Covariance Estimation for Small Sample Data with Applications to Forensic Glass

This study is motivated by a trace element analysis procedure often used in forensic science for the analysis of glass and other trace evidence. Existing glass data sets used to analyze error rates are of small sample size (n) relative to dimension (p); and potential benefits may arise from the consolidation of multiple small data sets into one. However, while forensic data is generally not readily available, information concerning their covariance (or correlation) matrices may be. This research proposes two methods for detecting similarity between covariance matrices and their true effective dimension as well as a method for combining covariance matrices based on the subspaces they span. These metrics and methods may apply to similar situations where data is inaccessible or otherwise unavailable due to privacy or other reasons. They may also apply if the data is believed or known to contain many outliers, as analysis of the covariance matrix may be more robust in reducing some of the effect of outliers. Although motivated by small n, we believe these methods may scale to the case where both n and p are large or p > n.