Unknown Covariance Structure Research Articles

Mixed-Effects Contextual Bandits

We study a novel variant of a contextual bandit problem with multi-dimensional reward feedback formulated as a mixed-effects model, where the correlations between multiple feedback are induced by sharing stochastic coefficients called random effects. We propose a novel algorithm, Mixed-Effects Contextual UCB (ME-CUCB), achieving tildeO(d sqrt(mT)) regret bound after T rounds where d is the dimension of contexts and m is the dimension of outcomes, with either known or unknown covariance structure. This is a tighter regret bound than that of the naive canonical linear bandit algorithm ignoring the correlations among rewards. We prove a lower bound of Omega(d sqrt(mT)) matching the upper bound up to logarithmic factors. To our knowledge, this is the first work providing a regret analysis for mixed-effects models and algorithms involving weighted least-squares estimators. Our theoretical analysis faces a significant technical challenge in that the error terms do not constitute martingales since the weights depend on the rewards. We overcome this challenge by using covering numbers, of theoretical interest in its own right. We provide numerical experiments demonstrating the advantage of our proposed algorithm, supporting the theoretical claims.

Tensor-based methods for blind spatial signature estimation under arbitrary and unknown source covariance structure

Spatial signature estimation is a problem encountered in several applications in signal processing such as mobile communications, sonar, radar, astronomy and seismology. In this paper, we propose higher-order tensor methods to solve the blind spatial signature estimation problem using planar arrays. By assuming that sources' powers vary between successive time blocks, we recast the spatial and spatiotemporal covariance models for the received data as third-order PARATUCK2 and fourth-order Tucker4 tensor decompositions, respectively. Firstly, by exploiting the multilinear algebraic structure of the proposed tensor models, new iterative algorithms are formulated to blindly estimate the spatial signatures. Secondly, in order to achieve a better spatial resolution, we propose an expanded form of spatial smoothing that returns extra spatial dimensions in comparison with the traditional approaches. Additionally, by exploiting the higher-order structure of the resulting expanded tensor model, a multilinear noise reduction preprocessing step is proposed via higher-order singular value decomposition. We show that the increase on the tensor order provides a more efficient denoising, and consequently a better performance compared to existing spatial smoothing techniques. Finally, a solution based on a multi-stage Khatri–Rao factorization procedure is incorporated as the final stage of our proposed estimators. Our results demonstrate that the proposed tensor methods yield more accurate spatial signature estimates than competing approaches while operating in a challenging scenario where the source covariance structure is unknown and arbitrary (non-diagonal), which is actually the case when sample covariances are computed from a limited number of snapshots.

Estimation equations for multivariate linear models with Kronecker structured covariance matrices

ABSTRACTThe aim of the paper is to determine maximum-likelihood estimation equations. Observations follow a multivariate normal distribution, , where , and describe the unknown covariance structure between rows and columns of , respectively. Imposing restrictions on and four types of covariance structures will be considered.

A Bayesian approach to estimating linear mixtures with unknown covariance structure

In this paper, we study a new Bayesian approach for the analysis of linearly mixed structures. In particular, we consider the case of hyperspectral images, which have to be decomposed into a collection of distinct spectra, called endmembers, and a set of associated proportions for every pixel in the scene. This problem, often referred to as spectral unmixing, is usually considered on the basis of the linear mixing model (LMM). In unsupervised approaches, the endmember signatures have to be calculated by an endmember extraction algorithm, which generally relies on the supposition that there are pure (unmixed) pixels contained in the image. In practice, this assumption may not hold for highly mixed data and consequently the extracted endmember spectra differ from the true ones. A way out of this dilemma is to consider the problem under the normal compositional model (NCM). Contrary to the LMM, the NCM treats the endmembers as random Gaussian vectors and not as deterministic quantities. Existing Bayesian approaches for estimating the proportions under the NCM are restricted to the case that the covariance matrix of the Gaussian endmembers is a multiple of the identity matrix. The self-evident conclusion is that this model is not suitable when the variance differs from one spectral channel to the other, which is a common phenomenon in practice. In this paper, we first propose a Bayesian strategy for the estimation of the mixing proportions under the assumption of varying variances in the spectral bands. Then we generalize this model to handle the case of a completely unknown covariance structure. For both algorithms, we present Gibbs sampling strategies and compare their performance with other, state of the art, unmixing routines on synthetic as well as on real hyperspectral fluorescence spectroscopy data.

Impact of unknown covariance structures in semiparametric models for longitudinal data: An application to Wisconsin diabetes data

Semiparametric models are becoming increasingly attractive for longitudinal data analysis. Often there is lack of knowledge of the covariance structure of the response variable. Although it is still possible to obtain consistent estimators for both parametric and nonparametric components of a semipatrametric model by assuming an identity structure for the covariance matrix, the resulting estimators may not be efficient. We conducted extensive simulation studies to investigate the impact of an unknown covariance structure on estimators in semiparametric models for longitudinal data. In some situations the loss of efficiency could be substantial. A two-step estimator is thus proposed to improve the efficiency. Our study was motivated by a population based data analysis to examine the temporal relationship between systolic blood pressure and urinary albumin excretion.

Noncircularity-exploitation in direction estimation of noncircular signals with an acoustic vector-sensor

The main motivation of using an acoustic vector-sensor in direction-of-arrival (DOA) estimation applications has been its unambiguous two-dimensional directivity, insensitivity to the range of sources, and independence of signal frequency. The main objection lies in its lack of geometry-redundancy and limited degree of freedom. Four thus emerged challenging tasks and the corresponding solutions by recurring to the redundancies in the nonvanishing conjugate correlations of noncircular signals are described in the paper: (1) fulfilling source decorrelation in a multipath propagation environment; (2) enhancing processing capacity to accommodate more signals; (3) suppressing colored-noise with unknown covariance structure; and (4) deriving closed-form approaches to avoid iteration and manifold storage. Simulation experiments are carried out to examine the associated DOA estimators termed as: (1) phase-smoothing MUSIC (multiple signal classification); (2) virtual-MUSIC; (3) conjugate-MUSIC; and (4) noncircular-ESPRIT (estimation of signal parameters via rotational invariance techniques), respectively.

Consistent estimator in multivariate errors-in-variables model in the case of unknown error covariance structure

We consider a linear multivariate errors-in-variables model AX ≈ B, where the matrices A and B are observed with errors and the matrix parameter X is to be estimated. In the case of lack of information about the error covariance structure, we propose an estimator that converges in probability to X as the number of rows in A tends to infinity. Sufficient conditions for this convergence and for the asymptotic normality of the estimator are found.

A CFAR adaptive subspace detector for second-order Gaussian signals

We study the problem of detecting subspace signals described by the Second-Order Gaussian (SOG) model in the presence of noise whose covariance structure and level are both unknown. Such a detection problem is often called Gauss-Gauss problem in that both the signal and the noise are assumed to have Gaussian distributions. We propose adaptive detectors for the SOG model signals based on a single observation and multiple observations. With a single observation, the detector can be derived in a manner similar to that of the generalized likelihood ratio test (GLRT), but the unknown covariance structure is replaced by sample covariance matrix based on training data. The proposed detectors are constant false alarm rate (CFAR) detectors. As a comparison, we also derive adaptive detectors for the First-Order Gaussian (FOG) model based on multiple observations under the same noise condition as for the SOG model. With a single observation, the seemingly ad hoc CFAR detector for the SOG model is a true GLRT in that it has the same form as the GLRT CFAR detector for the FOG model. We give an approximate closed form of the probability of detection and false alarm in this case. Furthermore, we study the proposed CFAR detectors and compute the performance curves.

Bayesian multiple testing for two-sample multivariate endpoints.

In clinical studies involving multiple variables, simultaneous tests are often considered where both the outcomes and hypotheses are correlated. This article proposes a multivariate mixture prior on treatment effects, that allows positive probability of zero effect for each hypothesis, correlations among effect sizes, correlations among binary outcomes of zero versus nonzero effect, and correlations among the observed test statistics (conditional on the effects). We develop a Bayesian multiple testing procedure, for the multivariate two-sample situation with unknown covariance structure, and obtain the posterior probabilities of no difference between treatment regimens for specific variables. Prior selection methods and robustness issues are discussed in the context of a clinical example.

Subspace-based adaptive generalized likelihood ratio detection

Subspace-based adaptive detection performance is examined for the generalized likelihood ratio detector based on Wilks' /spl Lambda/ statistic. The problem considered here is detecting the presence of one or more signals of known shape embedded in Gaussian distributed noise with unknown covariance structure. The data is mapped into a subspace prior to detection. The probability of false alarm is independent of the subspace transformation and depends only on subspace dimension. The probability of detection depends on the subspace transformation through a nonadaptive signal-to-noise ratio (SNR) parameter. Subspace processing results in an SNR loss that tends to decrease performance and a gain in statistical stability that tends to increase performance. It is shown that the statistical stability effect dominates the SNR loss for short data records, and subspace detectors can require substantially less SNR than full space detectors for equivalent performance. A method for designing the subspace transformation to minimize the SNR loss is proposed and illustrated through simulations.

Estimating the Mean and Covariance Structure Nonparametrically When the Data are Curves

SUMMARY We develop methods for the analysis of a collection of curves which are stochastically modelled as independent realizations of a random function with an unknown mean and covariance structure. We propose a method of estimating the mean function non-parametrically under the assumption that it is smooth. We suggest a variant on the usual form of cross-validation for choosing the degree of smoothing to be employed. This method of cross-validation, which consists of deleting entire sample curves, has the advantage that it does not require that the covariance structure be known or estimated. In the estimation of the covariance structure, we are primarily concerned with models in which the first few eigenfunctions are smooth and the eigenvalues decay rapidly, so that the variability is predominantly of large scale. We propose smooth nonparametric estimates of the eigenfunctions and a suitable method of cross-validation to determine the amount of smoothing. Our methods are applied to data on the gaits of a group of 5-year-old children.

An Analysis of International Exchange Rates Using Multivariate DLM's

New models for multiple time series are introduced and illustrated in an application to international currency exchange rate data. The models, based on matrix-variate normal extensions of the dynamic linear model (DLM), provide a tractable, sequential procedure for estimation of unknown covariance structure between series. A principal components analysis is carried out providing a basis for easy model assessment. A practically important elaboration of the model incorporates time- variation in covariance matrices.

Estimation in regression models with stationary, dependent errors

Often the unknown covariance structure of a stationary, dependent, Gaussian error sequence can be simply parametrised. The error sequence can either be directly observed or observed only through a random sequence containing a deterministic regression model. The method of scoring is used here, in conjunction with recursive estimation techniques, to effect the maximum likelihood estimation of the covariance parameters. Sequences of recursive residuals, useful in model diagnostics and data analysis, are obtained in the estimation procedure.

Adaptive prediction of stationary time series by modified conjugate direction methods

The problem considered is that of obtaninig, rocursive estimates of the best finite cemory linear one-step predictor for a non-deterministic weakly stationary stochastic process ut+k a o, Z 2, ± 2, ± 2, ± … with unknown covariance structure. if u(k) x 3ut ut+k is the covariance function for the process this problem reduces to solring iteratively a ‘coving’ system of linear equations Pn x = Pn where Pn and Pn are consistent estimates of p = (0(1),p(2), …,p(d)) and p = (0(i-j)),i,j=1,2…, based on the first n observations of the process. Two iterative procedure are developed for producing a sequence fo estimators (-)n=1 which converge almost surely to 3 the solution of P x = 3 Both are modifications of conjugate direction methods originally proposed by Estenses and Stiefel (1952).

Passive sonar processing for noise with unknown covariance structure

The piedmont dry forest of northwestern Argentina has been under intensive and unplanned forest logging focused on 12 tree species, without any attempts having been made to ensure their regeneration or long-term conservation. In this study, we assessed the conservation status of these timber species in the piedmont dry forest of northwestern Argentina and compared our results with the IUCN assessment. We considered an inadequate conservation status if the species: (1) occurred in less than 50 % of the sampled plots, (2) had a density of large trees (more than 40 cm DBH) lower than one in. ha−1, or (3) had a density of saplings (more than 30 cm in height and less than 9.9 cm in DBH) lower than 30 in. ha−1. Our results showed an inadequate conservation status for eight of the 12 studied timber tree species. Additionally, only seven species were previously assessed and categorized on the IUCN Red List, with our categorization agreeing with only four of them. The data provided in this work can serve as a baseline to monitor the population trends of these species; information can also help to prioritize conservation efforts, which is essential considering the high number of tree species that have potential extinction risk in the short-term.

Passive Sonar Processing for Noise with Unknown Covariance Structure

Passive processing is interpretation of the noise and noise like signals received by an array of hydrophones. Our model assumes that there is an important noise component about which very little is known, that the data are stationary, that a long data sample is available, that signal wavefronts are planar, and that the response of the array to plane waves is known. We propose a processor and compare it with high-resolution frequency-wavenumber spectrum analysis. Let F̂ denote an estimate of the spectral density matrix, and let H denote any spectral density matrix that corresponds to a noise field composed of uncorrelated plane waves. A reasonable estimate of the spectral matrix of the signals and ambient noise is a value of H for which F̂-H is positive semidefinite and trace H is maximized. The rationale for this estimate is closely related to a rationale for the estimate obtained by high-resolution frequency-wavenumber spectrum analysis (which is also called adaptive beamforming). The difference is that the former extracts the contribution from all directions at once, whereas the latter extracts the contribution from each direction separately. The former estimate rejects more unwanted noise than the latter.