The Multigrid Beta Function Approach for Modeling of Background Error Covariance in the Real-Time Mesoscale Analysis (RTMA)

We describe a method for the efficient generation of the covariance operators of a variational data assimilation scheme which is suited to implementation on a massively parallel computer. The elementary components of this scheme are what we call ‘beta filters’, since they are based on the same spatial profiles possessed by the symmetric beta distributions of probability theory. These approximately Gaussian (bell-shaped) polynomials blend smoothly to zero at the ends of finite intervals, which makes them better suited to parallelization than the present quasi-Gaussian ‘recursive filters’ used in operations at NCEP. These basic elements are further combined at a hierarchy of different spatial scales into an overall multigrid structure formulated to preserve the necessary self-adjoint attribute possessed by any valid covariance operator. This paper describes the underlying idea of the beta filter and discusses how generalized Helmholtz operators can be enlisted to weight the elementary contributions additively in such a way that the covariance operators may exhibit realistic negative sidelobes, which are not easily obtained through the recursive filter paradigm. The main focus of the paper is on the basic logistics of the multigrid structure by which more general covariance forms are synthesized from the basic quasi-Gaussian elements. We describe several ideas on how best to organize computation, which led us to a generalization of this structure which made it practical so that it can efficiently perform with any rectangular arrangement of processing elements. Some simple idealized examples of the applications of these ideas are given.

Quadratic-exponential functionals of Gaussian quantum processes

This paper is concerned with exponential moments of integral-of-quadratic functions of quantum processes with canonical commutation relations of position-momentum type. Such quadratic-exponential functionals (QEFs) arise as robust performance criteria in control problems for open quantum harmonic oscillators (OQHOs) driven by bosonic fields. We develop a randomised representation for the QEF using a Karhunen–Loeve expansion of the quantum process on a bounded time interval over the eigenbasis of its two-point commutator kernel, with noncommuting position-momentum pairs as coefficients. This representation holds regardless of a particular quantum state and employs averaging over an auxiliary classical Gaussian random process whose covariance operator is specified by the commutator kernel. This allows the QEF to be related to the moment-generating functional of the quantum process and computed for multipoint Gaussian states. For stationary Gaussian quantum processes, we establish a frequency-domain formula for the QEF rate in terms of the Fourier transform of the quantum covariance kernel in composition with trigonometric functions. A differential equation is obtained for the QEF rate with respect to the risk sensitivity parameter for its approximation and numerical computation. The QEF is also applied to large deviations and worst-case mean square cost bounds for OQHOs in the presence of statistical uncertainty with a quantum relative entropy description.

Partial Separability and Functional Graphical Models for Multivariate Gaussian Processes

The covariance structure of multivariate functional data can be highly complex, especially if the multivariate dimension is large, making extensions of statistical methods for standard multivariate data to the functional data setting challenging. For example, Gaussian graphical models have recently been extended to the setting of multivariate functional data by applying multivariate methods to the coefficients of truncated basis expansions. However, a key difficulty compared to multivariate data is that the covariance operator is compact, and thus not invertible. The methodology in this paper addresses the general problem of covariance modelling for multivariate functional data, and functional Gaussian graphical models in particular. As a first step, a new notion of separability for the covariance operator of multivariate functional data is proposed, termed partial separability, leading to a novel Karhunen–Loève-type expansion for such data. Next, the partial separability structure is shown to be particularly useful in order to provide a well-defined functional Gaussian graphical model that can be identified with a sequence of finite-dimensional graphical models, each of identical fixed dimension. This motivates a simple and efficient estimation procedure through application of the joint graphical lasso. Empirical performance of the method for graphical model estimation is assessed through simulation and analysis of functional brain connectivity during a motor task.

Generalized partially functional linear model

In this paper, we propose a generalized functional linear regression model with scalar and functional multiple predictors. We develop maximum likelihood estimators for the regression coefficients. For the functional predictors, we adopt the method of functional principal component analysis to reduce their dimensions. We then propose the generalized auto-covariance operator, based on which an appropriate measure quantifies the difference between the estimators and their true values is established. The asymptotic joint distribution of estimated regression functions is proved. For the scalar predictors, we establish a distance between the estimated value and the true value, and prove the asymptotic property of the estimated regression coefficients. Extensive simulation experiment results are consistent with the theoretical result. Finally, two application examples of the model are given. One is sleep quality study where we studied the effects of heart rate, percentage of sleep time on total sleep in bed, wake after sleep onset and number of wakening during the night on sleep quality in 22 healthy people. The other one is mortality rate where we studied the effects of air quality index, temperature, relative humidity , GDP per capita and the number of beds per thousand people on the mortality rate across 80 major cities in China.

A Conditional Simulation Method for Predicting Wind Pressure Fields of Large-Span Spatial Structures

Wind load is among the control loads for large-span spatial structures. Wind tunnel test is one of the commonly used methods for measuring wind pressure fields of different kinds of structures. However, due to the limited wind pressure data obtained from wind tunnel testing, it is quite meaningful to employ the limited measured data to predict the unknown wind pressure at target points. Considering the complexity of wind pressure fields of large-span spatial structures, a simplified nonparametric method based on conditional simulation is proposed to predict the unknown pressures using the existing data. The Karhunen–Loève (KL for short) expansion is employed to represent wind pressure random variants as eigenfunctions of the covariance operator. To reduce the variant dimensionality, the nearest neighboring estimator is given for the transition distribution of the KL expansion. The targeted wind pressure fields are obtained by expanding the Fourier basis of the eigenfunction and estimating its expansion coefficients. The proposed method is applied to estimate wind pressures on a gable roof building. The relevant parameters of the wind pressure field are obtained, and the results compare well with those from wind tunnel testing, with higher efficiency. The proposed method effectively reduces the dimensionality of the predicted wind pressures, with reduced errors, higher accuracy, and increased efficiency.

An Enhanced Joint Hilbert Embedding-Based Metric to Support Mocap Data Classification with Preserved Interpretability

Motion capture (Mocap) data are widely used as time series to study human movement. Indeed, animation movies, video games, and biomechanical systems for rehabilitation are significant applications related to Mocap data. However, classifying multi-channel time series from Mocap requires coding the intrinsic dependencies (even nonlinear relationships) between human body joints. Furthermore, the same human action may have variations because the individual alters their movement and therefore the inter/intraclass variability. Here, we introduce an enhanced Hilbert embedding-based approach from a cross-covariance operator, termed EHECCO, to map the input Mocap time series to a tensor space built from both 3D skeletal joints and a principal component analysis-based projection. Obtained results demonstrate how EHECCO represents and discriminates joint probability distributions as kernel-based evaluation of input time series within a tensor reproducing kernel Hilbert space (RKHS). Our approach achieves competitive classification results for style/subject and action recognition tasks on well-known publicly available databases. Moreover, EHECCO favors the interpretation of relevant anthropometric variables correlated with players’ expertise and acted movement on a Tennis-Mocap database (also publicly available with this work). Thereby, our EHECCO-based framework provides a unified representation (through the tensor RKHS) of the Mocap time series to compute linear correlations between a coded metric from joint distributions and player properties, i.e., age, body measurements, and sport movement (action class).

Double-scale diffusive wave model dedicated to spatial river observation and associated covariance kernel for variational data assimilation

<p>The spatial altimetry provides an important amount of water surface height data from multi-missions satellites (especially Jason-3, Sentinel-3A/B and the forthcoming NASA-CNES SWOT mission). To exploit at best the potential of spatial altimetry, the present study proposes on the derivation of a model adapted to spatial observations scale; a diffusive-wave type model but adapted to a double scale [1].</p><p>Moreover, Green-like kernel can be employed to derived covariance operators, therefore they may provide an approximation of the covariance kernel of the background error in Variational Data Assimilation processes. Following the derivation of the aforementioned original flow model, we present the derivation of a Green kernel which provides an approximation of the covariance kernel of the background error for the bathymetry (i.e. the control variable) [2].</p><p>This approximation of the covariance kernel is used to infer the bathymetry in the classical Saint-Venant’s (Shallow-Water) equations with better accuracy and faster convergence than if not introducing an adequate covariance operator [3].</p><p>Moreover, this Green kernel helps to analyze the sensitivity of the double-scale diffusive waves (or even the Saint-Venant’s equations) with respect to the bathymetry.</p><p>Numerical results are analyzed on real like datasets (derived from measurements of the Rio Negro, Amazonia basin).</p><p>The double-scale diffusive wave provide more accurate results than the classical version. Next, in terms of inversions, the derived physically-based covariance operators enable to improve the inferences, compared to the usual exponential one.</p><p>[1] T. Malou, J. Monnier "Double-scale diffusive wave equations dedicated to spatial river observations". In prep.</p><p>[2] T. Malou, J. Monnier "Physically-based covariance kernel for variational data assimilation in spatial hydrology". In prep.</p><p>[3] K. Larnier, J. Monnier, P.-A. Garambois, J. Verley. "River discharge and bathymetry estimations from SWOT altimetry measurements". Inv. Pb. Sc. Eng (2020).</p>

STOCHASTIC DIFFERENTIAL GAMES IN DISTRIBUTED SYSTEMS WITH DELAY

We study a differential game of approach in a delay stochastic system. The evolution of the system is described by Ito`s linear stochastic differential equation in Hilbert space. The considered Hilbert spaces are assumed to be real and separable. The Wiener process takes values in a Hilbert space and has a nuclear symmetric positive covariance operator. The pursuer and evader controls are non-anticipating random processes, taking on values, generally, in different Hilbert spaces. The operator multiplying the system state is the generator of an analytic semigroup. Solutions of the equation are represented with the help of a formula of variation of constants by the initial data and the control block. The delay effect is taken into account by summing shift type operators. To study the differential game, the method of resolving functions is extended to case of delay stochastic systems in Hilbert spaces. The technique of set-valued mappings and their selectors is used. We consider the application of obtained results in abstract Hilbert spaces to systems described by stochastic partial differential equations with time delay. By taking into account a random external influence and time delay, we study the heat propagation process with controlled distributed heat source and leak.

Kernel regression, minimax rates and effective dimensionality: Beyond the regular case

We investigate if kernel regularization methods can achieve minimax convergence rates over a source condition regularity assumption for the target function. These questions have been considered in past literature, but only under specific assumptions about the decay, typically polynomial, of the spectrum of the the kernel mapping covariance operator. In the perspective of distribution-free results, we investigate this issue under much weaker assumption on the eigenvalue decay, allowing for more complex behavior that can reflect different structure of the data at different scales.