scalar quantization for estimation from an asymptotic design to a practical solution in this paper we study an asymptotic approximation of the fisher information for the estimation of a scalar parameter using quantized measurements we show that as the number of quantization intervals tends to infinity the loss of fisher information induced by quantization decreases exponentially as a function of the number of quantization bits a characterization of the optimal quantizer through its interval density and an analytical expression for the fisher information are obtained a comparison between optimal uniform and nonuniform quantization for the location and scale estimation problems shows that nonuniform quantization is only slightly better than uniform quantization as the optimal quantization intervals are shown to depend on the unknown parameters by applying adaptive algorithms that jointly estimate the parameter and set the thresholds in the location and scale estimation problems we show that the asymptotic results can be approximately reached in practice using only 4 or 5 quantization bits

on the gibbs states of the noncritical potts model on z 2 we prove that all gibbs states of the q state nearest neighbor potts model on mathbb z 2 below the critical temperature are convex combinations of the q pure phases in particular they are all translation invariant to achieve this goal we consider such models in large finite boxes with arbitrary boundary condition and prove that the center of the box lies deeply inside a pure phase with high probability our estimate of the finite volume error term is of essentially optimal order which stems from the brownian scaling of fluctuating interfaces the results hold at any supercritical value of the inverse temperature beta beta c q log left 1 sqrt q right

a theory of nonsubtractive dither a detailed mathematical investigation of multibit quantizing systems using nonsubtractive dither is presented it is shown that by the use of dither having a suitably chosen probability density function moments of the total error can be made independent of the system input signal but that statistical independence of the error and the input signals is not achievable similarly it is demonstrated that values of the total error signal cannot generally be rendered statistically independent of one another but that their joint moments can be controlled and that in particular the error sequence can be rendered spectrally white the properties of some practical dither signals are explored and recommendations are made for dithering in audio video and measurement applications the paper collects all of the important results on the subject of nonsubtractive dithering and introduces important new ones with the goal of alleviating persistent and widespread misunderstandings regarding the technique

blind mimo identification using the second characteristic function we propose a new approach for the blind identification of a multi input multi output mimo system as a substitute to using classical high order statistics hos in the form of time lagged joint cumulants or polyspectra we use the estimated hessian matrices of the second joint generalized characteristic function of time lagged observations evaluated at several preselected processing points these matrices admit straightforward consistent estimates whose statistical stability can be finely tuned by proper selection of the processing points in contrast to classical hos transforming the obtained matrix sequence into the frequency domain we obtain and solve a sequence of frequency dependent joint diagonalization problems this yields a set of estimated frequency response matrices which are transformed back into the time domain after resolving frequency dependent phase and permutation ambiguities the performance of the proposed algorithm depends on the choice of processing points yet compares favorably with other algorithms especially at moderate signal to noise ratio conditions as we demonstrate in simulation results

dithered quantizers a theory of overall quantization noise for nonsubtractive dither was originally developed in unpublished work by j n wright and by t j stockham and subsequently expanded by l k brinton s p lipshitz j vanderkooy and r a wannamaker it is suggested that since these latter results are not as well known as the original results misunderstanding persists in the literature new proofs of the properties of quantizer dither both subtractive and nonsubtractive are provided the new proofs are based on elementary fourier series and rice s characteristic function method and do not require the use of generalized functions impulse trains of dirac delta functions and sampling theorem arguments the goal is to provide a unified derivation and presentation of the two forms of dithered quantizer noise based on elementary fourier techniques

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

first order phase transitions in lattice and continuous systems extension of pirogov sinai theory we generalize the notion of ground states in the pirogov sinai theory of first order phase transitions at low temperatures applicable to lattice systems with a finite number of periodic ground states to that of restricted ensembles with equal free energies a restricted ensemble is a gibbs ensemble i e equilibrium probability measure on a restricted set of configurations in the phase space of the system when a restricted ensemble contains only one configuration it coincides with a ground state in the more general case the entropy is also important

performance limit for distributed estimation systems with identical one bit quantizers full precision crame r rao lower bound crlb where no quantization is assumed is often employed to evaluate and compare distributed estimation performance even though the sensor observations are quantized before any further processing however as it completely disregards quantization and often does not exist when the sensor observation noise is bounded full precision crlb is often too optimistic or not applicable in this work we determine the performance limit of a distributed estimation system with identical one bit quantizers in terms of the metric minimax crlb the performance limit that a distributed estimation scheme with identical quantizers can achieve is found as well as the set of optimal noise distribution functions and quantizers compared to the full precision crlb the performance limit is shown to be a much tighter bound when the parameter range is relatively large and reveals the important role of the quantization system

forbidden gap argument for phase transitions proved by means of chessboard estimates chessboard estimates are one of the standard tools for proving phase coexistence in spin systems of physical interest in this note we show that the method not only produces a point in the phase diagram where more than one gibbs states coexist but that it can also be used to rule out the existence of shift ergodic states that differ significantly from those proved to exist for models depending on a parameter say the temperature this shows that the values of the conjugate thermodynamic quantity the energy inside the transitional gap are forbidden in all shift ergodic gibbs states we point out several models where our result provides useful additional information concerning the set of possible thermodynamic equilibria

multivariate signal parameter estimation under dependent noise from 1 bit dithered quantized data motivated by applications in sensor networks and communications we consider multivariate signal parameter estimation when only dithered 1 bit quantized samples are available the observation noise is taken to be a stationary strongly mixing process which covers a wide range of processes including autoregressive moving average arma models the noise is allowed to be gaussian or to have a heavy tail with possibly infinite variance an estimate of the signal parameters is proposed and is shown to be weakly consistent joint asymptotic normality of the parameters estimate is also established and the asymptotic mean and covariance matrices are identified

selective measurement transmission in distributed estimation with data association in distributed multisensor estimation tracking the problem of fusion is complicated by that of data association i e with false alarms and missed detections not only is it of concern to provide an estimation efficient sensor level quantization of the target originated measurement but it is also unclear which among each sensor s measurements this might be if any at all the former issue has been studied previously in this paper we address only the latter concern at first we assume that each sensor is tasked to communicate exactly one of its observations to a fusion center fc for a global estimate and we work in one dimension via order statistics we show that surprisingly the nearest neighbor nn is not always the most appropriate measurement to share we also expand our bandwidth to allow for transmission of multiple measurements for example the nearest and third nearest it turns out that a single measurement transmission is more bandwidth efficient than multiple the analysis and results are further extended to two dimensions but the moral that sharing of the nns is not always a good idea remains

noise enhancement in robust estimation of location in this paper we investigate the noise benefits to maximum likelihood type estimators m estimator for the robust estimation of a location parameter two distinct noise benefits are shown to be accessible under these conditions with symmetric heavy tailed noise distributions the asymptotic efficiency of the estimation can be enhanced by injecting extra noise into the m estimators with an asymmetric contaminated noise model having a convex cumulative distribution function we demonstrate that addition of noise can reduce the maximum bias of the median estimator these findings extend the analysis of stochastic resonance effects for noise enhanced signal and information processing

an alternate version of pirogov sinai theory a new approach to the pirogov sinai theory of phase transitions is developed not employing the contour models with a parameter the completeness of the phase diagram is proven

translation invariant gibbs states in the q state potts model we describe the set of all translation invariant gibbs states in theq state potts model for the case ofq large enough and the other parameters to be arbitrary

bandwidth constrained distributed estimation for wireless sensor networks part i gaussian case we study deterministic mean location parameter estimation when only quantized versions of the original observations are available due to bandwidth constraints when the dynamic range of the parameter is small or comparable with the noise variance we introduce a class of maximum likelihood estimators that require transmitting just one bit per sensor to achieve an estimation variance close to that of the clairvoyant sample mean estimator when the dynamic range is comparable or larger than the noise standard deviation we show that an optimum quantization step exists to achieve the best possible variance for a given bandwidth constraint we will also establish that in certain cases the sample mean estimator formed by quantized observations is preferable for complexity reasons we finally touch upon algorithm implementation issues and guarantee that all the numerical maximizations required by the proposed estimators are concave

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

a fundamental limitation on maximum parameter dimension for accurate estimation with quantized data it is revealed that there is a link between the quantization approach employed and the dimension of the vector parameter which can be accurately estimated by a quantized estimation system a critical quantity called inestimable dimension for quantized data idqd is introduced which does not depend on the quantization regions and the statistical models of the observations but instead depends only on the number of sensors and on the precision of the vector quantizers employed by the system it is shown that the idqd describes a quantization induced fundamental limitation on the estimation capabilities of the system to be specific if the dimension of the desired vector parameter is larger than the idqd of the quantized estimation system then the fisher information matrix for estimating the desired vector parameter is singular and moreover there exist infinitely many nonidentifiable vector parameter points in the vector parameter space furthermore it is shown that under some common assumptions on the statistical models of the observations and the quantization system a smaller idqd can be obtained which can specify an even more limiting quantization induced fundamental limitation on the estimation capabilities of the system

recursive algorithms for parameter estimation with adaptive quantizer this paper studies a parameter estimation problem of networked linear systems with fixed rate quantization under the minimum mean square error criterion we propose a recursive estimator of stochastic approximation type and derive a necessary and sufficient condition for its asymptotic unbiasedness this motivates to design an adaptive quantizer for the estimator whose strong consistency asymptotic unbiasedness and asymptotic normality are rigorously proved using the newton based and averaging techniques we obtain two accelerated recursive estimators with the fastest convergence speed of o 1 k and exactly evaluate the quantization effect on the estimation accuracy if the observation noise is gaussian an optimal quantizer and the accelerated estimators are co designed to asymptotically approach the minimum cramer rao lower bound all the estimators share almost the same computational complexity as the gradient algorithms with un quantized observations and can be easily implemented finally the theoretical results are validated by simulations

signal parameter estimation using 1 bit dithered quantization motivated by the estimation of spatio temporal events with cheap simple sensors we consider the problem of estimation of a parameter thetas of a signal s x thetas corrupted by noise assuming that only 1 bit precision dithered quantized samples are available an estimate that does not require the knowledge of the dither signal and the noise distribution is proposed and it is analyzed in detail under variety of nonidealities the consistency and asymptotic normality of the estimate is established for deterministic and random sampling imprecise knowledge of sampling locations gaussian and non gaussian noise with possibly infinite variance a wide class of dither distributions and under erroneous transmission of the binary observations via binary symmetric channels bscs it is also shown that if approximation to the log likelihood equation in the full precision case yields a good estimate then there is a corresponding good estimate based on 1 bit dithered samples the proposed estimate requires no more computation than the maximum likelihood estimate for the full precision case and suffers only a logarithmic rate loss compared to the full precision case when uniform dithering is used it is shown that uniform dithering leads to the best rate among a broad class of dither distributions a condition under which no dithering leads to a better estimate is also given

design of quantizers for decentralized estimation systems the authors consider parameter estimation in decentralized systems with distributed processors they restrict the local processors to be quantizers and consider the optimal design of the systems to minimize the estimation error they present necessary conditions for the optimal system based on the bayes distortion functions and fisher s information the numerical results compare the resulting quantizers obtained by different distortion criteria

universal decentralized estimation in a bandwidth constrained sensor network consider a situation where a set of distributed sensors and a fusion center wish to cooperate to estimate an unknown parameter over a bounded interval u u each sensor collects one noise corrupted sample performs a local estimation and transmits a message to the fusion center while the latter combines the received messages to produce a final estimate this correspondence investigates optimal local estimation and final fusion schemes under the constraint that the communication from each sensor to the fusion center must be a one bit message such a binary message constraint is well motivated by the bandwidth limitation of the communication links fusion center and by the limited power budget of local sensors in the absence of bandwidth constraint and assuming the noises are bounded to the interval u u additive independent but otherwise unknown the classical estimation theory suggests that a total of o u2 epsi2 sensors are necessary and sufficient in order for the sensors and the fusion center to jointly estimate the unknown parameter within epsi root mean squared error mse it is shown in this correspondence that the same remains true even with the binary message constraint furthermore the optimal decentralized estimation scheme suggests allocating 1 2 of the sensors to estimate the first bit of the unknown parameter 1 4 of the sensors to estimate the second bit and so on

marginal likelihood for estimation and detection theory this paper derives and analyzes the asymptotic performances of the maximum likelihood ml estimator and the generalized likelihood ratio test glrt derived under the assumption of independent identically distribution i i d samples where in the actual model the signal samples are m dependent the ml and glrt under such a modeling mismatch are based on the marginal likelihood function and they are referred to as marginal maximum likelihood mml and generalized sum marginal log likelihood ratio test gmlrt respectively under some regularity conditions the asymptotical distributions of the mml and gmlrt are derived the asymptotical distributions in some signal processing examples are analyzed simulation results support the theory via several examples

bandwidth constrained distributed estimation for wireless sensor networks part ii unknown probability density function wireless sensor networks wsns deployed to perform surveillance and monitoring tasks have to operate under stringent energy and bandwidth limitations these motivate well distributed estimation scenarios where sensors quantize and transmit only one or a few bits per observation for use in forming parameter estimators of interest in a companion paper we developed algorithms and studied interesting tradeoffs that emerge even in the simplest distributed setup of estimating a scalar location parameter in the presence of zero mean additive white gaussian noise of known variance herein we derive distributed estimators based on binary observations along with their fundamental error variance limits for more pragmatic signal models i known univariate but generally non gaussian noise probability density functions pdfs ii known noise pdfs with a finite number of unknown parameters iii completely unknown noise pdfs and iv practical generalizations to multivariate and possibly correlated pdfs estimators utilizing either independent or colored binary observations are developed and analyzed corroborating simulations present comparisons with the clairvoyant sample mean estimator based on unquantized sensor observations and include a motivating application entailing distributed parameter estimation where a wsn is used for habitat monitoring

distributed estimation and quantization an algorithm is developed for the design of a nonlinear n sensor distributed estimation system subject to communication and computation constraints the algorithm uses only bivariate probability distributions and yields locally optimal estimators that satisfy the required system constraints it is shown that the algorithm is a generalization of the classical lloyd max results

on quantizer design for distributed bayesian estimation in sensor networks we consider the problem of distributed estimation under the bayesian criterion and explore the design of optimal quantizers in such a system we show that for a conditionally unbiased and efficient estimator at the fusion center and when local observations have identical distributions it is optimal to partition the local sensors into groups with all sensors within a group using the same quantization rule when all the sensors use identical number of decision regions use of identical quantizers at the sensors is optimal when the network is constrained by the capacity of the wireless multiple access channel over which the sensors transmit their quantized observations we show that binary quantizers at the local sensors are optimal under certain conditions based on these observations we address the location parameter estimation problem and present our optimal quantizer design approach we also derive the performance limit for distributed location parameter estimation under the bayesian criterion and find the conditions when the widely used threshold quantizer achieves this limit we corroborate this result using simulations we then relax the assumption of conditionally independent observations and derive the optimality conditions of quantizers for conditionally dependent observations using counter examples we also show that the previous results do not hold in this setting of dependent observations and therefore identical quantizers are not optimal index terms

effects of sampling and quantization on single tone frequency estimation the effects of sampling and quantization on frequency estimation for a single sinusoid are investigated the cramer rao bound for 1 bit quantization is derived and compared with the limit of infinite quantization it is found that 1 bit quantization gives a slightly worse performance however with a dramatic increase of variance at certain frequencies this can be avoided by using four times oversampling the effect of sampling when using nonideal antialiasing lowpass filters is therefore investigated through derivation of the cramer rao lower bounds finally fast estimators for 1 bit quantization in particular correlation based estimators are derived and their performance is investigated the paper is concluded with simulation results for four times oversampled 1 bit quantization

nonparametric one bit quantizers for distributed estimation in this paper we consider the nonparametric distributed parameter estimation problem using one bit quantized data from peripheral sensors assuming that the sensor observations are bounded nonparametric distributed estimators are proposed based on the knowledge of the first n moments of sensor noises these estimators are shown to be either unbiased or asymptotically unbiased with bounded and known estimation variance further the uniformly optimal quantizer based only on the first moment information and the optimal minimax quantizer with the knowledge of the first two moments are determined the proposed estimators are shown to be consistent even when local sensor noises are not independent but m dependent the relationship between the proposed approaches and dithering in quantization is also investigated the superiority of the proposed quantization estimation schemes is illustrated via illustrative examples

the stochastic random cluster process and the uniqueness of random cluster measures the random cluster model is a generalization of percolation and ferromagnetic potts models due to fortuin and kasteleyn not only is the random cluster model a worthwhile topic for study in its own right but also it provides much information about phase transitions in the associated physical models this paper serves two functions first we introduce and survey random cluster measures from the probabilist s point of view giving clear statements of some of the many open problems second we present new results for such measures as follows we discuss the relationship between weak limits of random cluster measures and measures satisfying a suitable dlr condition using an argument based on the convexity of pressure we prove the uniqueness of random cluster measures for all but at most countably many values of the parameter p related results concerning phase transition in two or more dimensions are included together with various stimulating conjectures the uniqueness of the infinite cluster is employed in an intrinsic way in part of these arguments in the second part of this paper is constructed a markov process whose level sets are reversible markov processes with random cluster measures as unique equilibrium measures this construction enables a coupling of random cluster measures for all values of p furthermore it leads to a proof of the semicontinuity of the percolation probability and provides a heuristic probabilistic justification for the widely held belief that there is a first order phase transition if and only if the cluster weighting factor q is sufficiently large

phases coexistence and surface tensions for the potts model theq states potts model exhibits a first order phase transition at some inverse temperature t between ordered and disordered phases forq large as proved in 1 in space dimension 2 we use theduality transformation as aninternal symmetry of the partition function at t to derive an estimate on the probability of a contour this enables us to prove the preceding result and the following new results i the discontinuity of the mass gap at t ii the existence of astrictly positive surface tension between two ordered phases up to t iii the existence of a non zero surface tension between an ordered and the disordered phase at t