Process Covariance Research Articles

Stability analysis and stabilization of linear symmetric matrix-valued continuous, discrete, and impulsive dynamical systems — A unified approach for the stability analysis and the stabilization of linear systems

Matrix-valued dynamical systems are an important class of systems that can describe important processes such as covariance/second-order moment processes, or processes on manifolds and Lie Groups. We address here the case of processes that leave the cone of positive semidefinite matrices invariant, thereby including covariance and second-order moment processes. Both the continuous-time and the discrete-time cases are first considered. In the LTV case, the obtained stability and stabilization conditions are expressed as differential and difference Lyapunov conditions which are equivalent, in the LTI case, to some spectral conditions for the generators of the processes. Convex stabilization conditions are also obtained in both the continuous-time and the discrete-time setting. It is proven that systems with constant delays are stable provided that the systems with zero-delays are stable—which mirrors existing results for linear positive systems. The results are then extended and unified into an impulsive formulation for which similar results are obtained. The proposed framework is very general and can recover and/or extend many of the existing results in the literature on linear systems related to (mean-square) exponential (uniform) stability. Several examples are discussed to illustrate this claim by deriving stability conditions for stochastic systems driven by Brownian motion and Poissonian jumps, Markov jump systems, (stochastic) switched systems, (stochastic) impulsive systems, (stochastic) sampled-data systems, and all their possible combinations.

A Simultaneous Estimation of the Baseline Intensity and Parameters for Modulated Renewal Processes

This paper proposes a semiparametric solution to estimate the intensity (hazard) function of modulated renewal processes: a nonparametric estimate for the baseline intensity function together with a parametric estimate of the model parameters of the covariate processes. Based on the martingale property associated with the conditional intensity, we construct a statistic from a residual analysis to estimate the baseline renewal intensity function, when the model parameters of the covariate processes are known. In addition, when the baseline intensity is obtained, the model parameters can be estimated using the usual maximum likelihood estimation. In practice, both the baseline intensity and model parameters are suggested to be estimated simultaneously via an expectation–maximization (E–M)-type iterative algorithm. A more important feature of the newly proposed algorithm is that, given n events in the observation dataset, its computation time is of order O(n2), while the Nelson–Aalen–Breslow estimator takes a computation time of order O(n3). For illustration, we apply the proposed estimation procedure to a set of data simulated from a modulated gamma renewal process and the aftershock sequence following the Ms8 Wenchuan earthquake, which occurred in Sichuan Province, China on 12 May 2008.

Multilevel hybrid principal components analysis for region-referenced functional electroencephalography data.

Electroencephalography experiments produce region-referenced functional data representing brain signals in the time or the frequency domain collected across the scalp. The data typically also have a multilevel structure with high-dimensional observations collected across multiple experimental conditions or visits. Common analysis approaches reduce the data complexity by collapsing the functional and regional dimensions, where event-related potential (ERP) features or band power are targeted in a pre-specified scalp region. This practice can fail to portray more comprehensive differences in the entire ERP signal or the power spectral density (PSD) across the scalp. Building on the weak separability of the high-dimensional covariance process, the proposed multilevel hybrid principal components analysis (M-HPCA) utilizes dimension reduction tools from both vector and functional principal components analysis to decompose the total variation into between- and within-subject variance. The resulting model components are estimated in a mixed effects modeling framework via a computationally efficient minorization-maximization algorithm coupled with bootstrap. The diverse array of applications of M-HPCA is showcased with two studies of individuals with autism. While ERP responses to match vs mismatch conditions are compared in an audio odd-ball paradigm in the first study, short-term reliability of the PSD across visits is compared in the second. Finite sample properties of the proposed methodology are studied in extensive simulations.

A hybrid repair-replacement policy in the proportional hazards model

Cox's proportional hazards model is widely used to describe the hazard rate of a system deteriorating with age and diagnostic covariates. Existing maintenance in the proportional hazards model was primarily concerned with replacement, resulting in excessive maintenance for many repairable systems. This paper develops a novel hybrid repair-replacement model in the proportional hazards model with a stochastically increasing Markovian covariate process. Preventive repair reduces both age and covariate, and the reduction rate decreases as the number of repairs increases. At an inspection epoch where the system age, covariate state, and repair number are available, the decision-maker considers three possible actions, i.e., no maintenance, preventive repair, and preventive replacement. The objective is to derive the optimal policy that minimizes the long-run average maintenance cost rate. The optimization problem is formulated in the semi-Markov decision process (SMDP) framework. The structural properties of the optimal policy are examined to reduce the policy space. Then a policy-iteration algorithm with a backward policy-improvement step is developed for efficiently finding the optimization results. A practical numerical example with sensitivity analysis is conducted to illustrate the effectiveness of the proposed approach. A comparison with two heuristic policies confirms the superiority of the proposed policy in reducing maintenance costs.

Advances in Noise Modeling for Stochastic Systems in Optimal Control

In this paper some noise models for stochastic control systems are described that differ from the well known model of Brownian motion. Some of the processes are Gaussian such as the family of fractional Brownian motions and other processes are non-Gaussian, especially Rosenblatt processes. These processes have a long range dependence property that can be described by a slow decay of the covariance process. A stochastic calculus for these processes exists that is more limited than the calculus for Brownian motion that is inherited from the martingale property but nevertheless is sufficient for addressing some stochastic control problems. In many physical situations the data demonstrates a long range dependence which can justify a choice from these non-Brownian processes. Explicit solutions of stochastic control problems with quadratic cost functionals having driving noise from the Rosenblatt processes are given.

Bandit and covariate processes, with finite or non-denumerable set of arms

We introduce herein a new approach to nonparametric multi-armed bandit theory involving both the bandit and the covariate processes. Following Berry et al. (1997), we assume a non-denumerable set of arms for the bandit process. The approach we develop herein can be readily extended to continuous-time processes by using ε-greedy randomization and arm elimination instead of dynamic allocation indices. It also carries out a stochastic search with O(1) expected time for a nearly optimal arm at covariate values in a given set B before applying ε-greedy randomization and arm elimination. The procedure is shown to attain the asymptotically minimal rates for the regret over B.

An infinite‐dimensional affine stochastic volatility model

AbstractWe introduce a flexible and tractable infinite‐dimensional stochastic volatility model. More specifically, we consider a Hilbert space valued Ornstein–Uhlenbeck‐type process, whose instantaneous covariance is given by a pure‐jump stochastic process taking values in the cone of positive self‐adjoint Hilbert–Schmidt operators. The tractability of our model lies in the fact that the two processes involved are jointly affine, that is, we show that their characteristic function can be given explicitly in terms of the solutions to a set of generalized Riccati equations. The flexibility lies in the fact that we allow multiple modeling options for the instantaneous covariance process, including state‐dependent jump intensity. Infinite dimensional volatility models arise, for example, when considering the dynamics of forward rate functions in the Heath–Jarrow–Morton–Musiela (HJMM) modeling framework using the Filipović space. In this setting, we discuss various examples: an infinite‐dimensional version of the Barndorf–Nielsen–Shephard stochastic volatility model, as well as covariance processes with a state dependent intensity.

A recursive method for the health assessment of systems using the proportional hazards model

The failure of many practical systems is dependent on both age and a diagnostic covariate process. Cox's proportional hazards model is widely adopted to describe the failure rate of such systems. If the covariate state space is large, it is computationally not feasible to use an analytical method for health assessment at inspection epochs. Existing approximation methods, although can address the above problem, fail to satisfy the critical requirements of modern health management in terms of accuracy, memory storage, and computational speed. This paper develops a novel recursive method to approximately assess the health indices of the proportional hazards model with a Markovian covariate process. The method discretizes age into equidistant and small subintervals. Over each subinterval, an incomplete state transition matrix is constructed with each element measured by its upper and lower bounds. The consideration of dual bounds makes our model more robust than previous methods considering only an upper bound. Then the recursive formulas of various health indices are derived based on the matrixes of consecutive subintervals. Two practical examples demonstrate that the proposed method can produce accurate assessment results with higher efficiency and less memory compared with existing approximation methods.

Feasible Panel GARCH Models: Variance-Targeting Estimation and Empirical Application

For the panel generalized autoregressive conditional heteroskedasticity (GARCH) model, the conditions for the stationarity and positive definiteness of conditional covariance processes are examined. A new feasible specification is constructed for the class of panel GARCH models, and a three-step estimation technique is developed based on a variance-targeting (VT) approach. The consistency and asymptotic normality of the VT estimator are shown when the time dimension tends to infinity and the cross-sectional dimension is fixed. The stationarity and asymptotic properties are discussed for both time and cross-sectional dimensions tend to infinity. The results of Monte Carlo experiments indicate that the finite sample property of the VT estimator is satisfactory, implying that increasing the cross-sectional dimension does not affect the speed of convergence, but shrinks the asymptotic covariance matrix. The empirical results of the analysis of the inflation rates of G7 countries and growth rates for the value of trade in four economic regions indicate that the feasible specification provides a competitive alternative to the class of panel GARCH models. The empirical results indicate that the global financial crisis affects the growth rates of trades, while the influence of the COVID-19 pandemic shows that its effect on inflation rates is insignificant.

Factorized MVDR Deep Beamforming for Multi-Channel Speech Enhancement

Traditionally, adaptive beamformers such as the minimum-variance distortionless response (MVDR) beamformer and generalized eigenvalue beamformer have been widely used for multi-channel speech enhancement with a single-channel postfilter. Recently, several approaches have been proposed to enhance the signals used to estimate speech and noise spatial covariance matrices (SCMs) and process the outputs of the beamformers using deep neural networks (DNNs). However, the preprocessing of the signals for SCMs estimation may disrupt phase relations among input signals and the time-averages used to estimate speech and noise SCMs may not be optimal for beamformer performance even though the estimated signals are close to the ground truth. In this letter, we propose a deep beamforming approach which estimates factors of the MVDR beamformer using a DNN to circumvent the difficulty of the speech and noise SCM estimation. We formulate the MVDR beamformer as a factorized form related to two complex factors and estimate them using a DNN with a cost function comparing beamformed signal and the original clean speech. Experimental results showed that the proposed factorized MVDR beamformer could mimic the characteristics of the MVDR beamformer with true relative transfer function and noise SCM and outperformed the MVDR beamformer with deep learning-based pre- and postprocessing in terms of the perceptual evaluation of speech quality scores.

Covariate-adjusted hybrid principal components analysis for region-referenced functional EEG data.

Electroencephalography (EEG) studies produce region-referenced functional data via EEG signals recorded across scalp electrodes. The high-dimensional data can be used to contrast neurodevelopmental trajectories between diagnostic groups, for example between typically developing (TD) children and children with autism spectrum disorder (ASD). Valid inference requires characterization of the complex EEG dependency structure as well as covariate-dependent heteroscedasticity, such as changes in variation over developmental age. In our motivating study, EEG data is collected on TD and ASD children aged two to twelve years old. The peak alpha frequency, a prominent peak in the alpha spectrum, is a biomarker linked to neurodevelopment that shifts as children age. To retain information, we model patterns of alpha spectral variation, rather than just the peak location, regionally across the scalp and chronologically across development. We propose a covariate-adjusted hybrid principal components analysis (CA-HPCA) for EEG data, which utilizes both vector and functional principal components analysis while simultaneously adjusting for covariate-dependent heteroscedasticity. CA-HPCA assumes the covariance process is weakly separable conditional on observed covariates, allowing for covariate-adjustments to be made on the marginal covariances rather than the full covariance leading to stable and computationally efficient estimation. The proposed methodology provides novel insights into neurodevelopmental differences between TD and ASD children.

Consistent and Contrasting Aspects of the Geomagnetic Field Across Epochs With Distinct Reversal Frequencies Revealed by Modeling the Kiaman Superchron

AbstractThis work presents an extensive directional paleomagnetic database of the Kiaman reversed superchron. It is composed of 1,459 paleomagnetic directions from igneous rocks corresponding to 91 data sets (or paleomagnetic poles). An almost constant behavior of more concentrated and circular distributions for latitudes higher than 10° was found, which contrasts strongly with predictions of the representative models for the past few million years. We searched for simplified and spatially covariant Giant Gaussian Process (GGP) models that best explain the directional distribution of the Kiaman database. We used the mean strength based on the mean of virtual dipole moment (VDM) results for the period drawn from the available databases. Among the tested models, the one that best explains the directional paleosecular variation of the Kiaman database is the covariant type. According to this model, the correlations between the Gaussian coefficients are valid for the last 10 Myr and the Kiaman superchron. The resulting GGP models have parameters similar to the 0–10 Ma models, which indicates that the relation between symmetric and antisymmetric families appears unchanged in the geological past. The relative variability of the Kiaman field, as inferred from the ratio from GGP models, is lower than for the past 10 Myr. Thus, as well as the paleointensity, seems to be a proxy that can be used for evaluating the geomagnetic development along the geological time.

Evaluation of the image quality and dose reduction in digital radiography with an advanced spatial noise reduction algorithm in pediatric patients.

To evaluate whether the advanced spatial noise reduction (ASNR) algorithm installed in a digital radiography system generates acceptable images at a lower dose than a conventional denoising algorithm in pediatric patients. Nine sets of 30 images of pediatric patients, classified under three protocols and three age groups, were retrospectively selected. Different levels of low-dose image sets of these 270 images were generated by a noise simulation tool after validation testing using phantoms. Each image set was obtained with both the ASNR and conventional algorithm, and grouped randomly and blinded. Three experienced pediatric radiologists were asked to pick the "image with optimum dose" among images of different dose levels with an ALARA (as low as reasonably achievable) perspective. Dose reduction rates for each protocol and age group were calculated, and entrance skin exposure (ESE) was calculated using the values of kVp and mAs, assuming a standard body depth for each age group. With the ASNR algorithm, estimated dose reduction rates were highest for abdominal radiographs (45.0%, 27.3%, and 24.3% in infants, children, and adolescents, respectively, p < 0.001). The mean dose reductions for all age groups in the abdominal, chest, and skull radiographs were 32.8%, 12.9%, and 23.2%, respectively (p < 0.001). Average of the calculated ESE was lower with the ASNR algorithm than with the conventional algorithm group (p < 0.001). The ASNR algorithm facilitated optimization of image quality with a higher reduction in radiation dose than the conventional algorithm, making it more acceptable for use in pediatric patients. •ASNR algorithm in DR system improves image quality via enhanced contrast and noise removal by estimating actual noise distribution based on a multi-scale noise covariance and frequency processing. •Noise simulation tool (NST) generating images of different dose levels can be used for evaluation of the optimum dose without unnecessary additional radiation exposure to pediatric patients. •Retrospective clinical study using NST showed that the ASNR algorithm enabled a higher reduction in radiation dose than the conventional algorithm in pediatric patients.

Targeting Predictors Via Partial Distance Correlation With Applications to Financial Forecasting

High-dimensional time series datasets are becoming increasingly common in various fields of economics and finance. Given the ubiquity of time series data, it is crucial to develop efficient variable screening methods that use the unique features of time series. This article introduces several model-free screening methods based on partial distance correlation and developed specifically to deal with time-dependent data. Methods are developed both for univariate models, such as nonlinear autoregressive models with exogenous predictors (NARX), and multivariate models such as linear or nonlinear VAR models. Sure screening properties are proved for our methods, which depend on the moment conditions, and the strength of dependence in the response and covariate processes, amongst other factors. We show the effectiveness of our methods via extensive simulation studies and an application on forecasting U.S. market returns.

Joint optimization of lot sizing and condition-based maintenance for a production system using the proportional hazards model

This paper optimizes economic production quantity (EPQ) and condition-based maintenance (CBM) simultaneously for a production system subject to aging and deterioration. Different from previous models jointly determining EPQ and CBM, the proposed model is developed based on the proportional hazards model with a continuous-state covariate process, and on a CBM policy with multiple maintenance actions and dynamic control limits. Taking advantage of the opportunity of the downtime when a production run completes, condition monitoring is performed to reveal the system deterioration and then a suitable action is selected from preventive replacement, preventive repair, and no maintenance. Random failures during a production phase can be fixed by corrective replacement, corrective repair, or minimal repair. Considering that in many practical situations condition monitoring cannot be conducted at failure, corrective maintenance actions are determined based on (1) the system age at failure and the deterioration at the beginning of the production run (Scenario 1); or (2) both the system age and deterioration at the beginning of the production run (Scenario 2). The objective is to jointly optimize the production lot size and the CBM policy by minimizing the long-run average cost rate. The optimization problem is solved based on the policy-iteration algorithm in the semi-Markov decision process (SMDP) framework. A numerical example is provided to illustrate the proposed approach. The results show that Scenario 1 produces a more cost-effective but more complex maintenance strategy than Scenario 2 does.

A multivariate 4/2 stochastic covariance model: properties and applications to portfolio decisions

This paper introduces a multivariate 4/2 stochastic covariance process generalizing the one-dimensional counterparts presented in [Grasselli, M., The 4/2 stochastic volatility model: A unified approach for the Heston and the 3/2 model. Math. Finance, 2017, 27(4), 1013–1034]. Our construction permits stochastic correlation not only among stocks but also among volatilities, also known as co-volatility movements, both driven by more convenient 4/2 stochastic structures. The parametrization is flexible enough to separate these types of correlation, permitting their individual study. Conditions for proper changes of measure and closed-form characteristic functions under risk-neutral and historical measures are provided, allowing for applications of the model to risk management and derivative pricing. We apply the model to an expected utility theory problem in incomplete markets. Our analysis leads to closed-form solutions for the optimal allocation and value function. Conditions are provided for well-defined solutions together with a verification theorem. Our numerical analysis highlights and separates the impact of key statistics on equity portfolio decisions, in particular, volatility, correlation, and co-volatility movements, with the latter being the least important in an incomplete market.

Recurrent Events Analysis With Data Collected at Informative Clinical Visits in Electronic Health Records

Although increasingly used as a data resource for assembling cohorts, electronic health records (EHRs) pose many analytic challenges. In particular, a patient’s health status influences when and what data are recorded, generating sampling bias in the collected data. In this article, we consider recurrent event analysis using EHR data. Conventional regression methods for event risk analysis usually require the values of covariates to be observed throughout the follow-up period. In EHR databases, time-dependent covariates are intermittently measured during clinical visits, and the timing of these visits is informative in the sense that it depends on the disease course. Simple methods, such as the last-observation-carried-forward approach, can lead to biased estimation. On the other hand, complex joint models require additional assumptions on the covariate process and cannot be easily extended to handle multiple longitudinal predictors. By incorporating sampling weights derived from estimating the observation time process, we develop a novel estimation procedure based on inverse-rate-weighting and kernel-smoothing for the semiparametric proportional rate model of recurrent events. The proposed methods do not require model specifications for the covariate processes and can easily handle multiple time-dependent covariates. Our methods are applied to a kidney transplant study for illustration. Supplementary materials for this article are available online.

Coupling and perturbation techniques for categorical time series

We present a general approach for studying autoregressive categorical time series models with dependence of infinite order and defined conditional on an exogenous covariate process. To this end, we adapt a coupling approach, developed in the literature for bounding the relaxation speed of a chain with complete connections and from which we derive a perturbation result for non-homogenous versions of such chains. We then study stationarity, ergodicity and dependence properties of some chains with complete connections and exogenous covariates. As a consequence, we obtain a general framework for studying some observation-driven time series models used both in statistics and econometrics but without theoretical support.

Covariant Giant Gaussian Process Models With Improved Reproduction of Palaeosecular Variation

AbstractA commonly used family of statistical magnetic field models is based on a giant Gaussian process (GGP), which assumes each Gauss coefficient can be realized from an independent normal distribution. GGP models are capable of generating suites of plausible Gauss coefficients, allowing for palaeomagnetic data to be tested against the expected distribution arising from a time‐averaged geomagnetic field. However, existing GGP models do not simultaneously reproduce the distribution of field strength and palaeosecular variation estimates reported for the past 10 million years and tend to underpredict virtual geomagnetic pole (VGP) dispersion at high latitudes unless trade‐offs are made to the fit at lower latitudes. Here we introduce a new family of GGP models, BB18 and BB18.Z3 (the latter includes non‐zero‐mean zonal terms for spherical harmonic degrees 2 and 3). Our models are distinct from prior GGP models by simultaneously treating the axial dipole variance separately from higher degree terms, applying an odd‐even variance structure, and incorporating a covariance between certain Gauss coefficients. Covariance between Gauss coefficients, a property both expected from dynamo theory and observed in numerical dynamo simulations, has not previously been included in GGP models. Introducing covariance between certain Gauss coefficients inferred from an ensemble of “Earth‐like” dynamo simulations and predicted by theory yields a reduced misfit to VGP dispersion, allowing for GGP models which generate improved reproductions of the distribution of field strengths and palaeosecular variation observed for the last 10 million years.

Covariance and Spinorial Statistical Description of Simple Relativistic Stochastic Kinematics

It is shown that Generalized Poisson–Kac processes are closed with respect to Lorentz transformations, providing a class of covariant kinematic processes. The transformation properties of the associated partial probability densities (waves) display spinorial character in a probability space, and their spinorial character is intrinsically related to the parametrization of the internal degrees of freedom of the process. Parity function analysis associated with the bias induced in the partial-wave recombination process by a Lorentz boost, indicates a symmetry breaking in the recombination dynamics. In an inertial reference frame moving with constant velocity [Formula: see text] with respect to the rest frame of the process, stochastic fluctuations are progressively damped out till complete suppression in the limit for [Formula: see text].