# Exponential Smoothing

Regular moving average forecasts use the mean of the past k observations as the forecast. This implies equal weight (*1/k*)
to all *k* data points. With the data we have, it is reasonable to assume that the most recent observations are probably
the best approximations for forecasts. The NHL has changed tremendously over the past 83 years and its actions in the recent past
have a much larger impact on its present and future state than the going further back in time. Exponential smoothing methods
provide a weight scheme that incorporates decreasing weights as observations grow older. Simple exponential smoothing assumes
that there are no trend or seasonal aspects in the data and that the level of the series changes slowly over time.

### Simple Exponential Smoothing

Since the alpha value that minimizes SSE is 1.0 (*Appendix C-1*), we see that the simple exponential smoothing is just
the Naive forecast. This assumes that the forecast for the next time period will be equal to the previous observed value. The
graph below shows this result. As we can see the graph shows that there is no smoothing in the forecast, and shows that
exponential smoothing is equivalent to using the last observation as a forecast.

### Holt's Method

Double exponential smoothing allows forecasting data with trends. Since our data could possibly contain an underlying trend
component, we conduct an analysis using Holt's Method. The results from this analysis can be viewed in *Appendix C-2*.
We notice that the Sum of Squared Errors is minimized when alpha is 1.00 and gamma is 0.00. This is a similar result to the one
we saw in the single exponential method. Here we get the same result as a Naive Forecast Method 1. The graph shows that the line
(blue) representing the fitted games per game using Holt's method and the line (red) representing the fitted games per game using
simple exponential smoothing, are identical. It shows us that the forecast for the next time period is the observed value of the
previous time period. Thus, we conclude that there is no identifiable trend in the data.