Comparison of spectra using a Bayesian approach. An argument using oil spills as an example

The problem of assigning a probability of matching a number of spectra is addressed. The context is in environmental spills when an EPA needs to show that the material from a polluting spill (e.g., oil) is likely to have originated at a particular site (factory, refinery) or from a vehicle (road tanker or ship). Samples are taken from the spill, and candidate sources and are analyzed by spectroscopy (IR, fluorescence) or chromatography (GC or GC/MS). A matching algorithm is applied to pairs of spectra giving a single statistic (R). This can be a point-to-point match giving a correlation coefficient or a Euclidean distance or a derivative of these parameters. The distributions of R for same and different samples are established from existing data. For matching statistics with values in the range {0,1} corresponding to no match (0) to a perfect match (1) a beta distribution can be fitted to most data. The values of R from the match of the spectrum of a spilled oil and of each of a number of suspects are calculated and Bayes' theorem is applied to give a probability of matches between spill sample and each candidate and the probability of no match at all. The method is most effective when simple inspection of the matching parameters does not lead to an obvious conclusion; i.e., there is overlap of the distributions giving rise to dubiety of an assignment. The probability of finding a matching statistic if there were a match to the probability of finding it if there were no match, expressed as a ratio (called the likelihood ratio), is a sensitive and useful parameter to guide the analyst. It is proposed that this approach may be acceptable to a court of law and avoid challenges of apparently subjective opinion of an analyst. Examples of matching the fluorescence and infrared spectra of diesel oils are given.