The basic idea of the PDE method is to classify events as "signal" or "background" events depending on the values of several variables simultaneously. An "event" is a single collision producing a set of final state particles, and events can be characterized by some selected variables (energies, angles, ...) which we combine into a "feature vector" x. If events are characterized by N variables, then each event is a point in an N-dimensional "feature space".

A distribution of many such events in feature space can be approximated by a smooth function f(x) which is formed by placing an N-dimensional Gaussian ( or a "kernel") at each data point and adding them all up. Such a function is called a (probability) density estimate for the entire distribution. The more data there is, the better the accuracy of the estimate. If we generate signal events by a Monte Carlo, we can then form a signal density estimate s(x) in this manner. Similarly, we can form a background density estimate b(x), using either Monte Carlo events or perhaps real events. We can then construct a discriminant function D(x), defined as D(x) = s(x)/[s(x)+b(x)]. If the signal and background events are different in character, i.e. if they lie in different regions of feature space, then the values of D are near 1 for signal events and near 0 for background events.

We can now test the character of "candidate" events by simply computing D(x) for each such event. If the values of D are near 1, the events are likely to be signal. Note that the equation D(x) = const. defines an N-dimensional contour in feature space (e.g. in two dimensions the equation D(x,y) = const. defines a contour in the xy-plane). If we select a set of events with a cut of e.g. D>0.9, we are making a graphical cut in N dimensions, selecting a region where the signal density exceeds the background density by a factor of 10. Thus the PDE method has an intuitive graphical interpretation.

The PDE method was developed in collaboration with Lasse Holmstrom (Univ. of Helsinki) and Steve Sain (Rice), and it is published in Comp. Phys. Comm. 88, 195 (1995).  The method was used successfully in the top quark search, and it is now embedded in Quaero, a Web-based program which allows users to test any theoretical model against D0 data.

aPDE is a multivariate technique designed for parameter estimation. The probability densities are estimated using Gaussian kernels just as in PDE, but this time the parameters to be estimated are treated on an equal footing with the variables x, and the joint probability density f(a,x) is constructed. For example, in case of the top quark a could be the top mass, and we could generate top events with random mass m, then generate x associated with each m using our knowledge of top decays, and then finally construct the joint density f(m,x).

If we have a real top event with a given feature vector x, then the  posterior probability density f(m|x) gives us the probability that the top mass is m, given the vector x. The most likely value of m for that event is the value which maximizes f(m|x). But for a fixed x the posterior density is proportional to the joint density, and the best estimate of m for a given x is therefore obtained by finding the maximum of the joint density f(m,x), which we have constructed explicitly.

aPDE was developed with Lasse Holmstrom and Bruce Knuteson, a former Rice undergraduate who is now a faculty member at MIT. Bruce deserves most of the credit for this work.

dPDE is a multivariate method for comparing two N-dimensional distributions. It is often interesting to compare two data sets A and B directly, and to quantify the degree of similarity between them. In one dimension the Kolmogorov-Smirnov (K-S) test works well, but in higher dimensions the comparison is not that straightforward.

In dPDE the N-dimensional probability densities for the two data sets are again formed by kernel estimation. We then define two discriminant functions, one appropriate for the null hypothesis (i.e. the hypothesis that A and B identical, apart from statistical fluctuations) and another appropriate for the actual data. The values of the two discriminant functions are calculated at randomly selected test points, and the distributions of these values are then compared using the standard K-S test. Thus the method combines the rich information content of N-dimensional data with the simplicity of the K-S test.

This method was developed with Jim Loudin, a Rice undergraduate. Jim gave a talk about our method in a conference at SLAC, and the paper is published in the proceedings. We are currently putting together a Web-based version of the method which anyone could use. In the near future we hope to apply dPDE to D0 data.