Calculadora online: Triple exponential smoothing

So we've reached triple exponential smoothing.

Those who were not aware - it may be simple Exponential smoothing, there is also Double exponential smoothing and now let me introduce you triple exponential smoothing.

A bit of history first.
Exponential smoothing was first proposed in 1957 by C. C. Holt and intended for non-recurrent (no seasonality) time series that doesn't show any trends.

In 1958, he also proposed a modification of this method, which takes into account the trend - double exponential smoothing.

In 1965, Winters generalized this method to a seasonally adjusted one. Therefore, the triple exponential smoothing method is also called the Holt-Winters (Holt-Winters method).

Since I've already said all sorts of introductions in previous articles, let's get straight to the formulas.

Triple exponential smoothing:

General equation:

$S_t = \alpha \frac{y_t}{I_{t-L}} + (1-\alpha)(S_{t-1}+b_{t-1})$ ,

Trend smoothing

$b_t = \gamma (S_t - S_{t-1}) + (1-\gamma)b_{t-1}$ ,

Seasonality smoothing

$I_t = \beta \frac{y_t}{S_t} + (1-\beta)I_{t-L}$ ,

Forecast

$F_{t+m} = (S_t + mb_t)I_{t-L+m}$

where,

$\alpha$ , $\beta$ , $\gamma$ takes the value from the range [0;1]
y - observation
S - smoothed observation value
b - trend rate
I - seasonality index
F - forecast for m periods ahead
t - current observation index

As well as for the other exponential smoothing, $\alpha$ , $\beta$ , $\gamma$ are chosen by trial and error so as to minimize the mean square error.

The special thing here - the existence of L, determining the number of periods. By the number of periods, it's needed to construct the corresponding starting seasonality indices. Thus, the method, in terms of the seasonality indices calculation, requires minimum L of observations. It is clear that the more full seasons available, the better - the initial seasonality indices will be more precise.

Seasonality indices are calculated as follows - suppose there is observational evidence for n seasons by L periods.
Then
1) mean value is calculated for each season

$A_j=\frac{\sum_{i=1}^{L}y_{ji}}{L}$ , j varies from 1 to n

2) seasonality index is calculated for each period

$I_i=\frac{\sum_{j=1}^{n}\frac{y_{ji}}{A_j}}{n}$ , i varies from 1 to L

where
$y_{ji}$ - observation, corresponding to i period of season j.

Next - to correctly calculate the initial trend, we must take the impact of the seasonal fluctuations into account. If we only have data for a single season (e.g., year - L = 12), the trend will be difficult to distinguish from the seasonal fluctuations. Thus, the method, in terms of the initial coefficient trend calculation, requires minimum 2L of observations. With the data for two seasons ( L = 24), it is clear that it is already possible to identify a trend, comparing the respective season periods (for example, in January last year to January this year).

The commonly used formula for trend estimation

$b=\frac{1}{L}(\frac{y_{L+1}-y_1}{L} + \frac{y_{L+2}-y_2}{L} + ... + \frac{y_{L+L}-y_L}{L})$

As we can see, the data of two seasons is used.

Hence the moral - it is best to use a triple exponential smoothing for the data showing a strong trend and seasonal fluctuations. Simultaneously, it is necessary to have the results of 2L and more observations.

The calculator below is the quintessence of all three articles - it builds a simple exponential smoothing, double exponential smoothing, and triple exponential smoothing. Also, it builds forecasted values at the specified distance.
Set the parameters $\alpha$ , $\beta$ , $\gamma$ , data frequency L (4 by default - 4 quarters of a year) and forecast range m (also 4).

Note: The calculator will work only if there is at least 2L`` observations.

P.S. By the way, if the default date will be replaced with the data really have a strong trend and frequency, the mean square error of triple smoothing will be much less than the mean square error of single and double smoothing. That even surprised me. The default data, perhaps, not very indicative for demonstration.

Triple exponential smoothing

Time series

Alpha

Beta

Gamma

Number of periods

Forecast range

Calculation precision

Digits after the decimal point: 2

Single exponential mean-square error

Double smoothing mean-square error

Triple smoothing mean-square error

Double exponential smoothing

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