In most of the examples used in the paper the "bump" or kernel used to construct the KDE is the normal probability density function. The spread of this kernel, which affects the appearance of the KDE, is determined by the standard deviation of the normal distribution. In the context of kernel density estimation we shall call this the window-width and denote it by h. For distributions other than the normal, with finite spread, the window width is directly related to this spread; however we shall not consider such cases here.

*Figure 28: The effect on the KDE of different choices of window-width, h*.

That the choice of h can dramatically affect the appearance of the KDE is easily illustrated by the example in Figure 28, from Beardah and Baxter (1996), which uses data on the diameters of sixty Bronze Age Italian cups, and was also used in Figure 1. This shows the appearance of the KDE for a normal kernel with four choices of h. The appearance for the smallest value of h is rather jagged, and the data is under-smoothed with too many peaks. As h increases the appearance of the KDE becomes smoother until, in the final over-smoothed diagram, any structure in the data has disappeared.

Clearly the choice of h is important, and this is considered in the rest of this section.

If the true density to be estimated is normal, and if a normal kernel is used, theory exists to determine the optimal choice of h (Silverman, 1986, p.45). It is approximately (actually 1.06 times) the standard deviation of the true density, divided by the fifth root of the sample size. This is the normal scale rule. In practice the standard deviation would need to be estimated. For non-normal, multi-modal data, the normal scale rule will over-smooth the data. For the Bronze Age cup diameters used in Figures 1 and 28 the normal scale rule gives a value of h = 2.5. This gives the third diagram in Figure 28, where the second mode is almost completely smoothed out.

A legitimate way of deciding on a value of h is subjective choice. Starting, perhaps, with the normal scale rule and systematically reducing the value of h allows the change in the appearance of the KDE to be monitored. The appearance will often be qualitatively similar over a wide range of h, before spurious "bumps" begin to suggest under-smoothing, so that a precise choice may not be important for presentational purposes.

*Figure 29: A further illustration to show the effect of varying h*.

This is illustrated in Figure 29, using the Bronze Age cup diameters, where a broadly similar picture is obtained for h in the range 1 to 2, with spurious bumps beginning to appear for h = 1.

There are many methods of automatically selecting h that improve on the normal scale rule. A good recent account is given by Wand and Jones (1995), and a summary of a subset of these is presented in a technical fashion in Baxter and Beardah (1996). A "good" KDE is one which is close to the true but unknown density in some sense. One possible measure of "closeness" can be shown to be a function of a term that can be interpreted as the "roughness" of the true density, which is also unknown. Direct plug-in (DPI) and solve-the-equation (STE) rules are obtained from different ways of estimating this roughness. In the STE approach, which is the one that will be used in this paper, an equation is established that relates h to the unknown density. An initial choice of h allows the density to be estimated and h is then recalculated. This process is repeated until h converges. Using the Bronze Age cup diameter data, Figure 30 contrasts the STE with the normal scale estimator. The bimodal structure is much more apparent, although it is possible that the STE estimate is slightly under-smoothing.

*Figure 30: A comparison of the normal scale and STE rules for generating h*.

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Last updated: Tue Sep 10 1996