3,362 research outputs found
Detection of an anomalous cluster in a network
We consider the problem of detecting whether or not, in a given sensor
network, there is a cluster of sensors which exhibit an "unusual behavior."
Formally, suppose we are given a set of nodes and attach a random variable to
each node. We observe a realization of this process and want to decide between
the following two hypotheses: under the null, the variables are i.i.d. standard
normal; under the alternative, there is a cluster of variables that are i.i.d.
normal with positive mean and unit variance, while the rest are i.i.d. standard
normal. We also address surveillance settings where each sensor in the network
collects information over time. The resulting model is similar, now with a time
series attached to each node. We again observe the process over time and want
to decide between the null, where all the variables are i.i.d. standard normal,
and the alternative, where there is an emerging cluster of i.i.d. normal
variables with positive mean and unit variance. The growth models used to
represent the emerging cluster are quite general and, in particular, include
cellular automata used in modeling epidemics. In both settings, we consider
classes of clusters that are quite general, for which we obtain a lower bound
on their respective minimax detection rate and show that some form of scan
statistic, by far the most popular method in practice, achieves that same rate
to within a logarithmic factor. Our results are not limited to the normal
location model, but generalize to any one-parameter exponential family when the
anomalous clusters are large enough.Comment: Published in at http://dx.doi.org/10.1214/10-AOS839 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Searching for a trail of evidence in a maze
Consider a graph with a set of vertices and oriented edges connecting pairs
of vertices. Each vertex is associated with a random variable and these are
assumed to be independent. In this setting, suppose we wish to solve the
following hypothesis testing problem: under the null, the random variables have
common distribution N(0,1) while under the alternative, there is an unknown
path along which random variables have distribution , , and
distribution N(0,1) away from it. For which values of the mean shift can
one reliably detect and for which values is this impossible? Consider, for
example, the usual regular lattice with vertices of the form and oriented edges , where . We show that for paths of length starting at
the origin, the hypotheses become distinguishable (in a minimax sense) if
, while they are not if . We derive
equivalent results in a Bayesian setting where one assumes that all paths are
equally likely; there, the asymptotic threshold is . We
obtain corresponding results for trees (where the threshold is of order 1 and
independent of the size of the tree), for distributions other than the Gaussian
and for other graphs. The concept of the predictability profile, first
introduced by Benjamini, Pemantle and Peres, plays a crucial role in our
analysis.Comment: Published in at http://dx.doi.org/10.1214/07-AOS526 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Clustering comparison of point processes with applications to random geometric models
In this chapter we review some examples, methods, and recent results
involving comparison of clustering properties of point processes. Our approach
is founded on some basic observations allowing us to consider void
probabilities and moment measures as two complementary tools for capturing
clustering phenomena in point processes. As might be expected, smaller values
of these characteristics indicate less clustering. Also, various global and
local functionals of random geometric models driven by point processes admit
more or less explicit bounds involving void probabilities and moment measures,
thus aiding the study of impact of clustering of the underlying point process.
When stronger tools are needed, directional convex ordering of point processes
happens to be an appropriate choice, as well as the notion of (positive or
negative) association, when comparison to the Poisson point process is
considered. We explain the relations between these tools and provide examples
of point processes admitting them. Furthermore, we sketch some recent results
obtained using the aforementioned comparison tools, regarding percolation and
coverage properties of the Boolean model, the SINR model, subgraph counts in
random geometric graphs, and more generally, U-statistics of point processes.
We also mention some results on Betti numbers for \v{C}ech and Vietoris-Rips
random complexes generated by stationary point processes. A general observation
is that many of the results derived previously for the Poisson point process
generalise to some "sub-Poisson" processes, defined as those clustering less
than the Poisson process in the sense of void probabilities and moment
measures, negative association or dcx-ordering.Comment: 44 pages, 4 figure
- …