A likelihood ratio model for three-way data coupled with a Tucker3 model

In forensic science, the analysis of diesel fuel is particularly important in fire investigations. The task is usually to compare the original accelerant found among the suspect's belongings with the fire debris to find out if the fire could have been caused by the use of that particular diesel fuel (called source). The major problem when comparing the original accelerant with the fire debris is the weathering process of the accelerant taking place during the fire. The weathering process makes the composition of the accelerant change with the weathering state and may differ from the composition of the original accelerant. In this context the question arises if samples of the fire debris containing the accelerant weathered to different degrees are still so similar to the original accelerant that they can be regarded as coming from the same source (this particular accelerant) and whether samples of fire debris with accelerants from different sources are easily identified as such regardless of their weathering state. The hybrid likelihood ratio (LR) model which takes into account the information about the similarity and the frequency of observing the compared features in the samples was used for answering the above issues. Hybrid LR models use the new set of a limited number of variables that is generated using a variety of chemometric tools to summarise the data as well as possible and highlight the features that make each source of samples uniquely defined. The model was built for three-way GC–MS data of diesel fuel samples. Tucker3 model decomposed the three-dimensional array of the database into three matrices referring to GC, MS and samples (concentration) modes. The scores on the linear discriminant functions for the concentration mode served as an input for LR models. True origins for the majority of samples were indicated despite different weathering.