79 research outputs found
Data-Efficient Minimax Quickest Change Detection with Composite Post-Change Distribution
The problem of quickest change detection is studied, where there is an
additional constraint on the cost of observations used before the change point
and where the post-change distribution is composite. Minimax formulations are
proposed for this problem. It is assumed that the post-change family of
distributions has a member which is least favorable in some sense. An algorithm
is proposed in which on-off observation control is employed using the least
favorable distribution, and a generalized likelihood ratio based approach is
used for change detection. Under the additional condition that either the
post-change family of distributions is finite, or both the pre- and post-change
distributions belong to a one parameter exponential family, it is shown that
the proposed algorithm is asymptotically optimal, uniformly for all possible
post-change distributions.Comment: Submitted to IEEE Transactions on Info. Theory, Oct 2014. Preliminary
version presented at ISIT 2014 at Honolulu, Hawai
Data-efficient quickest change detection
In the classical problem of quickest change detection, a decision maker observes a sequence of random variables. At some point of time, the distribution of the random variables changes abruptly. The objective is to detect this change in distribution with minimum possible delay, subject to a constraint on the false alarm rate.
In many applications of quickest change detection, changes are rare and there is a cost associated with taking observations or acquiring data. For such applications, the classical quickest change detection model is no longer applicable. In this dissertation we extend the classical formulations by adding an
additional penalty on the cost of observations used before the change point. The objective is to find a causal on-off observation control policy and a stopping time, to minimize the detection delay, subject to constraints on the false
alarm rate and the cost of observations used before the change point. We show that two-threshold generalizations of the classical
single-threshold tests are asymptotically optimal for the proposed formulations. The nature of optimality is strong in the sense that the false alarm rates of the two-threshold tests are at least as good as the false alarm rates of their classical counterparts. Also, the delays of the two-threshold tests are within a constant of the delays of their classical counterparts. These results indicate that an arbitrary but fixed fraction of observations can be skipped before change without any loss in asymptotic
performance. A detailed performance analysis of these algorithms is provided, and guidelines are given for the design of the proposed tests, on the basis of the performance analysis. An important result obtained through this analysis is that the two constraints, on the false alarm rate and the cost of observations used before the change, can be met independent of each other.
Numerical studies of these two-threshold algorithms also reveal that they have good trade-off curves, and perform significantly better than the approach of fractional sampling, where classical single threshold tests are used and the constraint on the cost of observations is met by skipping observations randomly.
We first study the problem in Bayesian and minimax settings and then extend the results to more general quickest change detection models, namely, model with
unknown post-change distribution,
a sensor network model, and a multi-channel model
Sequential change-point detection when unknown parameters are present in the pre-change distribution
In the sequential change-point detection literature, most research specifies
a required frequency of false alarms at a given pre-change distribution
and tries to minimize the detection delay for every possible
post-change distribution . In this paper, motivated by a number of
practical examples, we first consider the reverse question by specifying a
required detection delay at a given post-change distribution and trying to
minimize the frequency of false alarms for every possible pre-change
distribution . We present asymptotically optimal procedures for
one-parameter exponential families. Next, we develop a general theory for
change-point problems when both the pre-change distribution and
the post-change distribution involve unknown parameters. We also
apply our approach to the special case of detecting shifts in the mean of
independent normal observations.Comment: Published at http://dx.doi.org/10.1214/009053605000000859 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Quickest Change Detection in Autoregressive Models
The problem of quickest change detection (QCD) in autoregressive (AR) models
is investigated. A system is being monitored with sequentially observed
samples. At some unknown time, a disturbance signal occurs and changes the
distribution of the observations. The disturbance signal follows an AR model,
which is dependent over time. Before the change, observations only consist of
measurement noise, and are independent and identically distributed (i.i.d.).
After the change, observations consist of the disturbance signal and the
measurement noise, are dependent over time, which essentially follow a
continuous-state hidden Markov model (HMM). The goal is to design a stopping
time to detect the disturbance signal as quickly as possible subject to false
alarm constraints. Existing approaches for general non-i.i.d. settings and
discrete-state HMMs cannot be applied due to their high computational
complexity and memory consumption, and they usually assume some asymptotic
stability condition. In this paper, the asymptotic stability condition is
firstly theoretically proved for the AR model by a novel design of forward
variable and auxiliary Markov chain. A computationally efficient Ergodic CuSum
algorithm that can be updated recursively is then constructed and is further
shown to be asymptotically optimal. The data-driven setting where the
disturbance signal parameters are unknown is further investigated, and an
online and computationally efficient gradient ascent CuSum algorithm is
designed. The algorithm is constructed by iteratively updating the estimate of
the unknown parameters based on the maximum likelihood principle and the
gradient ascent approach. The lower bound on its average running length to
false alarm is also derived for practical false alarm control. Simulation
results are provided to demonstrate the performance of the proposed algorithms
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