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. 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. In this paper an r dimensional biplot which maps the predicted value of a variable for every point in the plot, is introduced. Using such maps it is shown that even with continuous data, predicted values do not always vary continuously across the biplot plane. Prediction trajectories that are appropriate for summarising such non-continuous prediction maps are also introduced. These prediction trajectories allow information about two or more variables to be estimated even when the underlying predicted values do not vary continuously.