Curve Fitting Toolbox

Moving Average Filtering

A moving average filter smooths data by replacing each data point with the average of the neighboring data points defined within the span. This process is equivalent to lowpass filtering with the response of the smoothing given by the difference equation

where y_s(i) is the smoothed value for the ith data point, N is the number of neighboring data points on either side of y_s(i), and 2N+1 is the span.

The moving average smoothing method used by the Curve Fitting Toolbox follows these rules:

• The span must be odd.
• The data point to be smoothed must be at the center of the span.
• The span is adjusted for data points that cannot accommodate the specified number of neighbors on either side.
• The end points are not smoothed because a span cannot be defined.

Note that you can use MATLAB's filter function to implement difference equations such as the one shown above. However, because of the way that the end points are treated, the toolbox moving average result will differ from the result returned by filter. Refer to Difference Equations and Filtering in the MATLAB documentation for more information.

For example, suppose you smooth data using a moving average filter with a span of 5. Using the rules described above, the first four elements of ys are given by

• ys(1) = y(1)
  ys(2) = (y(1)+y(2)+y(3))/3
  ys(3) = (y(1)+y(2)+y(3)+y(4)+y(5))/5
  ys(4) = (y(2)+y(3)+y(4)+y(5)+y(6))/5
  

Note that y_s(1), y_s(2), ... ,y_s(end) refer to the order of the data after sorting, and not necessarily the original order.

The smoothed values and spans for the first four data points of a generated data set are shown below.

Plot (a) indicates that the first data point is not smoothed because a span cannot be constructed. Plot (b) indicates that the second data point is smoothed using a span of three. Plots (c) and (d) indicate that a span of five is used to calculate the smoothed value.

Smoothing Data

Lowess and Loess: Local Regression Smoothing