Partition Decoupling for Multi-gene Analysis of Gene Expression Profiling Data

We present the extention and application of a new unsupervised statistical learning technique--the Partition Decoupling Method--to gene expression data. Because it has the ability to reveal non-linear and non-convex geometries present in the data, the PDM is an improvement over typical gene expression analysis algorithms, permitting a multi-gene analysis that can reveal phenotypic differences even when the individual genes do not exhibit differential expression. Here, we apply the PDM to publicly-available gene expression data sets, and demonstrate that we are able to identify cell types and treatments with higher accuracy than is obtained through other approaches. By applying it in a pathway-by-pathway fashion, we demonstrate how the PDM may be used to find sets of mechanistically-related genes that discriminate phenotypes.Comment: Revise

Orienteering in Knowledge Spaces: The Hyperbolic Geometry of Wikipedia Mathematics

In this paper we show how the coupling of the notion of a network with directions with the adaptation of the four-point probe from materials testing gives rise to a natural geometry on such networks. This four-point probe geometry shares many of the properties of hyperbolic geometry wherein the network directions take the place of the sphere at infinity, enabling a navigation of the network in terms of pairs of directions: the geodesic through a pair of points is oriented from one direction to another direction, the pair of which are uniquely determined. We illustrate this in the interesting example of the pages of Wikipedia devoted to Mathematics, or “The MathWiki.” The applicability of these ideas extends beyond Wikipedia to provide a natural framework for visual search and to prescribe a natural mode of navigation for any kind of “knowledge space” in which higher order concepts aggregate various instances of information. Other examples would include genre or author organization of cultural objects such as books, movies, documents or even merchandise in an online store

Characterizing the Delaunay decompositions of compact hyperbolic surfaces

Given a Delaunay decomposition of a compact hyperbolic surface, one may record the topological data of the decomposition, together with the intersection angles between the `empty disks' circumscribing the regions of the decomposition. The main result of this paper is a characterization of when a given topological decomposition and angle assignment can be realized as the data of an actual Delaunay decomposition of a hyperbolic surface.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol6/paper12.abs.htm

A simple computational method for the identification of disease-associated loci in complex, incomplete pedigrees

We present an approach, called the Shadow Method, for the identification of disease loci from dense genetic marker maps in complex, potentially incomplete pedigrees. Shadow is a simple method based on an analysis of the patterns of obligate meiotic recombination events in genotypic data. This method can be applied to any high density marker map and was specifically designed to explore the fact that extremely dense marker maps are becoming more readily available. We also describe how to interpret and associated meaningful P-Values to the results. Shadow has significant advantages over traditional parametric linkage analysis methods in that it can be readily applied even in cases in which the topology of a pedigree or pedigrees can only be partially determined. In addition, Shadow is robust to variability in a range of parameters and in particular does not require prior knowledge of mode of inheritance, penetrance, or clinical misdiagnosis rate. Shadow can be used for any SNP data, but is especially effective when applied to dense samplings. Our primary example uses data from Affymetrix 100k SNPChip samples in which we illustrate our approach by analyzing simulated data as well as genome-wide SNP data from two pedigrees with inherited forms of kidney failure, one of which is compared with a typical LOD score analysis

A simple computational method for the identification of disease-associated loci in complex, incomplete pedigrees

We present an approach, called the Shadow Method, for the identification of disease loci from dense genetic marker maps in complex, potentially incomplete pedigrees. Shadow is a simple method based on an analysis of the patterns of obligate meiotic recombination events in genotypic data. This method can be applied to any high density marker map and was specifically designed to explore the fact that extremely dense marker maps are becoming more readily available. We also describe how to interpret and associated meaningful P-Values to the results. Shadow has significant advantages over traditional parametric linkage analysis methods in that it can be readily applied even in cases in which the topology of a pedigree or pedigrees can only be partially determined. In addition, Shadow is robust to variability in a range of parameters and in particular does not require prior knowledge of mode of inheritance, penetrance, or clinical misdiagnosis rate. Shadow can be used for any SNP data, but is especially effective when applied to dense samplings. Our primary example uses data from Affymetrix 100k SNPChip samples in which we illustrate our approach by analyzing simulated data as well as genome-wide SNP data from two pedigrees with inherited forms of kidney failure, one of which is compared with a typical LOD score analysis

Improved IBD Detection Using Incomplete Haplotype Information

The availability of high density genetic maps and genotyping platforms has transformed human genetic studies. The use of these platforms has enabled population-based genome-wide association studies. However, in inheritance-based studies, current methods do not take full advantage of the information present in such genotyping analyses. In this paper we describe an improved method for identifying genetic regions shared identical-by-descent (IBD) from recent common ancestors. This method improves existing methods by taking advantage of phase information even if it is less than fully accurate or missing. We present an analysis of how using phase information increases the accuracy of IBD detection compared to using only genotype information