Triple Exponential Smoothing
A Triple Exponential Smoothing is a Hadamard Conjecture that ...
- AKA: Exponential Smoothing.
- See: Hadamard Conjecture, Management Science: A Journal of The Institute For Operations Research And The Management Sciences.
References
2017
- (Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/Exponential_smoothing#Triple_exponential_smoothing Retrieved:2017-8-23.
- Triple exponential smoothing takes into account seasonal changes as well as trends (all of which are trends). Seasonality is defined to be the tendency of time-series data to exhibit behavior that repeats itself every L periods, much like any harmonic function. The term season is used to represent the period of time before behavior begins to repeat itself. There are different types of seasonality: 'multiplicative' and 'additive' in nature, much like addition and multiplication are basic operations in mathematics.
If every month of December we sell 10,000 more apartments than we do in November the seasonality is additive in nature. Can be represented by an 'absolute' increase. However, if we sell 10% more apartments in the summer months than we do in the winter months the seasonality is multiplicative in nature. Multiplicative seasonality can be represented as a constant factor, not an absolute amount.
Triple exponential smoothing was first suggested by Holt's student, Peter Winters, in 1960 after reading a signal processing book from the 1940s on exponential smoothing. Holt's novel idea was to repeat filtering an odd number of times (ignoring 1) . While recursive filtering had been used previously, it was applied twice and four times to coincide with the Hadamard conjecture, while triple application required more than double the operations of singular convolution.
Suppose we have a sequence of observations {xt}, beginning at time t = 0 with a cycle of seasonal change of length L.
The method calculates a trend line for the data as well as seasonal indices that weight the values in the trend line based on where that time point falls in the cycle of length L.
{st} represents the smoothed value of the constant part for time t. {bt} represents the sequence of best estimates of the linear trend that are superimposed on the seasonal changes. {ct} is the sequence of seasonal correction factors. ct is the expected proportion of the predicted trend at any time t mod L in the cycle that the observations take on. As a rule of thumb, a minimum of two full seasons (or 2L periods) of historical data is needed to initialize a set of seasonal factors.
The output of the algorithm is again written as Ft+m, an estimate of the value of x at time t+m, m>0 based on the raw data up to time t. Triple exponential smoothing with multiplicative seasonality is given by the formulas : [math]\displaystyle{ \begin{align} s_0& = x_0\\ s_{t}& = \alpha \frac{x_{t}}{c_{t-L}} + (1-\alpha)(s_{t-1} + b_{t-1})\\ b_{t}& = \beta (s_t - s_{t-1}) + (1-\beta)b_{t-1}\\ c_{t}& = \gamma \frac{x_{t}}{s_{t}}+(1-\gamma)c_{t-L}\\ F_{t+m}& = (s_t + mb_t)c_{t-L+1+(m-1)\mod L}, \end{align} }[/math] where α is the data smoothing factor, 0 < α < 1, β is the trend smoothing factor, 0 < β < 1, and γ is the seasonal change smoothing factor, 0 < γ < 1.
The general formula for the initial trend estimate b0 is: : [math]\displaystyle{ \begin{align} b_0& = \frac{1}{L} \left(\frac{x_{L+1}-x_1}{L} + \frac{x_{L+2}-x_2}{L} + \ldots + \frac{x_{L+L}-x_L}{L}\right) \end{align} }[/math] Setting the initial estimates for the seasonal indices ci for i = 1,2,...,L is a bit more involved. If N is the number of complete cycles present in your data, then: : [math]\displaystyle{ \begin{align} \\ c_i& = \frac{1}{N} \sum_{j=1}^{N} \frac{x_{L(j-1)+i}}{A_j} \quad \forall i& = 1,2,\ldots,L \\ \end{align} }[/math] where : [math]\displaystyle{ \begin{align} A_j& = \frac{\sum_{i=1}^{L} x_{L(j-1)+i}}{L} \quad \forall j& = 1,2,\ldots,N \end{align} }[/math] Note that Aj is the average value of x in the jth cycle of your data.
Triple exponential smoothing with additive seasonality is given by: [math]\displaystyle{ \begin{align} s_0& = x_0\\ s_{t}& = \alpha (x_{t}-c_{t-L}) + (1-\alpha)(s_{t-1} + b_{t-1})\\ b_{t}& = \beta (s_t - s_{t-1}) + (1-\beta)b_{t-1}\\ c_{t}& = \gamma (x_{t}-s_{t-1}-b_{t-1})+(1-\gamma)c_{t-L}\\ F_{t+m}& = s_t + mb_t+c_{t-L+1+(m-1)\mod L}, \end{align} }[/math]
- Triple exponential smoothing takes into account seasonal changes as well as trends (all of which are trends). Seasonality is defined to be the tendency of time-series data to exhibit behavior that repeats itself every L periods, much like any harmonic function. The term season is used to represent the period of time before behavior begins to repeat itself. There are different types of seasonality: 'multiplicative' and 'additive' in nature, much like addition and multiplication are basic operations in mathematics.