# Brown’s Simple Exponential Smoothing

Simple exponential smoothing is similar to the WMA except that the window size is infinite, and the weighting factors decrease exponentially.

$$Y_1=X_1$$ $$Y_2=(1-\alpha)Y_1+\alpha X_1=X_1$$ $$Y_3=(1-\alpha)Y_2+\alpha X_2=(1-\alpha)X_1+\alpha X_2$$ $$Y_4=(1-\alpha)Y_3+\alpha X_3=(1-\alpha)^2 X_1+\alpha (1-\alpha) X_2+\alpha X_3$$ $$Y_5=(1-\alpha)^3 X_1+\alpha (1-\alpha)^2 X_2+\alpha (1-\alpha) X_3 + \alpha X_4$$ $$Y_{T+1}=(1-\alpha)^T X_1+\alpha \sum_{i=1}^T (1-\alpha)^{T-i}X_{i+1}$$ $$\cdots$$ $$Y_{T+m}=Y_{T+1}$$

Where:

• $\alpha$ is the smoothing factor ($0 \prec \alpha \prec 1$)

As we have seen in the WMA, the simple exponential is suited for time series with a stable mean or a very slow-moving mean.

Example 1:

Let’s use the monthly sales data (as we did in the WMA example).

In the example above, we chose the smoothing factor to be 0.8, which begs the question: What is the best value for the smoothing factor?

Estimating the best $\alpha$ value from the data

In practice, the smoothing parameter is often chosen by a grid search of the parameter space; that is, different solutions for $\alpha$ are tried, starting with, for example, $\alpha=0.1$ to $\alpha = 0.9$, with increments of 0.1. Then $\alpha$ is chosen to produce the smallest sums of squares (or mean squares) for the residuals (i.e., observed values minus one-step-ahead forecasts; this means squared the error is also referred to as ex-post mean squared error (ex-post MSE for short).

Using the TSSUB function (to compute the error), SUMSQ, and Excel data tables, we computed the sum of the squared errors (SSE) and plotted the results:

The SSE reaches its minimum value around 0.8, so we picked this value for our smoothing.