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

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

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

On Testing for Impropriety of Complex-Valued Gaussian Vectors

Published versio

Quaternion Matrices : Statistical Properties and Applications to Signal Processing and Wavelets

Similarly to how complex numbers provide a possible framework for extending scalar signal processing techniques to 2-channel signals, the 4-dimensional hypercomplex algebra of quaternions can be used to represent signals with 3 or 4 components. For a quaternion random vector to be suited for quaternion linear processing, it must be (second-order) proper. We consider the likelihood ratio test (LRT) for propriety, and compute the exact distribution for statistics of Box type, which include this LRT. Various approximate distributions are compared. The Wishart distribution of a quaternion sample covariance matrix is derived from first principles. Quaternions are isomorphic to an algebra of structured 4x4 real matrices. This mapping is our main tool, and suggests considering more general real matrix problems as a way of investigating quaternion linear algorithms. A quaternion vector autoregressive (VAR) time-series model is equivalent to a structured real VAR model. We show that generalised least squares (and Gaussian maximum likelihood) estimation of the parameters reduces to ordinary least squares, but only if the innovations are proper. A LRT is suggested to simultaneously test for quaternion structure in the regression coefficients and innovation covariance. Matrix-valued wavelets (MVWs) are generalised (multi)wavelets for vector-valued signals. Quaternion wavelets are equivalent to structured MVWs. Taking into account orthogonal similarity, all MVWs can be constructed from non-trivial MVWs. We show that there are no non-scalar non-trivial MVWs with short support [0,3]. Through symbolic computation we construct the families of shortest non-trivial 2x2 Daubechies MVWs and quaternion Daubechies wavelets.Open Acces

The Precursors and Products of Justice Climates: Group Leader Antecedents and Employee Attitudinal Consequences

Drawing on the organizational justice, organizational climate, leadership and personality, and social comparison theory literatures, we develop hypotheses about the effects of leader personality on the development of three types of justice climates (e.g., procedural, interpersonal, and informational), and the moderating effects of these climates on individual level justice- attitude relationships. Largely consistent with the theoretically-derived hypotheses, the results showed that leader (a) agreeableness was positively related to procedural, interpersonal and informational justice climates, (b) conscientiousness was positively related to a procedural justice climate, and (c) neuroticism was negatively related to all three types of justice climates. Further, consistent with social comparison theory, multilevel data analyses revealed that the relationship between individual justice perceptions and job attitudes (e.g., job satisfaction, commitment) was moderated by justice climate such that the relationships were stronger when justice climate was high