Covariance Fitting Research Articles

Covariance fitting of highly-correlated data in lattice QCD

We address a frequently-asked question on the covariance fitting of highly-correlated data such as our B K data based on the SU(2) staggered chiral perturbation theory. Basically, the essence of the problem is that we do not have a fitting function accurate enough to fit extremely precise data. When eigenvalues of the covariance matrix are small, even a tiny error in the fitting function yields a large chi-square value and spoils the fitting procedure. We have applied a number of prescriptions available in the market, such as the cut-off method, modified covariance matrix method, and Bayesian method. We also propose a brand new method, the eigenmode shift (ES) method, which allows a full covariance fitting without modifying the covariance matrix at all. We provide a pedagogical example of data analysis in which the cut-off method manifestly fails in fitting, but the rest work well. In our case of the B K fitting, the diagonal approximation, the cut-off method, the ES method, and the Bayesian method work reasonably well in an engineering sense. However, interpreting the meaning of χ 2 is easier in the case of the ES method and the Bayesian method in a theoretical sense aesthetically. Hence, the ES method can be a useful alternative optional tool to check the systematic error caused by the covariance fitting procedure.

Gaussian mixture model approximation of total spatial power spectral density for multiple incoherently distributed sources

Practically, the spatial power spectral density (PSD) of single or multiple incoherently distributed (ID) sources is often unknown, and the total spatial PSD is suitable to model the spatial distribution characteristic of signals if the number of multiple ID sources is also unknown. In this study, the Gaussian mixture model (GMM) is employed to characterise the total spatial PSD of multiple ID sources, and two algorithms are proposed to estimate the parameters of the GMM. The first one is the covariance fitting method for multiple ID sources with Gaussian PSD, and the other is the iterative expectation maximisation (EM) algorithm. Simulation studies demonstrate that the EM algorithm outperforms other methods in approximating the shape of the total spatial PSD, especially for small spatial spread.

A Data-Adaptive Compressed Sensing Approach to Polarimetric SAR Tomography of Forested Areas

Super-resolution imaging via compressed sensing (CS)-based spectral estimators has been recently introduced to synthetic aperture radar (SAR) tomography. In the case of partial scatterers, the mainstream has so far been twofold, in that the tomographic reconstruction is conducted by either directly working with multiple looks and/or polarimetric channels or by exploiting the corresponding single-channel second-order statistics. In this letter, we unify these two methodologies in the context of covariance fitting. In essence, we exploit the fact that both vertical structures and the unknown polarimetric signatures can be approximated in a low-dimensional subspace. For this purpose, we make use of a wavelet basis in order to sparsely represent vertical structures. Additionally, we synthesize a data-adaptive orthonormal basis that spans the space of polarimetric signatures. Finally, we validate this approach by using fully polarimetric L-band data acquired by the E-SAR sensor of the German Aerospace Center (DLR).

Sparse covariance fitting for direction of arrival estimation

This article proposes a new algorithm for finding the angles of arrival of multiple uncorrelated sources impinging on a uniform linear array of sensors. The method is based on sparse signal representation and does not require either the knowledge of the number of the sources or a previous initialization. The proposed technique considers a covariance matrix model based on overcomplete basis representation and tries to fit the unknown signal powers to the sample covariance matrix. Sparsity is enforced by means of a l1-norm penalty. The final problem is reduced to an objective function with a non-negative constraint that can be solved efficiently using the LARS/homotopy algorithm. The method described herein is able to provide high resolution with a low computational burden. It proceeds in an iterative fashion solving at each iteration a small linear system of equations until a stopping condition is fulfilled. The proposed stopping criterion is based on the residual spectrum and arises in a natural way when the LARS/homotopy is applied to the considered objective function.

Robust Adaptive Beamforming Using Multidimensional Covariance Fitting

Over the last decade, several set-based worst-case beamformers have been proposed. It has been shown that some of these beamformers can be formulated equivalently as one-dimensional (ID) covariance fitting problems. Based on this formulation, we show that these beamformers lead to inherently nonoptimum results in the presence of interferers. To mitigate the detrimental effect of interferers, we extend the ID covariance fitting approach to multidimensional (MD) covariance fitting, modeling the source steering vectors by means of uncertainty sets. The proposed MD covariance fitting approach leads to a nonconvex optimization problem. We develop a convex approximation of this problem, which can be solved, for example, by means of the logarithmic barrier method. The complexity required to compute the barrier function and its first- and second-order derivatives is derived. Simulation results show that the proposed beamformer based on MD covariance fitting achieves an improved performance as compared to the state-of-the-art narrowband beamformers in scenarios with large sample support.

SPICE and LIKES: Two hyperparameter-free methods for sparse-parameter estimation

SPICE (SParse Iterative Covariance-based Estimation) is a recently introduced method for sparse-parameter estimation in linear models using a robust covariance fitting criterion that does not depend on any hyperparameters. In this paper we revisit the derivation of SPICE to streamline it and to provide further insights into this method. LIKES (LIKelihood-based Estimation of Sparse parameters) is a new method obtained in a hyperparameter-free manner from the maximum-likelihood principle applied to the same estimation problem as considered by SPICE. Both SPICE and LIKES are shown to provide accurate parameter estimates even from scarce data samples, with LIKES being more accurate than SPICE at the cost of an increased computational burden.

Reliability and validity of the attitudes to ageing questionnaire for Canadian and Norwegian older adults

Scand J Caring Sci; 2010; 24; 75–85 Reliability and validity of the attitudes to ageing questionnaire for Canadian and Norwegian older adultsThe aim of this study was to examine the reliability and validity of the Attitudes to Ageing Questionnaire (AAQ), a new scale designed for cross‐cultural comparisons of older adults. There are currently very few measures intended to measure the attitudes of older adults to their own ageing. The instrument was designed as part of a larger study to develop a measure of quality of life (QOL) of older adults and to assess factors related to QOL; it was hypothesized that attitudes to ageing would be related to QOL. Canadian (n = 202) and Norwegian (n = 490) data were used. In both study samples, a series of correlation analyses indicated that item scores correlated most strongly with their parent subscales (r=0.42–0.79; p < 0.01). Acceptable internal consistency was shown (Cronbach’s alpha of 0.70 or greater for all subscales). In a confirmatory factor analyses (CFA), all 24 items in the AAQ were retained (p < 0.001); the observed lack of goodness of fit and residual covariance patterns provided empirical support, in part, for the construct validity of the AAQ. Patterns of correlations of the AAQ subscales with WHOQOL‐OLD facets, WHOQOL‐BREF domains, a global QoL item and GDS scores provided evidence of convergent and divergent validity. Nonsignificant correlations were found between psychological growth and two facets of the WHOQOL‐OLD in the Canadian sample. Subscale scores also significantly discriminated between healthy and unhealthy groups. Further validation of the scale among older people, across countries, is recommended.

A covariance fitting approach for correlated acoustic source mapping

Microphone arrays are commonly used for noise source localization and power estimation in aeroacoustic measurements. The delay-and-sum (DAS) beamformer, which is the most widely used beamforming algorithm in practice, suffers from low resolution and high sidelobe level problems. Therefore, deconvolution approaches, such as the deconvolution approach for the mapping of acoustic sources (DAMAS), are often used for extracting the actual source powers from the contaminated DAS results. However, most deconvolution approaches assume that the sources are uncorrelated. Although deconvolution algorithms that can deal with correlated sources, such as DAMAS for correlated sources, do exist, these algorithms are computationally impractical even for small scanning grid sizes. This paper presents a covariance fitting approach for the mapping of acoustic correlated sources (MACS), which can work with uncorrelated, partially correlated or even coherent sources with a reasonably low computational complexity. MACS minimizes a quadratic cost function in a cyclic manner by making use of convex optimization and sparsity, and is guaranteed to converge at least locally. Simulations and experimental data acquired at the University of Florida Aeroacoustic Flow Facility with a 63-element logarithmic spiral microphone array in the absence of flow are used to demonstrate the performance of MACS.

CoCoa: a software tool for estimating the coefficient of coancestry from multilocus genotype data

Phenotypic data collected in breeding programs and marker-trait association studies are often analyzed by means of linear mixed models. In these models, the covariance between the genetic background effects of all genotypes under study is modeled by means of pairwise coefficients of coancestry. Several marker-based coancestry estimation procedures allow to estimate this covariance matrix, but generally introduce a certain amount of bias when the examined genotypes are part of a breeding program. CoCoa implements the most commonly used marker-based coancestry estimation procedures and as such, allows to select the best fitting covariance structure for the phenotypic data at hand. This better model fit translates into an increased power and improved type I error control in association studies and an improved accuracy in phenotypic prediction studies. The presented software package also provides an implementation of the new Weighted Alikeness in State (WAIS) estimator for use in hybrid breeding programs. Besides several matrix manipulation tools, CoCoa implements two different bending heuristics, in case the inverse of an ill-conditioned coancestry matrix estimate is needed. The software package CoCoa is freely available at http://webs.hogent.be/cocoa. Source code, manual, binaries for 32 and 64-bit Linux systems and an installer for Microsoft Windows are provided. The core components of CoCoa are written in C++, while the graphical user interface is written in Java.

Partial and ecological correlation: a common three-term covariance decomposition

Let (X, Y, Z) be a trivariate statistical variable observed at individual level. We propose a three-term decomposition of covariance between variables X and Y conditionally on the effects induced by the existence of a variable Z. The three terms are called residual covariance, covariance lack of fit and covariance fit, respectively. Partial covariance, between X and Y after removing the linear effects of Z, σZ (X, Y), is the sum of the first two terms while ecological covariance, between the two regression functions μX (Z) and μY (Z), Cov(μX (Z), μY (Z)), is the sum of the last two terms and, consequently, covariance lack of fit is the common additive term. Simple examples are given in two contexts: in ecological fallacy problems arising in linear modelling with aggregate level analysis contrasted with partial correlation step-wise procedures performed at individual level and in the special case of a two-level nested model. Previous basic decomposition is extended to a multivariate-multiple framework. Distinction between descriptive and stochastic approaches is not essential.

Selection of covariance patterns for longitudinal data in semi-parametric models

The use of patterned covariance structures in the parametric analysis of longitudinal data is both elegant and efficient. However, this strategy has not been well studied for semi-parametric models for analysing such data. We propose to estimate the covariance matrix in the semi-parametric model by rearranging the non-parametric component as a profiled linear function of the data and using a local smoothing technique. This results in a parametric regression formulation that enables us to construct likelihood functions and use various information criteria to select the best fitting covariance matrix. We apply our method to reanalyse data from a two-armed clinical trial for Scleroderma patients and show our method is more efficient for estimating the parametric components in the semi-parametric model.

SEM of another flavour: Two new applications of the supplemented EM algorithm

The supplemented EM (SEM) algorithm is applied to address two goodness-of-fit testing problems in psychometrics. The first problem involves computing the information matrix for item parameters in item response theory models. This matrix is important for limited-information goodness-of-fit testing and it is also used to compute standard errors for the item parameter estimates. For the second problem, it is shown that the SEM algorithm provides a convenient computational procedure that leads to an asymptotically chi-squared goodness-of-fit statistic for the 'two-stage EM' procedure of fitting covariance structure models in the presence of missing data. Both simulated and real data are used to illustrate the proposed procedures.

A Two-Stage Approach to Estimate the Angles of Arrival and the Angular Spreads of Locally Scattered Sources

We propose a new two-stage approach to estimate the nominal angles of arrival (AoAs) and the angular spreads (ASs) of multiple locally scattered sources using a uniform linear array (ULA) of sensors. In contrast to earlier works, we consider both long- and short-term channel variations, typically encountered in wireless links. In the first stage, we exploit sources independence to blindly estimate the channel over several data blocks regularly spaced by intervals larger than the coherence time but each, short enough in length, to make time variations negligible within the block duration. We, thereby, decouple the multisource channel parameters estimation problem in hand into parallel and independent single-source channel parameters estimation subproblems. In the second stage, for each spatially scattered source, we process the corresponding sequence of quasi-independent channel realization estimates as a new single-scattered-source observation over which we apply Taylor series expansions to transform the estimation of the nominal AoA and the AS of the corresponding scattered source into a simple localization of two closely spaced, equi-powered, and uncorrelated rays (i.e., point sources). To localize both rays, we propose new accurate and computationally simple closed-form expressions for the mean value of the spatial harmonics and their separation by means of covariance fitting. An asymptotic performance analysis is also provided to prove the efficiency of the proposed estimators. Then, the AS and the nominal AoA of every source are directly deduced. The whole proposed framework takes advantage of the capabilities of the preprocessing channel identification step (to reduce the noise effect and decouple the estimation of the channel parameters of every source from the others) and the new simple and accurate closed-form estimators to accurately retrieve the channel parameters even in the most adverse conditions, mainly low signal-to-noise ratio (SNR), few sensors, no prior knowledge of the angular distribution, and closely spaced sources, as supported by simulations.

Growth Curves of Turkish Saanen Goats’ Kids Grouped for Weight and Body Mass Index

Abstract. Growth curves and weaning stress differences were investigated in Turkish Saanen kids grouped for their weight and Body Mass Index (BMI = weight/height2). Data included 884 records collected from 61 Turkish Saanen kids raised in Canakkale, Turkey. Slopes were calculated for each kid and were analyzed as data using ordinary least squares, and repeated weights for each kid were analyzed using various covariance structures. Differences between male and female kids were small and non-significant before weaning (P = 0.55), while larger (P = 0.06) after weaning. Heavy animals grew faster than light animals before weaning (P < 0.01) but lost their advantage after weaning (P > 0.05). Fat animals (high BMI) grew faster than lean animals (low BMI) before weaning (P < 0.01) while the differences slightly diminished after weaning (P = 0.04). This may be because light and lean kids were not affected from the weaning shock as much as the heavy and fat kids, which had stuttered growth after weaning. Fit statistics indicated that first-order ante dependence was the best fit covariance structure for these data. Analyses specifying random model and covariance structures did not have an advantage over an approach where slopes were manually calculated and analyzed as data using ordinary least squares.

Chemical mass balance when an unknown source exists

AbstractA chemical mass balance method is proposed for the case where the existence of an unknown source is suspected. In general, when the existence of an unknown source is assumed in statistical receptor modeling, unknown quantities such as the composition of an unknown source and the contributions of assumed sources become unidentifiable. To estimate these unknown quantities avoiding the identification problem, a Bayes model for chemical mass balance is constructed in the form of composition without using prior knowledge on the unknown quantities except for natural constraints. The covariance of ambient observations given in the form of composition is defined in several ways. Markov chain Monte Carlo is used for evaluating the posterior means and variances of the unknown quantities as well as the likelihood for the proposed model. The likelihood is used for selecting the best fit covariance model. A simulation study is carried out to check the performance of the proposed method. Copyright © 2004 John Wiley & Sons, Ltd.

Use of the Covariance Matrix in Directly Fitting Kinetic Parameters: Application to GABA A Receptors

A new method of analysis is described that begins to explore the relationship between the phases of ion channel desensitization and the underlying states of the channel. The method, referred to as covariance fitting (CVF), couples Q-matrix calculations with a maximum likelihood algorithm to fit macroscopic desensitization data directly to kinetic models. Unlike conventional sum-of-squares minimization, CVF fits both the magnitude of the recorded current and the strength of the correlations between different time points. When applied to simulated data generated using various kinetic models with up to 11 free parameters, CVF leads to reasonable parameter estimates. Coupled with the likelihood ratio test, it accurately discriminates between models with different numbers of states, discriminates between most models with the same number but a different arrangement of states, and extracts meaningful information on the relationship between the desensitized states and the phases of macroscopic desensitization. When applied to GABA A receptor traces (outside out patches, α1 β2 γ2S, 1 mM GABA, >2.5 s), a model with two open states and three desensitized states is favored. When applied to simulated data generated using a consensus model, CVF leads to reasonable parameter estimates and accurately discriminates between this and other models.

A Covariance Fitting Approach to Parametric Localization of Multiple Incoherently Distributed Sources

In this paper, a new algorithm for parametric localization of multiple incoherently distributed sources is presented. This algorithm is based on an approximation of the array covariance matrix using central and noncentral moments of the source angular power densities. Based on this approximation, a new computationally simple covariance fitting-based technique is proposed to estimate these moments. Then, the source parameters are obtained from the moment estimates. Compared with earlier algorithms, our technique has lower computational cost and obtains the parameter estimates in a closed form. In addition, it can be applied to scenarios with multiple sources that may have different angular power densities, while other known methods are not applicable to such scenarios.

Soil efflux and total emission rates of magmatic CO 2 at the Horseshoe Lake tree kill, Mammoth Mountain, California, 1995–1999

We report the results of eight soil CO 2 efflux surveys by the closed circulation chamber method at the Horseshoe Lake tree kill (HLTK) — the largest tree kill on Mammoth Mountain. The surveys were undertaken from 1995 to 1999 to constrain total HLTK CO 2 emissions and to evaluate occasional efflux surveys as a surveillance tool for the tree kills. HLTK effluxes range from 1 to >10,000 g m −2 day −1 (grams CO 2 per square meter per day); they are not normally distributed. Station efflux rates can vary by 7–35% during the course of the 8- to 16-h surveys. Disturbance of the upper 2 cm of ground surface causes effluxes to almost double. Semivariograms of efflux spatial covariance fit exponential or spherical models; they lack nugget effects. Efflux contour maps and total CO 2 emission rates based on exponential, spherical, and linear kriging models of survey data are nearly identical; similar results are also obtained with triangulation models, suggesting that the kriging models are not seriously distorted by the lack of normal efflux distributions. In addition, model estimates of total CO 2 emission rates are relatively insensitive to the measurement precision of the efflux rates and to the efflux value used to separate magmatic from forest soil sources of CO 2. Surveys since 1997 indicate that, contrary to earlier speculations, a termination of elevated CO 2 emissions at the HLTK is unlikely anytime soon. The HLTK CO 2 efflux anomaly fluctuated greatly in size and intensity throughout the 1995–1999 surveys but maintained a N–S elongation, presumably reflecting fault control of CO 2 transport from depth. Total CO 2 emission rates also fluctuated greatly, ranging from 46 to 136 t day −1 (metric tons CO 2 per day) and averaging 93 t day −1. The large inter-survey variations are caused primarily by external (meteorological) processes operating on time scales of hours to days. The externally caused variations can mask significant changes occurring at depth; a striking example is the masking of a degassing event generated at depth and detected by a soil gas sensor network in September 1997 while an efflux survey was in progress. Thus, occasional efflux surveys are not an altogether effective surveillance tool for the HLTK, and making them effective by greatly increasing their frequency may not be practical.

On the parameterization of positive real sequences and MA parameter estimation

An algorithm for moving average (MA) parameter estimation was proposed by Stoica et al. (see ibid. vol.48, p.1999-2012, 2000). Its key step (covariance fitting) is a semidefinite programming (SDP) problem with two convex constraints: one reflecting the real positiveness of the desired covariance sequence and the other having a second-order cone form. We analyze two parameterizations of a positive real sequence and show that there is a one-to-one correspondence between them. We also show that the dual of the covariance fitting problem has a significantly smaller number of variables and, thus, a much reduced computational complexity. We discuss in detail the formulations that are best suited for the currently available semidefinite quadratic programming packages. Experimental results show that the execution times of the newly proposed algorithms scale well with the MA order, which are therefore convenient for large-order MA signals.

Vector ARMA estimation: a reliable subspace approach

A parameter estimation method for finite-dimensional multivariate linear stochastic systems, which is guaranteed to produce valid models approximating the true underlying system in a computational time of a polynomial order in the system dimension, is presented. This is achieved by combining the main features of certain stochastic subspace identification techniques with sound matrix Schur restabilizing procedures and multivariate covariance fitting, both of which are formulated as linear matrix inequality problems. All aspects of the identification method are discussed, with an emphasis on the two issues mentioned above, and examples of the overall performance are provided for two different systems.