Uniform error control in sequential change diagnosis
Warner, Austin
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Permalink
https://hdl.handle.net/2142/120411
Description
Title
Uniform error control in sequential change diagnosis
Author(s)
Warner, Austin
Issue Date
2023-04-25
Director of Research (if dissertation) or Advisor (if thesis)
This work considers the problem of sequential change diagnosis, in which a sequence of random variables is monitored online, a change occurs in their distribution, and the goal is to quickly detect the change, and also to identify the post-change distribution from a finite set of possibilities, while obeying a specified false alarm rate. The contributions of this work are separated by theme into three parts.
In the first part, a simple algorithm is studied, which raises an alarm as soon as the CuSum statistic that corresponds to one of the possible post-change distributions exceeds a certain threshold. We conduct a theoretical analysis which applies to the general sequential change diagnosis problem, and we consider two special cases: the multichannel problem, and the two-sided problem. In the former—where the data are generated by independent channels and the change occurs in only one of them—the conditional probability of misidentification, given that there was not a false alarm, is shown to decay exponentially fast in the detection threshold, uniformly over all possible change points. As a result, this simple algorithm is shown to asymptotically minimize Lorden’s delay criterion simultaneously for every possible post-change distribution within the class of schemes that satisfy prescribed bounds on the false alarm rate and the worst-case conditional probability
of misidentification, as the former goes to zero sufficiently faster than the latter. The theoretical results are supported by simulation studies.
In the second part, a novel, recursive algorithm is proposed and shown to shown to control the worst-case probability of misidentification for a family of change points without the use of additional tuning parameters and without sacrificing control of the worst-case delay in Lorden’s sense. A theoretical analysis is conducted for a general family of sequential change diagnosis procedures, which supports the proposed algorithm and revises certain state-of-the-art results. Additionally, a novel, comprehensive method is proposed for the design and evaluation of sequential change diagnosis algorithms. This method is illustrated with simulation studies, where existing procedures are compared to the proposed.
Lastly, we show that the ideas presented throughout this work can be extended into more complex settings. Specifically, we present several algorithms for the problem of sequential change diagnosis in the presence of parametric uncertainty in the post-change distributions, and illustrate how the ideas presented in the second part of this work can be used to improve them. In particular, we develop an analog of the method proposed in the preceding part, which has a lower computational requirement than the other algorithms proposed for this problem. These algorithms are analyzed and compared through numerical studies, where we specifically consider the multichannel problem under parametric uncertainty.
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