In the sequential change-point detection literature, most research specifies a required frequency of false alarms at a given pre-change distribution $f_θ$ and tries to minimize the detection delay for every possible post-change distribution $g_λ$. 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 $f_θ$. 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 $f_θ$ and the post-change distribution $g_λ$ involve unknown parameters. We also apply our approach to the special case of detecting shifts in the mean of independent normal observations.
We consider the online monitoring of multivariate streaming data for changes that are characterized by an unknown subspace structure manifested in the covariance matrix. In particular, we consider the covariance structure changes from an identity matrix to an unknown spiked covariance model. We assume the post-change distribution is unknown, and propose two detection procedures: the Largest-Eigenvalue Shewhart chart and the Subspace-CUSUM detection procedure. We present theoretical approximations to the average run length (ARL) and the expected detection delay (EDD) for the Largest-Eigenvalue …
We consider the sequential change-point detection problem of detecting changes that are characterized by a subspace structure. Such changes are frequent in high-dimensional streaming data altering the form of the corresponding covariance matrix. In this work we present a Subspace-CUSUM procedure and demonstrate its first-order asymptotic optimality properties for the case where the subspace structure is unknown and needs to be simultaneously estimated. To achieve this goal we develop a suitable analytical methodology that includes a proper parameter optimization for the proposed detection sche…
For estimating the unknown parameters in an unstable autoregressive AR(p), the paper proposes sequential least squares estimates with a special stopping time defined by the trace of the observed Fisher information matrix. The limiting distribution of the sequential LSE is shown to be normal for the parameter vector lying both inside the stability region and on some part of its boundary in contrast to the ordinary LSE. The asymptotic normality of the sequential LSE is provided by a new property of the observed Fisher information matrix which holds both inside the stability region of AR(p) proce…
We consider the problem of quickly detecting a signal in a sensor network when the subset of sensors in which signal may be present is completely unknown. We formulate this problem as a sequential hypothesis testing problem with a simple null (signal is absent everywhere) and a composite alternative (signal is present somewhere). We introduce a novel class of scalable sequential tests which, for any subset of affected sensors, minimize the expected sample size for a decision asymptotically, that is as the error probabilities go to 0. Moreover, we propose sequential tests that require minimal t…
Sequential change-point detection when the distribution parameters are unknown is a fundamental problem in statistics and machine learning. When the post-change parameters are unknown, we consider a set of detection procedures based on sequential likelihood ratios with non-anticipating estimators constructed using online convex optimization algorithms such as online mirror descent, which provides a more versatile approach to tackle complex situations where recursive maximum likelihood estimators cannot be found. When the underlying distributions belong to a exponential family and the estimator…
It is commonly required to detect change points in sequences of random variables. In the most difficult setting of this problem, change detection must be performed sequentially with new observations being constantly received over time. Further, the parameters of both the pre- and post- change distributions may be unknown. In a recent paper by Hawkins and Zamba (2005), the sequential generalised likelihood ratio test was introduced for detecting changes in this context, under the assumption that the observations follow a Gaussian distribution. However, we show that the asymptotic approximation …
Under some mild Markov assumptions it is shown that the problem of designing optimal sequential tests for two simple hypotheses can be formulated as a linear program. The result is derived by investigating the Lagrangian dual of the sequential testing problem, which is an unconstrained optimal stopping problem, depending on two unknown Lagrangian multipliers. It is shown that the derivative of the optimal cost function with respect to these multipliers coincides with the error probabilities of the corresponding sequential test. This property is used to formulate an optimization problem that is…
We consider the problem of sequential signal detection in a multichannel system where the number and location of signals is a priori unknown. We assume that the data in each channel are sequentially observed and follow a general non-i.i.d. stochastic model. Under the assumption that the local log-likelihood ratio processes in the channels converge r-completely to positive and finite numbers, we establish the asymptotic optimality of a generalized sequential likelihood ratio test and a mixture-based sequential likelihood ratio test. Specifically, we show that both tests minimize the first r mom…
Corrected confidence intervals are developed for the mean of the second component of a bivariate normal process when the first component is being monitored sequentially. This is accomplished by constructing a first approximation to a pivotal quantity, and then using very weak expansions to determine the correction terms. The asymptotic sampling distribution of the renormalised pivotal quantity is established in both the case where the covariance matrix is known and when it is unknown. The resulting approximations have a simple form and the results of a simulation study of two well-known sequen…
We study the problem of detecting an abrupt change to the signal covariance matrix. In particular, the covariance changes from a "white" identity matrix to an unknown spiked or low-rank matrix. Two sequential change-point detection procedures are presented, based on the largest and the smallest eigenvalues of the sample covariance matrix. To control false-alarm-rate, we present an accurate theoretical approximation to the average-run-length (ARL) and expected detection delay (EDD) of the detection, leveraging the extreme eigenvalue distributions from random matrix theory and by capturing a non…
The high structural deficient rate poses serious risks to the operation of many bridges and buildings. To prevent critical damage and structural collapse, a quick structural health diagnosis tool is needed during normal operation or immediately after extreme events. In structural health monitoring (SHM), many existing works will have limited performance in the quick damage identification process because 1) the damage event needs to be identified with short delay and 2) the post-damage information is usually unavailable. To address these drawbacks, we propose a new damage detection and localiza…
In engineering systems, it is usually assumed that lifetimes of components are independent and identically distributed (iid). But, the failure of a component results in a higher load on the remaining components and hence causes the distribution of the surviving components change. For modeling this kind of systems, the theory of sequential order statistics (SOS) can be used. Assuming Weibull distribution for lifetimes of components and conditionally proportional hazard rates model as a special case of the SOS theory, the maximum likelihood estimates of the unknown parameters are obtained in dif…
In treatment allocation problems the individuals to be treated often arrive sequentially. We study a problem in which the policy maker is not only interested in the expected cumulative welfare but is also concerned about the uncertainty/risk of the treatment outcomes. At the outset, the total number of treatment assignments to be made may even be unknown. A sequential treatment policy which attains the minimax optimal regret is proposed. We also demonstrate that the expected number of suboptimal treatments only grows slowly in the number of treatments. Finally, we study a setting where outcome…
We develop new efficient online algorithms for detecting transient sparse signals in TEM video sequences, by adopting the recently developed framework for sequential detection jointly with online convex optimization [1]. We cast the problem as detecting an unknown sparse mean shift of Gaussian observations, and develop adaptive CUSUM and adaptive SSRS procedures, which are based on likelihood ratio statistics with post-change mean vector being online maximum likelihood estimators with $\ell_1$. We demonstrate the meritorious performance of our algorithms for TEM imaging using real data.
Carnegie Mellon University（卡内基梅隆大学）；University of Chicago（芝加哥大学）
We present algorithms for nonparametric regression in settings where the data are obtained sequentially. While traditional estimators select bandwidths that depend upon the sample size, for sequential data the effective sample size is dynamically changing. We propose a linear time algorithm that adjusts the bandwidth for each new data point, and show that the estimator achieves the optimal minimax rate of convergence. We also propose the use of online expert mixing algorithms to adapt to unknown smoothness of the regression function. We provide simulations that confirm the theoretical results,…
We consider sequential hypothesis testing between two quantum states using adaptive and non-adaptive strategies. In this setting, samples of an unknown state are requested sequentially and a decision to either continue or to accept one of the two hypotheses is made after each test. Under the constraint that the number of samples is bounded, either in expectation or with high probability, we exhibit adaptive strategies that minimize both types of misidentification errors. Namely, we show that these errors decrease exponentially (in the stopping time) with decay rates given by the measured relat…
We address the computational challenge of finding the robust sequential change-point detection procedures when the pre- and post-change distributions are not completely specified. Earlier works [veeravalli 1994] and [Unnikrishnan 2011] establish the general conditions for robust procedures which include finding a pair of least favorable distributions (LFDs). However, in the multi-dimensional setting, it is hard to find such LFDs computationally. We present a method based on convex optimization that addresses this issue when the distributions are Gaussian with unknown parameters from pre-specif…
University of California, San Diego（加州大学圣地亚哥分校）；University of California, Berkeley（加州大学伯克利分校）
We propose a new algorithmic framework for sequential hypothesis testing with i.i.d. data, which includes A/B testing, nonparametric two-sample testing, and independence testing as special cases. It is novel in several ways: (a) it takes linear time and constant space to compute on the fly, (b) it has the same power guarantee as a non-sequential version of the test with the same computational constraints up to a small factor, and (c) it accesses only as many samples as are required - its stopping time adapts to the unknown difficulty of the problem. All our test statistics are constructed to b…
We introduce a framework for online changepoint detection and simultaneous model learning which is applicable to highly parametrized models, such as deep neural networks. It is based on detecting changepoints across time by sequentially performing generalized likelihood ratio tests that require only evaluations of simple prediction score functions. This procedure makes use of checkpoints, consisting of early versions of the actual model parameters, that allow to detect distributional changes by performing predictions on future data. We define an algorithm that bounds the Type I error in the se…
Given $iid$ observations from an unknown absolute continuous distribution defined on some domain $Ω$, we propose a nonparametric method to learn a piecewise constant function to approximate the underlying probability density function. Our density estimate is a piecewise constant function defined on a binary partition of $Ω$. The key ingredient of the algorithm is to use discrepancy, a concept originates from Quasi Monte Carlo analysis, to control the partition process. The resulting algorithm is simple, efficient, and has a provable convergence rate. We empirically demonstrate its efficiency a…
Georgia Institute of Technology（佐治亚理工学院）；University of Oxford（牛津大学）；Rensselaer Polytechnic Institute（伦斯勒理工学院）
We study sequential change-point detection procedures based on linear sketches of high-dimensional signal vectors using generalized likelihood ratio (GLR) statistics. The GLR statistics allow for an unknown post-change mean that represents an anomaly or novelty. We consider both fixed and time-varying projections, derive theoretical approximations to two fundamental performance metrics: the average run length (ARL) and the expected detection delay (EDD); these approximations are shown to be highly accurate by numerical simulations. We further characterize the relative performance measure of th…
We consider robust optimization problems, where the goal is to optimize an unknown objective function against the worst-case realization of an uncertain parameter. For this setting, we design a novel sample-efficient algorithm GP-MRO, which sequentially learns about the unknown objective from noisy point evaluations. GP-MRO seeks to discover a robust and randomized mixed strategy, that maximizes the worst-case expected objective value. To achieve this, it combines techniques from online learning with nonparametric confidence bounds from Gaussian processes. Our theoretical results characterize …
The goal of the paper is to design sequential strategies which lead to efficient optimization of an unknown function under the only assumption that it has a finite Lipschitz constant. We first identify sufficient conditions for the consistency of generic sequential algorithms and formulate the expected minimax rate for their performance. We introduce and analyze a first algorithm called LIPO which assumes the Lipschitz constant to be known. Consistency, minimax rates for LIPO are proved, as well as fast rates under an additional Hölder like condition. An adaptive version of LIPO is also introd…
The problem of sequentially detecting an abrupt change in a sequence of independent and identically distributed (IID) random variables is addressed. Whereas previous approaches assume a known probability density function (PDF) at the start of the sequence, the problem addressed is the detection of a single change in distribution among a finite number of known 'equal-energy' PDFs, but where the initial and final distributions are not known a priori. A Bayesian multiple hypothesis approach is proposed where (i) unlike previous threshold policies, the minimum cost hypothesis is tracked through ti…
Moscow Institute of Physics and Technology（莫斯科物理技术学院）；AGT StatConsult（AGT统计咨询公司）；Université de Technologie de Troyes（特鲁瓦技术大学）
The paper addresses a sequential changepoint detection problem, assuming that the duration of change may be finite and unknown. This problem is of importance for many applications, e.g., for signal and image processing where signals appear and disappear at unknown points in time or space. In contrast to the conventional optimality criterion in quickest change detection that requires minimization of the expected delay to detection for a given average run length to a false alarm, we focus on a reliable maximin change detection criterion of maximizing the minimal probability of detection in a giv…
Consider the following multi-phase project management problem. Each project is divided into several phases. All projects enter the next phase at the same point chosen by the decision maker based on observations up to that point. Within each phase, one can pursue the projects in any order. When pursuing the project with one unit of resource, the project state changes according to a Markov chain. The probability distribution of the Markov chain is known up to an unknown parameter. When pursued, the project generates a random reward depending on the phase and the state of the project and the unkn…
Motivated by sequential detection of transient signals in high dimensional data stream, we study the performance of EWMA, MA, CUSUM, and GLRT charts for detecting a transient signal in multivariate data streams in terms of the power of detection (POD) under the constraint of false detecting probability (FDP) at the stationary state. Approximations are given for FDP and POD. Comparisons show that the EWMA chart performs equally well as the GLRT chart when the signal strength is unknown, while its design is free of signal length and easy to update. In addition, the MEWMA chart with hard-threshol…
Let $X_1,X_2,..., X_n,...$ be a stochastic process with independent values whose distribution $P_θ$ depends on an unknown parameter $θ$, $θ\inΘ$, where $Θ$ is an open subset of the real line. The problem of testing $H_0:$ $θ=θ_0$ vs. a composite alternative $H_1:$ $θ>θ_0$ is considered, where $θ_0\inΘ$ is a fixed value of the parameter. The main objective of this work is the characterization of the structure of the locally most powerful (in the sense of Berk) sequential tests in this problem.
In the multisensor sequential change detection problem, a disruption occurs in an environment monitored by multiple sensors. This disruption induces a change in the observations of an unknown subset of sensors. In the Byzantine version of this problem, which is the focus of this work, it is further assumed that the postulated change-point model may be misspecified for an unknown subset of sensors. The problem then is to detect the change quickly and reliably, for any possible subset of affected sensors, even if the misspecified sensors are controlled by an adversary. Given a user-specified upp…
Given a Wiener process with unknown and unobservable drift, we try to estimate this drift as effectively but also as quickly as possible, in the presence of a quadratic penalty for the estimation error and of a fixed, positive cost per unit of observation time. In a Bayesian framework, where the unobservable drift is assumed to have a known "prior" distribution, this question reduces to choosing judiciously a stopping time for an appropriate diffusion process in natural scale. We establish structural properties of the solution for the corresponding problem of optimal stopping. In particular, w…
Suppose a process yields independent observations whose distributions belong to a family parameterized by θ\inΘ. When the process is in control, the observations are i.i.d. with a known parameter value θ_0. When the process is out of control, the parameter changes. We apply an idea of Robbins and Siegmund [Proc. Sixth Berkeley Symp. Math. Statist. Probab. 4 (1972) 37-41] to construct a class of sequential tests and detection schemes whereby the unknown post-change parameters are estimated. This approach is especially useful in situations where the parametric space is intricate and mixture-type…
We consider a design problem where experimental conditions (design points $X_i$) are presented in the form of a sequence of i.i.d.\ random variables, generated with an unknown probability measure $μ$, and only a given proportion $α\in(0,1)$ can be selected. The objective is to select good candidates $X_i$ on the fly and maximize a concave function $Φ$ of the corresponding information matrix. The optimal solution corresponds to the construction of an optimal bounded design measure $ξ_α^*\leq μ/α$, with the difficulty that $μ$ is unknown and $ξ_α^*$ must be constructed online. The construction p…
In this paper we consider a fractional stochastic volatility model, that is a model in which the volatility may exhibit a long-range dependent or a rough/antipersistent behavior. We propose a dynamic sequential Monte Carlo methodology that is applicable to both long memory and antipersistent processes in order to estimate the volatility as well as the unknown parameters of the model. We establish a central limit theorem for the state and parameter filters and we study asymptotic properties (consistency and asymptotic normality) for the filter. We illustrate our results with a simulation study …
We study sequential Bayesian inference in stochastic kinetic models with latent factors. Assuming continuous observation of all the reactions, our focus is on joint inference of the unknown reaction rates and the dynamic latent states, modeled as a hidden Markov factor. Using insights from nonlinear filtering of continuous-time jump Markov processes we develop a novel sequential Monte Carlo algorithm for this purpose. Our approach applies the ideas of particle learning to minimize particle degeneracy and exploit the analytical jump Markov structure. A motivating application of our methods is m…
The problem of sequential change diagnosis is considered, where observations are obtained on-line, an abrupt change occurs in their distribution, and the goal is to quickly detect the change and accurately identify the post-change distribution, while controlling the false alarm rate. A finite set of alternatives is postulated for the post-change regime, but no prior information is assumed for the unknown change-point. A drawback of many algorithms that have been proposed for this problem is the implicit use of pre-change data for determining the post-change distribution. This can lead to very …
The Bradley-Terry-Luce (BTL) model is a classic and very popular statistical approach for eliciting a global ranking among a collection of items using pairwise comparison data. In applications in which the comparison outcomes are observed as a time series, it is often the case that data are non-stationary, in the sense that the true underlying ranking changes over time. In this paper we are concerned with localizing the change points in a high-dimensional BTL model with piece-wise constant parameters. We propose novel and practicable algorithms based on dynamic programming that can consistentl…
What advantage do \emph{sequential} procedures provide over batch algorithms for testing properties of unknown distributions? Focusing on the problem of testing whether two distributions $\mathcal{D}_1$ and $\mathcal{D}_2$ on $\{1,\dots, n\}$ are equal or $ε$-far, we give several answers to this question. We show that for a small alphabet size $n$, there is a sequential algorithm that outperforms any batch algorithm by a factor of at least $4$ in terms sample complexity. For a general alphabet size $n$, we give a sequential algorithm that uses no more samples than its batch counterpart, and po…
Suppose that at any stage of a statistical experiment a control variable $X$ that affects the distribution of the observed data $Y$ can be used. The distribution of $Y$ depends on some unknown parameter $θ$, and we consider the classical problem of testing a simple hypothesis $H_0: θ=θ_0$ against a simple alternative $H_1: θ=θ_1$ allowing the data to be controlled by $X$, in the following sequential context. The experiment starts with assigning a value $X_1$ to the control variable and observing $Y_1$ as a response. After some analysis, we choose another value $X_2$ for the control variable, a…
Joint detection and estimation refers to deciding between two or more hypotheses and, depending on the test outcome, simultaneously estimating the unknown parameters of the underlying distribution. This problem is investigated in a sequential framework under mild assumptions on the underlying random process. We formulate an unconstrained sequential decision problem, whose cost function is the weighted sum of the expected run-length and the detection/estimation errors. Then, a strong connection between the derivatives of the cost function with respect to the weights, which can be interpreted as…
