

## Usage

Winters(x, period = NULL, trend = 2, lead = 0, plot = TRUE,
seasonal = c("additive", "multiplicative"), damped = FALSE, alpha = 0.2,
beta = 0.1057, gamma = 0.07168, phi = 0.98, a.start = NA,
b.start = NA, c.start = NA)

## Arguments

x
a univariate time series.
period
seasonal period. The default is NULL.
trend
the type of trend component, can be one of 1,2,3 which represents constant, linear and quadratic trend, respectively. The default is trend = 2.
the number of steps ahead for which prediction is required. The default is 0.
plot
a logical value indicating to display the smoothed graph. The default is TRUE.
seasonal
character string to select an "additive" or "multiplicative" seasonal model. The default is "additive".
damped
a logical value indicating to include the damped trend, only valid for trend = 2. The default is FALSE.
alpha
the parameter of level smoothing. The default is 0.2.
beta
the parameter of trend smoothing. The default is 0.1057.
gamma
the parameter of season smoothing. The default is 0.07168.
phi
the parameter of damped trend smoothing, only valid for damped = TRUE. The default is 0.98.
a.start
the starting value for level smoothing. The default is NA.
b.start
the starting value for trend smoothing. The default is NA.
c.start
the starting value for season smoothing. The default is NA.

## Value

• A list with class "Winters" containing the following components:
• seasonthe seasonal factors.
• estimatethe smoothed values.
• predthe prediction, only available with lead > 0.
• accuratethe accurate measurements.

## Details

The Winters filter is used to decompose the trend and seasonal components by updating equations. This is similar to the function HoltWinters in stats package but may be in different perspective. To be precise, it uses the updating equations similar to exponential smoothing to fit the parameters for the following models when seasonal = "additive". If the trend is constant (trend = 1): $$x[t] = a[t] + s[t] + e[t].$$ If the trend is linear (trend = 2): $$x[t] = (a[t] + b[t]*t) + s[t] + e[t].$$ If the trend is quadratic (trend = 3): $$x[t] = (a[t] + b[t]*t + c[t]*t^2) + s[t] + e[t].$$ Here $a[t],b[t],c[t]$ are the trend parameters, $s[t]$ is the seasonal parameter for the season corresponding to time $t$. For the multiplicative season, the models are as follows. If the trend is constant (trend = 1): $$x[t] = a[t] * s[t] + e[t].$$ If the trend is linear (trend = 2): $$x[t] = (a[t] + b[t]*t) * s[t] + e[t].$$ If the trend is quadratic (trend = 3): $$x[t] = (a[t] + b[t]*t + c[t]*t^2) * s[t] + e[t].$$ When seasonal = "multiplicative", the updating equations for each parameter can be seen in page 606-607 of PROC FORECAST document of SAS. Similarly, for the additive seasonal model, the 'division' (/) for $a[t]$ and $s[t]$ in page 606-607 is changed to 'minus' (-).

The default starting values for $a,b,c$ are computed by a time-trend regression over the first cycle of time series. The default starting values for the seasonal factors are computed from seasonal averages. The default smoothing parameters (weights) alpha, beta, gamma are taken from the equation 1 - 0.8^{1/trend} respectively. You can also use the HoltWinters function to get the optimal smoothing parameters and plug them back in this function.

The prediction equation is $x[t+h] = (a[t] + b[t]*t)*s[t+h]$ for trend = 2 and seasonal = "additive". Similar equations can be derived for the other options. When the damped = TRUE, the prediction equation is $x[t+h] = (a[t] + (\phi + ... + \phi^(h))*b[t]*t)*s[t+h]$. More details can be referred to R. J. Hyndman and G. Athanasopoulos (2013).

## References

P. R. Winters (1960) Forecasting sales by exponentially weighted moving averages, Management Science 6, 324-342.

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

HoltWinters, Holt, expsmooth
Winters(co2)
Winters(AirPassengers, seasonal = "multiplicative")