529 research outputs found
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
Tapering promotes propriety for Fourier transforms of real-valued time series
We examine Fourier transforms of real-valued stationary time series from the point of view of the statistical propriety. Processes with a large dynamic range spectrum have transforms that are very significantly improper for some frequencies; the real and imaginary parts can be highly correlated, and the periodogram will not have the standard chi-square distribution at these frequencies, nor have two degrees of freedom. Use of a taper reduces impropriety to just frequencies close to zero and Nyquist only, and frequency ranges where the propriety breaks down can be quite accurately and easily predicted by half the autocorrelation width of |H * H(2f)|, denoted by c, where H(f) is the Fourier transform of the taper and * denotes convolution. For vector time series we derive an improved distributional approximation for minus twice the log of the generalized likelihood ratio test statistic for testing for propriety of the Fourier transform at any frequency, and compare frequency range cutoffs for propriety determined by the hypothesis test with those determined by c
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 Kolmogorov–Smirnov goodness of fit and Ansari-Bradley non-parametric dispersion tests. Impropriety of the MT responses at both sites is pervasive, hence the improper Cramér-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
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 → ∞, 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
On Testing for Impropriety of Complex-Valued Gaussian Vectors
Published versio
Testing quaternion properness: generalized likelihood ratios and locally most powerful invariants
This paper considers the problem of determining whether a quaternion random vector is proper or not, which is an important problem because the structure of the optimal linear processing depends on the specific kind of properness. In particular, we focus on the Gaussian case and consider the two main kinds of quaternion properness, which yields three different binary hypothesis testing problems. The testing problems are solved by means of the generalized likelihood ratio tests (GLRTs) and the locally most powerful invariant tests (LMPITs), which can be derived even without requiring an explicit expression for the maximal invariant statistics. Some simulation examples illustrate the performance of the proposed tests, which allows us to conclude that, for moderate sample sizes, it is advisable to use the LMPITs.This work was supported by the Spanish Government, Ministerio de Ciencia e Innovación (MICINN), under projects COSIMA (TEC2010-19545-C04-03) and COMONSENS (CSD2008-00010, CONSOLIDERINGENIO 2010 Program)
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