Linear Poisson Models: A Pattern Recognition Solution to the Histogram Composition Problem
P. Tar, N. A. Thacker · Research Explorer (The University of Manchester) · 2014
The use of histogram data is ubiquitous within the sciences and histograms composed of linear combinations of sub-processes are common. The statistical properties of recorded frequencies are well understood, i.e. Poisson statistics can model the probabilities of ob-serving discrete events, from which an in-depth theoretical analysis of histogram compo-nents and their errors can be derived. Despite this, linear decomposition techniques such as Independent Component Analysis, Principal Component Analysis, and their variants, have largely focused on continuous data, with little or no account of error characteris-tics on estimated parameters. We argue that the properties of histograms, i.e. Poisson, non-negative discrete frequencies, requires a linear decomposition technique for mak-ing quantitative measurements. We also argue that a solution appropriate for scientific analysis tasks must involve a good understanding of noise so that uncertainties in mea-surements can be reported to end users. We present an approach to the analysis of his-tograms capable of summarising data in terms of Probability Mass Functions, weighting quantities and associated covariance matrices which we believe is suitable for scientific applications. The end result is a quantitative pattern recognition system capable of learn-ing distributions from training data, then estimating the quantity of similar distributions in new incoming data. 1