robust data fusion for multisensor detection systems minimax robust data fusion schemes for multisensor detection systems with discrete time observations characterized by statistical uncertainty are developed and analyzed block sequential and serial fusion rules are considered the performance measures used and made robust with respect to the uncertainties include the error probabilities of the hypothesis testing problem in the block fusion case and the error probabilities and expected numbers of samples or sensors in the sequential and serial fusion cases for different sensor observation statistics the minimax robust fusion rules are derived for two asymptotic cases of interest when the number of sensors is large and when the number of times the fusion center collects the local sensor decisions is large moreover for the case of identical sensor observation statistics and a large number of sensors it is shown that there is no loss in optimality if local tests using likelihood ratios and equal thresholds are used in the sequential fusion rule in all situations the robust decision rules at the sensors and the fusion center are shown to make use of likelihood ratios and thresholds that depend on the least favorable probability distributions of the uncertainty class describing the statistics of sensor observations

conformal iterated function systems with applications to the geometry of continued fractions in this paper we obtain some results about general conformal iterated function systems we obtain a simple characterization of the packing dimension of the limit set of such systems and introduce some special systems which exhibit some interesting behavior we then apply these results to the set of values of real continued fractions with restricted entries we pay special attention to the hausdorff and packing measures of these sets we also give direct interpretations of these measure theoretic results in terms of the arithmetic density properties of the set of allowed entries

the multifractal spectra of certain planar recurrence sets in the continued fraction dynamical system abstract in this paper we will introduce and study the planar recurrence level set s x y 0 1 2 r x y r x y 0 where r x y and r x y are the lower and upper quantitative hitting time indicators of y by x respectively in the continued fraction dynamical system it is shown that the multifractal spectrum is always degenerate in a sense that it is constant independent of the choice of the level set

sequential testing of sorted and transformed data as an efficient way to implement long glrts it is often required to detect a long weak signal in gaussian noise and frequently the exact form of that signal is parameterized but not known a bank of matched filters provides an appropriate detector however in some practical applications there are very many matched filters and most are quite long the consequent computational needs may render the classical bank of filters approach infeasibly expensive one example and our original motivation is the detection of chirp gravitational waves by an earth based interferometer in this paper we provide a computational approach to this problem via sequential testing since the sequential tests to be used are not for constant signals we develop the theory in terms of average sample number asn for this case specifically we propose two easily calculable expressions for the asn one a bound and the other an approximation the sequential approach does yield moderate computational savings but we find that by preprocessing the data using short medium fast fourier transforms ffts and an appropriate sorting of these fft outputs such that the most informative samples are entered to a sequential test first quite high numerical efficiency can be realized the idea is simple but appears to be quite successful examples are presented in which the computational load is reduced by several orders of magnitude the fft is an example of an energy agglomerating transform but of course there are many others the point here is that the transform need not match the sought signal exactly in the sense that all energy becomes confined to a single sample it is enough that the energy becomes concentrated and the more concentrated the better

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

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

a new numeric analytical principle for nonlinear deterministic and stochastic dynamical systems a new implicit local linearization method termed as the locally transversal linearization ltl is developed in the present paper for a numeric analytical integration of nonlinear dynamical systems under deterministic and or stochastic excitations the ltl procedure is developed in such a way that at a chosen time instant the ltl solution manifold transversally intersects the nonlinear solution manifold at that particular point or cross section in the state space where the solution vector needs to be obtained in general construction of the ltl equation is non unique and thus it is generally needed to substitute the conditionally linear ltl based solution into the given nonlinear ordinary differential equation ode at a specified time instant to ensure a transversal intersection in the present study however it is demonstrated that a transversal intersection at a given time instant may as well be ensured by just choosing a few specific forms of the ltl equation thereby eliminating the need for substitution this new variant of ltl is particularly suitable for discretization of stochastic differential equations sdes in that one simultaneously avoids dealing with formal derivatives of wiener processes while satisfying the nonlinear sde at discrete time points the stochastic ltl procedure is adaptable for obtaining strong pathwise solutions of a general nonlinear dynamical system with continuous but not necessarily differentiable vector fields under additive and or multiplicative excitations a specific advantage of the ltl method is that for a class of response regime herein referred to as the phase independent regime it can obtain an accurate description of response irrespective of the choice of the time step and without a propagation of local errors presently the method has been numerically illustrated for strong solutions of the hardening duffing oscillator under deterministic sinusoidal or additive stochastic white noise excitations or their combination limited numerical results on periodic solutions chaos with or without noise and stochastic resonance are included in this study the results presented are indicative of the sample path accuracy consistency stochastic numerical stability and a partial insensitivity to time step sizes of the proposed method

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

sequential cfar detectors using a dead zone limiter the performances of some proposed sequential constant false alarm rate cfar detectors are evaluated the observations are passed through a dead zone limiter the output of which is 1 0 or 1 depending on whether the input is less than c between c and c or greater than c where c is a constant the test statistic is the sum of the outputs the test is performed on a reduced set of data those with absolute value larger than c with the test statistic being the sum of the signs of the reduced set of data both constant and linear boundaries are considered numerical results show a significant reduction of the average number of observations needed to achieve the same false alarm and detection probabilities as compared to a fixed sample size cfar detector using the same kind of test statistic

multi step tangential versus transversal linearizations in non linear dynamics a family of multi step tangential linearization mtnl techniques is developed for numeric analytic and geometric integration of non linear oscillators under deterministic loading a family of multi step transversal linearization mtrl schemes is also explored and a comparison between the two linearization families is provided with a view of bringing out their relative advantages and deficiencies the mtnl methodology essentially works through an application of the concept of mtrl to non linear rate equations different variants of the mtnl are obtained and it turns out that they are specific cases of a generalized mtnl method local error estimates are also provided moreover higher order mtrl and mtnl methods are successfully implemented through chebyshev nodes instead of equidistant nodes this helps to improve accuracy even with larger step sizes a comparison of relative errors through the mtrl and mtnl techniques consistently indicates a superior quality of approximations via the mtrl route whilst the usage of the tangential linearization is very common in non linear computational mechanics it is demonstrated that the transversal version offers in addition to higher numerical accuracy an easier implementation and avoids computation of jacobian matrices this allows the mtrl family to be applicable even to non linear oscillators having c0 vector fields copyright 2007 john wiley sons ltd

constrained partially observed markov decision processes with probabilistic criteria for adaptive sequential detection dynamic programming equations are derived which characterize the optimal value functions for a partially observed constrained markov decision process problem with both total cost and probabilistic criteria more specifically the goal is to minimize an expected total cost subject to a constraint on the probability that another total cost exceeds a prescribed threshold the markov decision process is partially observed but it is assumed that the constraint costs are available to the controller i e they are fully observed the problem is motivated by an adaptive sequential detection application the application of the dynamic programming results to optimal adaptive truncated sequential detection is demonstrated using an example involving the optimization of a radar detection process

opportunistic detection under a fixed sample size setting with a finite number of samples drawn from one of two possible distributions sequentially revealed an opportunistic detection rule is proposed which possibly makes an early decision in favor of the alternative hypothesis while always deferring the decision of the null hypothesis until collecting all the samples properties of this opportunistic detection rule are discussed and its key asymptotic behavior in the large sample size limit is established specifically a chernoff stein lemma type of characterization of the exponential decay rate of the miss probability under the neyman pearson criterion is established and consequently a performance metric of asymptotic exponential efficiency loss is proposed and discussed which is exactly the ratio between the kullback leibler distance and the chernoff information of the two hypotheses analytical results are corroborated by numerical experiments

quantitative recurrence properties in conformal iterated function systems abstract we consider a quantitative version of poincare s recurrence theorem in a conformal iterated function system let i i be a conformal iterated function system on 0 1 d with a countable index set denote by j the attractor of let f 0 1 d r be a positive function s n f x be the sum f x f w 1 1 x f w 1 w n 1 1 x analogous to an ergodic sum and consider the set of points for which the inequality x w 1 w n 0 1 d x w 1 w n 1 x e s n f x is realized for infinitely many n n with w 1 w n n the set above contains the points x whose orbit returns very close to x infinitely many times with the quality of approximation depending on the time n and the point x this can be viewed as a quantitative version of poincare s recurrence theorem it is shown that its hausdorff dimension is the solution to some pressure function the setting considered in this paper includes the quantitative recurrence properties in the dynamical systems of b adic expansions continued fraction expansion as well as some dynamical systems defined on fractal sets applying the main result to diophantine approximation gives a partial answer to a question by k mahler

local dimensions of measures on infinitely generated self affine sets abstract we show the existence of the local dimension of an invariant probability measure on an infinitely generated self affine set for almost all translations this implies that an ergodic probability measure is exactly dimensional furthermore the local dimension equals the minimum of the local lyapunov dimension and the dimension of the space

counterexamples in distributed detection specific counterexamples violating three very plausible conjectures about optimal design of distributed detection and decision systems are reported the numerical magnitude of the violation is small in each case

near optimality of chebyshev interpolation for elementary function computations a common practice for computing an elementary transcendental function in an libm implementation nowadays has two phases reductions of input arguments to fall into a tiny interval and polynomial approximations for the function within the interval typically the interval is made tiny enough so that polynomials of very high degree aren t required for accurate approximations often approximating polynomials as such are taken to be the best polynomials or any others such as the chebyshev interpolating polynomials the best polynomial of degree n has the property that the biggest difference between it and the function is smallest among all possible polynomials of degrees no higher than n thus it is natural to choose the best polynomials over others in this paper it is proven that the best polynomial can only be more accurate by at most a fractional bit than the chebyshev interpolating polynomial of the same degree in computing elementary functions or in other words the chebyshev interpolating polynomials will do just as well as the best polynomials similar results were obtained in 1967 by powell who however did not target elementary function computations in particular and placed no assumption on the function and remarkably whose results imply accuracy differences of no more than 2 to 3 bits

autocorrelation based decentralized sequential detection of ofdm signals in cognitive radios this paper introduces a simple and computationally efficient spectrum sensing scheme for orthogonal frequency division multiplexing ofdm based primary user signal using its autocorrelation coefficient further it is shown that the log likelihood ratio test llrt statistic is the maximum likelihood estimate of the autocorrelation coefficient in the low signal to noise ratio snr regime performance of the local detector is studied for the additive white gaussian noise awgn and multipath channels using theoretical analysis obtained results are verified in simulation the performance of the local detector in the face of shadowing is studied by simulations a sequential detection sd scheme where many secondary users cooperate to detect the same primary user is proposed user cooperation provides diversity gains as well as facilitates using simpler local detectors the sequential detection reduces the delay and the amount of data needed in identification of the underutilized spectrum the decision statistics from individual detectors are combined at the fusion center fc the statistical properties of the decision statistics are established the performance of the scheme is studied through theory and validated by simulations a comparison of the sd scheme with the neyman pearson fixed sample size fss test for the same false alarm and missed detection probabilities is also carried out

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

fractal measures for parabolic ifs abstract let h be the hausdorff dimension of the limit set of a conformal parabolic iterated function system in dimension d 2 in case the system of maps is finite we provide necessary and sufficient conditions for the h dimensional hausdorff measure to be positive and finite and also assuming the strong open set condition holds characterize when the h dimensional packing measure of the limit set is positive and finite we also prove that the upper ball box counting dimension and the hausdorff dimension of this limit set coincide as a byproduct we include a compact analysis of the behaviour of parabolic conformal diffeomorphisms in dimension 2 and separately in any dimension greater than or equal to 3

new forms of extended kalman filter via transversal linearization and applications to structural system identification two novel forms of the extended kalman filter ekf are proposed for parameter estimation in the context of problems of interest in structural mechanics these filters are based on variants of the derivative free locally transversal linearization ltl and multi step transversal linearization mtrl schemes thus unlike the conventional ekf the proposed filters do not need computing jacobian matrices at any stage while the ltl based filter provides a single step procedure the mtrl based filter works over multiple time steps and finds the system transition matrix of the conditionally linearized vector field through magnus expansion other major advantages of the new filters over the conventional ekf are in their superior numerical accuracy and considerably less sensitivity to the choice of timestep sizes a host of numerical illustrations covering single as well as multi degree of freedom oscillators with time invariant parameters and those with discontinuous temporal variations are presented to confirm the numerical advantages of the novel forms of ekf over the conventional one

detecting small moving objects in image sequences using sequential hypothesis testing an algorithm is proposed for the solution of the class of multidimensional detection problems concerning the detection of small barely discernible moving objects of unknown position and velocity in a sequence of digital images a large number of candidate trajectories organized into a tree structure are hypothesized at each pixel in the sequence and tested sequentially for a shift in mean intensity the practicality of the algorithm is facilitated by the use of multistage hypothesis testing mht for simultaneous inference as well as the existence of exact closed form expressions for mht test performance in gaussian white noise gwn these expressions predict the algorithm s computation and memory requirements where it is shown theoretically that several orders of magnitude of processing are saved over a brute force approach based on fixed sample size tests the algorithm is applied to real data by using a robust preprocessing procedure to eliminate background structure and transform the image sequence into a residual representation modeled as gwn results are verified experimentally on a variety of video image sequences

generalized integrating factor methods for stiff pdes the integrating factor if method for numerical integration of stiff nonlinear pdes has the disadvantage of producing large error coefficients when the linear term has large norm we propose a generalization of the if method and in particular construct multistep type methods with several orders of magnitude improved accuracy we also consider exponential time differencing etd methods and point out connections with a particular application of the commutator free lie group methods we present a new fourth order etdrk method with improved accuracy the methods considered are compared in several numerical examples

numerical integrators that preserve symmetries and reversing symmetries we consider properties of flows the relationships between them and whether numerical integrators can be made to preserve these properties this is done in the context of automorphisms and antiautomorphisms of a certain group generated by maps associated to vector fields this new framework unifies several known constructions we also use the concept of covariance of a numerical method with respect to a group of coordinate transformations the main application is to explore the relationship between spatial symmetries reversing symmetries and time symmetry of flows and numerical integrators

dimension spectra of hyperbolic flows for flows with a conformal hyperbolic set we establish a conditional variational principle for the dimension spectra of holder continuous functions we consider simultaneously birkhoff averages into the future and into the past we emphasize that the description of the spectra is not a consequence of the existing results for birkhoff averages into the future or into the past the main difficulty is that even though the local product structure is bi lipschitz the level sets of the birkhoff averages are never compact our proof is based on the use of markov systems and is inspired in earlier arguments in the case of discrete time

repeated significance tests for exponential families abstract this report considered the significance levels and powers of repeates significance test rts typically these quantities cannot be calculated exactly and some sort of approximations are required satisfactory approximations of significance levels of rst for exponential families have been obtained by lalley 1983 and woodroofe 1978 in this report another method due to siegmund 1985 in the special case of normal observations with known variance is developed the main advantages that are claimed for this method are two fold first it can be used to approximate the power of the rst second it enables one to estimate both powers and significance levels of the modified rst the approximations are also useful in determining confidence intervals the proof of a nonlinear renewas theorem for conditional random walks and some numerical results are also given additional keywords distribution functions tables data random walks null hypothesis case author

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

on transfer operators for continued fractions with restricted digits for any non empty subset i of the natural numbers let lambda i denote those numbers in the unit interval whose continued fraction digits all lie in i define the corresponding transfer operator formula for formula where re rs theta i is the abscissa of convergence of the series formula when acting on a certain hilbert space hi rs we show that the operator li rs is conjugate to an integral operator ki rs if furthermore rs is real then ki rs is selfadjoint so that li rs hi rs hi rs has purely real spectrum it is proved that li rs also has purely real spectrum when acting on various hilbert or banach spaces of holomorphic functions on the nuclear space c omega 0 1 and on the frechet space c infty 0 1 the analytic properties of the map rs m li rs are investigated for certain alphabets i of an arithmetic nature for example i primes i squares i an arithmetic progression i the set of sums of two squares it is shown that rs m li rs admits an analytic continuation beyond the half plane re rs theta i

asymptotic efficiencies of truncated sequential tests truncation of a sequential test with constant boundaries is considered for the problem of testing a location hypothesis f x theta 0 versus f x theta 1 a test design procedure is developed by using bounds for the error probabilities under the hypothesis and alternative by viewing the truncated sequential test as a mixture of a sequential probability ratio test and a fixed sample size test its boundaries and truncation point can be obtained once the degree of mixture is specified asymptotically correct approximations for the operating characteristic function and the average sample number function of the resulting test are derived numerical results show that an appropriately designed truncated sequential test performs favorably as compared to both the fixed sample size test and the sequential probability ratio test with the same error probabilities the average sample number function of the truncated test is uniformly smaller than that of the fixed sample size test and the truncated test maintains average sample sizes under the hypothesis and the alternative that are close to those optimum values achieved by wald s sequential probability ratio test moreover the truncated test is more favorable than the sequential probability ratio test in the sense that is has smaller average sample size when the actual location parameter is between theta 0 and theta 1 this behavior becomes more pronounced as the error probabilities become smaller implying that the truncated sequential test becomes more favorable as the error probabilities become smaller

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

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

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

collocation and relaxed collocation for the fer and the magnus expansions we consider the fer and the magnus expansions for the numerical solution of the nonlinear matrix lie group ode y gamma t y y y 0 y 0 where y evolves in a matrix lie group g and gamma t y is in the lie algebra hbox fr g ghe departing from a geometrical approach that distinguishes between those operations performed in the group and those performed in the tangent space we construct lie group invariant methods based on collocation we prove that as long as the two expansions are correctly truncated the collocation nodes c 1 c 2 ldots c nu yield numerical methods whose order is the same as in the classical setting we also relax the collocation conditions thereby devising explicit methods of order 3 to conclude we discuss the proposed methods in a numerical experiment that arises in hamiltonian mechanics comparing the results with projection techniques

dimension and measures for a curvilinear sierpinski gasket or apollonian packing in this paper we apply some results about general conformal iterated function systems to a the residual set of a standard apollonian packing or a curvilinear sierpinski gasket within this context it is straightforward to show that h the hausdorff dimension of a is greater than 1 and the packing dimension and the upper and lower box counting dimensions are all the same as the hausdorff dimension among other things we verify sullivan s result that 0 h h a p h a

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

asymptotic locally optimal detector for large scale sensor networks under the poisson regime we consider distributed detection with a large number of identical binary sensors deployed over a region where the phenomenon of interest poi has spatially varying signal strength each sensor makes a binary decision based on its own measurement and the local decision of each sensor is sent to a fusion center using a random access protocol the fusion center decides whether the event has occurred under a global size constraint in the neyman pearson formulation assuming homogeneous poisson distributed sensors we show that the distribution of alarmed sensors satisfies the local asymptotic normality lan we then derive an asymptotically locally most powerful almp detector optimized jointly over the fusion form and the local sensor threshold under the poisson regime we establish conditions on the spatial signal shape that ensure the existence of the almp detector we show that the almp test statistic is a weighted sum of local decisions the optimal weights being the shape of the spatial signal the exact value of the signal strength is not required we also derive the optimal threshold for each sensor for the case of independent identically distributed iid sensor observations we show that the counting based detector is also almp under the poisson regime the performance of the proposed detector is evaluated through analytic results and monte carlo simulations and compared with that of the counting based detector the effect of mismatched signal shapes is also investigated

integrable expansions for posterior distributions for a two parameter exponential family asymptotic expansions of posterior distributions are derived for a two dimensional exponential family which includes normal gamma inverse gamma and inverse gaussian distributions reparameterization allows us to use a data dependent transformation convert the likelihood function to the two dimensional standard normal density and apply a version of stein s identity to assess the posterior distributions applications are given to characterize optimal noninformative priors in the sense of stein to suggest the form of a high order correction to the distribution function of a sequential likelihood ratio statistic and to provide confidence intervals for one parameter in the presence of other nuisance parameters

multi step transversal and tangential linearization methods applied to a class of nonlinear beam equations a novel and continuously parameterized form of multi step transversal linearization mtrl method is developed and numerically explored for solving nonlinear ordinary differential equations governing a class of boundary value problems bvps of relevance in structural mechanics a similar family of multi step tangential linearization mtnl methods is also developed and applied to such bvp s within the framework of mtrl and mtnl a bvp is treated as a constrained dynamical system i e a constrained initial value problem ivp while the mtrl requires the linearized solution manifold to transversally intersect the nonlinear solution manifold at a chosen set of points across the axis of the independent variable the essential difference of the present mtrl method from its previous version roy d kumar r 2005 a multistep transversal linearization mtl method in nonlinear structural dynamics j sound vib 17 829 852 is that it has the flexibility of treating nonlinear damping and stiffness terms as time variant damping and stiffness terms in the linearized system the resulting time variant linearized system is then solved using magnus characterization magnus w 1954 on the exponential solution of differential equations for a linear operator commun pure appl math 7 649 673 towards numerical illustrations response of a tip loaded cantilever beam elastica is first obtained next the response of a simply supported nonlinear timoshenko beam is obtained using a variationally correct vc model for the beam marur s prathap g 2005 nonlinear beam vibration problems and simplification in finite element model comput mech 35 5 352 360 the new model does not involve any simplifications commonly employed in the finite element formulations in order to ease the computation of nonlinear stiffness terms from nonlinear strain energy terms a comparison of results through mtrl and mtnl techniques consistently indicate a superior quality of approximations via the transversal linearization technique while the usage of tangential system matrices is common in nonlinear finite element practices it is demonstrated that the transversal version of linearization offers an easier and more general implementation requires no computations of directional derivatives and leads to a consistently higher level of numerical accuracy it is also observed that higher order versions of mtrl mtnl with lagrangian interpolations may not work satisfactorily and hence spline interpolations are suggested to overcome this problem

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

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

the decentralized quickest detection problem two detectors making independent observations must decide when a markov chain jumps from state 0 to state 1 the decisions are coupled through a common cost function it is shown that the optimal decision is characterized by thresholds as in the decoupled case however the thresholds are time varying and their computation requires the solution of two coupled sets of dynamic programming equations a comparison to the decoupled case shows the structure of the coupling

explicit eigenvalue estimates for transfer operators acting on spaces of holomorphic functions abstract we consider transfer operators acting on spaces of holomorphic functions and provide explicit bounds for their eigenvalues more precisely if is any open set in c d and l is a suitable transfer operator acting on bergman space a 2 its eigenvalue sequence n l is bounded by n l a exp a n 1 d where a a 0 are explicitly given

chains of large gaps between consecutive primes where log n log n log n log log i n this result has so far not been improved it implies in particular that lim sup d log n co since by the prime number theorem the average size of d is log n this means that the largest gaps exceed the average by a ratio approaching infinity westzynthius lo in 1949 erdgs 4 proposed the problem for a fixed k and n co can k consecutive gaps be simultaneously much larger than log n or more precisely does lim sup min d d log n co erdos proved this for the case k 2 4 in this paper we prove erdos conjecture for all k moreover we obtain the same lower bound for k consecutive gaps as that given by rankin s theorem for single gaps the only difference is that as k increases the constant implied by o decreases

conformal geometric and invariant measures for transcendental expanding functions we describe the fractal structure of expanding meromorphic maps of the form h circ exp circ q where h and q are rational functions whose most transparent examples are among the functions of the form frac a exp z p b exp z p c exp z p d exp z p with ad bc ne 0 in particular we show that depending upon whether the hausdorff dimension of the julia set is greater or less than 1 the finite non zero geometric measure is provided by the hausdorff or packing measure in order to describe this fractal structure we introduce and explore in detail walters expanding conformal maps and jump like conformal maps

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

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

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

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

approximate tail probabilities for the maxima of some random fields abstract hogan and siegmund 1986 adapt the method developed by pickands 1969 qualls and watanabe 1973 and bickel and rosenblatt 1973 to obtain explicit large deviation approximations for the maxima of several gaussian random fields arising in statistics using a special argument for one particular case they suggest a heuristic second order approximation for that case and they show by a monte carlo experiment that the second order approximation frequently gives considerably better numerical results the purpose of this paper is to show that the method developed by woodroofe 1976 1982 for problems in one dimensional time can be adapted to study maxima of random fields overall it involves simpler computations than the previous method and consequently seems potentially capable of delivering a genuine second order approximation should one seem desirable

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

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

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

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

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

a semi analytical locally transversal linearization method for non linear dynamical systems an numeric analytical implicit and local linearization methodology called the locally transversal linearization ltl is developed in the present paper for analyses and simulations of non linear oscillators the ltl principle is based on deriving the locally linearized equations in such a way that the tangent space of the linearized equations transversally intersects that of the given non linear dynamical system at that particular point in the state space where the solution vector is sought for purposes of numerical implementation two different numerical schemes namely ltl 1 and ltl 2 schemes based on the ltl methodology are presented both ltl 1 and ltl 2 procedures finally reduce the given set of non linear ordinary differential equations odes to a set of transcendental algebraic equations valid over a short interval of time or over a short segment of the evolving trajectories as projected on the phase space while in the ltl 1 scheme the desired solution vector at a forward time point enters the linearized differential equations as an unknown parameter in the ltl 2 scheme a set of unknown residues enters the linearized system as parameters a limited set of examples involving a few well known single degree of freedom sdof non linear oscillators indicate that the ltl methodology is capable of accurately predicting many complicated non linear response patterns including limit cycles quasi periodic orbits and even strange attractors copyright 2001 john wiley sons ltd

scaling properties of hausdorff and packing measures let m in mathbb n let theta be a continuous increasing function defined on mathbb r for which theta 0 0 and theta t t m is a decreasing function of t let cdot be a norm on mathbb r m and let rho mathcal h theta mathcal h theta rho mathcal p theta mathcal p theta rho denote the corresponding metric and hausdorff and packing measures respectively we characterize those functions theta such that the corresponding hausdorff or packing measure scales with exponent alpha by showing it must be of the form theta t t alpha l t where l is slowly varying we also show that for continuous increasing functions theta and eta defined on mathbb r for which theta 0 eta 0 0 mathcal h theta mathcal p eta is either trivially true or false we show that if mathcal h theta mathcal p eta then mathcal h theta mathcal p eta c cdot lambda for a constant c where lambda is the lebesgue measure on mathbb r m

a 2 sprt based approach to multiple model hypothesis testing for multi distribution detection this paper presents an approach to multiple model hypothesis testing mmht based on 2 sprt 2 mmsrpt for detecting unknown events that may have multiple possible distributions the sequential probability ratio test sprt based mmht method mmsprt is promising because of its efficiency and theoretical validity however it may suffer from sprt s lack of an upper bound on its stopping time especially in the mis specified case the proposed 2 mmsprt algorithm not only copes with this problem but also is in a setting that can provide efficient detection in the sense of minimizing the maximum expected sample size subject to error probability constraints specifically we prove the theoretical validity of 2 sprt for the problem of testing hypotheses with multivariate normal densities moreover we present a technique of forced independence and identical distribution i i d to map the non i i d likelihood ratio sequence to an i i d one which enables us to apply the sprt and 2 sprt effectively to the dynamic case under the linear gaussian assumption with a non identical distribution also 2 mmsprt s detection efficiency under some assumptions approximations is verified performance of 2 mmsprt is evaluated for signal detection and model set selection problems in several scenarios simulation results demonstrate the detection efficiency of the proposed 2 mmsprt compared with the mmsprt and some traditional tests

a multi step transversal linearization mtl method in non linear structural dynamics an implicit family of multi step transversal linearization mtl methods is proposed for efficient and numerically stable integration of nonlinear oscillators of interest in structural dynamics the presently developed method is a multi step extension and further generalization of the locally transversal linearization ltl method proposed earlier by roy proceedings of the academy of the royal society london 457 2001 539 566 roy and ramachandra journal of sound and vibration 41 2001a 653 679 international journal for numerical methods in engineering 51 2001b 203 224 and roy international journal of numerical methods in engineering 61 2004 764 the mtl based linearization is achieved through a non unique replacement of the nonlinear part of the vector field by a conditionally linear interpolating expansion of known accuracy whose coefficients contain the discretized state variables defined at a set of grid points in the process the nonlinear part of the vector field becomes a conditionally determinable equivalent forcing function the mtl based linearized differential equations thus become explicitly integrable based on the linearized solution a set of algebraic constraint equations are so formed that transversal intersections of the linearized and nonlinearized solution manifolds occur at the multiple grid points the discretized state vectors are thus found as the zeros of the constraint equations simple error estimates for the displacement and velocity vectors are provided and in particular it is shown that the formal accuracy of the mtl methods as a function of the time step size depends only on the error of replacement of the nonlinear part of the vector field presently only two different polynomial based interpolation schemes are employed for transversal linearization viz the taylor like interpolation and the lagrangian interpolation while the taylor like interpolation leads to numerical ill conditioning as the order of interpolation increases the lagrangian interpolation is shown to overcome this numerical problem finally the family of mtl methods is illustrated through limited numerical results for a couple of harmonically driven workhorse oscillators viz the hardening duffing and the duffing holmes oscillators comparisons with results obtained via a sixth order runge kutta method with adaptive time step sizes are also provided

hausdorff dimension of certain sets arising in continued fraction expansions this paper is concerned with the fractional dimensions of some sets of points with their partial quotients obeying some restrictions in their continued fraction expansions the hausdorff dimension of the following set which shares a dichotomy law according to borel bernstein s theorem is completely determined our result significantly strengthens the results of good and luczak

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

noncoherent sequential acquisition of pn sequences for ds ss communications with without channel fading studies sequential acquisition schemes of m sequences based on the sequential probability ratio test sprt and a truncated sprt tsprt with noncoherent demodulation most reported results on sequential acquisition schemes assume that independent samples are available for the decision process the assumption of independent and identically distributed i i d samples requires the integrator in the receiver to be reset periodically this introduces loss in the effective signal to noise ratio degrading the performance in the present paper on the contrary the authors two sequential schemes use continuous integration it can be shown that the likelihood ratio is monotonic consequently the tests can be easily implemented in realtime methods are proposed for designing the decision thresholds to achieve the desired false alarm and miss probabilities performances of the proposed schemes are obtained and they suggest that the tsprt is more desirable the effect of slowly varying channel fading is also investigated results show that fading does not affect the false alarm probabilities but it can drastically reduce the probability of detecting the alignment of the two pn sequences especially when the fading is severe

global normal form and asymptotic spectral gap for open partially expanding maps we consider a mathbb r extension of one dimensional uniformly expanding open dynamical systems and prove a new explicit estimate for the asymptotic spectral gap to get these results we use a new application of a global normal form for the dynamical system a semiclassical expression beyond the ehrenfest time that expresses the transfer operator at large time as a sum over rank one operators each is associated to one orbit in this paper we establish the validity of the so called diagonal approximation up to twice the local ehrenfest time

a multivariate change point model for statistical process control multivariate statistical process control spc carries out ongoing checks to ensure that a process is in control these checks on the process are traditionally done by t2 multivariate cusum and multivariate exponentially weighted moving average control charts these traditional spc charts assume that the in control true parameters are known exactly and use these assumed true values to set the control limits the reality however is that true parameter values are seldom if ever known exactly rather they are commonly estimated from a phase i sample it is increasingly recognized that this phase i study needs to involve large samples if the parameter estimates are to provide run behavior matching that of the known parameter situation but apart from the general undesirability of large and thus expensive studies preliminary to actual charting some industrial settings have a paucity of relevant data for estimating the process parameters an attractive alternative to traditional charting methods when mon

dimensions and measures in infinite iterated function systems the hausdorff and packing measures and dimensions of the limit sets of iterated function systems generated by countable families of conformal contractions are investigated conformal measures for such systems reflecting geometric properties of the limit set are introduced proved to exist and to be unique the existence of a unique invariant probability equivalent to the conformal measure is derived our methods employ the concepts of the perron frobenius operator symbolic dynamics on a shift space with an infinite alphabet and the properties of the above mentioned ergodic invariant measure a formula for the hausdorff dimension of the limit set in terms of the pressure function is derived fractal phenomena not exhibited by finite systems are shown to appear in the infinite case in particular a variety of conditions are provided for hausdorff and packing measures to be positive or finite and a number of examples are described showing the appearance of various possible combinations for these quantities one example given special attention is the limit set associated to the complex continued fraction expansion in particular lower and upper estimates for its hausdorff dimension are given a large natural class of systems whose limit sets are dimensionless in the restricted sense is described

robustness of decentralized tests with spl epsiv contamination prior we consider a decentralized detection problem where the prior density is not completely known but is assumed to belong to an spl epsiv contamination class the expressions for the infimum and the supremum of the posterior probability that the parameter under question is in a given region as the prior varies over the spl epsiv contamination class are derived numerical results are obtained for a specific case of an exponentially distributed observation and an exponentially distributed nominal prior asymptotic as number of sensors tends to a large value results are also obtained the results illustrate the degree of robustness achieved with quantized observations as compared to unquantized observations

opportunistic detection rules finite and asymptotic analysis opportunistic detection rules odrs are variants of fixed sample size detection rules in which the statistician is allowed to make an early decision on the alternative hypothesis opportunistically based on the sequentially observed samples from a sequential decision perspective odrs are also mixtures of one sided and truncated sequential detection rules several results regarding odrs are established in this paper in the finite regime the maximum sample size is modeled either as a fixed finite number or a geometric random variable with a fixed finite mean for both cases the corresponding bayesian formulations are investigated the former case is a slight variation of the well known finite length sequential hypothesis testing procedure in the literature whereas the latter case is new for which the bayesian optimal odr is shown to be a sequence of likelihood ratio threshold tests with two different thresholds a running threshold which is determined by solving a stationary state equation is used when future samples are still available and a terminal threshold simply the ratio between the priors scaled by costs is used when the statistician reaches the final sample and thus has to make a decision immediately in the asymptotic regime the tradeoff among the exponents of the false alarm and miss error probabilities and the normalized expected stopping time under the alternative hypothesis is completely characterized and proved to be tight via an information theoretic argument within the tradeoff region one noteworthy fact is that the performance of the stein chernoff lemma is attainable by odrs