multiple solutions for 2mth order sturm liouville boundary value problems growth conditions are imposed on f such that the following boundary value problem 1 my 2m f t y i 1y 2i 0 i 1y 2i 1 0 i 1y 2i 1 i 1y 2i 1 1 0 0 i m 1 has an arbitrary number of positive solutions

brief paper an efficient sensor quantization algorithm for decentralized estimation fusion in this paper we consider the design problem of optimal sensor quantization rules quantizers and an optimal linear estimation fusion rule in bandwidth constrained decentralized random signal estimation fusion systems first we derive a fixed point type necessary condition for both optimal sensor quantization rules and an optimal linear estimation fusion rule a fixed point of an integral operation then we can motivate an iterative gauss seidel algorithm to simultaneously search for both optimal sensor quantization rules and an optimal linear estimation fusion rule without gaussian assumptions on the joint probability density function pdf of the estimated parameter and observations moreover we prove that the algorithm converges to a person by person optimal solution in the discretized scheme after a finite number of iterations it is worth noting that the new method can be applied to vector quantization without any modification finally several numerical examples demonstrate the efficiency of our method and provide some reasonable and meaningful observations how the estimation performance is influenced by the observation noise power and numbers of sensors or quantization levels

comparison of eigenvalues for sturm liouville boundary value problems on a measure chain under consideration is a class of even ordered linear differential equations 1 m 2m t i 0m 1pi t 2i t with sturm liouville boundary conditions i 2i 0 i 2i 0 yi 1 2i 1 i 1 2i 1 0 for 0 i m 1 the derivative in this dynamic equation is the generalized delta derivative defined on a measure chain for a pair of eigenvalue problems for this dynamic equation we first verify the existence of smallest positive eigenvalues and then establish a comparison between the smallest eigenvalues of each eigenvalue problem

on optimal quantization of noisy sources problems in optimal multidimensional quantization of sources corrupted by noise are addressed expressions for the optimum quantizer values and the optimum quantization rule for the weighted squared error distortion measure are found and calculated for the gaussian signal in additive independent gaussian noise problem some properties of the optimum quantizer and its relations with the optimal estimator for the general problem are derived

identification of arma models using intermittent and quantized output observations this paper studies system identification of arma models whose outputs are subject to finite level quantization and random packet dropouts using the maximum likelihood criterion we propose a recursive identification algorithm which we show to be strongly consistent and asymptotically normal we also propose a simple adaptive quantization scheme which asymptotically achieves the minimum parameter estimation error covariance the joint effect of finite level quantization and random packet dropouts on identification accuracy are exactly quantified the theoretical results are verified by simulations

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

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

information driven distributed maximum likelihood estimation based on gauss newton method in wireless sensor networks in this paper we develop an energy efficient distributed estimation method that can be used in applications such as the estimation of a diffusive source and the localization and tracking of an acoustic target in wireless sensor networks we first propose a statistical measurement model in which we separate the linear and nonlinear parameters this modeling strategy reduce the processing complexity we then study the distributed implementation of the gauss newton method in the maximum likelihood estimation after that we propose a fully distributed estimation approach based on an incremental realization of the gauss newton method we derive three modifications of the basic algorithm to improve the distributed processing performance while still considering the energy restriction we implement the idea of information driven collaborative signal processing and provide a sensor node scheduling scheme in which the cramer rao bound crb is used as the performance and information utility measure to select the next sensor node numerical examples are used to study the performance of the distributed estimation and we show that of the methods considered here the proposed multiple iteration kalman filtering method has the most advantages for wireless sensor networks

comparison of eigenvalues for lidstone boundary value problems on a measure chain preprint the theory of u0 positive operators with respect to a cone in a banach space is applied to the class of even order differential equation on measure chains x 2m t 0m 1 pi t x 2i t t 0 1 satisfying the lidstone boundary conditions x 2i 0 0 0 i m 1 x 2j 2 1 0 0 j m 1 the existence of a smallest positive eigenvalue is obtained and then a theorem is established comparing the smallest positive eigenvalues for two problems of this type

multiple positive solutions of some boundary value problems abstract we study the existence of multiple positive solutions of the equations u t u subject to linear boundary conditions we show that there are at least two positive solutions if t u is superlinear at one end point zero or infinity and sublinear at the other applications of these results are provided to yield multiple positive solutions of some elliptic boundary value problems on an annulus

hyperplane based vector quantization for distributed estimation in wireless sensor networks this paper considers distributed estimation of a vector parameter in the presence of zero mean additive multivariate gaussian noise in wireless sensor networks due to stringent power and bandwidth constraints vector quantization is performed at each sensor to convert its local noisy vector observation into one bit of information which is then forwarded to a fusion center where a final estimate of the vector parameter is obtained within such a context this paper focuses on a class of hyperplane based vector quantizers which linearly convert the observation vector into a scalar by using a compression vector and then carry out a scalar quantization it is shown that the key of the vector quantization design is to find a compression vector for each sensor under the framework of the cramer rao bound crb analysis the compression vector design problem is formulated as an optimization problem that minimizes the trace of the crb matrix such an optimization problem is extensively studied in particular an efficient iterative algorithm is developed for the general case along with optimal and near optimal solutions for some specific but important noise scenarios performance analysis and simulation results are carried out to illustrate the effectiveness of the proposed scheme

networked fusion estimation with bounded noises most of time varying systems in fusion estimation are generally modeled without bounded noises in this paper we study the distributed fusion estimation problem for networked time varying systems with bounded noises where the resource constraints i e bandwidth or energy and quantization effect are described by a unified model a new local estimator with time varying gain is designed by solving a class of convex optimization problems such that the square error of the estimator is bounded when each local estimate is transmitted to the fusion center over communication networks the selection probability criterion is derived such that the mean square error of the each compensating state estimate is bounded then a convex optimization problem on the design of an optimal weighting fusion criterion is established in terms of linear matrix inequalities which can be solved by standard software packages target tracking system with time varying sampling period is given to show the effectiveness of the proposed method

necessary and sufficient conditions for complete blow up and extinction for one dimensional quasilinear heat equations we characterize the occurrence of complete or incomplete blow up and extinction for a general quasilinear heat equation of the form he text u t phi u xx pm f u text in r times 0 t in terms of the constitutive functions and f we assume that u 0 for u 0 and that f u 0 for the positive sign before f u in he with f u superlinear as u blow up occurs in finite time supxu x t as t t t i e after the singularity occurs if such continuation exists we say that the blow up extinction is incomplete otherwise it is called complete our characterization is based on the qualitative behaviour of the family of travelling wave solutions and a proper use of the intersection comparison argument the analysis applies to other nonlinear models like the equations with gradient dependent diffusivity

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

location estimation of a random signal source based on correlated sensor observations the problem of location estimation of a source of random signals using a network of sensors is considered a novel maximum likelihood estimation mle based approach using copula functions is proposed the measurements received at the sensors are often spatially correlated and characterized by a multivariate distribution using the theory of copulas the joint parametric density of sensor observations joint likelihood is approximated assuming only the knowledge of the marginal likelihood functions of the sensor observations the problem of selecting the best copula function to model the joint likelihood is approached as one of model selection and a model fusion strategy is used to reduce the effect of selection bias an example involving source localization of a poisson source is presented to illustrate the proposed approach and demonstrate its performance

distributed adaptive quantization for wireless sensor networks from delta modulation to maximum likelihood we consider distributed parameter estimation using quantized observations in wireless sensor networks wsns where due to bandwidth constraint each sensor quantizes its local observation into one bit of information a conventional fixed quantization fq approach which employs a fixed threshold for all sensors incurs an estimation error growing exponentially with the difference between the threshold and the unknown parameter to be estimated to address this difficulty we propose a distributed adaptive quantization aq approach which with sensors sequentially broadcasting their quantized data allows each sensor to adaptively adjust its quantization threshold three aq schemes are presented 1 aq fs that involves distributed delta modulation dm with a fixed stepsize 2 aq vs that employs dm with a variable stepsize and 3 aq ml that adjusts the threshold through a maximum likelihood ml estimation process the ml estimators associated with the three aq schemes are developed and their corresponding cramer rao bounds crbs are analyzed we show that our 1 bit aq approach is asymptotically optimum yielding an asymptotic crb that is only pi 2 times that of the clairvoyant sample mean estimator using unquantized observations

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

input design in worst case system identification with quantized measurements this paper addresses the problem of set membership system identification with quantized measurements following the work developed for binary measurements the problem of optimal input design with multiple sensor thresholds is tackled for a fir model of order n the problem is decomposed into n static gain problems the one step optimal input problem is solved both for equispaced and generic sensor threshold distribution moreover the n step optimal input problem for the case of equispaced thresholds is addressed and a solution is provided under a suitable assumption on the sensor range and resolution the obtained results allow us to construct an upper bound on the time complexity of the fir identification problem for the case of equispaced thresholds numerical application examples are reported to show the effectiveness of the proposed algorithms

statistical inference under multiterminal data compression this paper presents a survey of the literature on the information theoretic problems of statistical inference under multiterminal data compression with rate constraints significant emphasis is put on problems 1 multiterminal hypothesis testing 2 multiterminal parameter estimation and 3 multiterminal pattern classification in either case of positive rates or zero rates in addition the paper includes three new results i e the converse theorems for all problems of multiterminal hypothesis testing multiterminal parameter estimation and multiterminal pattern classification at the zero rate

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

statistical results for system identification based on quantized observations system identification based on quantized observations requires either approximations of the quantization noise leading to suboptimal algorithms or dedicated algorithms tailored to the quantization noise properties this contribution studies fundamental issues in estimation that relate directly to the core methods in system identification as a first contribution results from statistical quantization theory are surveyed and applied to both moment calculations mean variance etc and the likelihood function of the measured signal in particular the role of adding dithering noise at the sensor is studied the overall message is that tailored dithering noise can considerably simplify the derivation of optimal estimators the price for this is a decreased signal to noise ratio and a second contribution is a detailed study of these effects in terms of the cramer rao lower bound the common additive uniform noise approximation of quantization is discussed compared and interpreted in light of the suggested approaches

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

robust distributed maximum likelihood estimation with dependent quantized data in this paper we consider the distributed maximum likelihood estimation mle with dependent quantized data under the assumption that the structure of the joint probability density function pdf is known but it contains unknown deterministic parameters the parameters may include different vector parameters corresponding to marginal pdfs and parameters that describe the dependence of observations across sensors since mle with a single quantizer is sensitive to the choice of thresholds due to the uncertainty of pdf we concentrate on mle with multiple groups of quantizers which can be determined by the use of prior information or some heuristic approaches to fend off against the risk of a poor outlier quantizer the asymptotic efficiency of the mle scheme with multiple quantizers is proved under some regularity conditions and the asymptotic variance is derived to be the inverse of a weighted linear combination of fisher information matrices based on multiple different quantizers which can be used to show the robustness of our approach as an illustrative example we consider an estimation problem with a bivariate non gaussian pdf that has applications in distributed constant false alarm rate cfar detection systems simulations show the robustness of the proposed mle scheme especially when the number of quantized measurements is small

blow up results and localization of blow up points for the heat equation with a nonlinear boundary condition abstract in this paper we study the large time behavior of positive solutions of the heat equation under the nonlinear boundary condition u f u where is the outward normal and f is nondecreasing with f u 0 for u 0 we show that if b r and 1 f is integrable at infinity there is finite time blow up for any initial datum in the two dimensional case we show that this is true for any smooth simply connected domain in the radially symmetric case if f c2 is convex and satisfies the properties above we show that blow up occurs only at the boundary

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

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

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