Polygon Methods

By connecting the centre point of the top of each bin with a continuous line a frequency polygon is obtained. Scott (1985,1983) considered the problem of choosing amongst the collection of multivariate frequency polygons each with the same smoothing parameter but differing bin origins. Rather than choosing the smoothest¹ curve or surface, he proposed averaging a series of such polygons. As the average of piecewise linear curves is also piecewise linear² the result appears to be a frequency polygon.

A similar device that is just as general is the Averaged Shifted Histogram (ASH) in which several shifted histograms are averaged³. Again since the average of piecewise constant functions is itself a piecewise constant function, the resulting ASH appears to be a frequency polygon as well. In practice the ASH is made continuous using some form of linear interpolation. ASH is a useful tool for density estimation, however these methods are complex and do not yield the smooth style of density estimate preferred for the application here.

Figure:ASH of the Old Faithful data with different bin counts and kernels.