Statistical Analysis of Unlabeled Point Sets: Comparing Molecules in Chemoinformatics

Abstract

We consider Bayesian methodology for comparing two or more unlabeled point sets. Application of the technique to a set of steroid molecules illustrates its potential utility involving the comparison of molecules in chemoinformatics and bioinformatics. We initially match a pair of molecules, where one molecule is regarded as random and the other fixed. A type of mixture model is proposed for the point set coordinates, and the parameters of the distribution are a labeling matrix (indicating which pairs of points match) and a concentration parameter. An important property of the likelihood is that it is invariant under rotations and translations of the data. Bayesian inference for the parameters is carried out using Markov chain Monte Carlo simulation, and it is demonstrated that the procedure works well on the steroid data. The posterior distribution is difficult to simulate from, due to multiple local modes, and we also use additional data (partial charges on atoms) to help with this task. An approximation is considered for speeding up the simulation algorithm, and the approximating fast algorithm leads to essentially identical inference to that under the exact method for our data. Extensions to multiple molecule alignment are also introduced, and an algorithm is described which also works well on the steroid data set. After all the steroid molecules have been matched, exploratory data analysis is carried out to examine which molecules are similar. Also, further Bayesian inference for the multiple alignment problem is considered.

Alignment of Multiple Configurations Using Hierarchical Models

“…As far as we know, ours is the only available method which models multiple matchings of different types, in a general and mathematically rigorous manner. In terms of computation speed, the methodology of Dryden, Hirst, and Melville (2007) would probably be more efficient than the one proposed in this article when C is large. So the choice of method might depend upon the number of configurations to be aligned and the extent to which one wished to retain full statistical efficiency and control the prior match specifications.…”

“…The problem of aligning multiple molecules has also been treated by Dryden, Hirst, and Melville (2007); their approach is similar to ours, in that a hierarchical model is constructed and a hidden reference molecule defined. However, the hidden points are not integrated out, and the transformations are maximized out using Procrustean registration techniques.…”

“…In many contexts, additional information on the observations is available. For example, Dryden, Hirst, and Melville (2007) included "marks" on each atom of the molecules to be aligned; these marks may contain information influencing the matching, such as partial charge and van der Waals radius. In a similar vein, Green and Mardia (2006) included the possibility of coloring the observations, to model the possibility that points of the same color are more likely to be matched a priori.…”

“…The elegance of the pairwise model makes this extension natural and relatively straightforward. The problem of matching multiple configurations has also been addressed by Dryden, Hirst, and Melville (2007); see the Discussion section of our article for further details.…”

“…We will regroup the matches in a parameter M. How these matches are represented is irrelevant for the moment; one might wish to use binary matrices, as did Dryden, Hirst, and Melville (2007) and Green and Mardia (2006), or write each match as an index array containing the labels of the matched points and those of the configurations involved. For M to be consistent with the model described in Section 2.1, we must in particular ensure that a given point is involved in exactly one (maximal) match, with the convention that an unmatched point is itself a trivial match of size 1.…”