Statistical models for firearm and tool mark image comparisons based on the congruent matching cells (CMC) method

Abstract

In the branch of forensic science known as firearm evidence identification, various similarity scores have been proposed to compare firearm marks. Some similarity score comparisons, for example, congruent matching cells (CMC) method, are based on pass-or-fail tests. The CMC method compares the pairwise topography images of breech face impressions, from which the similarity score is derived for quantifying their topography similarity. For an image pair, the CMC method determines a certain number of correlated cell pairs. Next, each correlated pair is determined to be a congruent match cell (CMC) pair, or not based on several identification parameters. The number of CMC pairs as a threshold is required so that the two images of surface topographies can be either identified as matching or determined to be non-matching. To reliably estimate error rates or evaluate likelihood ratio (LR), the key is to find an appropriate probability distribution for the frequency distribution of the observed CMC results. This paper discusses four statistical models for CMC measurements, which are binomial and three binomial-related probability distributions. In previous studies, for a sequence of binomial distributed or other binomial-related distributed random variables (r.v.), the number of Bernoulli trials N for each r.v. is assumed to be the same. However, in practice, N(the number of cell pairs in an image pair) varies from one r.v. (or one image pair) to another. In that case, the term, frequency function, of the CMC results is not appropriate. In this paper, the generalized frequency function is introduced to depict the behavior of the CMC values and its limiting distribution is provided. Based on that, nonlinear regression models are used to estimate the model parameters. The methodology is applied to a set of actual CMC values of fired cartridge cases.