Introduction

Statistical data analysis with multivariate distributions is often performed with one-dimensional projections when variable distributions consist of essentially uncorrelated variables. This approach works well when observables are indeed uncorrelated. In some other cases, ignoring correlations for a given experiment leads to biases or event mis-classifications.

In this note, the joint probability distribution functions (PDFs) are approximated with three different techniques. Ensemble tests are then performed to identify the best general method for signal extraction with different classes of events. The limitation of each method are investigated with large statistics toy Monte Carlo simulations.

The three methods used for the evaluation of the joint PDFs are [1]:

1. The 1-dimensional distribution functions. This standard method used the product of the marginal distribution functions for the evaluation of the join PDFs. Monte Carlo samples are often used for the computation of the 1-dimensional distribution functions.

2. The Projection and Correlation Approximation (PCA). The PCA method is a modification to the standard joint PDF calculation. It accounts for correlations in the input variables with a canonical transformation of the correlated input variables into a set of uncorrelated variables [2].

3. The multi-dimensional distribution functions. This is a general purpose multivariate classification technique that is easy to understand and apply to binned data. Normally, one parametrized the data as a function of binned variables and the PDFs are extracted for each bin of the input variables. The multi-dimensional must then relies on very large Monte Carlo samples to insure the adequate representation of the joint PDFs in each bins (even those with low statistical population).

This SNO analysis note also describes how to use qSigEx for the proposed task and how to investigate biases and binned effects in multivariate data analysis for event classification.