7 research outputs found
Graph ranking for exploratory gene data analysis
<p>Abstract</p> <p>Background</p> <p>Microarray technology has made it possible to simultaneously monitor the expression levels of thousands of genes in a single experiment. However, the large number of genes greatly increases the challenges of analyzing, comprehending and interpreting the resulting mass of data. Selecting a subset of important genes is inevitable to address the challenge. Gene selection has been investigated extensively over the last decade. Most selection procedures, however, are not sufficient for accurate inference of underlying biology, because biological significance does not necessarily have to be statistically significant. Additional biological knowledge needs to be integrated into the gene selection procedure.</p> <p>Results</p> <p>We propose a general framework for gene ranking. We construct a bipartite graph from the Gene Ontology (GO) and gene expression data. The graph describes the relationship between genes and their associated molecular functions. Under a species condition, edge weights of the graph are assigned to be gene expression level. Such a graph provides a mathematical means to represent both species-independent and species-dependent biological information. We also develop a new ranking algorithm to analyze the weighted graph via a kernelized spatial depth (KSD) approach. Consequently, the importance of gene and molecular function can be simultaneously ranked by a real-valued measure, KSD, which incorporates the global and local structure of the graph. Over-expressed and under-regulated genes also can be separately ranked.</p> <p>Conclusion</p> <p>The gene-function bigraph integrates molecular function annotations into gene expression data. The relevance of genes is described in the graph (through a common function). The proposed method provides an exploratory framework for gene data analysis.</p
Cram\'er-Rao bounds for synchronization of rotations
Synchronization of rotations is the problem of estimating a set of rotations
R_i in SO(n), i = 1, ..., N, based on noisy measurements of relative rotations
R_i R_j^T. This fundamental problem has found many recent applications, most
importantly in structural biology. We provide a framework to study
synchronization as estimation on Riemannian manifolds for arbitrary n under a
large family of noise models. The noise models we address encompass zero-mean
isotropic noise, and we develop tools for Gaussian-like as well as heavy-tail
types of noise in particular. As a main contribution, we derive the
Cram\'er-Rao bounds of synchronization, that is, lower-bounds on the variance
of unbiased estimators. We find that these bounds are structured by the
pseudoinverse of the measurement graph Laplacian, where edge weights are
proportional to measurement quality. We leverage this to provide interpretation
in terms of random walks and visualization tools for these bounds in both the
anchored and anchor-free scenarios. Similar bounds previously established were
limited to rotations in the plane and Gaussian-like noise
On the pseudo-inverse of the Laplacian of a bipartite graph
We provide an efficient method to calculate the pseudo-inverse of the Laplacian of a bipartite graph, which is based on the pseudo-inverse of the normalized Laplacian. Keywords: Laplacian, Pseudo-inverse, Bipartite graph.