The likelihood ratio test for hidden Markov models in two-sample problems

Algorithms were developed to approximate the 3-dimensional internal packing density distributions of fibrous filter media at different levels of detail and complexity. Each algorithm uses certain input information derived from X-ray tomographic data of filter samples.

Algorithm 1 creates a binary media structure consisting of the true (i.e. tomographically determined) void distribution plus regions of uniform packing density. The average packing density of the media is maintained constant. Algorithm 2 creates a model fibrous structure of straight fibres of equal diameter and random length positioned randomly in space, while maintaining the true (i.e. tomographically determined) fibre orientation distribution. The number and length of the fibres on average adds up to the packing density of the filter. The model fibrous structure is recreated by a stationary Poisson process of convex bodies. Algorithm 3 distributes pores of random size and location within a homogeneous matrix, such that the average packing density again coincides with the true (i.e. tomographically determined) packing density. This algorithm is also based on a stationary Poisson process of convex bodies.
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The capability of each algorithm to recreate the essential structural features of the media was tested against ‘reality’ by computing the respective overall pressure drop of the filter as well as the velocity distribution in the filter interior, and comparing with the results obtained for the ‘true’ packing density distribution of a sample measured by tomography. Compared to the assumption of a completely homogenous filter (which gives roughly 2 times the actual ∆p), all algorithms are closer to reality. The binary algorithm deviates in ∆p by a factor of 1.8; algorithm 3 comes within a factor of 1.6 of the true ∆p. The best approximation is by algorithm 2 which narrows the difference in ∆p to a factor of 1.4