76 research outputs found
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
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
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
Topological structures in the equities market network
We present a new method for articulating scale-dependent topological
descriptions of the network structure inherent in many complex systems. The
technique is based on "Partition Decoupled Null Models,'' a new class of null
models that incorporate the interaction of clustered partitions into a random
model and generalize the Gaussian ensemble. As an application we analyze a
correlation matrix derived from four years of close prices of equities in the
NYSE and NASDAQ. In this example we expose (1) a natural structure composed of
two interacting partitions of the market that both agrees with and generalizes
standard notions of scale (eg., sector and industry) and (2) structure in the
first partition that is a topological manifestation of a well-known pattern of
capital flow called "sector rotation.'' Our approach gives rise to a natural
form of multiresolution analysis of the underlying time series that naturally
decomposes the basic data in terms of the effects of the different scales at
which it clusters. The equities market is a prototypical complex system and we
expect that our approach will be of use in understanding a broad class of
complex systems in which correlation structures are resident.Comment: 17 pages, 4 figures, 3 table
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
Curvature bounds for surfaces in hyperbolic 3-manifolds
We prove existence of thick geodesic triangulations of hyperbolic 3-manifolds
and use this to prove existence of universal bounds on the principal curvatures
of surfaces embedded in hyperbolic 3-manifolds.Comment: 21 pages, 9 figures, published version, added figures, fixed typo
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
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