13 research outputs found
On Testing for Impropriety of Complex-Valued Gaussian Vectors
Published versio
Asymptotic regime for impropriety tests of complex random vectors
Impropriety testing for complex-valued vector has been considered lately due
to potential applications ranging from digital communications to complex media
imaging. This paper provides new results for such tests in the asymptotic
regime, i.e. when the vector dimension and sample size grow commensurately to
infinity. The studied tests are based on invariant statistics named impropriety
coefficients. Limiting distributions for these statistics are derived, together
with those of the Generalized Likelihood Ratio Test (GLRT) and Roy's test, in
the Gaussian case. This characterization in the asymptotic regime allows also
to identify a phase transition in Roy's test with potential application in
detection of complex-valued low-rank subspace corrupted by proper noise in
large datasets. Simulations illustrate the accuracy of the proposed asymptotic
approximations.Comment: 11 pages, 8 figures, submitted to IEEE TS
Frequency-Domain Stochastic Modeling of Stationary Bivariate or Complex-Valued Signals
There are three equivalent ways of representing two jointly observed
real-valued signals: as a bivariate vector signal, as a single complex-valued
signal, or as two analytic signals known as the rotary components. Each
representation has unique advantages depending on the system of interest and
the application goals. In this paper we provide a joint framework for all three
representations in the context of frequency-domain stochastic modeling. This
framework allows us to extend many established statistical procedures for
bivariate vector time series to complex-valued and rotary representations.
These include procedures for parametrically modeling signal coherence,
estimating model parameters using the Whittle likelihood, performing
semi-parametric modeling, and choosing between classes of nested models using
model choice. We also provide a new method of testing for impropriety in
complex-valued signals, which tests for noncircular or anisotropic second-order
statistical structure when the signal is represented in the complex plane.
Finally, we demonstrate the usefulness of our methodology in capturing the
anisotropic structure of signals observed from fluid dynamic simulations of
turbulence.Comment: To appear in IEEE Transactions on Signal Processin
Testing blind separability of complex Gaussian mixtures
The separation of a complex mixture based solely on second-order statistics can be achieved using the Strong Uncorrelating Transform (SUT) if and only if all sources have distinct circularity coefficients. However, in most problems we do not know the circularity coefficients, and they must be estimated from observed data. In this work, we propose a detector, based on the generalized likelihood ratio test (GLRT), to test the separability of a complex Gaussian mixture using the SUT. For the separable case (distinct circularity coefficients), the maximum likelihood (ML) estimates are straightforward. On the other hand, for the non-separable case (at least one circularity coefficient has multiplicity greater than one), the ML estimates are much more difficult to obtain. To set the threshold, we exploit Wilks' theorem, which gives the asymptotic distribution of the GLRT under the null hypothesis. Finally, numerical simulations show the good performance of the proposed detector and the accuracy of Wilks' approximation
A frequency domain test for propriety of complex-valued vector time series
This paper proposes a frequency domain approach to test the hypothesis that a stationary complexvalued
vector time series is proper, i.e., for testing whether the vector time series is uncorrelated with its
complex conjugate. If the hypothesis is rejected, frequency bands causing the rejection will be identified
and might usefully be related to known properties of the physical processes. The test needs the associated
spectral matrix which can be estimated by multitaper methods using, say, K tapers. Standard asymptotic
distributions for the test statistic are of no use since they would require K â â, but, as K increases
so does resolution bandwidth which causes spectral blurring. In many analyses K is necessarily kept
small, and hence our efforts are directed at practical and accurate methodology for hypothesis testing for
small K. Our generalized likelihood ratio statistic combined with exact cumulant matching gives very
accurate rejection percentages. We also prove that the statistic on which the test is based is comprised of
canonical coherencies arising from our complex-valued vector time series. Frequency specific tests are
combined using multiple hypothesis testing to give an overall test. Our methodology is demonstrated on
ocean current data collected at different depths in the Labrador Sea. Overall this work extends results
on propriety testing for complex-valued vectors to the complex-valued vector time series setting
Locally Most Powerful Invariant Tests for Correlation and Sphericity of Gaussian Vectors
In this paper we study the existence of locally most powerful invariant tests
(LMPIT) for the problem of testing the covariance structure of a set of
Gaussian random vectors. The LMPIT is the optimal test for the case of close
hypotheses, among those satisfying the invariances of the problem, and in
practical scenarios can provide better performance than the typically used
generalized likelihood ratio test (GLRT). The derivation of the LMPIT usually
requires one to find the maximal invariant statistic for the detection problem
and then derive its distribution under both hypotheses, which in general is a
rather involved procedure. As an alternative, Wijsman's theorem provides the
ratio of the maximal invariant densities without even finding an explicit
expression for the maximal invariant. We first consider the problem of testing
whether a set of -dimensional Gaussian random vectors are uncorrelated or
not, and show that the LMPIT is given by the Frobenius norm of the sample
coherence matrix. Second, we study the case in which the vectors under the null
hypothesis are uncorrelated and identically distributed, that is, the
sphericity test for Gaussian vectors, for which we show that the LMPIT is given
by the Frobenius norm of a normalized version of the sample covariance matrix.
Finally, some numerical examples illustrate the performance of the proposed
tests, which provide better results than their GLRT counterparts
Component-wise conditionally unbiased widely linear MMSE estimation
AbstractBiased estimators can outperform unbiased ones in terms of the mean square error (MSE). The best linear unbiased estimator (BLUE) fulfills the so called global conditional unbiased constraint when treated in the Bayesian framework. Recently, the component-wise conditionally unbiased linear minimum mean square error (CWCU LMMSE) estimator has been introduced. This estimator preserves a quite strong (namely the CWCU) unbiased condition which in effect sufficiently represents the intuitive view of unbiasedness. Generally, it is global conditionally biased and outperforms the BLUE in a Bayesian MSE sense. In this work we briefly recapitulate CWCU LMMSE estimation under linear model assumptions, and additionally derive the CWCU LMMSE estimator under the (only) assumption of jointly Gaussian parameters and measurements. The main intent of this work, however, is the extension of the theory of CWCU estimation to CWCU widely linear estimators. We derive the CWCU WLMMSE estimator for different model assumptions and address the analytical relationships between CWCU WLMMSE and WLMMSE estimators. The properties of the CWCU WLMMSE estimator are deduced analytically, and compared by simulation to global conditionally unbiased as well as WLMMSE counterparts with the help of a parameter estimation example and a data estimation/channel equalization application
Magnetotelluric data, stable distributions and impropriety: an existential combination
Author Posting. © Author, 2014. This article is posted here by permission of The Royal Astronomical Society for personal use, not for redistribution. The definitive version was published in Geophysical Journal International 198 (2014): 622-636, doi: 10.1093/gji/ggu121.The robust statistical model of a Gaussian core contaminated by outlying data that underlies robust estimation of the magnetotelluric (MT) response function has been re-examined. The residuals from robust estimators are systematically long tailed compared to a distribution based on the Gaussian, and hence are inconsistent with the robust model. Instead, MT data are pervasively described by the alpha stable distribution family whose variance and sometimes mean are undefined. A maximum likelihood estimator (MLE) that exploits the stable nature of MT data is formulated, and its two-stage implementation in which stable parameters are first fit to the data and then the MT responses are solved for is described. The MLE is shown to be inherently robust, but differs from the conventional robust estimator because it is based on a model derived from the data, while robust estimators are ad hoc, being based on the robust model that is inconsistent with actual data. Propriety versus impropriety of the complex MT response was investigated, and a likelihood ratio test for propriety and its null distribution was established. The Cramér-Rao lower bounds for the covariance matrix of proper and improper MT responses were specified.
The MLE was applied to exemplar long period and broad-band data sets from South Africa. Both are shown to be significantly stably distributed using the KolmogorovâSmirnov goodness of fit and Ansari-Bradley non-parametric dispersion tests. Impropriety of the MT responses at both sites is pervasive, hence the improper CramĂ©r-Rao bound was used to estimate the MLE covariance. The MLE is shown to be nearly unbiased and well described by a Gaussian distribution based on bootstrap simulation. The MLE was compared to a conventional robust estimator, establishing that the standard errors of the former are systematically smaller than for the latter and that the standardized differences between them exhibit excursions that are both too frequent and too large to be described by a Gaussian model. This is ascribed to pervasive bias of the robust estimator that is to some degree obscured by their systematically large confidence bounds. Finally, a series of topics for further investigation is proposed.This work was supported by NSF grant EAR0809074