learning equivalence classes of bayesian network structures two bayesian network structures are said to be equivalent if the set of distributions that can be represented with one of those structures is identical to the set of distributions that can be represented with the other many scoring criteria that are used to learn bayesian network structures from data are score equivalent that is these criteria do not distinguish among networks that are equivalent in this paper we consider using a score equivalent criterion in conjunction with a heuristic search algorithm to perform model selection or model averaging we argue that it is often appropriate to search among equivalence classes of network structures as opposed to the more common approach of searching among individual bayesian network structures we describe a convenient graphical representation for an equivalence class of structures and introduce a set of operators that can be applied to that representation by a search algorithm to move among equivalence classes we show that our equivalence class operators can be scored locally and thus share the computational efficiency of traditional operators defined for individual structures we show experimentally that a greedy model selection algorithm using our representation yields slightly higher scoring structures than the traditional approach without any additional time overhead and we argue that more sophisticated search algorithms are likely to benefit much more

analysis of subspace fitting and ml techniques for parameter estimation from sensor array data it is shown that the multidimensional signal subspace method termed weighted subspace fitting wsf is asymptotically efficient this results in a novel compact matrix expression for the cramer rao bound crb on the estimation error variance the asymptotic analysis of the maximum likelihood ml and wsf methods is extended to deterministic emitter signals the asymptotic properties of the estimates for this case are shown to be identical to the gaussian emitter signal case i e independent of the actual signal waveforms conclusions concerning the modeling aspect of the sensor array problem are drawn

modified subspace algorithms for doa estimation with large arrays this paper proposes the use of a new generalized asymptotic paradigm in order to analyze the performance of subspace based direction of arrival doa estimation in array signal processing applications instead of assuming that the number of samples is high whereas the number of sensors antennas remains fixed the asymptotic situation analyzed herein assumes that both quantities tend to infinity at the same rate this asymptotic situation provides a more accurate description of a potential situation where these two quantities are finite and hence comparable in magnitude it is first shown that both music and ssmusic are inconsistent when the number of antennas sensors increases without bound at the same rate as the sample size this is done by analyzing and deriving closed form expressions for the two corresponding asymptotic cost functions by examining these asymptotic cost functions one can establish the minimum number of samples per antenna needed to resolve closely spaced sources in this asymptotic regime next two alternative estimators are constructed that are strongly consistent in the new asymptotic situation i e they provide consistent doa estimates not only when the number of snapshots goes to infinity but also when the number of sensors antennas increases without bound at the same rate these estimators are inspired by the theory of g estimation and are therefore referred to as g music and g ssmusic respectively simulations show that the proposed algorithms outperform their traditional counterparts in finite sample size situations although they still present certain limitations

detection of signals by information theoretic criteria a new approach is presented to the problem of detecting the number of signals in a multichannel time series based on the application of the information theoretic criteria for model selection introduced by akaike aic and by schwartz and rissanen mdl unlike the conventional hypothesis testing based approach the new approach does not requite any subjective threshold settings the number of signals is obtained merely by minimizing the aic or the mdl criteria simulation results that illustrate the performance of the new method for the detection of the number of signals received by a sensor array are presented

probability laws related to the jacobi theta and riemann zeta functions and brownian excursions this paper reviews known results which connect riemann s integral representations of his zeta function involving jacobi s theta function and its derivatives to some particular probability laws governing sums of independent exponential variables these laws are related to one dimensional brownian motion and to higher dimensional bessel processes we present some characterizations of these probability laws and some approximations of riemann s zeta function which are related to these laws

estimation of directed acyclic graphs through two stage adaptive lasso for gene network inference abstractgraphical models are a popular approach to find dependence and conditional independence relationships between gene expressions directed acyclic graphs dags are a special class of directed graphical models where all the edges are directed edges and contain no directed cycles the dags are well known models for discovering causal relationships between genes in gene regulatory networks however estimating dags without assuming known ordering is challenging due to high dimensionality the acyclic constraints and the presence of equivalence class from observational data to overcome these challenges we propose a two stage adaptive lasso approach called ns dist which performs neighborhood selection ns in stage 1 and then estimates dags by the discrete improving search with tabu dist algorithm within the selected neighborhood simulation studies are presented to demonstrate the effectiveness of the method and its computational efficiency two real data examples are used to demonstrate the p

spatial spectrum estimation in a location sector multiple narrowband source localization using arbitrarily configured arrays and spatial spectrum estimation is considered a new eigenspace based approach which uses projections onto a particular vector or vector set in the estimated noise only subspace is described several closest vector estimators are developed by using different measures of closeness first closest is a novel full dimensional element space approach to spatial spectrum estimation which has important performance advantages relative to pertinent established spatial spectrum estimators it incorporates a priori knowledge of the array manifold over a location sector of interest to provide snr spectral resolution thresholds which are lower than those of min norm for some arrays substantially lower second relationships between the closest approach and several established approaches to spatial spectrum estimation are established for a linear equispaced array min norm is shown to be a special case of the closest approach one which is based on projection onto a noise only subspace vector which is close to the array manifold over the entire field of view

statistical analysis of music and subspace rotation estimates of sinusoidal frequencies consideration is given to the analysis of the large sample second order properties of multiple signal classification music and subspace rotation sur methods such as esprit for sinusoidal frequency estimation explicit expressions for the covariance elements of the estimation errors associated with either method are derived these expressions of covariances are then used to analyze and compare the statistical performances of the music and sur estimation sure methods both music and sure are based on the eigendecomposition of a sample data covariance matrix the expressions for the estimation error variances derived are used to study the dependence of music and sure performances on the dimension of the data covariance matrix used

doa estimation of temporally and spatially correlated narrowband noncircular sources in spatially correlated white noise the main contribution of our work consists in developing for the first time a method of estimating the direction of arrival doa parameters assuming noncircular and temporally and spatially correlated signals this new approach based on a significant enhancement of the two sided instrumental variable signal subspace fitting iv ssf method outperforms its classical version in terms of lower bias and error variance moreover it will be shown that our new method is statistically more efficient than the mode method especially in the case of partly and fully coherent signals where only the extended and the classical two sided iv ssf methods are applicable we also derive an explicit expression for the stochastic cramer rao bound crb of the doa estimates from temporally and spatially correlated signals generated from noncircular sources the new crb is compared to those of circular temporally correlated and noncircular independent and identically distributed signals to show that the crb obtained assuming both noncircular sources and temporally correlated signals is lower than the crbs derived considering only one of these two assumptions this illustrates the potential gain that both noncircularity and temporal correlation provide when considered together it will also be proven that the difference between the three crbs increases with the number of snapshots however as the signal to noise ratio snr increases the crbs merge together and decrease linearly moreover at low snr values it will be shown that temporal correlation is more informative about the unknown doa parameters than noncircularity finally the crb derived assuming noncircular and temporally correlated signals depends on the noncircularity rate the circularity phase separation and the doa separation

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

estimating high dimensional directed acyclic graphs with the pc algorithm we consider the pc algorithm spirtes et al 2000 for estimating the skeleton and equivalence class of a very high dimensional directed acyclic graph dag with corresponding gaussian distribution the pc algorithm is computationally feasible and often very fast for sparse problems with many nodes variables and it has the attractive property to automatically achieve high computational efficiency as a function of sparseness of the true underlying dag we prove uniform consistency of the algorithm for very high dimensional sparse dags where the number of nodes is allowed to quickly grow with sample size n as fast as o na for any 0 a the sparseness assumption is rather minimal requiring only that the neighborhoods in the dag are of lower order than sample size n we also demonstrate the pc algorithm for simulated data

penalized likelihood methods for estimation of sparse high dimensional directed acyclic graphs directed acyclic graphs are commonly used to represent causal relationships among random variables in graphical models applications of these models arise in the study of physical and biological systems where directed edges between nodes represent the influence of components of the system on each other estimation of directed graphs from observational data is computationally np hard in addition directed graphs with the same structure may be indistinguishable based on observations alone when the nodes exhibit a natural ordering the problem of estimating directed graphs reduces to the problem of estimating the structure of the network in this paper we propose an efficient penalized likelihood method for estimation of the adjacency matrix of directed acyclic graphs when variables inherit a natural ordering we study variable selection consistency of lasso and adaptive lasso penalties in high dimensional sparse settings and propose an error based choice for selecting the tuning parameter we show that although the lasso is only variable selection consistent under stringent conditions the adaptive lasso can consistently estimate the true graph under the usual regularity assumptions

right inverses of nonsymmetric levy processes we analyze the existence and properties of right inverses k for nonsymmetric levy processes x extending recent work of evans 7 in the symmetric setting first both x and x have right inverses if and only if x is recurrent and has a nontrivial gaussian component our main result is then a description of the excursion measure n z of the strong markov process z x l reflected process where lt inf x 0 kx t specifically n z is essentially the restriction of n x to the excursions starting negative second when only asking for right inverses of x a certain strength of asymmetry is needed millar s 9 notion of creeping turns out necessary but not sufficient for the existence of right inverses we analyze this both in the bounded and unbounded variation case with a particular emphasis on results in terms of the levy khintchine characteristics 1 introduction consider a real valued levy process that is a continuous time process with stationary independent increments and cadlag paths right inverses of levy processes have first been studied in the symmetric setting by evans 7 he defines an increasing process k to be a right inverse of a levy process x if it satisfies x kx x for all x 0 the minimal such k turns out to be a subordinator i e an increasing levy process specifically evans characterized the symmetric levy processes that have right inverses he introduced lt inf x 0 kx t and the reflected process z x l and showed that z is strong markov with local time l at zero he gave some fluctuation type identities for z and l and proved formulae for the entrance laws to the excursion measure n z of z evans also showed that k is distributed like a linear time change of the inverse local time of x at zero in this work we basically answer three questions left open by evans first what happens when symmetry fails the only result of evans that is clearly based on symmetry is the coincidence of the laws of k and the inverse local time of x in zero also the class of processes possessing inverses is not restricted to those having a positive gaussian component in the symmetric setting this is evans s characterization result but those form again an important class we provide a further characterization in the bounded variation case and we also point

subordination in the wide sense for levy processes given t t 0 a transient convolution semigroup over rd we characterize the transient semi groups which are subordinate in the wide sense to the latter via some quotient relation between the two levy generators this is also equivalent to an inclusion between excessive cones thanks to a theorem of hirsch this allows us in big dimensions to give a probabilistic description of some measures satisfying the classical maximum principle

the multifractal nature of levy processes we show that the sample paths of most levy processes are multifractal functions and we determine their spectrum of singularities

cyclically stationary brownian local time processes local time processes parameterized by a circle defined by the occupation density up to time t of brownian motion with constant drift on the circle are studied for various random times t while such processes are typically non markovian their laplace functionals are expressed by series formulae related to similar formulae for the markovian local time processes subject to the ray knight theorems for bm on the line and for squares of bessel processes and their bridges for t the time that bm on the circle first returns to its starting point after a complete loop around the circle the local time process is cyclically stationary with same two dimensional distributions but not the same three dimensional distributions as the sum of squares of two i i d cyclically stationary gaussian processes this local time process is the infinitely divisible sum of a poisson point process of local time processes derived from brownian excursions the corresponding intensity measure on path space and similar levy measures derived from squares of bessel processes are described in terms of a 4 dimensional bessel bridge by williams decomposition of ito s law of brownian excursions

exact maximum likelihood parameter estimation of superimposed exponential signals in noise a unified framework for the exact maximum likelihood estimation of the parameters of superimposed exponential signals in noise encompassing both the time series and the array problems is presented an exact expression for the ml criterion is derived in terms of the linear prediction polynomial of the signal and an iterative algorithm for the maximization of this criterion is presented the algorithm is equally applicable in the case of signal coherence in the array problem simulation shows the estimator to be capable of providing more accurate frequency estimates than currently existing techniques the algorithm is similar to those independently derived by kumaresan et al in addition to its practical value the present formulation is used to interpret previous methods such as prony s pisarenko s and modifications thereof

design of two music like estimators based on bias minimization two classes of music like estimators are considered one class called weighted norm music possesses an optimizing functional or null spectrum which is the product of the music null spectrum and an angle dependent weight the second class which is denoted the d sub r estimator has an optimizing functional that is dependent on a parameter r and is a generalized distance between two particular vectors in the signal subspace it is shown that the asymptotic mean square errors of these estimators are the same as music by determining an appropriate weight based on a derived large sample expression for the estimator bias a weighted norm music estimator is found that gives zero bias of order n sup 1 where n is the sample size using an approximate relation between the two types of estimators under consideration a data dependent parameter r spl theta is derived for the d sub r estimator which results in small bias over a wide range of signal to noise ratios snrs for two closely spaced sources

estimation of sparse directed acyclic graphs for multivariate counts data the next generation sequencing data called high throughput sequencing data are recorded as count data which are generally far from normal distribution under the assumption that the count data follow the poisson log normal distribution this article provides an l1 penalized likelihood framework and an efficient search algorithm to estimate the structure of sparse directed acyclic graphs dags for multivariate counts data in searching for the solution we use iterative optimization procedures to estimate the adjacency matrix and the variance matrix of the latent variables the simulation result shows that our proposed method outperforms the approach which assumes multivariate normal distributions and the log transformation approach it also shows that the proposed method outperforms the rank based pc method under sparse network or hub network structures as a real data example we demonstrate the efficiency of the proposed method in estimating the gene regulatory networks of the ovarian cancer study

asymptotic behavior of maximum likelihood estimates of superimposed exponential signals strong consistency and asymptotic normality are derived for the maximum likelihood estimates mles of the unknown parameters omega sub 1 omega sub p alpha sub 1 alpha sub p and sigma sup 2 in the superimposed exponential model for signals y sub t sigma alpha exp it omega sub k e sub t where the summation is from k 1 to p t 0 1 n 1 and sigma sup 2 is the variance of the complex normal distribution of e sub t as a by product it is found that the mles of the parameters attain the cramer rao lower bound for the asymptotic covariance matrix

bias analysis of the music location estimator the authors present a rigorous bias analysis of the music location estimator and they derive an accurate and concise bias expression the analysis is based on the second order taylor series expansion of the derivative of the null spectrum properties of the null spectrum and statistics of the estimated signal eigenvectors it is proven that in the derivation the remainder term in the second order taylor series can be dropped but the second order terms cannot be simulations verify that the bias expression is valid over a wide range of signal to noise ratios snrs extending down into the resolution threshold region of music although asymptotic this expression can be accurately applied to a limited number of snapshot cases the utility of the expression is shown by using it in a study of music location estimator characteristics estimate bias and standard deviation are compared for variations in snr numbers of sensors and snapshots and source correlation music resolvability and estimator performance bounds are addressed accounting for bias

pseudo randomly generated estimator banks a new tool for improving the threshold performance of direction finding a new powerful tool for improving the threshold performance of direction finding is considered the main idea of our approach is to reduce the number of outliers in the doa estimates using a previously proposed joint estimation strategy jes for this purpose multiple different doa estimators are calculated in a parallel manner for the same batch of data i e for a single data record employing these estimators simultaneously the jes improves the threshold performance because it removes outliers and exploits only successful estimators that are sorted out using a hypothesis testing procedure we consider an efficient modification of the jes with application to the pseudo randomly generated eigenstructure estimator banks based on secondand higher order statistics weighted music estimators based on the covariance and contracted quadricovariance matrices are chosen as appropriate underlying techniques for the second and fourth order estimator banks respectively computer simulations with uncorrelated sources verify the dramatic improvements of threshold performance as compared with the conventional second and fourth order music algorithms simulations also show that in the second order case the threshold performance of our technique is close to that of the wsf method and stochastic deterministic ml methods which are known today as the most powerful in the sense of estimation performance and at the same time as the most computationally expensive doa estimation techniques the computational cost of our algorithm is much lower than that of the wsf and ml techniques because no multidimensional optimization is required

weighted subspace methods and spatial smoothing analysis and comparison the effect of using a spatially smoothed forward backward covariance matrix on the performance of weighted eigen based state space methods esprit and weighted music for direction of arrival doa estimation is analyzed expressions for the mean squared error in the estimates of the signal zeros and the doa estimates along with some general properties of the estimates and optimal weighting matrices are derived a key result is that optimally weighted music and weighted state space methods esprit have identical asymptotic performance moreover by properly choosing the number of subarrays the performance of unweighted state space methods can be significantly improved it is also shown that the mean squared error in the doa estimates is independent of the exact distribution of the source amplitudes this results in a unified framework for dealing with doa estimation using a uniformly spaced linear sensor array and the time series frequency estimation problems

uniform measure results for the image of subsets under brownian motion the paper obtains bounds on the hausdorff and packing measures of the imagex e of a borel sete by a transient strictly stable processx t which a s hold for alle and for every measure function h beta gamma s s beta left log s right gamma star in some cases examples are constructed to show that the bounds are sharp

large deviations for squares of bessel and ornstein uhlenbeck processes let x t t 0 be the besq process starting at x we are interested in large deviations as delta rightarrow infty for the family 1 x t t t or more generally for the family of squared radial ou process the main properties of this family allow us to develop three different approaches an exponential martingale method a cramer type theorem thanks to a remarkable additivity property and a wentzell freidlin method with the help of mckean results on the controlled equation we also derive large deviations for bessel bridges

analysis of the asymptotic relative efficiency of the music algorithm an analytical performance evaluation of the errors of the direction of arrival estimates obtained by the music algorithm for uncorrelated sources is provided explicit asymptotic formulas are derived for the means and the covariance of the estimates the covariances are then compared to the cramer rao lower bound it is shown that for a single course the music algorithm is asymptotically efficient for multiple sources the algorithm is not efficient in general however it approaches asymptotic efficiency when the snrs signal to noise ratios of all sources tend to infinity it is illustrated by several test cases that the relative efficiency of the music algorithm is nearly one under a wide range of parameter variations the analytic performance evaluation thus confirms empirical evidence to the excellent performance of the music algorithm for narrowband signals

penpc a two step approach to estimate the skeletons of high dimensional directed acyclic graphs estimation of the skeleton of a directed acyclic graph dag is of great importance for understanding the underlying dag and causal effects can be assessed from the skeleton when the dag is not identifiable we propose a novel method named penpc to estimate the skeleton of a high dimensional dag by a two step approach we first estimate the nonzero entries of a concentration matrix using penalized regression and then fix the difference between the concentration matrix and the skeleton by evaluating a set of conditional independence hypotheses for high dimensional problems where the number of vertices p is in polynomial or exponential scale of sample size n we study the asymptotic property of penpc on two types of graphs traditional random graphs where all the vertices have the same expected number of neighbors and scale free graphs where a few vertices may have a large number of neighbors as illustrated by extensive simulations and applications on gene expression data of cancer patients penpc has higher sensitivity and specificity than the state of the art method the pc stable algorithm

an algorithm for pole zero modeling and spectral analysis an explicit connection between fitting exponential models and pole zero models to observed data is made the fitting problem is formulated as a constrained nonlinear minimization problem this problem is then solved using a simplified iterative algorithm the algorithm is applied to simulated data and the performance of the algorithm is compared to previous results

measuring the range of an additive levy process the primary goal of this paper is to study the range of the random field x t nj 1xj tj where x1 xn vspace 1pt are independent levy processes in rd to cite a typical result of this paper let us suppose that i denotes the levy exponent of xi for each i 1 n then under certain mild conditions we show that a necessary and sufficient condition for x rn to have positive d dimensional lebesgue measure is the integrability of the function rd nj 1r 1 j 1 this extends a celebrated result of kesten and of bretagnolle in the one parameter setting furthermore we show that the existence of square integrable local times is yet another equivalent condition for the mentioned integrability criterion this extends a theorem of hawkes to the present random fields setting and completes the analysis of local times for additive levy processes initiated in a companion by paper khoshnevisan xiao and zhong

the statistical performance of the music and the minimum norm algorithms in resolving plane waves in noise this paper presents an asymptotic statistical analysis of the null spectra of two eigen assisted methods music 1 and minimum norm 2 for resolving independent closely spaced plane waves in noise particular attention is paid to the average deviation of the null spectra from zero at the true angles of arrival for the plane waves these deviations are expressed as functions of signal to noise ratios number of array elements angular separation of emitters and the number of snapshots in the case of music an approximate expression is derived for the resolution threshold of two plane waves with equal power in noise this result is validated by monte carlo simulations

consistency and mse performance of music based doa of a single source in white noise with randomly missing data music is a popular algorithm for estimating the direction of arrival doa in array signal processing applications in this paper we analyze the performance of the music algorithm for a single source system in the presence of noisy and missing data when only a random subset of the entries in the data matrix are observed we prove consistency of the doa estimate when signal from a single source is impinging on low coherence arrays and derive an analytic expression for the mean squared error mse performance of music for the case of uniform linear arrays in the large array and relatively large sample setting our analysis is mathematically justified in both the sample rich and deficient regimes the expression for the mse is a simple function of array geometry signal to noise ratio snr the fraction of entries observed and the ratio of the number of sensors to number of snapshots we derive a phase transition threshold which separates a regime where music is consistent from a regime where music is inconsistent this threshold depends upon the snr the probability of observing entries in the data matrix and number of sensors and snapshots in a simple manner which we make explicit

optimal structure identification with greedy search in this paper we prove the so called meek conjecture in particular we show that if a dag h is an independence map of another dag g then there exists a finite sequence of edge additions and covered edge reversals in g such that 1 after each edge modification h remains an independence map of g and 2 after all modifications g h as shown by meek 1997 this result has an important consequence for bayesian approaches to learning bayesian networks from data in the limit of large sample size there exists a two phase greedy search algorithm that when applied to a particular sparsely connected search space provably identifies a perfect map of the generative distribution if that perfect map is a dag we provide a new implementation of the search space using equivalence classes as states for which all operators used in the greedy search can be scored efficiently using local functions of the nodes in the domain finally using both synthetic and real world datasets we demonstrate that the two phase greedy approach leads to good solutions when learning with finite sample sizes

right inverses of levy processes we call a right continuous increasing process k x a partial right inverse pri of a given levy process x if x kx x for at least all x in some random interval 0 of positive length in this paper we give a necessary and sufficient condition for the existence of a pri in terms of the levy triplet

regression model selection a residual likelihood approach summary we obtain the residual information criterion ric a selection criterion based on the residual log likelihood for regression models including classical regression models box cox transformation models weighted regression models and regression models with autoregressive moving average errors we show that ric is a consistent criterion and that simulation studies for each of the four models indicate that ric provides better model order choices than the akaike information criterion corrected akaike information criterion final prediction error cp and r 2 except when the sample size is small and the signal to noise ratio is weak in this case none of the criteria performs well monte carlo results also show that ric is superior to the consistent bayesian information criterion bic when the signal to noise ratio is not weak and it is comparable with bic when the signal to noise ratio is weak and the sample size is large

some brownian functionals and their laws we develop some topics about brownian motion with a particular emphasis on the study of principal values of brownian local times we show some links between principal values and doob s h transforms of brownian motion for nonpositive harmonic functions h we also give a survey and complement some martingale approaches to ray knight theorems for local times

learning sparse causal gaussian networks with experimental intervention regularization and coordinate descent causal networks are graphically represented by directed acyclic graphs dags learning causal networks from data is a challenging problem due to the size of the space of dags the acyclicity constraint placed on the graphical structures and the presence of equivalence classes in this article we develop an l 1 penalized likelihood approach to estimate the structure of causal gaussian networks a blockwise coordinate descent algorithm which takes advantage of the acyclicity constraint is proposed for seeking a local maximizer of the penalized likelihood we establish that model selection consistency for causal gaussian networks can be achieved with the adaptive lasso penalty and sufficient experimental interventions simulation and real data examples are used to demonstrate the effectiveness of our method in particular our method shows satisfactory performance for dags with 200 nodes which have about 20 000 free parameters supplementary materials for this article are available online

time reversal acoustics and maximum entropy imaging a common problem in acoustics imaging is target location using either passive or active data inversion time reversal methods in acoustics have the important characteristic that they provide a means of determining the eigenfunctions and eigenvalues of the scattering operator for either of these two cases each eigenfunction may often be approximately associated with an individual scatterer the resulting decoupling of the scattered field from a collection of targets is a very useful aid to localizing the targets and suggests a number of imaging and localization algorithms two of these are linear subspace methods and maximum entropy imaging

right inverses of levy processes and stationary stopped local times if x is a symmetric levy process on the line then there exists a non decreasing cadlag process h such that x h x x for all x 0 if and only if x is recurrent and has a non trivial gaussian component the minimal such h is a subordinator k the law of k is identified and shown to be the same as that of a linear time change of the inverse local time at 0 of x when x is brownian motion k is just the usual ladder times process and this result extends the classical result of levy that the maximum process has the same law as the local time at 0 write g t for last point in the range of k prior to t in a parallel with classical fluctuation theory the process z x t x gt t 0 is markov with local time at 0 given by x gt t 0 the transition kernel and excursion measure of z are identified a similar programme is outlined for levy processes on the circle this leads to the construction of a stopping time such that the stopped local times constitute a stationary process indexed by the circle

music maximum likelihood and cramer rao bound further results and comparisons the problem of determining the direction of arrival of narrowband plane waves using sensor arrays and the related problem of estimating the parameters of superimposed signals from noisy measurements are studied a number of results have been recently presented by the authors on the statistical performance of the multiple signal characterization music and the maximum likelihood ml estimators for the above problems this work extends those results in several directions first it establishes that in the class of weighted music estimators the unweighted music achieves the best performance i e the minimum variance of estimation errors in large samples next it derives the covariance matrix of the ml estimator and presents detailed analytic studies of the statistical efficiency of music and ml estimators these studies include performance comparisons of music and mle with each other as well as with the ultimate performance corresponding to the cramer rao bound finally some numerical examples are given which provide a more quantitative study of performance for the problem of finding two directions with uniform linear sensor arrays

packing and covering indices for a general levy process there has been substantial interest in the indices 0 2 defined by blumenthal and getoor determined by a general levy process in r d pruitt defined an index which determines the covering dimension and taylor showed that an index first considered by hendricks determines the packing dimension for the trajectory in the present paper we prove that b 2 min d and give examples to show that the whole range is attainable however we cannot completely determine the set of values of which can be attained as indices of some levy process

model selection and estimation in the gaussian graphical model summary we propose penalized likelihood methods for estimating the concentration matrix in the gaussian graphical model the methods lead to a sparse and shrinkage estimator of the concentration matrix that is positive definite and thus conduct model selection and estimation simultaneously the implementation of the methods is nontrivial because of the positive definite constraint on the concentration matrix but we show that the computation can be done effectively by taking advantage of the efficient maxdet algorithm developed in convex optimization we propose a bic type criterion for the selection of the tuning parameter in the penalized likelihood methods the connection between our methods and existing methods is illustrated simulations and real examples demonstrate the competitive performance of the new methods

optimal utilization of signal free samples for array processing in unknown colored noise fields many algorithms for direction of arrival doa estimation require the noise covariance matrix to be known or to possess a known structure in many cases the noise covariance is in fact estimated from separate measurements this paper addresses the combined effects of finite sample sizes both in the estimated noise covariance matrix and in the data with signals present it is assumed that a batch of signal free samples is available in addition to the signal containing samples no assumption is made on the structure of the noise covariance in this paper the asymptotic covariance of the weighted subspace fitting wsf algorithm is derived for the case in which the data are whitened using an estimated noise covariance the expression obtained suggests an optimal weighting that improves performance compared to the standard choice in addition a new method based on covariance matching is proposed both methods are asymptotically statistically efficient the cramer rao lower bound crb on the covariance of the estimate for the data model is also derived monte carlo simulations show promising small sample performance for the two new methods and confirm the asymptotic results

smoothly clipped absolute deviation on high dimensions the smoothly clipped absolute deviation scad estimator proposed by fan and li has many desirable properties including continuity sparsity and unbiasedness the scad estimator also has the asymptotically oracle property when the dimension of covariates is fixed or diverges more slowly than the sample size in this article we study the scad estimator in high dimensional settings where the dimension of covariates can be much larger than the sample size first we develop an efficient optimization algorithm that is fast and always converges to a local minimum second we prove that the scad estimator still has the oracle property on high dimensional problems we perform numerical studies to compare the scad estimator with the lasso and sis scad estimators in terms of prediction accuracy and variable selectivity when the true model is sparse through the simulation we show that the variance estimator of fan and li still works well for some limited high dimensional cases where the true nonzero coeffic

music like estimation of direction of arrival for noncircular sources this paper examines the asymptotic performance of music like algorithms for estimating directions of arrival doa of narrowband complex noncircular sources using closed form expressions of the covariance of the asymptotic distribution of different projection matrices it provides a unifying framework for investigating the asymptotic performance of arbitrary subspace based algorithms valid for gaussian or non gaussian and complex circular or noncircular sources we also derive different robustness properties from the asymptotic covariance of the estimated doa given by such algorithms these results are successively applied to four algorithms to two attractive music like algorithms previously introduced in the literature to an extension of these algorithms and to an optimally weighted music algorithm proposed in this paper numerical examples illustrate the performance of the studied algorithms compared to the asymptotically minimum variance amv algorithms introduced as benchmarks

on the performance of signal subspace processing part i narrow band systems this paper presents an analytical evaluation of detection determination of the number of sources and estimation performances of narrow band signal subspace processing for multiple source direction finding the probabilities of underestimating and overestimating the number of sources are derived under asymptotic conditions and around the threshold regions in terms of the choice of a penalty function and signal noise and array parameters for the cases of at most two closely spaced sources in the spatially white noise a scalar measure is introduced for the evaluation of the quality of the estimated signal subspace based on the statistics of this measure performance thresholds are demonstrated for the signal to noise ratio angle separation and correlation between two equipowered sources

eigenvalues and eigenvectors of covariance matrices for signals closely spaced in frequency the eigenstructures of common covariance matrices are identified for the general case of m closely spaced signals it is shown that the largest signal space eigenvalue is relatively insensitive to signal separation by contrast the ith largest eigenvalue is proportional to delta omega sup 2 i 1 or delta omega sup 4 i 1 where delta omega is a measure of signal separation therefore matrix conditioning degrades rapidly as signal separation is reduced it is also shown that the limiting eigenvectors have remarkably simple structures the results are very general and apply to planar far field direction finding problems involving almost arbitrary scenarios and also to time series analysis of sinusoids exponentials and other signals

sensor array processing based on subspace fitting algorithms for estimating unknown signal parameters from the measured output of a sensor array are considered in connection with the subspace fitting problem the methods considered are the deterministic maximum likelihood method ml esprit and a recently proposed multidimensional signal subspace method these methods are formulated in a subspace fitting based framework which provides insight into their algebraic and asymptotic relations it is shown that by introducing a specific weighting matrix the multidimensional signal subspace method can achieve the same asymptotic properties as the ml method the asymptotic distribution of the estimation error is derived for a general subspace weighting and the weighting that provides minimum variance estimates is identified the resulting optimal technique is termed the weighted subspace fitting wsf method numerical examples indicate that the asymptotic variance of the wsf estimates coincides with the cramer rao bound the performance improvement compared to the other techniques is found to be most prominent for highly correlated signals

using markov blankets for causal structure learning we show how a generic feature selection algorithm returning strongly relevant variables can be turned into a causal structure learning algorithm we prove this under the faithfulness assumption for the data distribution in a causal graph the strongly relevant variables for a node x are its parents children and children s parents or spouses also known as the markov blanket of x identifying the spouses leads to the detection of the v structure patterns and thus to causal orientations repeating the task for all variables yields a valid partially oriented causal graph we first show an efficient way to identify the spouse links we then perform several experiments in the continuous domain using the recursive feature elimination feature selection algorithm with support vector regression and empirically verify the intuition of this direct but computationally expensive approach within the same framework we then devise a fast and consistent algorithm total conditioning tc and a variant tcbw with an explicit backward feature selection heuristics for gaussian data after running a series of comparative experiments on five artificial networks we argue that markov blanket algorithms such as tc tcbw or grow shrink scale better than the reference pc algorithm and provides higher structural accuracy

estimating evoked dipole responses in unknown spatially correlated noise with eeg meg arrays we present maximum likelihood ml methods for estimating evoked dipole responses using electroencephalography eeg and magnetoencephalography meg arrays which allow for spatially correlated noise between sensors with unknown covariance the electric source is modeled as a collection of current dipoles at fixed locations and the head as a spherical conductor we permit the dipoles moments to vary with time by modeling them as linear combinations of parametric or nonparametric basis functions we estimate the dipoles locations and moments and derive the cramer rao bound for the unknown parameters we also propose an ml based method for scanning the brain response data which can be used to initialize the multidimensional search required to obtain the true dipole location estimates numerical simulations demonstrate the performance of the proposed methods

music maximum likelihood and cramer rao bound the performance of the music and ml methods is studied and their statistical efficiency is analyzed the cramer rao bound crb for the estimation problems is derived and some useful properties of the crb covariance matrix are established the relationship between the music and ml estimators is investigated as well a numerical study is reported of the statistical efficiency of the music estimator for the problem of finding the directions of two plane waves using a uniform linear array an exact description of the results is included

random brownian scaling identities and splicing of bessel processes an identity in distribution due to knight for brownian motion is extended in two different ways first by replacing the supremum of a reflecting brownian motion by the range of an unreflected brownian motion and second by replacing the reflecting brownian motion by a recurrent bessel process both extensions are explained in terms of random brownian scaling transformations and brownian excursions the first extension is related to two different constructions of ito s law of brownian excursions due to williams and bismut each involving back to back splicing of fragments of two independent three dimensional bessel processes generalizations of both splicing constructions are described which involve bessel processes and bessel bridges of arbitrary positive real dimension

performance study of conditional and unconditional direction of arrival estimation a numerical and analytical study of conditional and unconditional direction of arrival doa estimation is presented explicit expressions for the unconditional cramer rao bounds on the doa estimation accuracy and the covariance matrix of the conditional maximum likelihood method are given it is shown that many doa estimation methods have the same asymptotic statistical properties under conditional and unconditional models the situation of two narrowband plane signals impinging on a uniformly spaced linear array is discussed

maximum likelihood methods for direction of arrival estimation five methods of direction of arrival doa estimation which can be derived from the maximum likelihood ml principle are considered the ml method mlm results from the application of the ml principle to the statistics of the observed raw data the standard multiple signal classification music procedure called music 1 is obtained as a brute force approximation of the mlm an improved music procedure named music 2 is obtained by applying the ml principle to the statistics of certain linear combinations of the sample noise space eigenvectors a procedure which compromises between the good performance of the mlm and the computational simplicity of music is a method of direction estimation mode 1 which is derived as a large sample realization of the mlm a fifth method called mode 2 is obtained by using the ml principle on the statistics of certain linear combinations of the sample eigenvectors mode 2 is computationally less demanding than the mlm it is of the same complexity as mode 1 and statistically more efficient a numerical comparison of these five doa estimation methods is presented it confirms the analytic results on their theoretical performance levels

the sde solved by local times of a brownian excursion or bridge derived from the height profile of a random tree or forest let b be a standard one dimensional brownian motion started at 0 let l t v b be the occupation density of b at level v up to time t the distribution of the process of local times l t v b v 0 conditionally given b 0 and l t 0 b l is shown to be that of the unique strong solution x of the ito sde dx v 4 x 2 a t 0 x v du 1 dv 2 x v dbv on the interval 0 v t x where v t x inf v f v 0 x u du t and x v 0 for all v v t x this conditioned form of the ray knight description of brownian local times arises from study of the asymptotic distribution as n and 2k n l of the height profile of a uniform rooted random forest of k trees labeled by a set of n elements as obtained by conditioning a uniform random mapping of the set to itself to have k cyclic points the sde is the continuous analog of a simple description of a galton watson branching process conditioned on its total progeny for l 0 corresponding to asymptotics of a uniform random tree the sde gives a description of the process of local times of a brownian excursion which is equivalent to jeulin s description of these local times as a time change of twice a brownian excursion another corollary is the biane yor description of the local times of a reflecting brownian ridge as a time changed reversal of twice a brownian meander of the same length

optimality of high resolution array processing using the eigensystem approach in the classical approach to underwater passive listening the medium is sampled in a convenient number of look directions from which the signals are estimated in order to build an image of the noise field in contrast a modern trend is to consider the noise field as a global entity depending on few parameters to be estimated simultaneously in a gaussian context it is worthwhile to consider the application of likelihood methods in order to derive a detection test for the number of sources and estimators for their locations and spectral levels this paper aims to compute such estimators when the wavefront shapes are not assumed known a priori this justifies results previously found using the asymptotical properties of the eigenvalue eigenvector decomposition of the estimated spectral density matrix of the sensor signals they have led to a variety of high resolution array processing methods more specifically a covariance matrix test for equality of the smallest eigenvalues is presented for source detection for source localization a best fit method and a test of orthogonality between the smallest eigenvectors and the source vectors are discussed

noncoherent mimo radar for location and velocity estimation more antennas means better performance this paper presents an analysis of the joint estimation of target location and velocity using a multiple input multiple output mimo radar employing noncoherent processing for a complex gaussian extended target a mimo radar with m transmit and n receive antennas is considered to provide insight we focus on a simplified case first assuming orthogonal waveforms temporally and spatially white noise plus clutter and independent reflection coefficients under these simplifying assumptions the maximum likelihood ml estimate is analyzed and a theorem demonstrating the asymptotic consistency large mn of the ml estimate is provided numerical investigations given later indicate similar behavior for some reasonable cases violating the simplifying assumptions in these initial investigations we study unconstrained systems in terms of complexity and energy where each added transmit antenna employs a fixed energy so that the total transmitted energy is allowed to increase as we increase the number of transmit antennas following this we also look at constrained systems where the total system energy and complexity are fixed to approximate systems of fixed complexity in an abstract way we restrict the total number of antennas employed to be fixed here we show numerical examples which indicate a preference for receive antennas similar to mimo communications but where systems with multiple transmit antennas yield the smallest possible mean square error mse the joint cramer rao bound crb is calculated and the mse of the ml estimate is analyzed it is shown for some specific numerical examples that the signal to clutter plus noise ratio scnr threshold indicating the scnrs above which the mse of the ml estimate is reasonably close to the crb can be lowered by increasing mn the noncoherent mimo radar ambiguity function af is developed in two different ways and illustrated by examples it is shown for some specific examples that the size of the product mn controls the levels of the sidelobes of the af

the growth of random walks and levy processes let x i be a sequence of independent identically distributed non degenerate random variables taking values in mathbb r d and s n sum n i 1 x i m n max 1 leqq i leqq n s i define for x 0 g x p x 1 x k x x 2 e x 1 2 1 x 1 leq x m x x 1 e x 1 1 x 1 leq x and h x g x k x m x then if beta sup alpha lim sup x alpha h x 0 delta sup alpha lim inf x alpha h x 0 it is proved that n 1 alpha m n rightarrow 0 for alpha delta while the lim inf is 0 and the lim sup is infty for beta alpha delta some alternative characterizations of the indices beta delta are obtained as well as the analogous results for levy processes

stopped markov chains with stationary occupation times let e be a finite set equipped with a group g of bijective transformations and suppose that x is an irreducible markov chain on e that is equivariant under the action of g in particular if e g with the corresponding transformations being left or right multiplication then x is a random walk on g we show that when x is started at a fixed point there is a stopping time u such that the distribution of the random vector of pre u occupation times is invariant under the action of g when g acts transitively that is e is a homogeneous space any non zero finite expectation stopping time with this property can occur no earlier than the time s of the first return to the starting point after all states have been visited we obtain an expression for the joint laplace transform of the pre s occupation times for an arbitrary finite chain and show that even for random walk on the group of integers mod r the pre s occupation times do not generally have a group invariant distribution this appears to contrast with the brownian analog as there is considerable support for the conjecture that the field of local times for brownian motion on the circle prior to the counterpart of s is stationary under circular shifts