branch and bound search for experimental designs based on d optimality and other criteria this article presents a branch and bound algorithm that constructs a catalog of all d optimal n point designs for specified design region linear model and number of observations n while the primary design criterion is d optimality the algorithm may also be used to find designs performing well by other secondary criteria if a small sacrifice in ellkiency as measured by d optimahty is accepted finally some designs are supplied for a quadratic response surface model

bifurcation analysis of a generalized gause model with prey harvesting and a generalized holling response function of type iii abstract in this paper we study a generalized gause model with prey harvesting and a generalized holling response function of type iii p x m x 2 a x 2 b x 1 the goal of our study is to give the bifurcation diagram of the model for this we need to study saddle node bifurcations hopf bifurcation of codimension 1 and 2 heteroclinic bifurcation and nilpotent saddle bifurcation of codimension 2 and 3 the nilpotent saddle of codimension 3 is the organizing center for the bifurcation diagram the hopf bifurcation is studied by means of a generalized lienard system and for b 0 we discuss the potential integrability of the system the nilpotent point of multiplicity 3 occurs with an invariant line and can have a codimension up to 4 but because it occurs with an invariant line the effective highest codimension is 3 we develop normal forms in which the invariant line is preserved for studying of the nilpotent saddle bifurcation for b 0 the reversibility of the nilpotent saddle is discussed we study the type of the heteroclinic loop and its cyclicity the phase portraits of the bifurcations diagram partially conjectured via the results obtained allow us to give a biological interpretation of the behavior of the two species

estimation of nonstationary spatial covariance structure summary we introduce a method for estimating nonstationary spatial covariance structure from space time data and apply the method to an analysis of sydney wind patterns our method constructs a process honouring a given spatial covariance matrix at observing stations and uses one or more stationary processes to describe conditional behaviour given observing site values the stationary processes give a localised description of the spatial covariance structure the method is computationally attractive and can be extended to the assessment of covariance for multivariate processes the technique is illustrated for data describing the east west component of sydney winds for this example our own methods are contrasted with a geometrically appealing though computationally intensive technique which describes spatial correlation via an isotropic process and a deformation of the geographical space

deformed quantum calogero moser problems and lie superalgebras the deformed quantum calogero moser sutherland problems related to the root systems of the contragredient lie superalgebras are introduced the construction is based on the notion of the generalized root systems suggested by v serganova for the classical series a recurrent formula for the quantum integrals is found which implies the integrability of these problems the corresponding algebras of the quantum integrals are investigated the explicit formulas for their poincare series for generic values of the deformation parameter are presented

wiener disorder problem with observations at fixed discrete time epochs suppose that a wiener process gains a known drift rate at some unobservable disorder time with some zero modified exponential distribution the process is observed only at known fixed discrete time epochs which may not always be spaced in equal distances the problem is to detect the disorder time as quickly as possible by means of an alarm that depends only on the observations of wiener process at those discrete time epochs we show that bayes optimal alarm times which minimize expected total cost of frequent false alarms and detection delay time always exist optimal alarms may in general sound between observation times and when the space time process of the odds that disorder happened in the past hits a set with a nontrivial boundary the optimal stopping boundary is piecewise continuous and explodes as time approaches from left to each observation time on each observation interval if the boundary is not strictly increasing everywhere then it first decreases and then increases it is strictly monotone wherever it does not vanish its decreasing portion always coincides with some explicit function we develop numerical algorithms to calculate nearly optimal detection algorithms and their bayes risks and we illustrate their use on numerical examples the solution of wiener disorder problem with discretely spaced observation times will help reduce risks and costs associated with disease outbreak and production quality control where the observations are often collected and or inspected periodically

new integrable generalizations of calogero moser quantum problem a one parameter deformation of calogero moser quantum problem is introduced it is shown that corresponding schrodinger operator is integrable for any value of the parameter and algebraically integrable in case of integer value

quickest sequential multiband spectrum sensing with mixed observations spectrum sensing is a key technology enabling the cognitive radio system in this paper the problem of how to quickly and accurately find an unoccupied channel from a large amount of potential channels is considered the cognitive radio system under consideration is equipped with a narrow band sensor hence it can only sense those potential channels in a sequential manner in this scenario we propose a novel two stage mixed observation sensing strategy in the first stage which is named as scanning stage the sensor observes a linear combination of the signals from a pair of channels the purpose of the scanning stage is to quickly identify a pair of channels such that at least one of them is highly likely to be unoccupied in the second stage which is called refinement stage the sensor only observers the signal from one of those two channels identified from the first stage and selects one of them as the unoccupied channel the problem under this setup is an ordered two concatenated markov stopping time problem the optimal solution is solved using the tools from the multiple stopping time theory it turns out that the optimal solution has a rather complex structure hence a low complexity algorithm is proposed to facilitate the implementation in the proposed low complexity algorithm the cumulative sum test is adopted in the scanning stage and the sequential probability ratio test is adopted in the refinement stage the performance of this low complexity algorithm is analyzed when the presence of unoccupied channels is rare numerical simulation results show that the proposed sensing strategy can significantly reduce the sensing time when the majority of potential channels are occupied

finite cyclicity of graphics with a nilpotent singularity of saddle or elliptic type abstract in this paper we prove finite cyclicity of several of the most generic graphics through a nilpotent point of saddle or elliptic type of codimension 3 inside c families of planar vector fields in some cases our results are independent of the exact codimension of the point and depend only on the fact that the nilpotent point has multiplicity 3 by blowing up the family of vector fields we obtain all the limit periodic sets we calculate two different types of dulac maps in the blown up family and develop a general method to prove that some regular transition maps have a nonzero higher derivative at a point the finite cyclicity theorems are derived by a generalized derivation division method introduced by roussarie

optimal stopping of linear diffusions with random discounting we propose a new solution method for optimal stopping problems with random discounting for linear diffusions whose state space has a combination of natural absorbing or reflecting boundaries the method uses a concave characterization of excessive functions for linear diffusions killed at a rate determined by a markov additive functional and reduces the original problem to an undiscounted optimal stopping problem for a standard brownian motion the latter can be solved essentially by inspection the necessary and sufficient conditions for the existence of an optimal stopping rule are proved when the reward function is continuous the results are illustrated on examples

quantum integrability of the deformed elliptic calogero moser problem the integrability of the deformed quantum elliptic calogero moser problem introduced by chalykh feigin and veselov is proven explicit recursive formulas for the integrals are found for integer values of the parameter this implies the algebraic integrability of the systems

analyzing nonstationary spatial data using piecewise gaussian processes in many problems in geostatistics the response variable of interest is strongly related to the underlying geology of the spatial location in these situations there is often little correlation in the responses found in different rock strata so the underlying covariance structure shows sharp changes at the boundaries of the rock types conventional stationary and nonstationary spatial methods are inappropriate because they typically assume that the covariance between points is a smooth function of distance in this article we propose a generic method for the analysis of spatial data with sharp changes in the underlying covariance structure our method works by automatically decomposing the spatial domain into disjoint regions within which the process is assumed to be stationary but the data are assumed independent across regions uncertainty in the number of disjoint regions their shapes and the model within regions is dealt with in a fully bayesian fashion we illustrate our approach on a previously

bayesian prediction of deterministic functions with applications to the design and analysis of computer experiments abstract this article is concerned with prediction of a function y t over a multidimensional domain t given the function values at a set of sites t 1 t 2 t n in t and with the design that is with the selection of those sites the motivating application is the design and analysis of computer experiments where t determines the input to a computer model of a physical or behavioral system and y t is a response that is part of the output or is calculated from it following a bayesian formulation prior uncertainty about the function y is expressed by means of a random function y which is taken here to be a gaussian stochastic process the mean of the posterior process can be used as the prediction function t and the variance can be used as a measure of uncertainty this kind of approach has been used previously in bayesian interpolation and is strongly related to the kriging methods used in geostatistics here emphasis is placed on product linear and product cubic correlation func

minimax robust decentralized detection decentralized detection problems are studied where the sensor distributions are not specified completely the sensor distributions are assumed to belong to known uncertainty classes it is shown for a broad class of such problems that a set of least favorable distributions exists for minimax robust testing between the hypotheses it is hence established that the corresponding minimax robust tests are solutions to simple decentralized detection problems for which the sensor distributions are specified to be the least favorable distributions

maximum entropy sampling and optimal bayesian experimental design when shannon entropy is used as a criterion in the optimal design of experiments advantage can be taken of the classical identity representing the joint entropy of parameters and observations as the sum of the marginal entropy of the observations and the preposterior conditional entropy of the parameters following previous work in which this idea was used in spatial sampling the method is applied to standard parameterized bayesian optimal experimental design under suitable conditions which include non linear as well as linear regression models it is shown in a few steps that maximizing the marginal entropy of the sample is equivalent to minimizing the preposterior entropy the usual bayesian criterion thus avoiding the use of conditional distributions it is shown using this marginal formulation that under normality assumptions every standard model which has a two point prior distribution on the parameters gives an optimal design supported on a single point other results include a new asymptotic formula which applies as the error variance is large and bounds on support size

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

sequential sensor installation for wiener disorder detection we consider a centralized multisensor online quickest disorder detection problem where the observation from each sensor is a wiener process gaining a constant drift at a common unobservable disorder time the objective is to detect the disorder time as quickly as possible with small probability of false alarms unlike the earlier work on multisensor change detection problems we assume that the observer can apply a sequential sensor installation policy at any time before a disorder alarm is raised the observer can install new sensors to collect additional signals the sensors are statistically identical and there is a fixed installation cost per sensor we propose a bayesian formulation of the problem we identify an optimal policy consisting of a sequential sensor installation strategy and an alarm time which minimize a linear bayes risk of detection delay false alarm and new sensor installations we also provide a numerical algorithm and illustrate it on examples our numerical examples show that

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

handbook of brownian motion facts and formulae i theory i stochastic processes in general ii linear diffusions iii stochastic calculus iv brownian motion v local time as a markov process vi differential systems associated to brownian motion appendix 1 briefly on some diffusions ii tables of distributions of functionals of brownian motion and related processes 1 brownian motion 2 brownian motion with drift 3 reflecting brownian motion 4 bessel process of order 5 bessel process of order 1 2 6 bessel process of order zero 7 ornstein uhlenbeck process 8 radial ornstein uhlenbeck process 9 geometric brownian motion appendix 2 special functions appendix 3 inverse laplace transforms appendix 4 differential equations and their solutions appendix 5 formulae for n fold differentiation

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

poisson disorder problem with exponential penalty for delay we solve the poisson disorder problem when the delay is penalized exponentially our objective is to detect as quickly as possible the unobservable time of the change or disorder in the intensity of a poisson process the disorder time delimits two different regimes in which one employs distinct strategies e g investment advertising manufacturing we seek a stopping rule that minimizes the frequency of false alarms and an exponential unlike previous formulations which use a linear cost function of the detection delay in the financial applications the exponential penalty is a more apt measure for the delay cost because of the compounding of the investment growth the poisson disorder problem with a linear delay cost was studied by peskir and shiryaev 2002 solving the poisson disorder problem advances in finance and stochastics springer berlin germany 295 312 which is a limiting case of ours

connections between optimal stopping and singular stochastic control ii reflected follower problems the stochastic control problem of following a brownian path by a process of bounded variation and subject to a reflecting barrier at the origin reflected follower is reduced to a question of optimal stopping with absorption direct probabilistic arguments are used to establish the equivalence of the two problems under suitable conditions on the cost functions

parameter estimation the proper way to use bayesian posterior processes with brownian noise this paper studies a problem of bayesian parameter estimation for a sequence of scaled counting processes whose weak limit is a brownian motion with an unknown drift the main result of the paper is that the limit of the posterior distribution processes is in general not equal to the posterior distribution process of the mentioned brownian motion with the unknown drift instead it is equal to the posterior distribution process associated with a brownian motion with the same unknown drift and a different standard deviation coefficient the difference between the two standard deviation coefficients can be arbitrarily large the characterization of the limit of the posterior distribution processes is then applied to a family of stopping time problems we show that the proper way to find asymptotically optimal solutions to stopping time problems with respect to the scaled counting processes is by looking at the limit of the posterior distribution processes rather than by the naive approach of looking at

optimal multiple stopping of linear diffusions motivated by the analysis of financial instruments with multiple exercise rights of american type and mean reverting underlyers we formulate and solve the optimal multiple stopping problem for a general linear regular diffusion process and a general reward function instead of relying on specific properties of geometric brownian motion and call and put option payoffs as in most of the existing literature we use general theory of optimal stopping for diffusions and we illustrate the resulting optimal exercise policies by concrete examples and constructive recipes

a note on sequential detection with exponential penalty for the delay we study the continuous time analogue of the bayes problem of poor as usual the results on the optimal solution are more explicit than in the discrete time set up

nonparametric estimation of nonstationary spatial covariance structure abstract estimation of the covariance structure of spatial processes is a fundamental prerequisite for problems of spatial interpolation and the design of monitoring networks we introduce a nonparametric approach to global estimation of the spatial covariance structure of a random function z x t observed repeatedly at times ti i 1 t at a finite number of sampling stations xi i 1 2 n in the plane our analyses assume temporal stationarity but do not assume spatial stationarity or isotropy we analyze the spatial dispersions var z xi t z xj t as a natural metric for the spatial covariance structure and model these as a general smooth function of the geographic coordinates of station pairs xi xj the model is constructed in two steps first using nonmetric multidimensional scaling mds we compute a two dimensional representation of the sampling stations for which a monotone function of interpoint distances ij approximates the spatial dispersions mds transforms the problem

quickest detection in coupled systems this work considers the problem of quickest detection of signals in a coupled system of n sensors which receive continuous sequential observations from the environment it is assumed that the signals which are modeled by general ito processes are coupled across sensors but that their onset times may differ from sensor to sensor two main cases are considered in the first one signal strengths are the same across sensors while in the second one they differ by a constant the objective is the optimal detection of the first time at which any sensor in the system receives a signal the problem is formulated as a stochastic optimization problem in which an extended minimal kullback leibler divergence criterion is used as a measure of detection delay with a constraint on the mean time to the first false alarm the case in which the sensors employ cumulative sum cusum strategies is considered and it is proved that the minimum of n cusums is asymptotically optimal as the mean time to the first false

bayesian switching multiple disorder problems the switching multiple disorder problem seeks to determine an ordered infinite sequence of times of alarms which are as close as possible to the unknown times of disorders or change points at which the observable process changes its probability characteristics we study a bayesian formulation of this problem for an observable brownian motion with switching constant drift rates the method of proof is based on the reduction of the initial problem to an associated optimal switching problem for a three dimensional diffusion posterior probability process and the analysis of the equivalent coupled parabolic type free boundary problem we derive analytic form estimates for the bayesian risk function and the optimal switching boundaries for the components of the posterior probability process

second order asymptotic optimality in multisensor sequential change detection a generalized multisensor sequential change detection problem is considered in which a number of possibly correlated sensors monitor an environment in real time the joint distribution of their observations is determined by a global parameter vector and at some unknown time there is a change in an unknown subset of components of this parameter vector the goal is to detect the change as soon as possible while controlling the rate of false alarms we establish the second order asymptotic optimality with respect to lorden s criterion of various generalizations of the cusum rule that is we show that their additional expected worst case detection delay relative to the one that could be achieved if the affected subset was known remains bounded as the rate of false alarm goes to 0 for any possible subset of affected components this general framework incorporates the traditional multisensor setup in which only an unknown subset of sensors is affected by the change the latter problem has a special structure which we exploit in order to obtain feasible representations of the proposed schemes we present the results of a simulation study where we compare the proposed schemes with scalable detection rules that are only first order asymptotically optimal finally in the special case that the change affects exactly one sensor we consider the scheme that runs in parallel the local cusum rules and study the problem of specifying the local thresholds

a comparison of three methods for selecting values of input variables in the analysis of output from a computer code two types of sampling plans are examined as alternatives to simple random sampling in monte carlo studies these plans are shown to be improvements over simple random sampling with respect to variance for a class of estimators which includes the sample mean and the empirical distribution function

the usefulness of optimum experimental designs optimum experimental designs were originally developed by kiefer mainly for response surface models this survey of recent developments emphasizes potential or actual usefulness for linear models the construction of exact designs particularly over irregular design regions is stressed as is the blocking of response surface designs other important areas include systematic designs that are robust against trend and designs for mixtures with irregular design regions several industrial examples are mentioned both d and c optimum designs are found for a non linear model of the economic response of cereal production to fertilizer level the c optimum design being for the conditions of maximum economic return locally optimum and bayesian designs are both described similar results for generalized linear models lead to designs for the ld 95 in a logistic model in which male and female subjects respond differently designs with structure in the variance suggest alternatives to the potentially wasteful product designs of taguchi designs for sequential clinical trials to include random balance are presented the last section outlines some applications including life testing and models in which time is a factor

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

an exact algorithm for maximum entropy sampling we study the experimental design problem of selecting a most informative subset having prespecified size from a set of correlated random variables the problem arises in many applied domains such as meteorology environmental statistics and statistical geology in these applications observations can be collected at different locations and possibly at different times information is measured by entropy in the gaussian case the problem is recast as that of maximizing the determinant of the covariance matrix of the chosen subset we demonstrate that this problem is np hard we establish an upper bound for the entropy based on the eigenvalue interlacing property and we incorporate this bound in a branch and bound algorithm for the exact solution of the problem we present computational results for estimated covariance matrices that correspond to sets of environmental monitoring stations in the united states

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

on the optimal exercise boundaries of swing put options we use probabilistic methods to characterise time dependent optimal stopping boundaries in a problem of multiple optimal stopping on a finite time horizon motivated by financial applications we consider a payoff of immediate stopping of put type and the underlying dynamics follows a geometric brownian motion the optimal stopping region relative to each optimal stopping time is described in terms of two boundaries which are continuous monotonic functions of time and uniquely solve a system of coupled integral equations of volterra type finally we provide a formula for the value function of the problem

decentralized detection with censoring sensors in the censoring approach to decentralized detection sensors transmit real valued functions of their observations when informative and save energy by not transmitting otherwise we address several practical issues in the design of censoring sensor networks including the joint dependence of sensor decision rules randomization of decision strategies and partially known distributions in canonical decentralized detection problems involving quantization of sensor observations joint optimization of the sensor quantizers is necessary we show that under a send no send constraint on each sensor and when the fusion center has its own observations the sensor decision rules can be determined independently in terms of design and particularly for adaptive systems the independence of sensor decision rules implies that minimal communication is required we address the uncertainty in the distribution of the observations typically encountered in practice by determining the optimal sensor decision rules and fusion rule for three formulations a robust formulation generalized likelihood ratio tests and a locally optimum formulation examples are provided to illustrate the independence of sensor decision rules and to evaluate the partially known formulations

near optimal sensor placements in gaussian processes theory efficient algorithms and empirical studies when monitoring spatial phenomena which can often be modeled as gaussian processes gps choosing sensor locations is a fundamental task there are several common strategies to address this task for example geometry or disk models placing sensors at the points of highest entropy variance in the gp model and a d or e optimal design in this paper we tackle the combinatorial optimization problem of maximizing the mutual information between the chosen locations and the locations which are not selected we prove that the problem of finding the configuration that maximizes mutual information is np complete to address this issue we describe a polynomial time approximation that is within 1 1 e of the optimum by exploiting the submodularity of mutual information we also show how submodularity can be used to obtain online bounds and design branch and bound search procedures we then extend our algorithm to exploit lazy evaluations and local structure in the gp yielding significant speedups we also extend our approach to find placements which are robust against node failures and uncertainties in the model these extensions are again associated with rigorous theoretical approximation guarantees exploiting the submodularity of the objective function we demonstrate the advantages of our approach towards optimizing mutual information in a very extensive empirical study on two real world data sets

learning from minimum entropy queries in a large committee machine in supervised learning the redundancy contained in random examples can be avoided by learning from queries using statistical mechanics we study learning from minimum entropy queries in a large tree committee machine the generalization error decreases exponentially with the number of training examples providing a significant improvement over the algebraic decay for random examples the connection between entropy and generalization error in multi layer networks is discussed and a computationally cheap algorithm for constructing queries is suggested and analysed

a dirichlet process gaussian state machine model for change detection in transient processes abstractthe ability to detect incipient and critical changes in real world process esessential for system integrity assurance is currently impeded by the mismatch between the key assumption of stationarity underlying most change detection methods and the nonlinear and nonstationary transient dynamics of most real world processes the current approaches are slow or outright unable to detect qualitative changes in the behaviors that lead to anomalies we present a dirichlet process gaussian state machine dpgsm model to represent dynamic intermittency which is one of the most ubiquitous real world transient behaviors the dpgsm model treats a signal as a random walk among a dirichlet process mixture of gaussian clusters hypothesis tests and a numerical scheme based on this nonparametric representation were developed to detect subtle changes in the transient intermittent dynamics experimental investigations suggest that the dpgsm approach can consistently detect incipient critical changes in intermi

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

sprt and cusum in hidden markov models in this paper we study the problems of sequential probability ratio tests for parameterized hidden markov models we investigate in some detail the performance of the tests and derive corrected brownian approximations for error probabilities and expected sample sizes asymptotic optimality of the sequential probability ratio test for testing simple hypotheses based on hidden markov chain data is established next we consider the cumulative sum cusum procedure for change point detection in this model based on the renewal property of the stopping rule cusum can be regarded as a repeated one sided sequential probability ratio test asymptotic optimality of the cusum procedure is proved in the sense of lorden 1971 motivated by the sequential analysis in hidden markov models wald s likelihood ratio identity and wald s equation for products of markov random matrices are also given we apply these results to several types of hidden markov models i i d hidden markov models switch gaussian regression and switch gaussian autoregression which are commonly used in digital communications speech recognition bioinformatics and economics

organizing center for the bifurcation analysis of a generalized gause model with prey harvesting and holling response function of type iii abstract the present note is an addendum to the paper of etoua rousseau 2010 1 which presented a study of a generalized gause model with prey harvesting and a generalized holling response function of type iii p x m x 2 a x 2 b x 1 complete bifurcation diagrams were proposed but some parts were conjectural an organizing center for the bifurcation diagram was given by a nilpotent point of saddle type lying on an invariant line and of codimension greater than or equal to 3 this point was of codimension 3 when b 0 and was conjectured to be of infinite codimension when b 0 this conjecture was in line with a second conjecture that the hopf bifurcation of order 2 degenerates to a hopf bifurcation of infinite codimension when b 0 in this note we prove these two conjectures

a note on ritov s bayes approach to the minimax property of the cusum procedure 1 introduction page 1954 proposed the cusum procedures as a solu tion to the problem of sequential detection of a change in distribution lorden 1971 introduced a notion of minimax optimality for change point models andshowed that the cusum procedures are asymptotically optimal in his sense moustakides 1986 then showed the minimax optimality of the cusum pro cedures when the initial and nal distribution are both known ritov 1990 used a bayes approach and gave an interesting argument for moustakides result similar to that of lehmann 1959 on the optimality of the sequentialprobability ratio test we transfer his approach to a continuous time modeland discuss the structure of the bayes risk let be an arbitrary but known constant we will consider a process wwithw

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

active hypothesis testing for anomaly detection the problem of detecting a single anomalous process among a finite number m of processes is considered at each time a subset of the processes can be observed and the observations from each chosen process follow two different distributions depending on whether the process is normal or abnormal the objective is a sequential search strategy that minimizes the expected detection time subject to an error probability constraint this problem can be considered as a special case of active hypothesis testing first considered by chernoff where a randomized strategy referred to as the chernoff test was proposed and shown to be asymptotically as the error probability approaches zero optimal for the special case considered in this paper we show that a simple deterministic test achieves asymptotic optimality and offers better performance in the finite regime we further extend the problem to the case where multiple anomalous processes are present in particular we examine the case where only an upper bound on the number of anomalous processes is known

a sequential procedure for multihypothesis testing the sequential testing of more than two hypotheses has important applications in direct sequence spread spectrum signal acquisition multiple resolution element radar and other areas a useful sequential test which we term the msprt is studied in this paper the test is shown to be a generalization of the sequential probability ratio test under bayesian assumptions it is argued that the msprt approximates the much more complicated optimal test when error probabilities are small and expected stopping times are large bounds on error probabilities are derived and asymptotic expressions for the stopping time and error probabilities are given a design procedure is presented for determining the parameters of the msprt two examples involving gaussian densities are included and comparisons are made between simulation results and asymptotic expressions comparisons with bayesian fixed sample size tests are also made and it is found that the msprt requires two to three times fewer samples on average

connections between optimal stopping and singular stochastic control i monotone follower problems the stochastic control problem of tracking a brownian motion by a nondecreasing process monotone follower is related to a question of optimal stopping direct probabilistic arguments are employed to show that the two problems are equivalent and that both admit optimal solutions

optimal switching between a pair of brownian motions consider two brownian motions b 1 s1 and b 2 s2 each taking values on an interval 0 a i i 1 2 with absorption at the endpoints we can alternate between letting b 1 s1 run while freezing b 2 s2 and letting b 2 s2 run while freezing b 1 s1 this results in a swtiched process that evolves in the rectangle d 0 a 1 0 a 2 like a horizontal brownian motion when b 2 s2 freezes and like a vertical brownian motion when b 1 s1 freezes let f x 1 x 2 be a nonnegative continuous payoff function defined on the boundary d of d a control consists of a switching strategy and a stopping time we study the problem of finding an optimal control which maximizes the expected payoff obtained at time stopping in the interior results in zero reward

generalized lame operators we introduce a class of multidimensional schrodinger operators with elliptic potential which generalize the classical lame operator to higher dimensions one natural example is the calogero moser operator others are related to the root systems and their deformations we conjecture that these operators are algebraically integrable which is a proper generalization of the finite gap property of the lame operator using earlier results of braverman etingof and gaitsgory we prove this under additional assumption of the usual liouville integrability in particular this proves the chalykh veselov conjecture for the elliptic calogero moser problem for all root systems we also establish algebraic integrability in all known two dimensional cases a general procedure for calculating the bloch eigenfunctions is explained it is worked out in detail for two specific examples one is related to the b2 case another one is a certain deformation of the a2 case in these two cases we also obtain similar results for the discrete versions of these problems related to the difference operators of macdonald ruijsenaars type

on the one dimensional optimal switching problem we explicitly solve the optimal switching problem for one dimensional diffusions by directly using the dynamic programming principle and the excessive characterization of the value function the shape of the value function and the smooth fit principle then can be proved using the properties of concave functions