Results 1 
4 of
4
Changepoint detection over graphs with the spectral scan statistic. Arxiv preprint arXiv:1206.0773
, 2012
"... We consider the changepoint detection problem of deciding, based on noisy measurements, whether an unknown signal over a given graph is constant or is instead piecewise constant over two induced subgraphs of relatively low cut size. We analyze the corresponding generalized likelihood ratio (GLR) st ..."
Abstract

Cited by 10 (5 self)
 Add to MetaCart
(Show Context)
We consider the changepoint detection problem of deciding, based on noisy measurements, whether an unknown signal over a given graph is constant or is instead piecewise constant over two induced subgraphs of relatively low cut size. We analyze the corresponding generalized likelihood ratio (GLR) statistic and relate it to the problem of finding a sparsest cut in a graph. We develop a tractable relaxation of the GLR statistic based on the combinatorial Laplacian of the graph, which we call the spectral scan statistic, and analyze its properties. We show how its performance as a testing procedure depends directly on the spectrum of the graph, and use this result to explicitly derive its asymptotic properties on few graph topologies. Finally, we demonstrate both theoretically and by simulations that the spectral scan statistic can outperform naive testing procedures based on edge thresholding and χ2 testing. 1
Connected Subgraph Detection
"... We characterize the family of connected subgraphs in terms of linear matrix inequalities (LMI) with additional integrality constraints. We then show that convex relaxations of the integral LMI lead to parameterization of all weighted connected subgraphs. These developments allow for optimizing ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
We characterize the family of connected subgraphs in terms of linear matrix inequalities (LMI) with additional integrality constraints. We then show that convex relaxations of the integral LMI lead to parameterization of all weighted connected subgraphs. These developments allow for optimizing arbitrary graph functionals under connectivity constraints. For concreteness we consider the connected subgraph detection problem that arises in a number of applications including network intrusion, disease outbreaks, and video surveillance. In these applications feature vectors are associated with nodes and edges of a graph. The problem is to decide whether or not the null hypothesis is true based on the measured features. For simplicity we consider the elevated mean problem wherein feature values at various nodes are distributed IID under the null hypothesis. The nonnull (positive) hypothesis is distinguished from the null hypothesis by the fact that feature values on some unknown connected subgraph has elevated mean. 1
Graph Structured Normal Means Inference
, 2013
"... This thesis addresses statistical estimation and testing of signals over a graph when measurements are noisy and highdimensional. Graph structured patterns appear in applications as diverse as sensor networks, virology in human networks, congestion in internet routers, and advertising in social net ..."
Abstract
 Add to MetaCart
This thesis addresses statistical estimation and testing of signals over a graph when measurements are noisy and highdimensional. Graph structured patterns appear in applications as diverse as sensor networks, virology in human networks, congestion in internet routers, and advertising in social networks. We will develop asymptotic guarantees of the performance of statistical estimators and tests, by stating conditions for consistency by properties of the graph (e.g. graph spectra). The goal of this thesis is to demonstrate theoretically that by exploiting the graph structure one can achieve statistical consistency in extremely noisy conditions. We begin with the study of a projection estimator called laplacian eigenmaps, and find that eigenvalue concentration plays a central role in the ability to estimate graph structured patterns. We continue with the study of the edge lasso, a least squares procedure with total variation penalty, and determine combinatorial conditions under which changepoints (edges across which the underlying signal changes) on the graph are recovered. We will shift focus to testing for anomalous activations in the graph, using the scan statistic relaxations, the spectral scan statistic and the graph ellipsoid scan statistic. We will also show how one can form a decomposition of the graph from a spanning tree which will lead to a test for activity in the graph. This will lead to the construction of a spanning tree wavelet basis, which can be used to localize activations on the graph.
Graph Structured Statistical Inference
, 2012
"... This thesis addresses statistical estimation and testing of signals over a graph when measurements are noisy and highdimensional. Graph structured patterns appear in applications as diverse as sensor networks, virology in human networks, congestion in internet routers, and advertising in social net ..."
Abstract
 Add to MetaCart
This thesis addresses statistical estimation and testing of signals over a graph when measurements are noisy and highdimensional. Graph structured patterns appear in applications as diverse as sensor networks, virology in human networks, congestion in internet routers, and advertising in social networks. We will develop asymptotic guarantees of the performance of statistical estimators and tests, by stating conditions for consistency by properties of the graph (e.g. graph spectra). The goal of this thesis is to demonstrate theoretically that by exploiting the graph structure one can achieve statistical consistency in extremely noisy conditions. We begin with the study of a projection estimator called laplacian eigenmaps, and find that eigenvalue concentration plays a central role in the ability to estimate graph structured patterns. We continue with the study of the edge lasso, a least squares procedure with total variation penalty, and determine combinatorial conditions under which changepoints (edges across which the underlying signal changes) on the graph are recovered. We will shift focus to testing for anomalous activations in the graph, using a scan statistic relaxation and through the construction of a spanning tree wavelet basis. Finally, we study the consistency of kernel density estimation for vertex valued random variables with densities that are Lipschitz with respect to graph metrics.