Cagri Ozcaglar

We propose a novel optimization-based approach to embedding heterogeneous high-dimensional data characterized by a graph. The goal is to create a two-dimensional visualization of the graph structure such that edge-crossings are minimized while preserving proximity relations between nodes. This paper provides a fundamentally new approach for addressing the crossing minimization criteria that exploits Farkas' Lemma to re-express the condition for no edge-crossings as a system of nonlinear inequality constraints. The approach has an intuitive geometric interpretation closely related to support vector machine classification. While the crossing minimization formulation can be utilized in conjunction with any optimization-based embedding objective, here we demonstrate the approach on multidimensional scaling by modifying the stress majorization algorithm to include penalties for edge crossings. The proposed method is used to (1) solve a visualization problem in tuberculosis molecular epidemiology and (2) generate embeddings for a suite of randomly generated graphs designed to challenge the algorithm. Experimental results demonstrate the efficacy of the approach. The proposed edge-crossing constraints and penalty algorithm can be readily adapted to other supervised and unsupervised optimization-based embedding or dimensionality reduction methods. The constraints can be generalized to remove overlaps between any graph components represented as convex polyhedrons including node-edge and node-node intersections.

Biomarkers of Mycobacterium tuberculosis complex (MTBC) mutate over time. Among the biomarkers of MTBC, spacer oligonucleotide type (spoligotype) and mycobacterium interspersed repetitive unit (MIRU) patterns are commonly used to genotype clinical MTBC strains. In this study, we present an evolution model of spoligotype rearrangements using MIRU patterns to disambiguate the ancestors of spoligotypes. We use a large patient dataset from the United States Centers for Disease Control and Prevention (CDC) to generate this model. Based on the contiguous deletion assumption and rare observation of convergent evolution, we first generate the most parsimonious forest of spoligotypes, called a spoligoforest, using three genetic distance measures. An analysis of topological attributes of the spoligoforest and number of variations at the direct repeat (DR) locus of each strain reveals interesting properties of deletions in the DR region. First, we compare our mutation model to existing mutation models of spoligotypes and find that our mutation model produces as many within-lineage mutation events as other models, with slightly higher segregation accuracy. Second, based on our mutation model, the number of descendant spoligotypes follows a power law distribution. Third, contrary to prior studies, the power law distribution does not plausibly fit to the mutation length frequency. Moreover, we find that the total number of mutation events at consecutive spacers follows a spatially bimodal distribution. The two modes are spacers 13 and 40, which are hotspots for chromosomal rearrangements, and the change point is spacer 34, which is absent in most MTBC strains. Based on this observation, we built two alternative models for mutation length frequency: the Starting Point Model (SPM) and the Longest Block Model (LBM). Both models are plausibly good fits to the mutation length frequency distribution, as verified by the goodness-of-fit test. We also apply SPM and LBM to a dataset from Institut Pasteur de Guadeloupe and verify that these models hold for different strain datasets.

Biomarkers of Mycobacterium tuberculosis complex (MTBC) mutate over time. Among the biomarkers of MTBC, spacer oligonucleotide type (spoligotype) and Mycobacterium Interspersed Repetitive Unit (MIRU) patterns are commonly used to genotype clinical MTBC strains. In this study, we present an evolution model of spoligotype rearrangements using MIRU patterns to disambiguate the ancestors of spoligotypes, in a large patient dataset from the United States Centers for Disease Control and Prevention (CDC). Based on the contiguous deletion assumption and rare observation of convergent evolution, we first generate the most parsimonious forest of spoligotypes, called a spoligoforest, using three genetic distance measures. An analysis of topological attributes of the spoligoforest and number of variations at the direct repeat (DR) locus of each strain reveals interesting properties of deletions in the DR region. First, we compare our mutation model to existing mutation models of spoligotypes and find that our mutation model produces as many within-lineage mutation events as other models, with slightly higher segregation accuracy. Second, based on our mutation model, the number of descendant spoligotypes follows a power law distribution. Third, contrary to prior studies, the power law distribution does not plausibly fit to the mutation length frequency. Finally, the total number of mutation events at consecutive DR loci follows a bimodal distribution, which results in accumulation of shorter deletions in the DR region. The two modes are spacers 13 and 40, which are hotspots for chromosomal rearrangements. The change point in the bimodal distribution is spacer 34, which is absent in most MTBC strains. This bimodal separation results in accumulation of shorter deletions, which explains why a power law distribution is not a plausible fit to the mutation length frequency.

Strains of Mycobacterium tuberculosis complex (MTBC) can be classified into major lineages based on their genotype. Further subdivision of major lineages into sublineages requires multiple biomarkers along with methods to combine and analyze multiple sources of information in one unsupervised learning model. Typically, spacer oligonucleotide type (spoligotype) and mycobacterial interspersed repetitive units (MIRU) are used for TB genotyping and surveillance. Here, we examine the sublineage structure of MTBC strains with multiple biomarkers simultaneously, by employing a tensor clustering framework (TCF) on multiple-biomarker tensors.

Strains of the Mycobacterium tuberculosis complex (MTBC) can be classified into coherent lineages of similar traits based on their genotype. We present a tensor clustering framework to group MTBC strains into sublineages of the known major lineages based on two biomarkers: spacer oligonucleotide type (spoligotype) and mycobacterial interspersed repetitive units (MIRU). We represent genotype information of MTBC strains in a high-dimensional array in order to include information about spoligotype, MIRU, and their coexistence using multiple-biomarker tensors. We use multiway models to transform this multidimensional data about the MTBC strains into two-dimensional arrays and use the resulting score vectors in a stable partitive clustering algorithm to classify MTBC strains into sublineages. We validate clusterings using cluster stability and accuracy measures, and find stabilities of each cluster. Based on validated clustering results, we present a sublineage structure of MTBC strains and compare it to the sublineage structures of SpolDB4 and MIRU-VNTRplus.