Recommendations for Display of Projections in Multi-Dimensional Analyses

Feedback at ICHEP04, and subsequent discussion, suggest visiting the question of the graphical presentation of results of multi-dimensional analyses. The goal is to visualize the components present in the data, particularly the signal, and to have a visual impression whether our model (fit) for the data is adequate. Various approaches have been used so far, most notably “likelihood projections” and “sPlots”. The purpose of this note is to provide some guidance for BaBar authors both on procedures and how to describe them. It is perhaps worth stressing that the matter addressed here is one of presentation, not of classification. The classification problem is related to our topic in the sense that our fitting procedure is attempting to sort out how much of our data results from each of several possible categories. While there is a vast literature on the classification problem, we are not currently aware of immediately relevant literature on our presentation problem. This doesn’t mean it doesn’t exist, however, and as we become better educated our thinking could change. We are interested here in the situation of a multivariate analysis, in which it is difficult or impossible to visually present the full multi-dimensional space at once. To frame the discussion, suppose there is a set of n quantities x which are measured for each of N events in our dataset. For example, x could be x = {mES,ΔE,F}, in the usual BaBar notation. Several categories are assumed to contribute to the N sampled events, for example “signal” and “background”. Let the number of categories be denoted r. It is further assumed that the sampling distribution for x is known for each category, except possibly for a number of distribution parameters, {θi, i = 1, . . . , p} which must be estimated. A fit is performed to the observed x distribution in order to estimate the number of events, Nj , j = 1, . . . , r in each category. The parameters θ are also estimated in this fit. Estimates will always be denoted with “hats”, e.g., N̂j, in order to distinguish them from the actual quantities. We will call this the “full fit” in order to distinguish it from other fits introduced below.