563 research outputs found
Quickest Change-Point Detection with Sampling Right Constraints
The quickest change-point detection problems with sampling right constraints are considered. Specially, an observer sequentially takes observations from a random sequence, whose distribution will change at an unknown time. Based on the observation sequence, the observer wants to identify the change-point as quickly as possible. Unlike the classic quickest detection problem in which the observer can take an observation at each time slot, we impose a causal sampling right constraint to the observer. In particular, sampling rights are consumed when the observer takes an observation and are replenished randomly by a stochastic process. The observer cannot take observations if there is no sampling right left. The causal sampling right constraint is motivated by several practical applications. For example, in the application of sensor network for monitoring the abrupt change of its ambient environment, the sensor can only take observations if it has energy left in its battery. With this additional constraint, we design and analyze the optimal detection and sampling right allocation strategies to minimize the detection delay under various problem setups. As one of our main contributions, a greedy sampling right allocation strategy, by which the observer spends sampling rights in taking observations as long as there are sampling rights left, is proposed. This strategy possesses a low complexity structure, and leads to simple but (asymptotically) optimal detection algorithms for the problems under consideration. Specially, our main results include: 1) Non-Bayesian quickest change-point detection: we consider non-Bayesian quickest detection problem with stochastic sampling right constraint. Two criteria, namely the algorithm level average run length (ARL) and the system level ARL, are proposed to control the false alarm rate. We show that the greedy sampling right allocation strategy combined with the cumulative sum (CUSUM) algorithm is optimal for Lorden\u27s setup with the algorithm level ARL constraint and is asymptotically optimal for both Lorden\u27s and Pollak\u27s setups with the system level ARL constraint. 2) Bayesian quickest change-point detection: both limited sampling right constraint and stochastic sampling right constraint are considered in the Bayesian quickest detection problem. The limited sampling right constraint can be viewed as a special case of the stochastic sampling right constraint with a zero sampling right replenishing rate. The optimal solutions are derived for both sampling right constraints. However, the structure of the optimal solutions are rather complex. For the problem with the limited sampling right constraint, we provide asymptotic upper and lower bounds for the detection delay. For the problem with the stochastic sampling right constraint, we show that the greedy sampling right allocation strategy combined with Shiryaev\u27s detection rule is asymptotically optimal. 3) Quickest change-point detection with unknown post-change parameters: we extend previous results to the quickest detection problem with unknown post-change parameters. Both non-Bayesian and Bayesian setups with stochastic sampling right constraints are considered. For the non-Bayesian problem, we show that the greedy sampling right allocation strategy combined with the M-CUSUM algorithm is asymptotically optimal. For the Bayesian setups, we show that the greedy sampling right allocation strategy combined with the proposed M-Shiryaev algorithm is asymptotically optimal
Non-Bayesian Quickest Detection with Stochastic Sample Right Constraints
In this paper, we study the design and analysis of optimal detection scheme
for sensors that are deployed to monitor the change in the environment and are
powered by the energy harvested from the environment. In this type of
applications, detection delay is of paramount importance. We model this problem
as quickest change detection problem with a stochastic energy constraint. In
particular, a wireless sensor powered by renewable energy takes observations
from a random sequence, whose distribution will change at a certain unknown
time. Such a change implies events of interest. The energy in the sensor is
consumed by taking observations and is replenished randomly. The sensor cannot
take observations if there is no energy left in the battery. Our goal is to
design a power allocation scheme and a detection strategy to minimize the worst
case detection delay, which is the difference between the time when an alarm is
raised and the time when the change occurs. Two types of average run length
(ARL) constraint, namely an algorithm level ARL constraint and an system level
ARL constraint, are considered. We propose a low complexity scheme in which the
energy allocation rule is to spend energy to take observations as long as the
battery is not empty and the detection scheme is the Cumulative Sum test. We
show that this scheme is optimal for the formulation with the algorithm level
ARL constraint and is asymptotically optimal for the formulations with the
system level ARL constraint.Comment: 30 pages, 5 figure
Quickest detection in coupled systems
This work considers the problem of quickest detection of signals in a coupled
system of sensors, which receive continuous sequential observations from
the environment. It is assumed that the signals, which are modeled by general
It\^{o} processes, are coupled across sensors, but that their onset times may
differ from sensor to sensor. Two main cases are considered; in the first one
signal strengths are the same across sensors while in the second one they
differ by a constant. The objective is the optimal detection of the first time
at which any sensor in the system receives a signal. The problem is formulated
as a stochastic optimization problem in which an extended minimal
Kullback-Leibler divergence criterion is used as a measure of detection delay,
with a constraint on the mean time to the first false alarm. The case in which
the sensors employ cumulative sum (CUSUM) strategies is considered, and it is
proved that the minimum of CUSUMs is asymptotically optimal as the mean
time to the first false alarm increases without bound. In particular, in the
case of equal signal strengths across sensors, it is seen that the difference
in detection delay of the -CUSUM stopping rule and the unknown optimal
stopping scheme tends to a constant related to the number of sensors as the
mean time to the first false alarm increases without bound. Alternatively, in
the case of unequal signal strengths, it is seen that this difference tends to
zero.Comment: 29 pages. SIAM Journal on Control and Optimization, forthcomin
Data-Efficient Quickest Change Detection with On-Off Observation Control
In this paper we extend the Shiryaev's quickest change detection formulation
by also accounting for the cost of observations used before the change point.
The observation cost is captured through the average number of observations
used in the detection process before the change occurs. The objective is to
select an on-off observation control policy, that decides whether or not to
take a given observation, along with the stopping time at which the change is
declared, so as to minimize the average detection delay, subject to constraints
on both the probability of false alarm and the observation cost. By considering
a Lagrangian relaxation of the constraint problem, and using dynamic
programming arguments, we obtain an \textit{a posteriori} probability based
two-threshold algorithm that is a generalized version of the classical Shiryaev
algorithm. We provide an asymptotic analysis of the two-threshold algorithm and
show that the algorithm is asymptotically optimal, i.e., the performance of the
two-threshold algorithm approaches that of the Shiryaev algorithm, for a fixed
observation cost, as the probability of false alarm goes to zero. We also show,
using simulations, that the two-threshold algorithm has good observation
cost-delay trade-off curves, and provides significant reduction in observation
cost as compared to the naive approach of fractional sampling, where samples
are skipped randomly. Our analysis reveals that, for practical choices of
constraints, the two thresholds can be set independent of each other: one based
on the constraint of false alarm and another based on the observation cost
constraint alone.Comment: Preliminary version of this paper has been presented at ITA Workshop
UCSD 201
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