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

information bounds and quick detection of parameter changes in stochastic systems by using information theoretic bounds and sequential hypothesis testing theory this paper provides a new approach to optimal detection of abrupt changes in stochastic systems this approach not only generalizes previous work in the literature on optimal detection far beyond the relatively simple models treated but also suggests alternative performance criteria which are more tractable and more appropriate for general stochastic systems in addition it leads to detection rules which have manageable computational complexity for on line implementation and yet are nearly optimal under the different performance criteria considered

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

quantization effect on the log likelihood ratio and its application to decentralized sequential detection it is well known that quantization cannot increase the kullback leibler divergence which can be thought of as the expected value or first moment of the log likelihood ratio in this paper we investigate the quantization effects on the second moment of the log likelihood ratio it is shown via the convex domination technique that quantization may result in an increase in the case of the second moment but the increase is bounded above by 2 e the result is then applied to decentralized sequential detection problems not only to provide simpler sufficient conditions for asymptotic optimality theories in the simplest models but also to shed new light on more complicated models in addition some brief remarks on other higher order moments of the log likelihood ratio are also provided

information bounds and quickest change detection in decentralized decision systems the quickest change detection problem is studied in decentralized decision systems where a set of sensors receive independent observations and send summary messages to the fusion center which makes a final decision in the system where the sensors do not have access to their past observations the previously conjectured asymptotic optimality of a procedure with a monotone likelihood ratio quantizer mlrq is proved in the case of additive gaussian sensor noise if the signal to noise ratios snr at some sensors are sufficiently high this procedure can perform as well as the optimal centralized procedure that has access to all the sensor observations even if all snrs are low its detection delay will be at most pi 2 1ap 57 larger than that of the optimal centralized procedure next in the system where the sensors have full access to their past observations the first asymptotically optimal procedure in the literature is developed surprisingly the procedure has the same asymptotic performance as the optimal centralized procedure although it may perform poorly in some practical situations because of slow asymptotic convergence finally it is shown that neither past message information nor the feedback from the fusion center improves the asymptotic performance in the simplest model

chebyshev collocation methods for fast orbit determination we apply some chebyshev collocation methods for fast and accurate determination of ephemeris of artificial satellites this formulation is based on the approximation of the differential system by means of chebyshev polynomials in order to assess the convergence of the iterative process and to reduce the number of operations the special formulation of dziobek brouwer is used from several numerical tests we conclude that the method presents very good behaviour for geostationary and low eccentric orbits the most frequent cases the number of operations with this method in some cases is 20 times less than that with a standard integrator besides a typical dense output in a polynomial form is obtained

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

a study of the effects of grid non orthogonality on the solution of shallow water equations in boundary fitted coordinate systems in the present study an existing two dimensional boundary fitted model j hydraul eng asce 122 9 1996 512 is used to study the effect of grid non orthogonality on the solution of shallow water equations using boundary fitted grids the linearized two dimensional shallow water equations are expressed in terms of the grid angle and aspect ratio the truncation errors of the finite difference approximations used in the solution of the governing equations are shown to be dependent on the grid angle and the aspect ratio the coefficient of the truncation error was shown to increase with the decrease in the grid angle the rms errors in model predicted surface elevations and velocities for the case of seiching in a rectangular basin are found to increase gradually as the grid resolution decreases from 174 to 80 grid points per wavelength or as the grid angle decreases from 90 to 50 and increases rather sharply for a grid angle of 30 at grid resolutions less than 80 gridpoints per wavelength the model predicted surface elevations for the case of tidal forcing in a rectangular basin are found to be insensitive to the grid angle at grid resolutions higher than 600 gridpoints per wavelength the rms error in the model predicted velocities is found to increase gradually as the grid angle decreases from 90 to 30 or as the grid resolution decreases from 1400 gridpoints per wavelength to 400 gridpoints per wavelength and increases sharply as the grid resolution decreases from 400 to 150 gridpoints per wavelength two dimensional depth averaged hydrodynamic modeling of tidal circulation in narragansett bay using three different boundary fitted grids showed that the model predicted surface elevations are insensitive to the grid angle at grid resolutions as low as 200 gridpoints per wavelength however the model predicted velocities were found to increase as the grid resolution decreases from 600 to 200 gridpoints per wavelength we conclude from this study that grid angle and grid resolution affects the accuracy of the model predicted currents and the numerical dispersion increases with the decrease in grid angle or grid resolution and these are in agreement with that reached by sankaranarayanan and spaulding dispersion and stability analyses of shallow water equations in boundary fitted coordinates department of ocean engineering university of rhode island 2001 p 33 through a fourier analysis of the discretized equations in boundary fitted coordinates

the sensitivity and accuracy of fourth order finite difference schemes on nonuniform grids in one dimension abstract we construct local fourth order finite difference approximations of first and second derivatives on nonuniform grids in one dimension the approximations are required to satisfy symmetry relationships that come from the analogous higher dimensional fundamental operators the divergence the gradient and the laplacian for example we require that the discrete divergence and gradient be negative adjoint of each other div grad and the discrete laplacian is defined as lap divgrad the adjointness requirement on the divergence and gradient guarantees that the laplacian is a symmetric negative operator the discrete approximations we derive are fourth order on smooth grids but the approach can be extended to create approximations of arbitrarily high order we analyze the loss of accuracy in the approximations when the grid is not smooth and include a numerical example demonstrating the effectiveness of the higher order methods on nonuniform grids

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

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

analysis of stability and accuracy of finite difference schemes on a skewed mesh numerical stability and accuracy of finite difference schemes on a skewed non uniform mesh are investigated to provide guidelines for mesh design and for devising appropriate solution methods when mesh skewness is unavoidable in the current analysis a linear advection diffusion equation in a cartesian coordinate system is transformed into a curvilinear one corresponding to a skewed mesh a finite difference approximation of the transformed equation leads to the leading order error terms which are responsible for time step restrictions and numerical instability a truncation error analysis of central difference approximation reveals the effects of mesh non uniformity and skewness on the solution quality in addition a modified wavenumber analysis is performed for the central and upwind difference schemes coupled with time integration methods to examine the effects of mesh skewness angle and flow direction relative to the mesh angle on the numerical stability in general severe mesh skewness leads to restrictions on the allowable cell peclet number and time step size and also increases phase and amplitude errors

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

a formula for estimation of truncation errors of convection terms in a curvilinear coordinate system abstract a formula for truncation errors of convection terms arising from the navier stokes equation in a curvilinear coordinate system is derived these truncation error terms are interpreted in terms of grid size grid uniformity grid line angle flow angle and derivatives of flow properties to gain physical insights from the formula the role of these factors in determining truncation errors is discussed it is shown that an optimal grid arrangement cannot be obtained without considering the interactions between the grid and the flow field the effect of the grid orthogonality on truncation errors is also analyzed for simple cases the derived formula provides a useful indicator for truncation error distribution which yields guidelines for grid adaption

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

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

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

modifications of the method of variation of parameters in spite of being a classical method for solving differential equations the method of variation of parameters continues having a great interest in theoretical and practical applications as in astrodynamics in this paper we analyse this method providing some modifications and generalised theoretical results finally we present an application to the determination of the ephemeris of an artificial satellite showing the benefits of the method of variation of parameters for this kind of problems

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

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

quickest change detection of a markov process across a sensor array recent attention in quickest change detection in the multisensor setting has been on the case where the densities of the observations change at the same instant at all the sensors due to the disruption in this work a more general scenario is considered where the change propagates across the sensors and its propagation can be modeled as a markov process a centralized bayesian version of this problem is considered with a fusion center that has perfect information about the observations and a priori knowledge of the statistics of the change process the problem of minimizing the average detection delay subject to false alarm constraints is formulated in a dynamic programming framework insights into the structure of the optimal stopping rule are presented in the limiting case of rare disruptions it is shown that the structure of the optimal test reduces to thresholding the a posteriori probability of the hypothesis that no change has happened under a certain condition on the kullback leibler k l divergence between the post and the pre change densities it is established that the threshold test is asymptotically optimal in the vanishing false alarm probability regime it is shown via numerical studies that this low complexity threshold test results in a substantial improvement in performance over naive tests such as a single sensor test or a test that incorrectly assumes that the change propagates instantaneously

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

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

a priori grid quality estimation for high order finite differencing structured grids using the finite differencing method contain two sources of grid induced truncation errors the first is dependent on the solution field the second is related only to the metrics of the grid transformation the accuracy of the grid transformation metrics is affected by the inverse metrics which are spatial derivatives of the grid in the generalised coordinates the truncation errors contained in the inverse metrics are generated by the spatial schemes fourier analysis shows that the dispersion errors by spatial schemes have similarities to the transfer function of spatial filters this similarity is exploited to define a grid quality metric that can be used to identify areas in the mesh that are likely to generate significant grid induced errors an inviscid vortex convection benchmark case is used to quantify the correlation between the grid quality metric and the solution accuracy for three common geometric features found in grids abrupt changes in the grid metrics skewness and grid stretching a strong correlation is obtained provided that the grid transformation errors are the most significant sources of error

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

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