From a purely mathematical point of view the theory for bivariate KDEs extends directly to trivariate KDEs. There are, however, two serious practical problems in extending the methodology. The first is that large sample sizes are needed to get satisfactory results, and the second is that visualisation of the results is much more difficult.

For trivariate data each observation may be visualised as a point in three-dimensional space. In univariate problems the kernel is a two-dimensional bump, centered on a univariate point; in bivariate problems the bump and the density estimate obtained from it are three-dimensional. Contouring allows a representation of the three-dimensional estimate in two dimensions. By analogy the KDE for trivariate data is a four-dimensional object that cannot be visualised in three dimensions as it stands. A representation in three dimensions is possible using contouring, but the resultant contours can be difficult to represent and interpret. One reason is that a contour in three dimensions at some level of density can be thought of as a three-dimensional "envelope" (e.g. the surface of a balloon) that contains and conceals higher density contours.

The examples below (Figures 21 and 22) illustrate this. Here the data represents three aspects of the chemical composition of specimens of French Medieval glass reported in Barrera and Velde (1989) and further analysed in Baxter and Beardah (1995). The three variables are the Soda content; the ratio of Calcium Oxide content to Calcium and Potassium Oxide content, and the Manganese Oxide content respectively. In each figure the 25, 50 and 75% contours are shown. Figure 21 also shows the data points which lie outside the 75% contour (i.e. the least dense 25% of the data). Both figures show that there are three groupings within the data.

*Figure 21: Three dimensional contouring based upon a trivariate KDE showing 25, 50 and 75% contours, with data not included within these shown as red crosses.*

*Figure 22: Three dimensional contouring based upon a trivariate KDE, as figure 21, but omitting data not included in the contours*

The way in which the contours are built up is shown in the next sequence of figures. In Figure 23 the three-dimensional scatter plot of the data is shown. In Figures 24 to 27 the 25, 50, 75 and 100% contours are successively added, using different colours, and "gaps" through which the contouring at higher density levels may be viewed. The approach used here was inspired by that of Scott (1992). Examples of his (superior) graphics are given in his book and details of how to obtain his routines for the S-Plus package are given in the section 8.0.

*Figure 23: Three dimensional scatter plot*
*Figure 24: Three dimensional contouring (25% level)*
*Figure 25: Three dimensional contouring (25 and 50% level)*
*Figure 26: Three dimensional contouring (25, 50 and 75% levels)*
*Figure 27: Three dimensional contouring (25, 50, 75 and 100% levels)*

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URL: http://intarch.ac.uk/journal/issue1/beardah/kdeia7.html

Last updated: Tue Sep 10 1996