Evidence evaluation and functional data analysis

Abstract

In forensic contexts, evidence gathered are valuable in suggesting possible crimes. The problems forensic scientists are often interested in are whether evidence found at the scene of the crime match those that are found related to some suspect. Prosecution and defense propositions are often put forward assuming the evidence came from the same source or they are not from the same source. A popular and objective measure of the value of evidence is the use of the likelihood ratios that is calculated as the ratio between the probabilities of observing the evidence given each proposition.

In this thesis, we will provide methodologies for the evaluation of the likelihood ra- tio when evidence are characterised by functional data such as mass spectrophotometry data. Three models will be developed based on fundamental functional data analysis and use of systems of basis functions for the decomposition of means. Each of the three models considers a different covariance structure for between- and within-group variations. They are independent and constant variances across groups, independent and constant within-group variances and auto-covariance. Two models that only make use of the data after dimension reduction are also developed. One is multivariate nor- mal random-effects model with constant covariance matrix and the other one puts an inverse Wishart prior distribution on within-group covariance matrix. Both models consider two levels of variability, within- and between-group, for the mean.

All models will be used to calculate likelihood ratios for three sets of data and re- sults will be compared using different measures of performances such of rates of mis- leading evidence, Tippett plots and empirical cross-entropy (ECE) plots. Sensitivity analysis is then done to test the effect of using different estimations of the hyperpa- rameters on likelihood ratios. Furthermore, we also preprocessed the data in another way, that is taking first order differences and replace the original data to feed into the models. Conclusions will be drawn based on the performances of each model on each dataset, including sensitivity analysis and more data preprocessing. Finally, guidances on how to choose the model for the calculation of likelihood ratios for other kinds of data will be provided.