glauberman correspondence of p blocks of finite groups abstract let g and a be finite groups with coprime orders suppose that a is solvable and that it acts on g by automorphisms let c cg a by irr g and irra g we denote the set of all irreducible and all a invariant irreducible characters of g respectively let d c be a fixed p subgroup of g for a prime p using the glauberman correspondence g a irr a g irr c a watanabe j algebra216 1999 548 565 recently established a glauberman correspondence between a invariant p blocks b of g with defect group d and p blocks b1 of c with defect group d let br b and br b1 be the brauer correspondents of b and b1 in ng d and nc d respectively the main result of this article asserts that the block algebras br b and br b1 are morita equivalent furthermore if g is p solvable and d is abelian then the block algebras b and b1 are morita equivalent

asymptotically optimal quantizers for detection of i i d the asymptotic probability of error for quantization in maximum likelihood tests is analyzed the authors assume quantizers with large numbers of levels generated from a companding function a theorem that relates the companding function to the asymptotic probability of error is proved the companding function is then optimized

decentralized sequential detection a class of decentralized sequential detection problems is investigated under appropriate independence assumptions it is shown that at each time instant the optimal local strategies are given by threshold tests on the likelihood of ratios furthermore local decisions depend not only on the present and past observations but on the past local decision as well that is for each local processor a different threshold exists for every different sequence of past decisions examples computational techniques discussions of the difficulties at hand and suggestions for further exploration of the problem are presented

high rate vector quantization for detection we investigate high rate quantization for various detection and reconstruction loss criteria a new distortion measure is introduced which accounts for global loss in best attainable binary hypothesis testing performance the distortion criterion is related to the area under the receiver operating characteristic roc curve specifically motivated by sanov s theorem we define a performance curve as the trajectory of the pair of optimal asymptotic type i and type ii error rates of the most powerful neyman pearson test of the hypotheses the distortion measure is then defined as the difference between the area under the curve auc of the optimal pre encoded hypothesis test and the auc of the optimal post encoded hypothesis test as compared to many previously introduced distortion measures for decision making this distortion measure has the advantage of being independent of any detection thresholds or priors on the hypotheses which are generally difficult to specify in the code design process a high resolution zador gersho type of analysis is applied to characterize the point density and the inertial profile associated with the optimal high rate vector quantizer the analysis applies to a restricted class of high rate quantizers that have bounded cells with vanishing volumes the optimal point density is used to specify a lloyd type algorithm which allocates its finest resolution to regions where the gradient of the pre encoded likelihood ratio has greatest magnitude

classification error analysis in stereo vision abstract depth perception in humans is obtained by comparing images generated by the two eyes to each other given the highly stochastic nature of neurons in the brain this comparison requires maximizing the mutual information mi between the neuronal responses in the two eyes by distributing the coding information across a large number of neurons unfortunately mi is not an extensive quantity making it very difficult to predict how the accuracy of depth perception will vary with the number of neurons n in each eye to address this question we present a two arm distributed decentralized sensors detection model we demonstrate how the system can extract depth information from a pair of discrete valued stimuli represented here by a pair of random dot matrix stereograms using the theory of large deviations we calculated the rate at which the global average error probability of our detector and the mi between the two arms output vary with n we found that mi saturates exponentially with n at a rate which decays as 1 n the rate function approached the chernoff distance between the two probability distributions asymptotically our results may have implications in computer stereo vision that uses hebbian based algorithms for terrestrial navigation

randomized sensor selection in sequential hypothesis testing we consider the problem of sensor selection for time optimal detection of a hypothesis we consider a group of sensors transmitting their observations to a fusion center the fusion center considers the output of only one randomly chosen sensor at the time and performs a sequential hypothesis test we study a sequential multiple hypothesis test with randomized sensor selection strategy we incorporate the random processing times of the sensors to determine the asymptotic performance characteristics of this test for three distinct performance metrics we show that for a generic set of sensors and binary hypothesis the time optimal policy requires the fusion center to consider at most two sensors we also show that for the case of multiple hypothesis the time optimal policy needs at most as many sensors to be observed as the number of underlying hypotheses

asymptotic optimality theory for decentralized sequential hypothesis testing in sensor networks the decentralized sequential hypothesis testing problem is studied in sensor networks where a set of sensors receive independent observations and send summary messages to the fusion center which makes a final decision in the scenario where the sensors have full access to their past observations the first asymptotically bayes sequential test is developed having the same asymptotic performance as the optimal centralized test that has access to all sensor observations next in the scenario where the sensors do not have full access to their past observations a simple but asymptotically bayes sequential tests is developed in which sensor message functions are what we call tandem quantizer where each sensor only uses two different sensor quantizers with at most one switch between these two possibilities moreover a new minimax formulation of optimal stationary sensor quantizers is proposed and is studied in detail in the case of additive gaussian sensor noise finally our results show that in the simplest models feedback from the fusion center does not improve asymptotic performance in the scenario with full local memory however even a one shot one bit feedback can significantly improve performance in the case of limited local memory

high rate vector quantization for the neyman pearson detection of correlated processes this paper investigates the effect of quantization on the performance of the neyman pearson test it is assumed that a sensing unit observes samples of a correlated stationary ergodic multivariate process each sample is passed through an n point quantizer and transmitted to a decision device which performs a binary hypothesis test for any false alarm level it is shown that the miss probability of the neyman pearson test converges to zero exponentially as the number of samples tends to infinity assuming that the observed process satisfies certain mixing conditions the main contribution of this paper is to provide a compact closed form expression of the error exponent in the high rate regime i e when the number n of quantization levels tends to infinity generalizing previous results of gupta and hero to the case of nonindependent observations if d represents the dimension of one sample it is proved that the error exponent converges at rate n2 d to the one obtained in the absence of quantization as an application relevant high rate quantization strategies which lead to a large error exponent are determined numerical results indicate that the proposed quantization rule can yield better performance than existing ones in terms of detection error

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

on the optimality of the likelihood ratio test for local sensor decision rules in the presence of nonideal channels distributed detection has been intensively studied in the past in this correspondence we consider the design of local decision rules in the presence of nonideal transmission channels between the sensors and the fusion center under the conditional independence assumption among multiple sensor observations we show that the optimal local decisions that minimize the error probability at the fusion center amount to a likelihood ratio test lrt given a particular constraint on the fusion rule this constraint turns out to be quite general and is easily satisfied for most sensible fusion rules a design example using a parallel sensor fusion structure with binary symmetric channels bscs between local sensors and the fusion center is given to illustrate the usefulness of the result in obtaining optimal thresholds for local sensor observations the study that incorporates the transmission channel in the sensor system design may have potential applications in the emerging field of wireless sensor networks

sequential detection of markov targets with trajectory estimation the problem of detection and possible estimation of a signal generated by a dynamic system when a variable number of noisy measurements can be taken is here considered assuming a markov evolution of the system in particular the pair signal observation forms a hidden markov model hmm a sequential procedure is proposed wherein the detection part is a sequential probability ratio test sprt and the estimation part relies upon a maximum a posteriori probability map criterion gated by the detection stage the parameter to be estimated is the trajectory of the state evolution of the system itself a thorough analysis of the asymptotic behavior of the test in this new scenario is given and sufficient conditions for its asymptotic optimality are stated i e for almost sure minimization of the stopping time and for first order minimization of any moment of its distribution an application to radar surveillance problems is also examined

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

encoding of correlated observations an important class of engineering problems involves sensing an environment and making estimates based on the phenomena sensed in the traditional model of this problem the sensors observations are available to the estimator without alteration there is growing interest in em distributed sensing systems in which several observations are communicated to the estimator over channels of limited capacity the observations must be separately encoded so that the target can be estimated with minimum distortion two questions are addressed for a special case of this problem wherein there are two sensors which observe noisy data and communicate with a single estimator 1 if the encoder is unlimited in complexity what communication rates and distortions can be achieved 2 if the encoder must be a quantizer a mapping of a single observation sample into a digital output how can it be designed for good performance the first question is treated by the techniques of information theory it is proved that a given combination of rates and distortion is achievable if there exist degraded versions of the observations that satisfy certain formulas the second question is treated by two approaches in the first the outputs of the quantizers undergo a second stage of encoding which exploits their correlation to reduce the output rate algorithms which design the second stage are presented and tested the second approach is based on the em distributional distance a measure of dissimilarity between two probability distributions an algorithm to modify a quantizer for increased distributional distance is derived and tested

quantization for maximin are in distributed estimation we consider the design of optimal quantizers for the distributed estimation of a deterministic parameter in particular we design deterministic scalar quantizers to maximize the minimum asymptotic relative efficiency are between quantized and unquantized ml estimators we first design identical quantizers using the class of score function quantizers sfq we show that the structure of sfqs generally depend on the parameter value but can be expressed as thresholds on the sufficient statistic for a large class of distributions we provide a convergent iterative algorithm to obtain the best sfq that maximizes the minimum are for distributions of that class we compare the performance of the optimal sfq with a general quantizer designed without making any restrictions on the structure this general quantizer is hard to implement due to lack of structure but is optimal if the iterative design algorithm does not encounter local minima through numerical simulations we illustrate that the two quantizers designed are identical in other words the optimal quantizer structure is that of an sfq for a distributed estimation setup designing identical quantizers is shown to be suboptimal we therefore propose a joint multiple quantizer design algorithm based on a person by person optimization technique employing the sfq structure using numerical examples we illustrate the gain in performance due to designing nonidentical quantizers

directionality of coupling from bivariate time series how to avoid false causalities and missed connections we discuss some problems encountered in inference of directionality of coupling or in the case of two interacting systems in inference of causality from bivariate time series we identify factors and influences that can lead to either decreased test sensitivity or false detections and propose ways to cope with them in order to perform tests with high sensitivity and a low rate of false positive results

asymptotic optimality theory for decentralized sequential multihypothesis testing problems the bayesian formulation of sequentially testing m 3 hypotheses is studied in the context of a decentralized sensor network system in such a system local sensors observe raw observations and send quantized sensor messages to a fusion center which makes a final decision when stopping taking observations asymptotically optimal decentralized sequential tests are developed from a class of two stage tests that allows the sensor network system to make a preliminary decision in the first stage and then optimize each local sensor quantizer accordingly in the second stage it is shown that the optimal local quantizer at each local sensor in the second stage can be defined as a maximin quantizer which turns out to be a randomization of at most m 1 unambiguous likelihood quantizers ulq we first present in detail our results for the system with a single sensor and binary sensor messages and then extend to more general cases involving any finite alphabet sensor messages multiple sensors or composite hypotheses

optimal transmission rate control policies in a wireless link under partial state information we consider the problem of phy layer transmission rate control for maximum throughput on a wireless link over a finite time horizon for each link state there exists a most advantageous transmission rate for the controller the controller needs to rely on incomplete link state information in the form of positive acknowledgements acks and negative acknowledgements nacks for transmitted packets we cast the problem as a partially observed markov decision processes pomdp we focus on the special case of time invariant or slowly time varying link quality with two states and two control options in that case the optimal control problem is a state estimation one and the optimal strategy is of threshold structure where the thresholds are certain values of the belief of being in one of the two states the policy admits a simple interpretation increase rate when the number of successive acks exceeds a threshold and decrease rate when the number of successive nacks exceeds a threshold

glauberman watanabe corresponding p blocks of finite groups with normal defect groups are morita equivalent let g be a finite group and let a be a solvable finite group that acts on g such that the orders of g and a are relatively prime let b be a p block of g with normal defect group d such that a stabilizes b and d c g a then there is a morita equivalence between the block b and its watanabe correspondent block w b of c g a given by a bimodule m with vertex d and trivial source that on the character level induces the glauberman correspondence and which is an isotypy by a theorem of watanabe

on optimal quantization rules for some problems in sequential decentralized detection we consider the design of systems for sequential decentralized detection a problem that entails several interdependent choices the choice of a stopping rule specifying the sample size a global decision function a choice between two competing hypotheses and a set of quantization rules the local decisions on the basis of which the global decision is made this correspondence addresses an open problem of whether in the bayesian formulation of sequential decentralized detection optimal local decision functions can be found within the class of stationary rules we develop an asymptotic approximation to the optimal cost of stationary quantization rules and exploit this approximation to show that stationary quantizers are not optimal in a broad class of settings we also consider the class of blockwise stationary quantizers and show that asymptotically optimal quantizers are likelihood based threshold rules

asymptotic efficiency of a sequential multihypothesis test a sequential multihypothesis test known as the m ary sequential probability ratio test msprt is generalized to account for nonuniform decision costs bounds on error probabilities and asymptotic expressions for the stopping time and error probabilities are given a key result of this correspondence is a proof that the generalized msprt is asymptotically efficient

reward rate maximization in sequential identification under a stochastic deadline any intelligent system performing evidence based decision making under time pressure must negotiate a speed accuracy trade off in computer science and engineering this is typically modeled as minimizing a bayes risk functional that is a linear combination of expected decision delay and expected terminal decision loss in neuroscience and psychology however it is often modeled as maximizing the long term reward rate or the ratio of expected terminal reward and expected decision delay the two approaches have opposing advantages and disadvantages while bayes risk minimization can be solved with powerful dynamic programming techniques unlike reward rate maximization it also requires the explicit specification of the relative costs of decision delay and error which is obviated by reward rate maximization here we demonstrate that for a large class of sequential multihypothesis identification problems under a stochastic deadline the reward rate maximization is equivalent to a special case of bayes r

a stochastic sensor selection scheme for sequential hypothesis testing with multiple sensors we study the problem of binary sequential hypothesis testing using multiple sensors with associated observation costs an off line randomized sensor selection strategy in which a sensor is chosen at every time step with a given probability is considered the objective of this work is to find a sequential detection rule and a sensor selection probability vector such that the expected total observation cost is minimized subject to constraints on reliability and sensor usage first the sequential probability ratio test is shown to be the optimal sequential detection rule in this framework as well efficient algorithms for obtaining the optimal sensor selection probability vector are then derived in particular a special class of problems in which the algorithm has complexity that is linear in the number of sensors is identified an upper bound for the average sensor usage to estimate the error incurred due to wald s approximations is also presented this bound can be used to set a safety margin for guaranteed satisfaction of the constraints on the sensor usage

a repeated significance test with applications to sequential detection in sensor networks in this paper we introduce a randomly truncated sequential hypothesis test using the framework of a repeated significance test rst we study a sequential test with truncation time based on a random stopping time using the functional central limit theorem fclt for a sequence of statistics we derive a general result that can be employed in developing a repeated significance test with random sample size we present effective methods for evaluating accurate approximations for the probability of type i error and the power function numerical results are presented to evaluate the accuracy of these approximations we apply the proposed test to a decentralized sequential detection problem in sensor networks sns with communication constraints finally a sequential detection problem with measurements at random times is investigated

fine quantization in signal detection and estimation the performance lost to data quantization in signal detection and estimation procedures is considered the performance is measured by f divergences which are useful indices of discrimination information between statistical hypotheses some properties of the f divergences are briefly discussed and a general result for the loss in divergence due to uniform data quantization is obtained several applications of this result in specific problems of signal detection and estimation being developed and some numerical results that illustrate the asymptotic behavior of the divergence in these applications are given the divergence lost to finite nonuniform quantization through the use of a companding model is considered and the asymptotically optimum nonuniform quantizer within this class is derived some interesting problems that remain open are discussed

controlled sensing for multihypothesis testing the problem of multiple hypothesis testing with observation control is considered in both fixed sample size and sequential settings in the fixed sample size setting for binary hypothesis testing the optimal exponent for the maximal error probability corresponds to the maximum chernoff information over the choice of controls and a pure stationary open loop control policy is asymptotically optimal within the larger class of all causal control policies for multihypothesis testing in the fixed sample size setting lower and upper bounds on the optimal error exponent are derived it is also shown through an example with three hypotheses that the optimal causal control policy can be strictly better than the optimal open loop control policy in the sequential setting a test based on earlier work by chernoff for binary hypothesis testing is shown to be first order asymptotically optimal for multihypothesis testing in a strong sense using the notion of decision making risk in place of the overall probability of error another test is also designed to meet hard risk constrains while retaining asymptotic optimality the role of past information and randomization in designing optimal control policies is discussed

distributed structures sequential optimization and quantization for detection in the design of distributed quantization systems one inevitably confronts two types of constraints those imposed by a distributed system s structure and those imposed by how the distributed system is optimized structural constraints are inherent properties of any distributed quantization system and are normally summarized by functional relationships defining the inputs and outputs of their component quantizers the use of suboptimal optimization methods are often necessitated by the computational complexity encountered in distributed problems this correspondence briefly explores the impact and interplay of these two types of constraints in the context of distributed quantization for detection we introduce two structures that exploit interquantizer communications and that represent extremes in terms of their structural constraints we then develop a sequential optimization scheme that maximizes the kullback leibler divergence takes advantage of statistical dependencies in the distributed system s output variables and leads to simple parameterizations of the component quantization rules we present an illustrative example from which we draw insights into how these constraints influence the quantization boundaries and affect performance relative to lower and upper bounds

conditional lyapunov exponents from time series a method for estimating conditional lyapunov exponents from time series of two unidirectionally coupled chaotic systems is developed it uses two scalar data sets one taken from the driving and the other from the response system and enables one to detect a generalized synchronization in an experiment without recourse to an auxiliary response system the method is illustrated on coupled maps as well as coupled chaotic flow models

decentralized estimation in an inhomogeneous sensing environment we consider decentralized estimation of a noise corrupted deterministic parameter by a bandwidth constrained sensor network with a fusion center the sensor noises are assumed to be additive zero mean spatially uncorrelated but otherwise unknown and possibly different across sensors due to varying sensor quality and inhomogeneous sensing environment the classical best linear unbiased estimator blue linearly combines the real valued sensor observations to minimize the mean square error mse unfortunately such a scheme cannot be implemented in a practical bandwidth constrained sensor network due to its requirement to transmit real valued messages in this paper we construct a decentralized estimation scheme des where each sensor compresses its observation to a small number of bits with length proportional to the logarithm of its local signal to noise ratio snr the resulting compressed bits from different sensors are then collected and combined by the fusion center to estimate the unknown parameter the proposed des is universal in the sense that each sensor compression scheme requires only the knowledge of local snr rather than the noise probability distribution functions pdf while the final fusion step is also independent of the local noise pdfs we show that the mse of the proposed des is within a constant factor of 25 8 of that achieved by the classical centralized blue estimator

morita equivalences related to the glauberman correspondence abstract starting with p interior algebras where p is a finite p group we prove two theorems establishing certain group graded morita equivalences these apply in particular to the case of blocks with normal defect groups and defect zero blocks of normal subgroups respectively

detection of an anomalous cluster in a network we consider the problem of detecting whether or not in a given sensor network there is a cluster of sensors which exhibit an unusual behavior formally suppose we are given a set of nodes and attach a random variable to each node we observe a realization of this process and want to decide between the following two hypotheses under the null the variables are i i d standard normal under the alternative there is a cluster of variables that are i i d normal with positive mean and unit variance while the rest are i i d standard normal we also address surveillance settings where each sensor in the network collects information over time the resulting model is similar now with a time series attached to each node we again observe the process over time and want to decide between the null where all the variables are i i d standard normal and the alternative where there is an emerging cluster of i i d normal variables with positive mean and unit variance the growth models used to represent the emerging cluster are quite general and in particular include cellular automata used in modelling epidemics in both settings we consider classes of clusters that are quite general for which we obtain a lower bound on their respective minimax detection rate and show that some form of scan statistic by far the most popular method in practice achieves that same rate within a logarithmic factor our results are not limited to the normal location model but generalize to any one parameter exponential family when the anoma lous clusters are large enough

sequential decentralized parameter estimation under randomly observed fisher information we consider the problem of decentralized scalar parameter estimation using wireless sensor networks with gaussian noise specifically we propose a novel framework based on level triggered sampling a non uniform sampling strategy and sequential estimation the proposed estimator can be used as an asymptotically optimal fixed sample size decentralized estimator when the observed fisher information i e fisher information without expectation is deterministic as an alternative to the one shot estimators commonly found in the literature it can also be used as an asymptotically optimal sequential decentralized estimator when the observed fisher information is random we show that the optimal centralized estimator under gaussian noise which is the maximum likelihood estimator is characterized by two processes namely the observed fisher information ut and the observed correlation vt it is noted that vt is always random even when ut is not in the proposed scheme each sensor computes its local random processes and sends a single bit to the fusion center fc whenever the local random processes passes certain predefined levels the fc upon receiving a bit from a sensor updates its approximation to the corresponding global random process and accordingly its estimate the sequential estimation process terminates when ut or the approximation to it reaches a target value we provide an asymptotic analysis for the proposed estimator and the one based on conventional uniform in time sampling under both deterministic and random ut and determine the conditions under which they are asymptotically optimal consistent and asymptotically unbiased analytical results together with simulation results demonstrate the superiority of the proposed estimator based on level triggered sampling over the traditional decentralized estimator based on uniform sampling

optimum quantization for detection vector quantization is a procedure widely used in communications the problem has been solved for the basic detection situation using the locally optimal procedure and with some constraints on the quantization scheme it is stated in its widest sense which consists of finding the partition of the observation space and the values associated with it in such a way that the detection performance is optimal it is solved completely with the deflection criterion and the solution introduces the concept of quantization by a likelihood ratio procedure

linear precoders for the detection of a gaussian process in wireless sensors networks we investigate the performance of neyman pearson detection of a stationary gaussian process in noise using a large wireless sensor network wsn in our model each sensor compresses its observation sequence using a linear precoder and a final decision is taken by a fusion center fc based on the compressed information two families of precoders are studied random i i d precoders and orthogonal precoders we analyse their performance under a regime where both the number of sensors k and the number of samples n per sensor tend to infinity at the same rate that is k n c 0 1 contributions are as follows 1 using results from random matrix theory and large toeplitz matrices we prove that when the above families of precoders are used the miss probability of the neyman pearson detector converges exponentially to zero closed form expressions of the corresponding error exponents are derived 2 in particular we propose a practical orthogonal precoding strategy the principal frequencies strategy pfs which achieves the best error exponent among all orthogonal strategies and which requires very little signaling overhead between the central processor and the nodes of the network 3 when the pfs is used a simplified low complexity testing procedure can be implemented at the fc we show that the proposed suboptimal test enjoys the same error exponent as the neyman pearson test which indicates a similar asymptotic behavior of the performance we illustrate our findings by numerical experiments on several examples

dynamic sequential team multi hypothesis testing under uniformly distributed nonstationary observations abstract we consider a dynamic team of communicating decision makers performing bayesian sequential multi hypothesis testing under nonstationary observations a simple decision rule is shown to be member by member optimal if the observations are uniformly distributed if the observation costs are defined to reflect the quality of observations and if vague terminal decisions are allowed without additional penalty the vague terminal decisions can be made clear by performing a finite number of member by member optimal sequential decisions successively

multihypothesis sequential probability ratio tests ii accurate asymptotic expansions for the expected sample size for pt i see ibid vol 45 p 2448 61 1999 we proved in pt i that two specific constructions of multihypothesis sequential tests which we refer to as multihypothesis sequential probability ratio tests msprts are asymptotically optimal as the decision risks or error probabilities go to zero the msprts asymptotically minimize not only the expected sample size but also any positive moment of the stopping time distribution under very general statistical models for the observations in this paper based on nonlinear renewal theory we find accurate asymptotic approximations up to a vanishing term for the expected sample size that take into account the overshoot over the boundaries of decision statistics the approximations are derived for the scenario where the hypotheses are simple the observations are independent and identically distributed i i d according to one of the underlying distributions and the decision risks go to zero simulation results for practical examples show that these approximations are fairly accurate not only for large but also for moderate sample sizes the asymptotic results given here complete the analysis initiated by baum and veeravalli 1994 where first order asymptotics were obtained for the expected sample size under a specific restriction on the kullback leibler distances between the hypotheses

on optimum and nearly optimum data quantization for signal detection the application of companding approximation theory to the quantization of data for detection of coherent signals in a noisy environment is considered this application is twofold allowing for greater simplicity in both analysis and design of quantizers for detection systems most computational methods for designing optimum most efficient quantizers for signal detection systems are iterative and are extremely sensitive to initial conditions companding approximation theory is used here to obtain suitable initial conditions for this problem furthermore the companding approximation idea is applied to design suboptimum quantizers which are nearly as efficient as optimum quantizers when the number of levels is large in this design iteration is not needed to derive the parameters of the quantizer and the design procedure is very simple in this paper we explore this approach numerically and demonstrate its effectiveness for designing and analyzing quantizers in detection systems

decentralized sequential hypothesis testing using asynchronous communication an asymptotically optimum test for the problem of decentralized sequential hypothesis testing is presented the induced communication between sensors and fusion center is asynchronous and limited to 1 bit data when the sensors observe continuously stochastic processes with continuous paths the proposed test is order 2 asymptotically optimal in the sense that its inflicted performance loss is bounded when the sensors take discrete time observations the proposed test achieves order 1 asymptotic optimality i e the ratio of its performance over the optimal performance tends to 1 moreover we show theoretically and corroborate with simulations that the performance of the suggested test in discrete time can be significantly improved when the sensors sample their underlying continuous time processes more frequently a property which is not enjoyed by other centralized or decentralized tests in the literature

quantization in multisensor random signal detection optimum detection schemes based on quantized data are of great interest in radar and sonar applications the design and properties of multisensor schemes are considered here for detection of weak random signals in additive possibly non gaussian noise signal to noise ratios are assumed unknown and the signals at the different sensors may be statistically dependent analytical expressions describing the best way to fuse the quantized observations for cases with any given observation sample size are provided the best schemes for originally quantizing the observations are also studied for the case of asymptotically large observation sample sizes these schemes are shown to minimize the mean squared error between the best weak signal test statistic based on unquantized observations and the best weak signal test statistic based on quantized observations under signal absent numerical results indicate it is sometimes best for each quantizer to use different size alphabets when a quantizer is located at each sensor

quickest linear search over correlated sequences consider a set of random sequences each consisting of independent and identically distributed random variables drawn from one of the two known distributions f 0 and f 1 the underlying distributions of different sequences are correlated induced by an inherent physical coupling in the mechanisms generating these sequences the objective is to design the quickest data adaptive and sequential search procedure for identifying one sequence generated according to f 1 the optimal design involves striking a balance between the average delay in reaching a decision and the rate of false alarms as two opposing figures of merit optimal and asymptotically optimal decision rules are derived which can take radically different forms depending on the correlation structure performance and sampling complexity analyses are provided to delineate the tradeoff between decision delay and quality the generalization to parallel sampling in which multiple sequences are sampled at the same time is also investigated

sequential testing problems for poisson processes we present the explicit solution of the bayesian problem of sequential testing of two simple hypotheses about the intensity of an observed poisson process the method of proof consists of reducing the initial problem to a free boundary differential difference stephan problem and solving the latter by use of the principles of smooth and continuous fit a rigorous proof of the optimality of the wald s sequential probability ratio test in the variational formulation of the problem is obtained as a consequence of the solution of the bayesian problem

cooperative sequential spectrum sensing based on level triggered sampling we propose a new framework for cooperative spectrum sensing in cognitive radio networks that is based on a novel class of nonuniform samplers called the event triggered samplers and sequential detection in the proposed scheme each secondary user su computes its local sensing decision statistic based on its own channel output and whenever such decision statistic crosses certain predefined threshold values the secondary user will send one or several bit of information to the fusion center fc the fc asynchronously receives the bits from different sus and updates the global sensing decision statistic to perform a sequential probability ratio test sprt to reach a sensing decision we provide an asymptotic analysis for the above scheme and under different conditions we compare it against the cooperative sensing scheme that is based on traditional uniform sampling and sequential detection simulation results show that the proposed scheme using even 1 bit can outperform its uniform sampling counterpart that uses infinite number of bits under changing target error probabilities snr values and number of sus

on the glauberman and watanabe correspondences for blocks of finite p solvable groups if g is a finite p solvable group for some prime p a a solvable subgroup of the automorphism group of g of order prime to g such that a stabilises a p block b of g and acts trivially on a defect group p of b then there is a morita equivalence between the block b and its watanabe correspondent w b of c g a given by a bimodule m with vertex p and an endo permutation module as source which on the character level induces the glauberman correspondence and which is an isotypy by watanabe s results

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

on the brauer glauberman correspondence abstract the purpose of the present paper is two fold on the one hand to show the existence of a correspondence unifying brauer s and glauberman s ones see theorem 4 6 and on the other hand to give an alternative proof of watanabe s equivalence in atumi watanabe the glauberman character correspondence and perfect isometries for blocks of finite groups j algebra 216 1999 548 565 by the way we give a short proof of the coincidence of the clifford extensions associated with a pair of glauberman correspondent irreducible representations see corollary 4 16 a question that surprisingly enough has only been partially solved recently see morton harris markus linckelmann on the glauberman and watanabe correspondences for blocks of finite p solvable groups trans amer math soc 354 2002 3435 3453 and shigeo koshitani gerhard michler glauberman correspondence of p blocks of finite groups j algebra 243 2001 504 517

on board component fault detection and isolation using the statistical local approach we describe both the key principles and real application examples of a unified theory which allows us to perform the on board incipient fault detection and isolation tasks involved in monitoring for condition based maintenance this theory is known under the name of local approach and it is especially suited to component faults it aims at transforming complex detection problems concerning a parameterized stochastic process into the universal problem of monitoring the mean of a gaussian vector based on a small deviation assumption the key tools are the first order taylor expansion and the asymptotic gaussianity of a convenient parameter estimating function ml ls iv and subspace identification methods are addressed in this perspective in the case of dynamic processes given in state space form the approach may also call for observer based state estimation or state elimination experiments concerning both simulated and real data for linear and nonlinear dynamical processes are reported how the key principles and features of the local approach compare with those of other approaches is discussed

the basal ganglia optimize decision making over general perceptual hypotheses the basal ganglia are a subcortical group of interconnected nuclei involved in mediating action selection within cortex a recent proposal is that this selection leads to optimal decision making over multiple alternatives because the basal ganglia anatomy maps onto a network implementation of an optimal statistical method for hypothesis testing assuming that cortical activity encodes evidence for constrained gaussian distributed alternatives this letter demonstrates that this model of the basal ganglia extends naturally to encompass general bayesian sequential analysis over arbitrary probability distributions which raises the proposal to a practically realizable theory over generic perceptual hypotheses we also show that the evidence in this model can represent either log likelihoods log likelihood ratios or log odds all leading proposals for the cortical processing of sensory data for these reasons we claim that the basal ganglia optimize decision making over general perceptual hypotheses represented in cortex the relation of this theory to cortical encoding cortico basal ganglia anatomy and reinforcement learning is discussed

extremal properties of likelihood ratio quantizers m hypotheses and a random variable y with a different probability distribution under each hypothesis are considered a quantizer is applied to form a quantized random variable gamma y the extreme points of the set of possible probability distributions of gamma y as gamma ranges over all quantizers is characterized optimality properties of likelihood ratio quantizers are established for a very broad class of quantization problems including problems involving the maximization of an ali silvey 1966 distance measure and the neyman pearson variant of the decentralized detection problem

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

multihypothesis sequential probability ratio tests i asymptotic optimality the problem of sequential testing of multiple hypotheses is considered and two candidate sequential test procedures are studied both tests are multihypothesis versions of the binary sequential probability ratio test sprt and are referred to as msprts the first test is motivated by bayesian optimality arguments while the second corresponds to a generalized likelihood ratio test it is shown that both msprts are asymptotically optimal relative not only to the expected sample size but also to any positive moment of the stopping time distribution when the error probabilities or more generally risks associated with incorrect decisions are small the results are first derived for the discrete time case of independent and identically distributed i i d observations and simple hypotheses they are then extended to general possibly continuous time statistical models that may include correlated and nonhomogeneous observation processes it also demonstrated that the results can be extended to hypothesis testing problems with nuisance parameters where the composite hypotheses due to nuisance parameters can be reduced to simple ones by using the principle of invariance these results provide a complete generalization of the results given by veeravalli and baum see ibid vol 41 p 1994 97 1995 where it was shown that the quasi bayesian msprt is asymptotically efficient with respect to the expected sample size for i i d observations

error probability bounds for balanced binary relay trees we study the detection error probability associated with a balanced binary relay tree where the leaves of the tree correspond to n identical and independent sensors the root of the tree represents a fusion center that makes the overall detection decision each of the other nodes in the tree is a relay node that combines two binary messages to form a single output binary message only the leaves are sensors in this way the information from the sensors is aggregated into the fusion center via the relay nodes in this context we describe the evolution of the type i and type ii error probabilities of the binary data as it propagates from the leaves toward the root tight upper and lower bounds for the total error probability at the fusion center as functions of n are derived these characterize how fast the total error probability converges to 0 with respect to n even if the individual sensors have error probabilities that converge to 1 2

optimal decision making on the basis of evidence represented in spike trains experimental data indicate that perceptual decision making involves integration of sensory evidence in certain cortical areas theoretical studies have proposed that the computation in neural decision circuits approximates statistically optimal decision procedures e g sequential probability ratio test that maximize the reward rate in sequential choice tasks however these previous studies assumed that the sensory evidence was represented by continuous values from gaussian distributions with the same variance across alternatives in this article we make a more realistic assumption that sensory evidence is represented in spike trains described by the poisson processes which naturally satisfy the mean variance relationship observed in sensory neurons we show that for such a representation the neural circuits involving cortical integrators and basal ganglia can approximate the optimal decision procedures for two and multiple alternative choice tasks

on the degrees of freedom in shape restricted regression for the problem of estimating a regression function say subject to shape constraints like monotonicity or convexity it is argued that the divergence of the maximum likelihood estimator provides a useful measure of the effective dimension of the model inequalities are derived for the expected mean squared error of the maximum likelihood estimator and the expected residual sum of squares these generalize equalities from the case of linear regression as an application it is shown that the maximum likelihood estimator of the error variance 2 is asymptotically normal with mean 2 and variance 2 2 n for monotone regression it is shown that the maximum likelihood estimator of attains the optimal rate of convergence and a bias correction to the maximum likelihood estimator of 2 is derived

on nodes and modes in resting state fmri this paper examines intrinsic brain networks in light of recent developments in the characterisation of resting state fmri timeseries and simulations of neuronal fluctuations based upon the connectome its particular focus is on patterns or modes of distributed activity that underlie functional connectivity we first demonstrate that the eigenmodes of functional connectivity or covariance among regions or nodes are the same as the eigenmodes of the underlying effective connectivity provided we limit ourselves to symmetrical connections this symmetry constraint is motivated by appealing to proximity graphs based upon multidimensional scaling crucially the principal modes of functional connectivity correspond to the dynamically unstable modes of effective connectivity that decay slowly and show long term memory technically these modes have small negative lyapunov exponents that approach zero from below interestingly the superposition of modes whose exponents are sampled from a power law distribution produces classical 1 f scale free spectra we conjecture that the emergence of dynamical instability that underlies intrinsic brain networks is inevitable in any system that is separated from external states by a markov blanket this conjecture appeals to a free energy formulation of nonequilibrium steady state dynamics the common theme that emerges from these theoretical considerations is that endogenous fluctuations are dominated by a small number of dynamically unstable modes we use this as the basis of a dynamic causal model dcm of resting state fluctuations as measured in terms of their complex cross spectra in this model effective connectivity is parameterised in terms of eigenmodes and their lyapunov exponents that can also be interpreted as locations in a multidimensional scaling space model inversion provides not only estimates of edges or connectivity but also the topography and dimensionality of the underlying scaling space here we focus on conceptual issues with simulated fmri data and provide an illustrative application using an empirical multi region timeseries