So, if a = 0.9, S[t] = 0.9 ( y[t-1] + 0.1*y[t-2] + 0.01*y … 1e-6*y[t-7]) would be an implementation of EWMA. Instead, our implementation is be: S[t] = 0.9 ( y[t-1] + 0.9*y[t-2] + 0.81*y … 0.531441*y[t-7]) / (0.9+…). The critical difference is in the way past data is weighted - with EWMA data in remote past is heavily penalized, whereas, comparatively, in our implementation it is not.
Fig. 1 below shows a plot that forecasts our metric for Feb 8 using Weighted Moving Average akin to our implementation. The metric forecasted for Feb 8 was 68392533 whereas observed value was 63868927, giving an absolute difference of ~7.1%. The absolute hourly differences lie between 0.6-29%.
Fig. 1, “Weighted moving average”
characterize a time series. For example, we a time series may be represented by a linear sum of sinusoids:
Generally, a combination of Fourier transform and least-squares regression in used to find the harmonics who linear sum best fit training data. In my experiment, I used the top seven harmonics with highest power density (i.e. contribution to the model), and then used linear regression to find their amplitudes. Fig. 2 shows the model (in red) and actual data (in blue). The forecast for Feb 8 was 60476559 whereas the observed value was 63868927, giving an absolute difference of ~5.3%. The absolute hourly differences lie between 0.05-25%.
Fig. 2, “Harmonic regression” ](/assets/images/harmreg2.png)
Holt-Winters method augments EWMA to account for data trends and seasonality.
The model I used assumes that the seasonal component is additive (instead of being multiplicative). Fig. 3 illustrates the breakdown of seasonality, trend and randomness in the data, Fig. 4 contrasts observed data with predicted data (note that the prediction start after one period, and the phase lag in predicted data), while Fig. 5 shows Holt-Winters prediction for Feb 8 versus observed data. The forecasted metric for Feb 8 was 63801332 whereas observed value was 63868927, giving an absolute difference of ~0.11%. The absolute hourly differences lie between 1.3-20%.
Fig. 3 ](/assets/images/holtwintersdecomposition2.png)
Fig. 4 ](/assets/images/holtwintersfit2.png)
Fig. 5 ](/assets/images/holtwintersprediction2.png)