Modeling higher-order exponential smoothing

The concept can be further extended to higher-order exponential smoothing with an n^th order polynomial model:

Here, error ε_t ∼ N(0,σ²) is normally distributed with 0 mean and σ^₂ variance. The exponential smoothers used for higher order are as follows:

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Here, is weights for smoothers. Usually, higher-order exponential smoothing is not used in time as even for second order smoothing, the computation becomes very hard and approaches such as Autoregressive Integrated Moving Average (ARIMA) are utilized. It will be further discussed in Chapter 4, Auto Regressive Models.

Another very popular exponential smoothing is triple exponential smoothing. The triple exponential smoothing allows you to capture seasonality with level and trend. The relationship between levels, trends, and seasonality is defined using the following set of equations:

In these equations, F_t captures levels of observation at time t. Similarly, T_t and S_t captures trend and seasonality...