Cross-spectral Tensor Research Articles

EEG functional connectivity as a Riemannian mediator: An application to malnutrition and cognition.

Mediation analysis assesses whether an exposure directly produces changes in cognitive behavior or is influenced by intermediate "mediators". Electroencephalographic (EEG) spectral measurements have been previously used as effective mediators representing diverse aspects of brain function. However, it has been necessary to collapse EEG measures onto a single scalar using standard mediation methods. In this article, we overcome this limitation and examine EEG frequency-resolved functional connectivity measures as a mediator using the full EEG cross-spectral tensor (CST). Since CST samples do not exist in Euclidean space but in the Riemannian manifold of positive-definite tensors, we transform the problem, allowing for the use of classic multivariate statistics. Toward this end, we map the data from the original manifold space to the Euclidean tangent space, eliminating redundant information to conform to a "compressed CST." The resulting object is a matrix with rows corresponding to frequencies and columns to cross spectra between channels. We have developed a novel matrix mediation approach that leverages a nuclear norm regularization to determine the matrix-valued regression parameters. Furthermore, we introduced a global test for the overall CST mediation and a test to determine specific channels and frequencies driving the mediation. We validated the method through simulations and applied it to our well-studied 50+-year Barbados Nutrition Study dataset by comparing EEGs collected in school-age children (5-11 years) who were malnourished in the first year of life with those of healthy classmate controls. We hypothesized that the CST mediates the effect of malnutrition on cognitive performance. We can now explicitly pinpoint the frequencies (delta, theta, alpha, and beta bands) and regions (frontal, central, and occipital) in which functional connectivity was altered in previously malnourished children, an improvement to prior studies. Understanding the specific networks impacted by a history of postnatal malnutrition could pave the way for developing more targeted and personalized therapeutic interventions. Our methods offer a versatile framework applicable to mediation studies encompassing matrix and Hermitian 3D tensor mediators alongside scalar exposures and outcomes, facilitating comprehensive analyses across diverse research domains.

Harmonized-Multinational qEEG norms (HarMNqEEG)

This paper extends frequency domain quantitative electroencephalography (qEEG) methods pursuing higher sensitivity to detect Brain Developmental Disorders. Prior qEEG work lacked integration of cross-spectral information omitting important functional connectivity descriptors. Lack of geographical diversity precluded accounting for site-specific variance, increasing qEEG nuisance variance. We ameliorate these weaknesses. (i) Create lifespan Riemannian multinational qEEG norms for cross-spectral tensors. These norms result from the HarMNqEEG project fostered by the Global Brain Consortium. We calculate the norms with data from 9 countries, 12 devices, and 14 studies, including 1564 subjects. Instead of raw data, only anonymized metadata and EEG cross-spectral tensors were shared. After visual and automatic quality control, developmental equations for the mean and standard deviation of qEEG traditional and Riemannian DPs were calculated using additive mixed-effects models. We demonstrate qEEG “batch effects” and provide methods to calculate harmonized z-scores. (ii) We also show that harmonized Riemannian norms produce z-scores with increased diagnostic accuracy predicting brain dysfunction produced by malnutrition in the first year of life and detecting COVID induced brain dysfunction. (iii) We offer open code and data to calculate different individual z-scores from the HarMNqEEG dataset. These results contribute to developing bias-free, low-cost neuroimaging technologies applicable in various health settings.

Spatial correlations and degree of polarization in homogeneous electromagnetic fields.

We consider stationary electromagnetic fields modeled as superpositions of unpolarized and angularly uncorrelated plane waves and show that in anisotropic case the electric cross-spectral tensor is proportional to the imaginary part of the Green tensor. This is as for blackbody radiation, but here the field need not be in thermal equilibrium. We also evaluate the degree of polarization for a homogeneous but nonisotropic field for which the plane waves propagate within a cone of angles. The results are compared with the known polarization properties of blackbody radiation.

Examination of large-scale structures in a turbulent plane mixing layer. Part 1. Proper orthogonal decomposition

Large-scale structures in a plane turbulent mixing layer are studied through the use of the proper orthogonal decomposition (POD). Extensive experimental measurements are obtained in a turbulent plane mixing layer by means of two cross-wire rakes aligned normal to the direction of the mean shear and perpendicular to the mean flow direction. The measurements are acquired well into the asymptotic region. From the measured velocities the two-point spectral tensor is calculated as a function of separation in the cross-stream direction and spanwise and streamwise wavenumbers. The continuity equation is then used for the calculation of the non-measured components of the tensor. The POD is applied using the cross-spectral tensor as its kernel. This decomposition yields an optimal basis set in the mean square sense. The energy contained in the POD modes converges rapidly with the first mode being dominant (49% of the turbulent kinetic energy). Examination of these modes shows that the first mode contains evidence of both known flow organizations in the mixing layer, i.e. quasi-two-dimensional spanwise structures and streamwise aligned vortices. Using the shot-noise theory the dominant mode of the POD is transformed back into physical space. This structure is also indicative of the known flow organizations.

Polarization and space-frequency coherence in radiometric emission

New results are derived for the relationship between polarization and coherence in radiometric fields. We consider the vector theory of coherence for thermal radiation emission from a planar, opaque, specular surface held at fixed, uniform temperature, in the asymptotic limit in which a radiometric description of the field is applicable. Three dyadic equations are derived here, which relate the three cross-spectral tensors to the components of the Stokes vector characterizing the polarization state of the field in transport theory. The second-order coherence properties of the radiometric field are determined once the Stokes vector has been specified.

Radiometric coherence tensors for thermal radiation emission from an opaque specular surface

For the first time to the author's knowledge, the electric, magnetic, and mixed cross-spectral tensors are derived for the thermal radiation emitted into free space from an opaque, planar, specular surface of finite size, held at uniform temperature. The results are valid in the asymptotic limit in which the mean-field energy density and energy flux are in agreement with the laws of classical radiometry, while at the same time field correlation effects are included in the description of the radiated field. Furthermore, the results are applicable to both wide and narrow angular diameter sources. The derivation is based on the generalized Kirchhoff radiation law, and the coherence tensor components can be viewed as a generalization to electromagnetic waves of the van Cittert–Zernike theorem in the radiometric limit.

Spatial behavior of the coherence tensors for blackbody radiation emission

The paper discusses the tensor coherence properties of thermal radiation emitted into space by a blackbody source and presents numerical results for observation points restricted to a volume of space with linear dimensions small compared with the distance to the source, from which the source subtends a solid angle of circular cross section. For the first time, to my knowledge, surface and contour plots are given, showing the magnitude of the electric, magnetic, and mixed cross-spectral tensor components on selected planes as a function of the spatial separation between the observation points. Also, it is shown how these distributions change as the angular diameter of the source is varied. Small angular diameter sources are well described by the van Cittert–Zernike theorem of scalar coherence theory. For large angular diameter sources all nonzero components are of the same order of magnitude and are not properly described by the scalar theory. However, the trace of the electric tensor is in agreement with scalar theory.

On the cross-spectral tensors for black-body emission into space

Abstract The cross-spectral tensors for black-body emission into space are derived, and the results expressed in terms of Debye integrals which have recently been found to be important in the scalar theory of partial coherence. For the case of an infinite black-body surface, closed-form expressions can be obtained for the cross-spectral tensors which are exact within the constraints of the local thermodynamic equilibrium assumption and the idealization of a perfectly absorbing surface. The trace of the electric cross-spectral tensor agrees with the recent results in the scalar theory of emission from uniform planar Lambertian sources.

On the cross-spectral tensors for black-body emission into space

The cross-spectral tensors for black-body emission into space are derived, and the results expressed in terms of Debye integrals which have recently been found to be important in the scalar theory of partial coherence. For the case of an infinite black-body surface, closed-form expressions can be obtained for the cross-spectral tensors which are exact within the constraints of the local thermodynamic equilibrium assumption and the idealization of a perfectly absorbing surface. The trace of the electric cross-spectral tensor agrees with the recent results in the scalar theory of emission from uniform planar Lambertian sources.

The Effect of Line Averaging on Scalar Flux Measurements with a Sonic Anemometer near the Surface

Abstract We present a theoretical analysis of the effect of line averaging by a sonic anemometer on scalar fluxes and an observational study of this phenomenon in the atmospheric surface layer. The theoretical analysis rests on an axisymmetric model for the cross-spectral tensor of vertical velocity and scalar fluctuations, limiting the validity of the derived transfer function to a constant-flux layer. Observations of temperature flux in the unstable surface layer confirm the theoretical prediction that line averaging does not significantly affect the flux estimate down to heights only several times the sonic path length. However, the observations exhibit large scatter at small height, indicating that problems with the representativity of the measurement and not with line averaging may become a limiting factor. Based on the analysis of the Kansas data and the characteristics of the transfer function, we infer that temperature flux measurements will, in general, be affected by some line averaging under st...

Quantum theory of a one-dimensional optical cavity with output coupling. II. Thermal radiation field and the fluctuation-dissipation theorem

The normally ordered correlation function and the power spectrum (cross spectral tensor) of a thermal radiation field existing inside and outside a one-dimensional optical cavity which transmits radiation at one of its end surfaces are calculated on the basis of a multimode formulation of the field developed in a previous paper. The correlation function is found to consist of a series of trigamma functions resulting from retarded correlations due to reflections at the cavity end surfaces. The power spectrum is a product of the Planck distribution and the mode functions of the universe which have a periodic factor with the period being equal to the cavity mode separation. For the purpose of obtaining a description of the cavity which allows one to deduce its thermal noise property, a form of fluctuation-dissipation theorem is derived which connects, in a one-dimensional space with arbitrary structure, the normally ordered correlation function with the admittance for the electric field probed by a current. Application of this theorem to the model cavity leads to an explicit and simultaneous description of the fluctuation and dissipation associated with all the cavity modes. The dissipation is shown to be due to radiation of electromagnetic energy into the free space. The above relation is reduced, under the assumption of high cavity quality factor and of single-cavity-mode selection, to that of a Markovian noise associated with a single decay constant. The consistency of these results with the field commutation relation is examined.

Coherence Properties of Blackbody Radiation. III. Cross-Spectral Tensors

In a recent paper, cross-spectra tensors of quantized stationary electromagnetic fields were introduced and some of their properties were discussed. In the present paper, closed expressions for the cross-spectral tensors of blackbody radiation are derived and their behavior is illustrated by a number of diagrams. It is found that the correlation distance of a spectral component of wavelength $\ensuremath{\lambda}$ is itself of the order of $\ensuremath{\lambda}$ and is therefore independent of the temperature of the radiation.

Correlation Theory of Quantized Electromagnetic Fields. II. Stationary Fields and Their Spectral Properties

In Paper I of this investigation, dynamical laws which describe the space-time development of second-order coherence tensors of a quantized electromagnetic field in vacuo, and the associated conservation laws, were derived. In the first part of the present paper these results are specialized to stationary fields. Second-order cross-spectral tensors for such fields are then introduced, and some of their properties are discussed. A relation analogous to the Wiener-Khintchine theorem of the theory of stationary random processes is derived. Various non-negative-definiteness conditions obeyed by the cross-spectral tensors are established, and equations which govern the spatial variation of these tensors are deduced. Certain analytic properties of the correlation tensors are derived, and some of their consequences are examined. It is also shown that in the limiting case when the two space-time arguments of the coherence tensors coincide, two of our conservation laws reduce to the averaged form of the energy and the momentum conservation laws of the electromagnetic field.