Forecasting: Principles and Practice 3rd (Ch8 Exponential smoothing)

exponential smoothing 簡單來說就是過去觀察值的加權平均，而根據時間變化決定權重值。

Forecasts produced using exponential smoothing methods are weighted averages of past observations, with the weights decaying exponentially as the observations get older.

Python套件: https://www.statsmodels.org/dev/tsa.html

8.1 Simple exponential smoothing

\[\begin{align*} \hat{y}_{3|2} & = \alpha y_2 + (1-\alpha) \left[\alpha y_1 + (1-\alpha) \ell_0\right] \\ & = \alpha y_2 + \alpha(1-\alpha) y_1 + (1-\alpha)^2 \ell_0 \\ \hat{y}_{4|3} & = \alpha y_3 + (1-\alpha) [\alpha y_2 + \alpha(1-\alpha) y_1 + (1-\alpha)^2 \ell_0]\\ & = \alpha y_3 + \alpha(1-\alpha) y_2 + \alpha(1-\alpha)^2 y_1 + (1-\alpha)^3 \ell_0 \\ & ~~\vdots \\ \hat{y}_{T+1|T} & = \sum_{j=0}^{T-1} \alpha(1-\alpha)^j y_{T-j} + (1-\alpha)^T \ell_{0}. \end{align*}\]

8.2 Methods with trend

Holt’s linear trend method

\[\begin{align*} \text{Forecast equation}&& \hat{y}_{t+h|t} &= \ell_{t} + hb_{t} \\ \text{Level equation} && \ell_{t} &= \alpha y_{t} + (1 - \alpha)(\ell_{t-1} + b_{t-1})\\ \text{Trend equation} && b_{t} &= \beta^*(\ell_{t} - \ell_{t-1}) + (1 -\beta^*)b_{t-1}, \end{align*}\]

The forecast function is no longer flat but trending. The \(h\)-step-ahead forecast is equal to the last estimated level plus \(h\) times the last estimated trend value. Hence the forecasts are a linear function of \(h\).

Damped trend methods

In conjunction with the smoothing parameters \(\alpha\) and \(\beta^*\) (with values between 0 and 1 as in Holt’s method), this method also includes a damping parameter \(0<\phi<1\):

\[\begin{align*} \hat{y}_{t+h|t} &= \ell_{t} + (\phi+\phi^2 + \dots + \phi^{h})b_{t} \\ \ell_{t} &= \alpha y_{t} + (1 - \alpha)(\ell_{t-1} + \phi b_{t-1})\\ b_{t} &= \beta^*(\ell_{t} - \ell_{t-1}) + (1 -\beta^*)\phi b_{t-1}. \end{align*}\]

Example: Internet usage

In this example, the process of selecting a method was relatively easy as both MSE and MAE comparisons suggested the same method (damped Holt’s). However, sometimes different accuracy measures will suggest different forecasting methods, and then a decision is required as to which forecasting method we prefer to use. As forecasting tasks can vary by many dimensions (length of forecast horizon, size of test set, forecast error measures, frequency of data, etc.), it is unlikely that one method will be better than all others for all forecasting scenarios. What we require from a forecasting method are consistently sensible forecasts, and these should be frequently evaluated against the task at hand.

8.3 Methods with seasonality

There are two variations to this method that differ in the nature of the seasonal component. The additive method is preferred when the seasonal variations are roughly constant through the series, while the multiplicative method is preferred when the seasonal variations are changing proportional to the level of the series.

With the additive method, the seasonal component is expressed in absolute terms in the scale of the observed series, and in the level equation the series is seasonally adjusted by subtracting the seasonal component. Within each year, the seasonal component will add up to approximately zero.

With the multiplicative method, the seasonal component is expressed in relative terms (percentages), and the series is seasonally adjusted by dividing through by the seasonal component.

Python套件參考:

https://www.statsmodels.org/stable/generated/statsmodels.tsa.exponential_smoothing.ets.ETSModel.html#statsmodels.tsa.exponential_smoothing.ets.ETSModel

https://www.statsmodels.org/dev/examples/notebooks/generated/ets.html

懶得研究算法...

Holt-Winters’ damped method

Damping is possible with both additive and multiplicative Holt-Winters’ methods. A method that often provides accurate and robust forecasts for seasonal data is the Holt-Winters method with a damped trend and multiplicative seasonality.

8.4 A taxonomy of exponential smoothing methods

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8.5 Innovations state space models for exponential smoothing

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8.6 Estimation and model selection

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8.7 Forecasting with ETS models

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