Time Series - Exponential Smoothing

In this chapter, we will talk about the techniques involved in exponential smoothing of time series.

Simple Exponential Smoothing

Exponential Smoothing is a technique for smoothing univariate time-series by assigning exponentially decreasing weights to data over a time period.

Mathematically, the value of variable at time ‘t+1’ given value at time t, y_(t+1|t) is defined as −

$$y_{t+1|t}\:=\:\alpha y_{t}\:+\:\alpha\lgroup1 -\alpha\rgroup y_{t-1}\:+\alpha\lgroup1-\alpha\rgroup^{2}\:y_{t-2}\:+\:...+y_{1}$$

where,$0\leq\alpha \leq1$ is the smoothing parameter, and

$y_{1},....,y_{t}$ are previous values of network traffic at times 1, 2, 3, … ,t.

This is a simple method to model a time series with no clear trend or seasonality. But exponential smoothing can also be used for time series with trend and seasonality.

Triple Exponential Smoothing

Triple Exponential Smoothing (TES) or Holt's Winter method, applies exponential smoothing three times - level smoothing $l_{t}$, trend smoothing $b_{t}$, and seasonal smoothing $S_{t}$, with $\alpha$, $\beta^{*}$ and $\gamma$ as smoothing parameters with ‘m’ as the frequency of the seasonality, i.e. the number of seasons in a year.

According to the nature of the seasonal component, TES has two categories −

• Holt-Winter's Additive Method − When the seasonality is additive in nature.

• Holt-Winter’s Multiplicative Method − When the seasonality is multiplicative in nature.

For non-seasonal time series, we only have trend smoothing and level smoothing, which is called Holt’s Linear Trend Method.

Let’s try applying triple exponential smoothing on our data.

In [316]:

from statsmodels.tsa.holtwinters import ExponentialSmoothing

model = ExponentialSmoothing(train.values, trend= )
model_fit = model.fit()


In [322]:

predictions_ = model_fit.predict(len(test))


In [325]:

plt.plot(test.values)
plt.plot(predictions_[1:1871])


Out[325]:

[<matplotlib.lines.Line2D at 0x1eab00f1cf8>]


Here, we have trained the model once with training set and then we keep on making predictions. A more realistic approach is to re-train the model after one or more time step(s). As we get the prediction for time ‘t+1’ from training data ‘til time ‘t’, the next prediction for time ‘t+2’ can be made using the training data ‘til time ‘t+1’ as the actual value at ‘t+1’ will be known then. This methodology of making predictions for one or more future steps and then re-training the model is called rolling forecast or walk forward validation.