Normalized Covariance Matrices Research Articles

Evaluating the impact of correlated noise for time‐lapse transient electromagnetic (TEM) monitoring studies

AbstractTypically, transient electromagnetic (TEM) hydrogeological applications focus on spatial characterization. However, there is scope for utilizing time‐lapse TEM to monitor dynamic processes. Changes related to hydrogeological processes are subtle; consequently, measurements must be of exceptional quality, and the presence and influence of noise sources must be well understood. A potentially problematic noise source is radio waves operating in the 3–300 kHz range, as they introduce correlated error into TEM measurements. For instance, it was anticipated that such radio waves could introduce smooth perturbations in the data that could be erroneously interpreted in the TEM inversion results. This work evaluates the presence of correlated noise sources, their temporal dependence and their significance for TEM monitoring. This work combines fully sampled data and normalized covariance matrices of gated data to identify correlated noise sources. Radio signals operating in the low and very low frequency (LF and VLF) bands were found to vary across the investigation period; however, their influence on the TEM inversion results was minimal. Measurement stacking remains an effective way to improve the signal‐to‐noise ratio of TEM data affected by LF and VLF noise sources as they do not introduce bias, for example smooth perturbation, into the TEM data explored in this work.

Testing equality of two normal covariance matrices with monotone missing data

The problem of testing equality of two multivariate normal covariance matrices is considered. Assuming that the incomplete data are of monotone pattern, a quantity similar to the Likelihood Ratio Test Statistic is proposed. A satisfactory approximation to the distribution of the quantity is derived. Hypothesis testing based on the approximate distribution is outlined. The merits of the test are investigated using Monte Carlo simulation. Monte Carlo studies indicate that the test is very satisfactory even for moderately small samples. The proposed methods are illustrated using an example.

Correlation analysis of target echoes using distributed transmit array

Transmit diversity multiple-input multiple-output (MIMO) radars decorrelate target echoes on distributed transmit array, and then through closely spaced receivers array estimate the angles of multiple targets using the received target echoes without separating transmitted signals. In this paper, the target echoes correlation is analyzed under Swerling I and II target model assumptions. The results show that: (1) Target echoes are uncorrelated under Swerling II model; (2) under Swerling I model, in order to decrease the correlation of target echoes by increasing the number of transmitters, it is required that the normalized covariance matrices of the signals before and after the transmitter number increases should satisfy a relation given in this study. Finally, the results are verified by computer simulations of the target echoes correlation and angles estimation performance.

Estimation of Multivariate Complex Normal Covariance Matrices Under an Invariant Quadratic Loss

The problem of estimating multivariate complex normal covariance matrices is considered under an invariant quadratic loss function. Estimators which dominate the best scalar multiple of the empirical covariance matrix are presented. Improved estimators include the best triangular equivariant estimators and complex analogues of estimators for multivariate real normal covariance matrices by Haff (1980).

Estimation of normal covariance matrices parametrized by irreducible symmetric cones under Stein's loss

In this paper the problem of estimating a covariance matrix parametrized by an irreducible symmetric cone in a decision-theoretic set-up is considered. By making use of some results developed in a theory of finite-dimensional Euclidean simple Jordan algebras, Bartlett's decomposition and an unbiased risk estimate formula for a general family of Wishart distributions on the irreducible symmetric cone are derived; these results lead to an extension of Stein's general technique for derivation of minimax estimators for a real normal covariance matrix. Specification of the results to the multivariate normal models with covariances which are parametrized by complex, quaternion, and Lorentz types gives minimax estimators for each model.

Covariance Mapping in the Analysis of Ignitable Liquids by Gas Chromatography/Mass Spectrometry

The covariance matrix computed from the retention time-ion abundance data matrix from gas chromatography/mass spectrometry analysis of ignitable liquids is shown to be a useful tool for automated identification of ignitable liquids in a database. The absolute value of the element-by-element difference between two normalized covariance matrices is shown to quantitatively differentiate between ignitable liquids composed of complex mixtures of hydrocarbons and is amenable to automated searching of ignitable liquid databases. The covariance mapping method is applied to a matrix-contaminated postburn sample, allowing the determination of a high degree of similarity between the ignitable liquid and a heavily evaporated gasoline.

Estimation of multivariate normal covariance and precision matrices in a star-shape model with missing data

In this paper, we study the problem of estimating the covariance matrix Σ and the precision matrix Ω (the inverse of the covariance matrix) in a star-shape model with missing data. By considering a type of Cholesky decomposition of the precision matrix Ω = Ψ ′ Ψ , where Ψ is a lower triangular matrix with positive diagonal elements, we get the MLEs of the covariance matrix and precision matrix and prove that both of them are biased. Based on the MLEs, unbiased estimators of the covariance matrix and precision matrix are obtained. A special group G , which is a subgroup of the group consisting all lower triangular matrices, is introduced. By choosing the left invariant Haar measure on G as a prior, we obtain the closed forms of the best equivariant estimates of Ω under any of the Stein loss, the entropy loss, and the symmetric loss. Consequently, the MLE of the precision matrix (covariance matrix) is inadmissible under any of the above three loss functions. Some simulation results are given for illustration.

Estimation of the multivariate normal precision and covariance matrices in a star-shape model

In this paper, we introduce the star-shape models, where the precision matrix Ω (the inverse of the covariance matrix) is structured by the special conditional independence. We want to estimate the precision matrix under entropy loss and symmetric loss. We show that the maximal likelihood estimator (MLE) of the precision matrix is biased. Based on the MLE, an unbiased estimate is obtained. We consider a type of Cholesky decomposition of Ω, in the sense that Ω=Ψ′Ψ, where Ψ is a lower triangular matrix with positive diagonal elements. A special group\(\mathcal{G}\), which is a subgroup of the group consisting all lower triangular matrices, is introduced. General forms of equivariant estimates of the covariance matrix and precision matrix are obtained. The invariant Haar measures on\(\mathcal{G}\), the reference prior, and the Jeffreys prior of Ψ are also discussed. We also introduce a class of priors of Ψ, which includes all the priors described above. The posterior properties are discussed and the closed forms of Bayesian estimators are derived under either the entropy loss or the symmetric loss. We also show that the best equivariant estimators with respect to\(\mathcal{G}\) is the special case of Bayesian estimators. Consequently, the MLE of the precision matrix is inadmissible under either entropy or symmetric loss. The closed form of risks of equivariant estimators are obtained. Some numerical results are given for illustration.

Inadmissibility of the Maximum Likekihood Estimator of Normal Covariance Matrices with the Lattice Conditional Independence

Lattice conditional independence (LCI) models introduced by S. A. Andersson and M. D. Perlman (1993, Ann. Statist.21, 1318–1358) have the pleasant feature of admitting explicit maximum likelihood estimators and likelihood ratio test statistics. This is because the likelihood function and parameter space for a LCI model can be factored into products of conditional likelihood functions and parameter spaces, where the standard multivariate techniques can be applied. In this paper we consider the problem of estimating the covariance matrices under LCI restriction in a decision theoretic setup. The Stein loss function is used in this study and, using the factorization mentioned above, minimax estimators are obtained. Since the maximum likelihood estimator has constant risk and is different from the minimax estimator, this shows that the maximum likelihood estimator under LCI restriction inadmissible. These results extend those obtained by W. James and C. Stein (1960, in “Proceedings of the Fourth Berkeley Symposium on Mathematics, Statistics, and Probability,” Vol. 1, pp. 360–380, Univ. of California Press, Berkeley, CA) and D. K. Dey and C. Srinivasan (1985, Ann. Statist.13, 1581–1591) for estimating normal covariance matrices to the LCI models.

Estimation of normal covariance and precision matrices with incomplete data

Suppose that we have n independent observations from Np (0,∑) and, in addition, we have m independent observations available on the first q(q < P) coordinates. Assuming that X i's and Y i's are independent,we consider the problem of estimation of ∑ and ∑-1 respectively under the loss functions .We propose some new estimators that dominate the best lower triangular invariant minimax estimator under the loss L We also derive the best lower triangular invariant minimax estimator of ∑-1 under L 1 and suggest some estimators that dominate it.

Minimal complete classes of invariant tests for equality of normal covariance matrices and sphericity

The problem of testing equality of two normal covariance matrices, Σ 1 = Σ 2 is studied. Two alternative hypotheses, Σ 1 ≠ Σ 2 and Σ 1 − Σ 2 > 0 are considered. Minimal complete classes among the class of invariant tests are found. The group of transformations leaving the problems invariant is the group of nonsingular matrices. The maximal invariant statistic is the ordered characteristic roots of S 1 S 2 −1, where S i , i = 1, 2, are the sample covariance matrices. Several tests based on the largest and smallest roots are proven to be inadmissible. Other tests are examined for admissibility in the class of invariant tests. The problem of testing for sphericity of a normal covariance matrix is also studied. Again a minimal complete class of invariant tests is found. The popular tests are again examined for admissibility and inadmissibility in the class of invariant tests.

On Testing the Equality of Two Exponential Distributions

This paper derives the null distribution of the likelihood ratio (LR) criterion for testing the equality of location parameters and scale parameters of two exponential distributions, based on type II censored data. It is shown that this null distribution is the same as that of Bartlett's modified LR test for equality of two bivariate normal covariance matrices. Box's chi squared approximation to the criterion is found to be satisfactory both for equal and for unequal sample sizes. The power of the criterion is then compared with Hogg-Tanis's iterated procedure and Perng's test. A numerical example is also included.

Leg loading in snow skiing: Computer analyses

The complete excitation between the boot and ski was measured during snow skiing. Data from two six degree-of-freedom dynamometers mounted inside the ski were pulse-coded and FM transmitted to a remote receiving station. Data from both cruising skiing maneuvers (snowplow, stem christiana, and parallel christiana) and authentic falls were recorded. These data were subsequently analyzed both in real time and by digital computer. In the computer analyses, attention was focused on the resultant boot loads. This paper discusses two aspects of the data reduction by digital computer, computation of power spectral density estimates and normalized covariance matrices. Both of these analysis techniques not only characterize the excitation environment but also lead to important conclusions regarding safety release binding design.