an approach to time series analysis and arma spectral estimation this paper presents an approach to time series analysis and arma spectral estimation from only the output data corrupted by noise it is shown that the generalized not well known modified yule walker myw equations hold when the residual is some correlated noise to solve such equations a new version of the generalized least squares gls method is proposed yielding ar parameter estimates with higher accuracy this gls method can also be used to enhance the estimation accuracy of ar parameters for short and noisy data in case of the myw equation holding theoretically furthermore a simple procedure for improving ma parameter estimates is studied our approach as cadzow s singular value decomposition svd method has provided significantly higher performance spectral estimates for a low order arma model than those obtained via usual techniques which are based upon direct solution of the myw equations and which use a high order model without considering an additive noise

schauder estimates of the uniformly elliptic equation with a inverse square potential abstract in this paper we derive schauder estimates to the solutions of the following uniformly elliptic equation with a inverse square potential and nonhomogeneous term 0 1 a i j x i j u x a x 2 u x f x in b which lead to the existence and sharp regularity results of the classical solutions more precisely we prove that u c n 2 provided f c n a i j c n and a 2 n d n

the effect of monte carlo approximation on coverage error of double bootstrap confidence intervals a double bootstrap confidence interval must usually be approximated by a monte carlo simulation consisting of two nested levels of bootstrap sampling we provide an analysis of the coverage accuracy of the interval which takes account of both the inherent bootstrap and monte carlo errors the analysis shows that by a suitable choice of the number of resamples drawn at the inner level of bootstrap sampling we can reduce the order of coverage error we consider also the effects of performing a finite monte carlo simulation on the mean length and variability of length of two sided intervals an adaptive procedure is presented for the choice of the number of inner level resamples the effectiveness of the procedure is illustrated through a small simulation study

estimation of decay rates of transient signals the problem of estimating time constants of multiple exponentials in additive white gaussian noise is investigated it is shown that existing linear production based techniques which perform very well for decay rate estimation of signals with different frequencies fail when applied to the case of decaying exponentials at a single frequency an alternate minimum weighted norm method is proposed it is shown that this new technique achieves acceptable estimates of time constants at a signal to noise ratio of 20 db a significant improvement over existing techniques

recursive updating the eigenvalue decomposition of a covariance matrix the author addresses the problem of computing the eigensystem of the modified hermitian matrix given the prior knowledge of the eigensystem of the original hermitian matrix specifically an additive rank k modification corresponding to adding and deleting blocks of data to and from the covariance matrix is considered an efficient and parallel algorithm which makes use of a generalized spectrum slicing theorem is derived for computing the eigenvalues the eigenvector can be computed explicitly in terms of the solution of a much reduced k k homogeneous hermitian system the overall computational complexity is shown to be improved by an order of magnitude from o n sup 3 to o n sup 2 k where n n is the size of the covariance matrix it is pointed out that these ideas can be applied to adaptive signal processing applications such as eigen based techniques for frequency or angle of arrival estimation and tracking specifically adaptive versions of the principal eigenvector method and the total least squares method are derived

variance of least squares estimators for a damped sinusoidal process the variance of least squares estimators for the parameter estimation of a damped sinusoidal process is analyzed based on first order perturbation analytical expressions for the variances of the frequency damping factor amplitude and phase estimators are derived explicit expressions are given for both damped and undamped single mode cases the effect of mode separation on the accuracy is investigated through the two mode case the dependence of the variances on number of data points model order signal to noise ratio and mode separation is investigated both analytically and numerically for practical applications extensive monte carlo simulation results are given to verify enhance and illustrate the analytical expressions

breakdown theory for bootstrap quantiles a general formula for computing the breakdown point in robustness for the tth bootstrap quantile of a statistic t n is obtained the answer depends on t and the breakdown point of t n since the bootstrap quantiles are vital ingredients of bootstrap confidence intervals the theory has implications pertaining to robustness of bootstrap confidence intervals for certain l and m estimators a robustification of bootstrap is suggested via the notion of winsorization

harmonic retrieval via state space and fourth order cumulants the harmonic retrieval problem considered concerns the estimation of the frequencies in a sum of complex sinusoids in the presence of gaussian measurement noise the method developed for solving this problem relies on the use of a diagonal form state space model representation of the harmonic retrieval problem fourth order cumulants are employed as the signal processing technique it is shown that using this approach the unknown frequencies can be estimated by finding the eigenvalues of a symmetric matrix made up from the cumulants of the data simulation results show the effectiveness of this method when the signals are corrupted by gaussian noise white or colored

on the zeros of the linear prediction error filter for deterministic signals the zeros of a linear prediction error filter polynomial for a class of deterministic signals that are a sum of samples of m exponentially damped undamped sinusoids is studied it is assumed that n samples are available for processing and that they are uncorrupted by noise it is shown that the exponent parameters of the m signals can be determined from m zeros called signal zeros of an lth degree prediction error filter polynomial l m if l lies in between m and n m n m 2 in a special case the rest of the l m zeros of the filter polynomial switch are called extraneous zeros are shown to be approximately uniformly distributed with in the unit circle regardless of the type of exponentials in the signal if the prediction filter coefficients are chosen to have minimum euclidean length the results obtained provide insight into the estimation problem with noisy data

improved quantile inference via fixed smoothing asymptotics and edgeworth expansion to estimate a sample quantile s variance the quantile spacing method involves smoothing parameter m when m n the corresponding studentized test statistic is asymptotically n 0 1 holding m fixed instead the asymptotic distribution contains the edgeworth expansion term capturing the variance of the quantile spacing consequently the fixed m distribution is more accurate than the standard normal under both asymptotic frameworks a testing optimal m is proposed to maximize power subject to size control in simulations the new method controls size better than similar methods while maintaining good power throughout are results for two sample quantile treatment effect inference code is available online

regularity of weak solutions of elliptic and parabolic equations with some critical or supercritical potentials abstract we prove holder continuity of weak solutions of the uniformly elliptic and parabolic equations 0 1 i a i j x j u x a x 2 u x 0 a 0 0 0 2 i a i j x t j u x t a x 2 u x t t u x t 0 a 0 0 with critical or supercritical 0 order term coefficients which are beyond de giorgi nash moser s theory we also prove in some special cases weak solutions are even differentiable previously p baras and j a goldstein 3 treated the case when a 0 a i j i and 0 for which they show that there does not exist any regular positive solution or singular positive solutions depending on the size of a when a 0 0 and a i j i p d milman and y a semenov 7 8 obtain bounds for the heat kernel

adaptive deconvolution and identification of nonminimum phase fir systems based on cumulants a novel adaptive deconvolution and system identification scheme is introduced for a linear non minimum phase finite impulse response fir system driven by non gaussian white noise the adaptive scheme is based on approximating the fir system by noncausal autoregressive ar models and using higher order cumulants of the system output as such it is a blind equalization deconvolution scheme the set of updated ar parameters is obtained by using a gradient type algorithm and by using higher order cumulants instead of time samples of the output signal it is demonstrated by means of extensive simulations that the adaptive scheme works well for both stationary and nonstationary cases as expected it outperforms the autocorrelation based gradient method for nonminimum phase system identification and deconvolution performance comparisons to existing methods are given using as figures of merit the probability of errors in the restored input sequence computational complexity and convergence rate

statistical analysis of effective singular values in matrix rank determination a major problem in using svd singular value decomposition as a tool in determining the effective rank of a perturbed matrix is that of distinguishing between significantly small and significantly large singular values to the end conference regions are derived for the perturbed singular values of matrices with noisy observation data the analysis is based on the theories of perturbations of singular values and statistical significance test threshold bounds for perturbation due to finite precision and i i d random models are evaluated in random models the threshold bounds depend on the dimension of the matrix the noisy variance and predefined statistical level of significance results applied to the problem of determining the effective order of a linear autoregressive system from the approximate rank of a sample autocorrelation matrix are considered various numerical examples illustrating the usefulness of these bounds and comparisons to other previously known approaches are given

standard errors and covariance matrices for smoothed rank estimators a pseudo bayesian interpretation of standard errors yields a natural induced smoothing of statistical estimating functions when applied to rank estimation the lack of smoothness which prevents standard error estimation is remedied efficiency and robustness are preserved while the smoothed estimation has excellent computational properties in particular convergence of the iterative equation for standard error is fast and standard error calculation becomes asymptotically a one step procedure this property also extends to covariance matrix calculation for rank estimates in multi parameter problems examples and some simple explanations are given

statistical uses of the spatial median the spatial median called the mediancentre in an earlier paper by j c gower is defined as the bivariate location measure to minimize the sum of absolute distances to observations its asymptotic efficiency relative to the sample mean vector with normal data is shown to exceed the usual univariate 2 ir 0 637 its estimating equations have an angular aspect and are used to develop angle tests which are analogues of sign tests in one dimension

geometric methods for determining system poles from transient response the problem treated is that of identifying the poles of a finite order system by observing its transient decay after cessation of input for a limited time using possibly multiple observation points and experimental repetition various approaches are studied having the common characteristic that a homogeneous matrix equation must be solved several techniques that have been given scant attention in the literature are consolidated by a geometric treatment together with new results including an analytical treatment of the consequences of assuming an excessively high system order derivation of a statistically unbiased estimate for an intermediate parameter in the solution new theorems on error effects a recipe for effective use of the singular value decomposition and a new method for suppression of extraneous poles

estimating frequencies of exponentials in noise using joint diagonalization the problem of estimating frequencies of superimposed exponentials in white noise collected by a single sensor is considered we discuss a solution based on a modification of an analytic algorithm that was proposed for separation of constant modulus signals received by an antenna array by van der veen and paulraj see ibid vol vol 44 p 1136 55 1996 we show that with an appropriate implementation of certain constraints the algorithm produces excellent super resolution results an advantage of the algorithm is that it does not require multidimensional search and therefore it is free from problems associated with search algorithms like convergence to local extremum or proper initialization another advantage is its applicability in nonuniform sampling conditions where methods based on linear prediction cannot be applied we carry out a detailed performance analysis under the assumption of long data record and derive conditions for estimating all the frequencies the performance of the proposed algorithm compares favorably with the performance of the tufts kumaresan method 1982

singular value decomposition and improved frequency estimation using linear prediction linear prediction based lp methods for fitting multiple sinusoid signal models to observed data such as the forward backward fblp method of nuttall 5 and ulrych and clayton 6 are very ill conditioned the locations of estimated spectral peaks can be greatly affected by a small amount of noise because of the appearance of outliers lp estimation of frequencies can be greatly improved at low snr by singular value decomposition svd of the lp data matrix the improved performance at low snr is also better than that obtained by using the eigenvector corresponding to the minimum eigenvalue of the correlation matrix as is done in pisarenko s method and its variants

the heat equation with a singular potential of concern is the singular problem du dt au c x 2 iz x tz x 0 u0 x and its generalizations here c 0 x g rn t 0 and and t 0 are nonnegative and not both identically zero there is a dimension dependent constant c n such that the problem has no solution for c ct n for c s ct a necessary and sufficient conditions are found for and u0 so that a nonnegative solution exists

cumulant based approach to harmonic retrieval and related problems a frequently encountered problem in signal processing is that of estimating the frequencies and amplitudes of harmonics observed in additive colored gaussian noise in practice the observed signals are contaminated with spatially and temporally colored noise of unknown power spectral density a cumulant based approach to these problems is proposed the cumulants of complex processes are defined and it is shown that specific 1 d slices of the fourth order cumulant of the noisy signal for the direction of arrival doa and retrieval of harmonics in noise rhn problems are identical to the autocorrelation of a related noiseless signal hence correlation based high resolution methods may be used with fourth order cumulants as well the effectiveness of the proposed methods is demonstrated through standard simulation examples

the reconstruction of a band limited function and its fourier transform from a finite number of samples at arbitrary locations by singular value decomposition a method for the stable interpolation of a bandlimited function known at sample instants with arbitrary locations in the presence of noise is given singular value decomposition is used to provide a series expansion that in contrast to the method of sampling functions permits simple identification of vectors in the minimum norm space poorly represented in the sample values three methods miller regularization least squares estimation and maximum a posteriori estimation are given for obtaining regularized reconstructions when noise is present the singular value decomposition svd method is used to interrelate these methods examples illustrating the technique are given

heat kernel bounds and desingularizing weights abstract we study the parabolic operator t x v t x in r 1 r d d 1 with a potential v v v v 0 assumed to be from a parabolic kato class and obtain two sided gaussian bounds on the associated heat kernel the constraints on the kato norms of v and v are completely asymmetric as they should be further improvements to our heat kernel bounds are obtained in the case of time independent potentials if v has singularities of the type c x 2 with a suitably small constant c we obtain new lower and sharp upper weighted heat kernel bounds the rate of growth of the weights depends explicitly on the constant c the standard bounds and methods estimates in lp spaces without desingularizing weights fail for singular potentials

state space system identification approach to radar data processing space time adaptive processing stap algorithms typically consist of a data transformation step to reduce the number of degrees of freedom and a sampling step wherein radar returns from adjacent range bins are used to estimate interference statistics the reduction in the number of degrees of freedom inadequate sample support presence of target in sampled data and range dependence of interference are some of the main reasons for stap performance loss in this paper we present an approach to target detection and localization that mitigates these performance losses using the well known stochastic realization algorithm from system identification theory we first identify a state space model from the radar return data in range pulse domain for a given range bin and then perform detection and localization using the identified state space matrices as interference statistics are not directly computed and since there is no sampling from adjacent range bins this approach is more robust to sample support issues target in training and range dependence of clutter a numerical comparison of the approach with beam space post doppler stap using simulated data is given

time and lag recursive computation of cumulants from a state space model time and lag recursive algorithms for the computation of the cumulants of the state vector and the output process of a multiple input multiple output time varying state space model are derived using a kronecker product representation for the cumulants of vector processes the noise processes are not assumed to be stationary symmetry relations for the cumulants of vector processes are discussed computational aspects are examined in detail it is conjectured that the recursive equations derived will be useful in developing state estimators and even optimal controllers based on higher order statistics

correction to rotation and gray scale transform invariant texture classification using spiral resampling subband decomposition and hidden markov model this paper proposes a new texture classification algorithm that is invariant to rotation and gray scale transformation first we convert two dimensional 2 d texture images to one dimensional 1 d signals by spiral resampling then we use a quadrature mirror filter qmf bank to decompose sampled signals into subbands in each band we take high order autocorrelation functions as features features in different bands which form a vector sequence are then modeled as a hidden markov model bmm during classification the unknown texture is matched against all the models and the best match is taken as the classification result simulations showed that the highest correct classification rate for 16 kinds of texture was 95 14

the smoothed median and the bootstrap summary even in one dimension the sample median exhibits very poor performance when used in conjunction with the bootstrap for example both the percentile t bootstrap and the calibrated percentile method fail to give second order accuracy when applied to the median the situation is generally similar for other rank based methods particularly in more than one dimension some of these problems can be overcome by smoothing but that usually requires explicit choice of the smoothing parameter in the present paper we suggest a new implicitly smoothed version of the k variate sample median based on a particularly smooth objective function our procedure preserves many features of the conventional median such as robustness and high efficiency in fact higher than for the conventional median in the case of normal data it is however substantially more amenable to application of the bootstrap focusing on the univariate case we demonstrate these properties both theoretically and numerically