quickest detection of a random pulse in white gaussian noise a class of stochastic processes characterized by rapid transitions in their structure is considered and the quickest detection of such transitions is studied in a bayesian framework the emphasis is on stochastic processes consisting of a randomly arrived causal pulse possibly with a set of random parameters such as amplitude and duration and an additive white gaussian noise in this model the pulse shape and the prior joint density of the arrival time and other random parameters are assumed known the task of quickest detection in this paper is described mathematically by minimizing the expected detection error the detection error is represented by a nonlinear function of the distance between the actual transition time and its associated detection time the assumptions on this function are fairly mild and allow to flexibly design its shape for a desired trade off between the detection delay and the false alarm rate two special cases of such design are well known error measures mean squared and mean absolute error the quickest detection problem a subclass of optimal stopping time problems is formulated as a stochastic optimal control problem and is resolved using dynamic programming the optimal detection rule is determined in terms of the solution of an integral equation that cannot be directly solved due to its complexity this equation is later used to develop a class of suboptimal detection rules and a lower bound on the minimum error using this lower bound it is shown for a numerical example that the suboptimal detector is nearly optimal

a robust surveillance scheme for stochastically ordered alternatives we sequentially observe independent observations x 1 x 2 such that initially they have distribution g 0 at some unknown time v they become stochastically larger having distribution g 1 neither g 0 nor g 1 is fully specified we wish to detect that a change has taken place as soon as possible after its occurrence subject to a constraint on the rate of false alarms we derive a family of nonparametric sequential procedures based on ranks with noncontiguous alternatives in mind large sample approximations to the operating characteristics are obtained analytically the proposed procedures all possess robustness of validity because they are based on ranks near optimal sensitivity can be obtained for specific alternatives by choosing an appropriate procedure for example when observations are normally distributed with one standard deviation shift in mean postchange an appropriate nonparametric surveillance scheme yields 97 asymptotic relative efficiency as compared to the optimal procedure when all distributional parameters are known our procedures are computationally feasible monte carlo experiments confirm the applicability of the asymptotic calculations including high levels of efficiency for sample sizes met in practice

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 f and tries to minimize the detection delay for every possible post change distribution g 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 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 and the post change distribution g involve unknown parameters we also apply our approach to the special case of detecting shifts in the mean of independent normal observations

a new look at warning and action lines of surveillance schemes abstract some authors have suggested an efficient dynamic monitoring procedure for detection of a change in the drift of a brownian motion their procedure can be described as a sequence of extremely short sequential probability ratio tests sprt s done in zero time with infinitesimal time between consecutive sprt s the drawbacks of this procedure are that it can be described only as a limit of practical procedures and it does not take into account the cost of initiating a test in this article we suggest a procedure that takes both sampling and overhead costs into account and that can reasonably be carried out in practice in this procedure the process is monitored continuously with a constant sampling rate the accumulated data are analyzed by the standard cumulative sum cusum statistics whenever the cusum procedure raises an alarm data are accumulated as fast as possible until either the alarm is relaxed or the process is stopped

a dynamic sampling approach for detecting a change in distribution the problem of detecting a change in drift of brownian motion is considered in the bayesian framework with the time of change having a prior exponential distribution to the well known problem of finding an optimal stopping rule for declaring a change we add the option of continuously controlling the sampling rates resulting in controlling the variance coefficient of the process the combined problem of finding an optimal rate function dynamic sampling together with an optimal stopping rule is solved and explicit expressions for the quantities of interest are derived the dynamic sampling procedure is shown to be significantly superior to constant rate sampling the comparison is most favorable when the expected time until change tends to infinity where the relative efficiency between the two procedures tends to infinity

average run length to false alarm for surveillance schemes designed with partially specified pre change distribution observations are taken independently and sequentially detection of a change in distribution is studied when the problem has an invariance structure the prechange distribution is assumed to be a member of a specified family and is assumed known up to a nuisance parameter we provide a general method of constructing surveillance schemes in the presence of a nuisance parameter and give sufficient conditions for approximating their average run lengths to false alarm applications include detecting a change in scale of i i d gamma variates with unknown initial scale detecting a change in location of i i d normal variates with unknown initial mean and a non parametric scheme based on ranks for detecting a change to a stochastically larger distribution

contour dynamics of incompressible 3 d fluids in a porous medium with different densities we consider the problem of the evolution of the interface given by two incompressible fluids through a porous medium which is known as the muskat problem and in two dimensions it is mathematically analogous to the two phase hele shaw cell we focus on a fluid interface given by a jump of densities being the equation of the evolution obtained using darcy s law we prove local well posedness when the smaller density is above stable case and in the unstable case we show ill posedness

dynamic theory of quasilinear parabolic equations i abstracts evolution equations on etablit les fondements abstraits pour une theorie dynamique generale pour des problemes aux valeurs limites et initiales quasilineaires

asymptotically optimal prediction for time varying data generating processes we develop a methodology referred to as kinetic prediction for predicting time series undergoing unknown changes in their data generating distributions based on kolmogorov tikhomirov s varepsilon entropy we propose a concept called varepsilon predictability that quantifies the size of a model class which can be parametric or nonparametric and the maximal number of abrupt structural changes that guarantee the achievability of asymptotically optimal prediction moreover for parametric distribution families we extend the aforementioned kinetic prediction with discretized function spaces to its counterpart with continuous function spaces and propose a sequential monte carlo based implementation we also extend our methodology for predicting smoothly varying data generating distributions under reasonable assumptions we prove that the average predictive performance converges almost surely to the oracle bound which corresponds to the case that the data generating distributions are known in advance the results also shed some light on the so called prediction inference dilemma various examples and numerical results are provided to demonstrate the wide applicability of our methodology

a comparison of weighted cusum procedures that account for monotone changes in population size the cusum procedure has been popularly used for detecting a shift in the incidence rate of a rare health event many cusum methods are developed based on a poisson model with a constant mean number of events in practice the expected number of events is likely to vary over time as the population size at risk is not constant but often grows over time an increase in the baseline incidence rate tends to be masked by the population growth to efficiently detect an increase in the baseline incidence rate it is appealing to assign more weight to recent observations and less weight to older observations this paper compares weighted cusum wcusum and conventional cusum procedures in the presence of monotone changes in population size the simulation results show that the wcusum method may be more efficient than the conventional cusum methods in detecting increases in the incidence rate especially for small shifts an example based on mortality data from new mexico is used to illustrate the implementation of the wcusum method copyright 2010 john wiley sons ltd

turning waves and breakdown for incompressible flows we consider the evolution of an interface generated between two immiscible incompressible and irrotational fluids specifically we study the muskat and water wave problems we show that starting with a family of initial data given by f0 the interface reaches a regime in finite time in which is no longer a graph therefore there exists a time t where the solution of the free boundary problem parameterized as f t blows up f l t in particular for the muskat problem this result allows us to reach an unstable regime for which the rayleigh taylor condition changes sign and the solution breaks down

the state of statistical process control as we proceed into the 21st century 2000 the state of statistical process control as we proceed into the 21st century journal of the american statistical association vol 95 no 451 pp 992 998

on the optimality of bayesian change point detection by introducing suitable loss random variables of detection we obtain optimal tests in terms of the stopping time or alarm time for bayesian change point detection not only for a general prior distribution of change points but also for observations being a markov process moreover the optimal minimal average detection delay is proved to be equal to 1 for any possibly large average run length to false alarm if the number of possible change points is finite

well posedness and stability results for a quasilinear periodic muskat problem abstract we study the muskat problem describing the spatially periodic motion of two fluids with equal viscosities under the effect of gravity in a vertical unbounded two dimensional geometry we first prove that the classical formulation of the problem is equivalent to a nonlocal and nonlinear evolution equation expressed in terms of singular integrals and having only the interface between the fluids as unknown secondly we show that this evolution equation has a quasilinear structure which is at a formal level not obvious and we also disclose the parabolic character of the equation exploiting these aspects we establish the local well posedness of the problem for arbitrary initial data in h s s with s 3 2 2 determine a new criterion for the global existence of solutions and uncover a parabolic smoothing property besides we prove that the zero steady state solution is exponentially stable

quickest detection of parameter changes in stochastic regression nonparametric cusum we consider the problem of detecting abrupt parameter changes in a stochastic regression with unknown noise distribution the process changes at some unknown point of time under general conditions on the regression function and unknown distributions of observations before and after the disruption the paper develops a nonparametric cumulative sum procedure cusum unlike likelihood based cusum algorithms constructed mostly on log likelihood ratio statistics we use a special system of basic statistics in page s procedure by applying a sequential sampling scheme which measures time in terms of accumulated kullback leibler k l divergence we come to a system of statistics with the martingale properties close to those of the log likelihood ratios the proposed approach suggests also an alternative performance criterion in the analysis of the procedure by replacing the expected detection delay by the corresponding k l divergence we show that under the false alarm probability constraint the nonparametric cusum rule is optimal in the sense that it ensures the logarithmic asymptotic for the detection delay

on the average run length to false alarm in surveillance problems which possess an invariance structure surveillance can be based in some change point detection problems on a sequence of invariant statistics gordon and pollak prove that under certain conditions the average run length arl to false alarm of invariance based shiryayev roberts detection schemes is asymptotically the same as that of the dual classical scheme that is based on the original sequence of observations in this paper we give alternative conditions under which the two arls coincide and demonstrate that these conditions are satisfied in cases where gordon and pollak s conditions are difficult to check

information bounds and quick detection of parameter changes in stochastic systems by using information theoretic bounds and sequential hypothesis testing theory this paper provides a new approach to optimal detection of abrupt changes in stochastic systems this approach not only generalizes previous work in the literature on optimal detection far beyond the relatively simple models treated but also suggests alternative performance criteria which are more tractable and more appropriate for general stochastic systems in addition it leads to detection rules which have manageable computational complexity for on line implementation and yet are nearly optimal under the different performance criteria considered

surveillance of a simple linear regression this article considers an important aspect of the general sequential analysis problem where a process is in control up to some unknown point i 1 after which the distribution from which the observations are generated changes an extensive sequential analytic literature assumes that the change in distribution is abrupt for example from n 0 1 to n 1 there is also an extensive literature that deals with a gradual change in the case where the decision whether or not a change has occurred is based on a fixed set of observations rather than an ongoing process of decision making every time a new observation is obtained however there is virtually no literature on the practical case of sequentially detecting a gradual change in distribution visualize a machine deteriorating gradually this article considers solutions to this problem as a first approximation the gradual change problem can be modeled as a change from a fixed distribution to a model of simple linear regression with respect to ti

nonlinear analytic semiflows in this paper a local existence and regularity theory is given for nonlinear parabolic initial value problems x t f x t and quasilinear initial value problems x t a x t x t f x t this theory extends the theory of daprato and grisvard of 1979 and shows how various properties like analyticity of solutions can be derived as a direct corollary of the existence theorem

quantization effect on the log likelihood ratio and its application to decentralized sequential detection it is well known that quantization cannot increase the kullback leibler divergence which can be thought of as the expected value or first moment of the log likelihood ratio in this paper we investigate the quantization effects on the second moment of the log likelihood ratio it is shown via the convex domination technique that quantization may result in an increase in the case of the second moment but the increase is bounded above by 2 e the result is then applied to decentralized sequential detection problems not only to provide simpler sufficient conditions for asymptotic optimality theories in the simplest models but also to shed new light on more complicated models in addition some brief remarks on other higher order moments of the log likelihood ratio are also provided

information bounds and quickest change detection in decentralized decision systems the quickest change detection problem is studied in decentralized decision systems where a set of sensors receive independent observations and send summary messages to the fusion center which makes a final decision in the system where the sensors do not have access to their past observations the previously conjectured asymptotic optimality of a procedure with a monotone likelihood ratio quantizer mlrq is proved in the case of additive gaussian sensor noise if the signal to noise ratios snr at some sensors are sufficiently high this procedure can perform as well as the optimal centralized procedure that has access to all the sensor observations even if all snrs are low its detection delay will be at most pi 2 1ap 57 larger than that of the optimal centralized procedure next in the system where the sensors have full access to their past observations the first asymptotically optimal procedure in the literature is developed surprisingly the procedure has the same asymptotic performance as the optimal centralized procedure although it may perform poorly in some practical situations because of slow asymptotic convergence finally it is shown that neither past message information nor the feedback from the fusion center improves the asymptotic performance in the simplest model

minimax robust quickest change detection the popular criteria of optimality for quickest change detection procedures are the lorden criterion the pollak criterion and the bayesian criterion in this paper a robust version of these quickest change detection problems is considered when the pre change and post change distributions are not known exactly but belong to known uncertainty classes of distributions for uncertainty classes that satisfy a specific condition it is shown that one can identify least favorable distributions lfds from the uncertainty classes such that the detection rule designed for the lfds is optimal for the robust problem in a minimax sense the condition is similar to that required for the identification of lfds for the robust hypothesis testing problem originally studied by huber an upper bound on the delay incurred by the robust test is also obtained in the asymptotic setting under the lorden criterion of optimality this bound quantifies the delay penalty incurred to guarantee robustness when the lfds can be identified the proposed test is easier to implement than the cusum test based on the generalized likelihood ratio glr statistic which is a popular approach for such robust change detection problems the proposed test is also shown to give better performance than the glr test in simulations for some parameter values

optimal sequential detection in multi stream data consider a large number of detectors each generating a data stream the task is to detect online distribution changes in a small fraction of the data streams previous approaches to this problem include the use of mixture likelihood ratios and sum of cusums we provide here extensions and modifications of these approaches that are optimal in detecting normal mean shifts we show how the optimal detection delay depends on the fraction of data streams undergoing distribution changes as the number of detectors goes to infinity there are three detection domains in the first domain for moderately large fractions immediate detection is possible in the second domain for smaller fractions the detection delay grows logarithmically with the number of detectors with an asymptotic constant extending those in sparse normal mixture detection in the third domain for even smaller fractions the detection delay lies in the framework of the classical detection delay formula of lorden we show that the optimal detection delay is achieved by the sum of detectability score transformations of either the partial scores or cusum scores of the data streams

interface evolution the hele shaw and muskat problems we study the dynamics of the interface between two incompressible 2 d ows where the evolution equation is obtained from darcy s law the free boundary is given by the discontinuity among the densities and viscosities of the uids this physical scenario is known as the two dimensional muskat problem or the two phase hele shaw ow we prove local existence in sobolev spaces when initially the dierence of the gradients of the pressure in the normal direction has the proper sign an assumption which is also known as the rayleigh taylor condition

asymptotic minimax robust quickest change detection for dependent stochastic processes with parametric uncertainty in this paper we consider the problem of quickly detecting an unknown change in the conditional densities of a dependent stochastic process in contrast to the existing quickest change detection approaches for dependent stochastic processes we propose minimax robust versions of the popular lorden pollak and bayesian criteria for when there is uncertainty about the parameter of the post change conditional densities under an information theoretic pythagorean inequality condition on the uncertainty set of possible post change parameters we identify asymptotic minimax robust solutions to our lorden pollak and bayesian problems finally through simulation examples we illustrate that asymptotically minimax robust rules can provide detection performance comparable to the popular but more computationally expensive generalized likelihood ratio rule

a note on the run length to false alarm of a change point detection policy a new proof is given to a known result on the average run length to false alarm of the shiryayev roberts change point detection policy when the observations are nonlattice via the approach of this new proof the average run length to false alarm can be calculated in the lattice case and for the mixed type shiryayev roberts scheme

optimal stopping times for detecting changes in distributions it is shown that page s stopping time is optimal for the detection of changes in distributions in a well defined sense this work is a generalization of an existing result where it was shown that page s stopping time is optimal asymptotically

quickest detection in multiple on off processes we consider the quickest detection of idle periods in multiple on off processes at each time only one process can be observed and the observations are random realizations drawn from two different distributions depending on the current state on or off of the chosen process the objective is to catch an idle period in any of the on off processes as quickly as possible subject to a reliability constraint we show that this problem presents a fresh twist to the classic problem of quickest change detection that considers only one stochastic process a bayesian formulation of the problem is developed for both infinite and finite number of processes based on the theory of partially observable markov decision process pomdp while a general pomdp is pspace hard we show that the optimal decision rule has a simple threshold structure for the infinite case for the finite case basic properties of the optimal decision rule are established and a low complexity threshold policy is proposed which converges to the optimal decision rule for the infinite case as the number of processes increases this problem finds applications in spectrum sensing in cognitive radio networks where a secondary user searches for idle channels in the spectrum

asymptotic optimality of mixture rules for detecting changes in general stochastic models the paper addresses a sequential changepoint detection problem for a general stochastic model assuming that the observed data may be non i i d i e dependent and non identically distributed and prior distribution of the change point is arbitrary tartakovsky and veeravalli 2005 baron and tartakovsky 2006 and more recently tartakovsky 2017 developed a general asymptotic theory of changepoint detection for non i i d stochastic models assuming certain stability of the log likelihood ratio process in the case of simple hypotheses when both pre change and post change models are completely specified however in most applications the post change distribution is not completely known in the present paper we generalize previous results to the case of parametric uncertainty assuming that the parameter of the post change distribution is unknown we introduce two detection rules based on mixtures the mixture shiryaev rule and the mixture shiryaev roberts rule and study their asymptotic properties in the bayesian context in particular we provide sufficient conditions under which these rules are first order asymptotically optimal minimizing moments of the delay to detection as the probability of false alarm approaches zero

data efficient quickest change detection in minimax settings the classical problem of quickest change detection is studied with an additional constraint on the cost of observations used in the detection process the change point is modeled as an unknown constant and minimax formulations are proposed for the problem the objective in these formulations is to find a stopping time and an on off observation control policy for the observation sequence to minimize a version of the worst possible average delay subject to constraints on the false alarm rate and the fraction of time observations are taken before change an algorithm called de cusum is proposed and is shown to be asymptotically optimal for the proposed formulations as the false alarm rate goes to zero numerical results are used to show that the de cusum algorithm has good tradeoff curves and performs significantly better than the approach of fractional sampling in which the observations are skipped using the outcome of a sequence of coin tosses independent of the observation process this study is guided by the insights gained from an earlier study of a bayesian version of this problem

solving nonstiff higher order odes directly by the direct integration method the variable order and stepsize direct integration di method for solving systems of nonstiff higher order odes directly is reviewed mention is also made of the current inroads in solving stiff higher order odes directly the advantages of the di method over the corresponding adams method when used to solve the same problem as a first order system are pointed out numerical results are presented comparing the two methods in terms of execution times for each tolerance used and execution times for each solution of the same range of accuracy

statistical inference for time inhomogeneous volatility models this paper offers a new approach for estimating and forecasting the volatility of financial time series no assumption is made about the parametric form of the processes on the contrary we only suppose that the volatility can be approximated by a constant over some interval in such a framework the main problem consists of filtering this interval of time homogeneity then the estimate of the volatility can be simply obtained by local averaging we construct a locally adaptive volatility estimate lave which can perform this task and investigate it both from the theoretical point of view and through monte carlo simulations finally the lave procedure is applied to a data set of nine exchange rates and a comparison with a standard garch model is also provided both models appear to be capable of explaining many of the features of the data nevertheless the new approach seems to be superior to the garch method as far as the out of sample results are concerned

weighted cumulative sum technique a class of weighted control schemes that generalizes the basic cumulative sum cusum technique is introduced the schemes of the first type in which the weights represent information concomitant with the data prove to be especially useful when handling charts corresponding to samples of varying sizes the schemes of the second type are based on giving greater weight to more recent information representatives of this class are shown to have better run length characteristics with respect to drift in the level of a controlled process than does the classical cusum while maintaining good sensitivity with respect to shifts analogous to the classical cusum scheme they admit a dual graphical representation that is the scheme can be applied by means of a one or two sided decision interval or via a v mask a special case of this type of scheme designated the geometric cusum is considered in detail it can be viewed as a cusum type counterpart of the exponentially weighted moving average

capturing correlation changes by applying kernel change point detection on the running correlations abstract change point detection methods signal the occurrence of abrupt changes in a time series non parametric approaches such as the gaussian kernel based change point kcp detection arlot et al 2012 are especially attractive because they impose less assumptions on the data yet a drawback of these methods is that most of them are sensitive to changes in the mean the variance etc making them less sensitive to specific kinds of changes we show that kcp can be adapted to detect a particular type of change only we focus here on correlation change which has been put forward by different theories but proved hard to trace in multivariate time series we propose kcp corr which boils down to applying kcp on the running correlations to confirm that kcp corr is more sensitive than merely applying kcp on the raw data kcp raw a simulation study was conducted in which the number of noise variables and the size of the correlation change were varied kcp corr emerged as the better method especially in the more difficult but realistic settings where the correlation change is minimal and or noise variables are present kcp corr also outperforms cusum a non parametric method that specifically targets the detection of correlation changes

a diffusion process and its applications to detecting a change in the drift of brownian motion abstract the purpose of the present paper is to make a quantitative comparison of the shiryayev roberts and page procedures we do this in the context of continuous time in order to use the machinery of diffusion processes to perform explicitly certain calculations which seem impossible in discrete time although the continuous time results are not especially good approximations to the corresponding quantities in discrete time they provide very useful comparative information on which to base selection of a stopping rule this paper is organized as follows the shiryayev roberts process is defined and shown to be a novel diffusion process with some surprising properties we also specify more precisely the basis for our comparison of the two procedures and give the results of some elementary calculations these developments contain an asymptotic evaluation we define a modification of our basic procedure and give an asymptotic evaluation of its average run length numerical comparisons and a discussion of their significance are contained next our conclusions are roughly these in simple situations where the two procedures can be directly compared neither seems dramatically superior to the other however the shiryayev roberts procedure is more easily adapted to complex circumstances and consequently warrants additional study

multiscale local change point detection with applications to value at risk this paper offers a new approach to modeling and forecasting of non stationary time series with applications to volatility modeling for financial data the approach is based on the assumption of local homogeneity for every time point there exists a historical interval of homogerzeity in which the volatility parameter can be well approximated by a constant the proposed procedure recovers this interval from the data using the local change point lcp analysis afterward the estimate of the volatility can be simply obtained by local averaging the approach carefully addresses the question of choosing the tuning parameters of the procedure using the so called propagation condition the main result claims a new oracle inequality in terms of the modeling bias which measures the quality of the local constant approximation this result yields the optimal rate of estimation for smooth and piecewise constant volatility functions then the new procedure is applied to some data sets and a comparison with a standard garch model is also provided finally we discuss applications of the new method to the value at risk problem the numerical results demonstrate a very reasonable performance of the new method

sequential detection of a change in a normal mean when the initial value is unknown three stopping rules are proposed to detect a change in a normal mean when the initial value of the mean is unknown but an estimate obtained from a training sample is available asymptotic approximations are given for the average run length when there is no change under certain hypotheses about the length of time before the change occurs and the magnitude of the change we obtain asymptotic approximations for the expected delay in detection in terms of the corresponding expected delay in the much simpler case of a known initial value the results of a monte carlo experiment supplement our asymptotic theory to yield some general conclusions about the relative merits of the three stopping rules and guidelines for choosing the size of the training sample

sequential changepoint detection in quality control and dynamical systems after a brief survey of a large variety of sequential detection procedures that are widely scattered in statistical references on quality control and engineering references on fault detection and signal processing we study some open problems concerning these procedures and introduce a unified theory of sequential changepoint detection this theory leads to a class of sequential detection rules which are not too demanding in computational and memory requirements for on line implementation and yet are nearly optimal under several performance criteria

a suboptimal quadratic change detection scheme we address the problem of detecting changes in multivariate gaussian random signals with an unknown mean after the change the window limited generalized likelihood ratio glr scheme is a well known approach to solve this problem however this algorithm involves at least log spl gamma spl rho likelihood ratio computations at each stage where spl gamma spl gamma spl rarr spl infin is the mean time before a false alarm and spl rho is the kullback leibler information we establish a new suboptimal recursive approach which is based on a collection of l parallel recursive spl chi sup 2 tests instead of the window limited glr scheme this new approach involves only a fixed number l of likelihood ratio computations at each stage for any combinations of spl gamma and spl rho by choosing an acceptable value of nonoptimality the designer can easily find a tradeoff between the complexity of the quadratic change detection algorithm and its efficiency

well posedness of the muskat problem with h2 initial data abstract we study the dynamics of the interface between two incompressible fluids in a two dimensional porous medium whose flow is modeled by the muskat equations for the two phase muskat problem we establish global well posedness and decay to equilibrium for small h 2 perturbations of the rest state for the one phase muskat problem we prove local well posedness for h 2 initial data of arbitrary size finally we show that solutions to the muskat equations instantaneously become infinitely smooth

nonanticipating estimation applied to sequential analysis and changepoint detection suppose a process yields independent observations whose distributions belong to a family parameterized by theta e theta when the process is in control the observations are i i d with a known parameter value theta 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 rules are operationally or conceptually difficult to formulate we exemplify our approach by applying it to the problem of detecting a change in the shape parameter of a gamma distribution in both a univariate and a multivariate setting

a generalized change detection problem the purpose of this paper is to give a new statistical approach to the change diagnosis detection isolation problem the change detection problem has received extensive research attention however the change isolation problem has for the most part been ignored we consider a stochastic dynamical system with abrupt changes and investigate the multiple hypotheses extension of lorden s 1971 results we introduce a joint criterion of optimality for the detection isolation problem and then design a change detection isolation algorithm we also investigate the statistical properties of this algorithm we prove a lower bound for the criterion in a class of sequential change detection isolation algorithms it is shown that the proposed algorithm is asymptotically optimal in this class the theoretical results are applied to the case of additive changes in linear stochastic models

rayleigh taylor breakdown for the muskat problem with applications to water waves the muskat problem models the evolution of the interface between two different fluids in porous media the rayleigh taylor condition is natural to reach linear stability of the muskat problem we show that the rayleigh taylor condition may hold initially but break down in finite time as a consequence of the method used we prove the existence of water waves turning

optimal detection of a change in distribution abstract document discusses a stopping rule that is a limit of bayes rules derived then an almost minimax rule is presented

quickest detection of abrupt changes for a class of random processes we consider the problem of quickest detection of abrupt changes for processes that are not necessarily independent and identically distributed i i d before and after the change by making a very simple observation that applies to most well known optimum stopping times developed for this problem in particular cusum and shiryayev roberts 1963 stopping rule we show that their optimality can be easily extended to more general processes than the usual i i d case

quickest change detection and kullback leibler divergence for two state hidden markov models in this paper the quickest change detection problem is studied in two state hidden markov models hmm where the vector parameter theta of the hmm changes from theta 0 to theta 1 at some unknown time and one wants to detect the true change as quickly as possible while controlling the false alarm rate it turns out that the generalized likelihood ratio glr scheme while theoretically straightforward is generally computationally infeasible for the hmm to develop efficient but computationally simple schemes for the hmm we first discuss a subtlety in the recursive form of the generalized likelihood ratio glr scheme for the hmm then we show that the recursive cusum scheme proposed in fuh ann statist 2003 can be regarded as a quasi glr scheme for pseudo post change hypotheses with certain dependence structure between pre and postchange observations next we extend the quasi glr idea to propose recursive score schemes in the scenario when the postchange parameter theta 1 of the hmm involves a real valued nuisance parameter finally the kullback leibler kl divergence plays an essential role in the quickest change detection problem and many other fields however it is rather challenging to numerically compute it in hmms here we develop a non monte carlo method that computes the kl divergence of two state hmms via the underlying invariant probability measure which is characterized by the fredholm integral equation numerical study demonstrates an unusual property of the kl divergence for hmm that implies the severe effects of misspecifying the postchange parameter for the hmm

tracking stopping times through noisy observations a novel quickest detection setting is proposed generalizing the well known bayesian change point detection model suppose xi yi i ges 1 is a sequence of pairs of random variables and that s is a stopping time with respect to xi i ges 1 the problem is to find a stopping time t with respect to yi i ges 1 that optimally tracks s in the sense that t minimizes the expected reaction delay bbe t s while keeping the false alarm probability p t s e t s below a given threshold alpha isin 0 1 this problem formulation applies in several areas such as in communication detection forecasting and quality control

absence of splash singularities for surface quasi geostrophic sharp fronts and the muskat problem in this paper for both the sharp front surface quasi geostrophic equation and the muskat problem we rule out the splash singularity blow up scenario in other words we prove that the contours evolving from either of these systems cannot intersect at a single point while the free boundary remains smooth splash singularities have been shown to hold for the free boundary incompressible euler equation in the form of the water waves contour evolution problem our result confirms the numerical simulations in earlier work in which it was shown that the curvature blows up because the contours collapse at a point here we prove that maintaining control of the curvature will remove the possibility of pointwise interphase collapse another conclusion that we provide is a better understanding of earlier work in which squirt singularities are ruled out in this case a positive volume of fluid between the contours cannot be ejected in finite time

boundary crossing probabilities and statistical applications abstract this paper surveys recent results involving boundary crossing probabilities and related statistical applications the first part is concerned with problems of sequential analysis especially repeated significance tests and their application to sequential clinical trials involving survival data the second part develops the probability theory motivated by the problems of part 1 a method for computing first passage distributions of brownian motion to linear boundaries is introduced and then modified to handle problems in discrete time and those involving nonlinear boundaries the third part is concerned with fixed sample statistical problems especially change point problems which involve boundary crossing probabilities examples are given of problems for which methods of part 2 appear adequate and of problems which require new methods

detecting a change in regression first order optimality observations are generated according to a regression with normal error as a function of time when the process is in control the process potentially changes at some unknown point of time and then the ensuing observations are normal with the same mean function plus an arbitrary function under suitable regularity conditions the problem is to obtain a stopping rule that is optimal in the sense that the rule minimizes the expected delay in detecting a change subject to a constraint on the average run length to a false alarm a bound on the expected delay is first obtained it is then shown that the cusum and shiryayev roberts procedures achieve this bound to first order

an algorithm for detecting a change in a stochastic process the problem of detecting a change from one given stationary and ergodic stochastic process to another such process is considered it is assumed that both stochastic processes are processes with memory and that they are mutually independent a sequential test is proposed and analyzed it is proved that the proposed test is asymptotically optimal in a mathematically precise sense

quickest detection for changes in maximal knn coherence of random matrices this paper addresses the problem of quickest detection of a change in the maximal coherence between columns of an n times p random matrix based on a sequence of matrix observations having a single unknown change point the random matrix is assumed to have identically distributed rows and the maximal coherence is defined as the largest of the p atopwithdelims 2 correlation coefficients associated with any row likewise the k nearest neighbor knn coherence is defined as the k th largest of these correlation coefficients the forms of the pre and postchange distributions of the observed matrices are assumed to belong to the family of elliptically contoured densities with sparse dispersion matrices but are otherwise unknown a nonparametric stopping rule is proposed that is based on the maximal knn sample coherence between columns of each observed random matrix this is a summary statistic that is related to a test of existence of a hub vertex in a sample correlation graph having degree at least k performance bounds on the delay and false alarm performance of the proposed stopping rule are obtained in the purely high dimensional regime where p rightarrow infty and n is fixed when the prechange dispersion matrix is diagonal it is shown that among all functions of the proposed summary statistic the proposed stopping rule is asymptotically optimal under a minimax quickest change detection model as the stopping threshold approaches infinity the theory developed also applies to sequential hypothesis testing and fixed sample size tests

average run lengths of an optimal method of detecting a change in distribution abstract suppose one accumulates independent observations from a certain process initially the process is at state 0 at some unknown point in time something occurs e g a breakdown which puts the process in state 1 and consequently the stochastic behavior of the observations changes it is of interest to declare that a change took place to raise an alarm as soon as possible after its occurrence subject to a restriction on the rate of false detections it is assumed that the aforementioned observations are the only information one has about the process and the problem is to construct a good detection scheme practical examples of this problem arise in areas such as health quality control ecological monitoring etc for instance consider surveillance for congenital malformations in newborn infants under normal circumstances the percentage of babies born with a certain type of malformation has a known value should something occur such as an environmental change the introduction of a new drug to the market etc the percentage may increase one would want to raise an alarm as quickly as possible after a change would have taken place subject to an acceptable rate of false alarms

efficient scalable schemes for monitoring a large number of data streams the sequential changepoint detection problem is studied in the context of global online monitoring of a large number of independent data streams we are interested in detecting an occurring event as soon as possible but we do not know when the event will occur nor do we know which subset of data streams will be affected by the event a family of scalable schemes is proposed based on the sum of the local cumulative sum cusum statistics from each individual data stream and is shown to asymptotically minimize the detection delays for each and every possible combination of affected data streams subject to the global false alarm constraint the usefulness and limitations of our asymptotic optimality results are illustrated by numerical simulations and heuristic arguments the appendices contain a probabilistic result on the first epoch to simultaneous record values for multiple independent random walks

misspecified and asymptotically minimax robust quickest change detection we investigate the quickest detection of an unknown change in the distribution of a stochastic process generating independent and identically distributed observations we develop new bounds on the performance of misspecified cumulative sum cusum rules and pose minimax robust versions of the popular lorden and pollak criteria with polynomial or higher order moment detection delay penalties by exploiting our results for misspecified cusum rules we identify solutions to our robust quickest change detection problems in the asymptotic regime of few false alarms in contrast to previous robust quickest change detection treatments our asymptotic results hold under relaxed conditions on the uncertainty sets of possible prechange and postchange distributions we illustrate our results in simulations and apply them to the problem of detecting target manoeuvres in low signal to noise ratio settings i e dim target manoeuvre detection

a note on optimal detection of a change in distribution suppose x 1 x 2 dots x nu 1 are iid random variables with distribution f 0 and x nu x nu 1 dots are are iid with distributed f 1 the change point nu is unknown the problem is to raise an alarm as soon as possible after the distribution changes from f 0 to f 1 detect the change but to avoid false alarms pollak found a version of the shiryayev roberts procedure to be asymptotically optimal for the problem of minimizing the average run length to detection over all stopping times which satisfy a given constraint on the rate of false alarms here we find that this procedure is strictly optimal for a slight reformulation of the problem he considered explicit formulas are developed for the calculation of the average run length both before and after the change for the optimal stopping time

on optimality of the shiryaev roberts procedure for detecting a change in distribution in 1985 for detecting a change in distribution pollak introduced a specific minimax performance metric and a randomized version of the shiryaev roberts procedure where the zero initial condition is replaced by a random variable sampled from the quasi stationary distribution of the shiryaev roberts statistic pollak proved that this procedure is third order asymptotically optimal as the mean time to false alarm becomes large the question of whether pollak s procedure is strictly minimax for any false alarm rate has been open for more than two decades and there were several attempts to prove this strict optimality in this paper we provide a counterexample which shows that pollak s procedure is not optimal and that there is a strictly optimal procedure which is nothing but the shiryaev roberts procedure that starts with a specially designed deterministic point

on a hele shaw flow with a time dependent gap in the presence of surface tension the introduction of surface tension into a hele shaw problem makes it more realistic from the physical viewpoint but more difficult from the mathematical viewpoint in this paper we discuss a hele shaw flow with a time dependent gap taking into account the surface tension of the free boundary we use the schwarz function method to find asymptotic solutions for the interior problem in the case when the initial shape of the droplet is a weakly distorted circle

treating a single stiff second order ode directly abstract the stability of direct methods for second order systems is examined the methods suggested by krogh are generalized bdf methods the control of stepsize to ensure stability is examined and criteria for deciding when to switch between methods are discussed

quickest change detection of a markov process across a sensor array recent attention in quickest change detection in the multisensor setting has been on the case where the densities of the observations change at the same instant at all the sensors due to the disruption in this work a more general scenario is considered where the change propagates across the sensors and its propagation can be modeled as a markov process a centralized bayesian version of this problem is considered with a fusion center that has perfect information about the observations and a priori knowledge of the statistics of the change process the problem of minimizing the average detection delay subject to false alarm constraints is formulated in a dynamic programming framework insights into the structure of the optimal stopping rule are presented in the limiting case of rare disruptions it is shown that the structure of the optimal test reduces to thresholding the a posteriori probability of the hypothesis that no change has happened under a certain condition on the kullback leibler k l divergence between the post and the pre change densities it is established that the threshold test is asymptotically optimal in the vanishing false alarm probability regime it is shown via numerical studies that this low complexity threshold test results in a substantial improvement in performance over naive tests such as a single sensor test or a test that incorrectly assumes that the change propagates instantaneously

asymptotic operating characteristics of an optimal change point detection in hidden markov models let 0 1 1 be observations from the hidden markov model with probability distribution p 0 and let 1 be observations from the hidden markov model with probability distribution p 1 the parameters 0 and 1 are given while the change point is unknown the problem is to raise an alarm as soon as possible after the distribution changes from p 0 to p 1 but to avoid false alarms specifically we seek a stopping rule n which allows us to observe the s sequentially such that e n is large and subject to this constraint sup k e k n k n k is as small as possible here e k denotes expectation under the change point k and e denotes expectation under the hypothesis of no change whatever in this paper we investigate the performance of the shiryayev roberts pollak srp rule for change point detection in the dynamic system of hidden markov models by making use of markov chain representation for the likelihood function the structure of asymptotically minimax policy and of the bayes rule and sequential hypothesis testing theory for markov random walks we show that the srp procedure is asymptotically minimax in the sense of pollak ann statist 13 1985 206 227 next we present a second order asymptotic approximation for the expected stopping time of such a stopping scheme when w 1 motivated by the sequential analysis in hidden markov models a nonlinear renewal theory for markov random walks is also given

using the generalized likelihood ratio statistic for sequential detection of a change point we study sequential detection of a change point using the generalized likelihood ratio statistic for the special case of detecting a change in a normal mean with known variance we give approximations to the average run lengths and compare our procedure to standard cusum tests and combined cusum shewhart tests several examples indicating extensions to problems involving multiple parameters are discussed

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 a well defined 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 additional conditions on the post change family of distributions it is shown that the proposed algorithm is asymptotically optimal uniformly for all possible post change distributions

sequential detection and identification of a change in the distribution of a markov modulated random sequence the problem of detection and identification of an unobservable change in the distribution of a random sequence is studied via a hidden markov model hmm approach the formulation is bayesian on line discrete time allowing both single and multiple disorder cases dealing with both independent and identically distributed i i d and dependent observations scenarios allowing for statistical dependencies between the change time and change type in both the observation sequence and the risk structure and allowing for general discrete time disorder distributions several of these factors provide useful new generalizations of the sequential analysis theory for change detection and or hypothesis testing taken individually in this paper a unifying framework is provided that handles each of these considerations not only individually but also concurrently optimality results and optimal decision characterizations are given as well as detailed examples that illustrate the myriad of sequential change detection and identification problems that fall within this new framework

nonlinear stability of the muskat problem with capillary pressure at the free boundary the muskat problem models the motion of two immiscible fluids in a porous medium we assume that the medium occupies the exterior of a circle r that the fluids are incompressible and that the capillary pressure at the interface is nonnegligible we take any radially symmetric stationary solution with interface r rs rs meiml l any integer 2 the nonlinear constraints are under some assumptions on the solution also necessary

outlier resistant algorithms for detecting a change in a stochastic process outlier resistant algorithms that detect a change from a given nominal stationary process to another such process are given the nominal processes are assumed to be mutually independent and to satisfy some general regularity conditions the outlier sequences are assumed to be independently and identically distributed and independent of the nominal processes the proposed algorithms are sequential and consist of uniformly bounded steps the asymptotic performance of the algorithms is analyzed both in the absence and the presence of outliers breakdown points and influence functions are defined and analyzed the algorithms are studied in more detail for gaussian autoregressive nominal processes

an efficient sequential nonparametric scheme for detecting a change of distribution ble and remarkably efficient the method is based on the sequential vectors of signs and ranks of the observations the methodology is one of likelihood ratio a sequence of likelihood ratios for these vectors is computed and the shiryayev roberts approach to changepoint detection is then applied to yield a class of statistics and associated stopping rules inequalities and asymptotic approximations for the operating characteristics of these rules are developed these are found to be valid also for small average run lengths and early changepoints as well the relative efficiency of these schemes with respect to a normal parametric shift detection policy is very high making them a robust alternative to parametric methof s