The theory of trackability with applications to sensor networks

Abstract

In this paper, we formalize the concept of tracking in a sensor network and develop a rigorous theory of trackability that investigates the rate of growth of the number of consistent tracks given a sequence of observations made by the sensor network. The phenomenon being tracked is modelled by a nondeterministic nite automaton and the sensor network is modelled by an observer capable of detecting events related, typically ambiguously, to the states of the underlying automaton.More formally, an input string, Z t , of t + 1 symbols (the sensor network observations) that is presented to a nondeterministic nite automaton, M , (the model) determines a set,, of state sequences (the tracks or hypotheses) that are capable of generating the input string Z t . We study the growth of the size of this set, |H M (Z t )|, as a function of the length of the input string, t + 1. Our main result is that for a given automaton and sensor coverage, the worst-case rate of growth is either polynomial or exponential in t, indicating a kind of phase transition in tracking accuracy.The techniques we use include the Joint Spectral Radius, ρ(Σ), of a nite set, Σ, of (0, 1)-matrices derived from M . Specically, we construct a set of matrices, Σ, corresponding to M with the property that ρ(Σ) ≤ 1 if and only if |HM (Z t )| grows polynomially in t. We also prove that for (0, 1)-matrices, the decision problem ρ(Σ) ≤ 1 is Turing decidable and, therefore, so is the problem of deciding whether worst case state sequence growth for a given automaton is polynomial or exponential. These results have applications in sensor networks, computer network security and autonomic computing as well as various tracking problems of recent interest involving detecting phenomena using noisy observations of hidden states.

Polynomial-Time Computation of the Joint Spectral Radius for Some Sets of Nonnegative Matrices

“…(1.2) During the last decade, the JSR has proved useful in a number of application contexts, including wavelets [9,11], capacity of codes [6,17], switched and hybrid systems [14,23], sensor networks [10,16], combinatorics on words [8,15], autoregressive models, Markov chains, etc. Unfortunately, the JSR of a set of matrices is notoriously difficult to compute and to approximate.…”

Linear switching systems with slow growth of trajectories

“…In the study of wavelets, refinement functional equations, and affine fractal curves, marginal stability is responsible for Lipschitz continuity and for boundedness of variation of solutions [10,11,8]. It is important for trackability of autonomous agents in sensor networks [12], in classifications of finite semigroups of integer matrices [13], in the problem of asymptotic growth of some regular sequences [14], in the stability analysis of LSS [3,7,9], etc.…”

Analysis of Deterministic Tracking of Multiple Objects Using a Binary Sensor Network

“…Recently, Crespi et al [2008] proposed a theoretical analysis of trackability. They modeled the evolution of a nondeterministic automaton whose state transitions can be observed.…”