Predictive nonlinear biplots: Maps and trajectories

When the difference between samples is measured using a Euclidean-embeddable dissimilarity function, observations and the associated variables can be displayed on a nonlinear biplot (Gower and Harding, 1988). Furthermore, a nonlinear biplot is predictive if information on variables is added in such a way that it allows the values of the variables to be estimated for points in the biplot. I will introduce a predictive nonlinear biplot map, an r dimensional plot which displays the predicted value of a variable for every point in the plot. Using such maps I will show that when the dissimilarity function with respect to a new point is not smooth everywhere, the set of predicted values can appear discrete even though the data are assumed to be continuous. That is, on an r dimensional biplot the region of points that predict a given value might also be r dimensional. Prediction trajectories that approximate 2 dimensional predictive regions are also introduced. These prediction trajectories allow information about two or more variables to be displayed on such 2 dimensional biplots.