Covariance Parameters Research Articles

Refining analytic approximation based estimation of mixed multinomial probit models by parameter selection

Applying analytic approximations for computing multivariate normal cumulative distribution functions has led to a substantial improvement in the estimability of mixed multinomial probit models, both in terms of accuracy and especially in terms of computation time. This paper makes a contribution by presenting a possible way to improve the accuracy of estimating mixed multinomial probit model covariances based on the idea of parameter selection using cross-validation. Comparisons to the MACML approach indicate that the proposed parameter selection approach is able to recover covariance parameters more accurately, even when there is a moderate degree of independence between the random coefficients. The approach also estimates parameters efficiently, with standard errors tending to be smaller than those of the MACML approach, which can be observed by means of a real data case.

Bootstrap Probability Errors of the Whittle MLE for Linear Regression Processes with Strongly Dependent Disturbances

This paper determines bounds on the asymptotic orders of the coverage probability errors of parametric bootstrap confidence intervals (CIs) and tests for the covariance parameters of a time series generated by a regression model with Gaussian, stationary, and strongly dependent errors. The CIs and tests are based on the plug-in Whittle maximum
 likelihood (PWML) estimators. It is shown that, under some sets of conditions on the regression coefficients, the spectral density function, and the parameter values, the bounds on the coverage probability errors of symmetric two-sided and one-sided parametric bootstrap confidence intervals on the plug-in Whittle log-likelihood function are shown to be O(n^{-3/2}\ln{n}) and O(n^{-1}\ln{n}), respectively. Apart from the \ln{n} term, the magnitudes of the coverage probability errors of the one-sided bootstrap confidence intervals for our model is shown to be essentially the same as that of the independent and identically distributed (iid) data. The error for the two-sided confidence intervals is not as small as the error O(n^{-2}) that has been established for many confidence intervals in the literature, see Hall (1992), pp 102-108.

Collective endpoint of visual acuity and contrast sensitivity function from hierarchical Bayesian joint modeling.

Clinical trials typically analyze multiple endpoints for signals of efficacy. To improve signal detection for treatment effects using the high-dimensional data collected in trials, we developed a hierarchical Bayesian joint model (HBJM) to compute a five-dimensional collective endpoint (CE5D) of contrast sensitivity function (CSF) and visual acuity (VA). The HBJM analyzes row-by-row CSF and VA data across multiple conditions, and describes visual functions across a hierarchy of population, individuals, and tests. It generates joint posterior distributions of CE5D that combines CSF (peak gain, peak frequency, and bandwidth) and VA (threshold and range) parameters. The HBJM was applied to an existing dataset of 14 eyes, each tested with the quantitative VA and quantitative CSF procedures in four Bangerter foil conditions. The HBJM recovered strong correlations among CE5D components at all levels. With 15 qVA and 25 qCSF rows, it reduced the variance of the estimated components by 72% on average. Combining signals from VA and CSF and reducing noises, CE5D exhibited significantly higher sensitivity and accuracy in discriminating performance differences between foil conditions at both the group and test levels than the original tests. The HBJM extracts valuable information about covariance of CSF and VA parameters, improves precision of the estimated parameters, and increases the statistical power in detecting vision changes. By combining signals and reducing noise from multiple tests for detecting vision changes, the HBJM framework exhibits potential to increase statistical power for combining multi-modality data in ophthalmic trials.

Online estimation of colored observation‐noise parameters within an ensemble Kalman filtering framework

AbstractThis work addresses the problem of data assimilation in large‐dimensional systems with colored observation noise of unknown statistics, a scenario that will become more common in the near future with the deployment of denser observational networks of high spatio‐temporal coverage. Here, we are interested in the ensemble Kalman filter (EnKF) framework, which has been derived around a white observation noise assumption. Recently, colored observation‐noise aware EnKFs in which the noise was modeled as a first‐order autoregressive (AR) model were introduced. This work generalizes the above‐mentioned filters to learn the statistics of the AR model further online. We follow the state augmentation approach first to estimate the state and the AR model transfer matrix simultaneously, then the variational Bayesian approach to estimate the AR model noise covariance parameters further. We accordingly derive two filtering EnKF‐like algorithms, which estimate those statistics together with the system state. We demonstrate the effectiveness of the proposed colored observation‐noise aware filtering schemes and compare their performance based on several numerical experiments conducted with the Lorenz‐96 model.

Simultaneous Optimization of 20 Key Parameters of the Integrated Forecasting System of ECMWF Using OpenIFS. Part I: Effect on Deterministic Forecasts

Abstract Numerical weather prediction models contain parameters that are inherently uncertain and cannot be determined exactly. It is thus desirable to have reliable objective approaches for estimation of optimal values and uncertainties of these parameters. Traditionally, the parameter tuning has been done manually, which can lead to the tuning process being a maze of subjective choices. In this paper we present how to optimize 20 key physical parameters in the atmospheric model Open Integrated Forecasting System (OpenIFS) that have a strong impact on forecast quality. The results show that simultaneous optimization of O(20) parameters is possible with O(100) algorithm steps using an ensemble of O(20) members; the results also show that the optimized parameters lead to substantial enhancement of predictive skill. The enhanced predictive skill can be attributed to reduced biases in low-level winds and upper-tropospheric humidity in the optimized model. We find that the optimization process is dependent on the starting values of the parameters that are optimized (starting from better-suited values results in a better model). The results show also that the applicability of the tuned parameter values across different model resolutions is somewhat limited because of resolution-dependent model biases, and we also found that the parameter covariances provided by the tuning algorithm seem to be uninformative. Significance Statement The purpose of this work is to show how to use algorithmic methods to optimize a weather model in a computationally efficient manner. Traditional manual model tuning is an extremely laborious and time-consuming process, so algorithmic methods have strong potential for saving the model developers’ time and accelerating development. This paper shows that algorithmic optimization is possible and that weather forecasts can be improved. However, potential issues related to the use of the optimized parameter values across different model resolutions are discussed as well as other shortcomings related to the tuning process.

Probabilistic Assessment of Antarctic Thermomechanical Structure: Impacts on Ice Sheet Stability

AbstractUncertainty in present‐day glacial isostatic adjustment (GIA) rates represents at least 44% of the total gravity‐based ice mass balance signal over Antarctica. Meanwhile, physical couplings between solid Earth, sea level and ice dynamics enhance the dependency of the spatiotemporally varying GIA signal on three‐dimensional variations in mantle rheology. Improved knowledge of thermomechanical mantle structure is therefore required to refine estimates of current and projected ice mass balance. Here, we present a Bayesian inverse method for self‐consistently mapping shear‐wave velocities from high‐resolution adjoint tomography into thermomechanical structure using calibrated parameterisations of anelasticity at seismic frequency. We constrain the model using regional geophysical data sets containing information on upper mantle temperature, attenuation and viscosity structure. Our treatment allows formal quantification of parameter covariances, and naturally permits propagation of material parameter uncertainties into thermomechanical structure estimates. We find that uncertainty in steady‐state viscosity structure at 150 km depth can be reduced by 4–5 orders of magnitude compared with a forward‐modeling approach neglecting covariance between viscoelastic parameters. By accounting for the dependence of apparent viscosity on loading timescale, we find good agreement between our estimates of mantle viscosity beneath West Antarctica, and those derived from satellite GPS. Direct access to temperature structure allows us to estimate lateral variations in lithosphere‐asthenosphere boundary (LAB) depth, geothermal heat flow (GHF), and associated uncertainties. We find evidence for shallow LAB depths (63 ± 13 km), and high GHF (76 ± 7 mW m−2) beneath West Antarctica that, combined with low asthenospheric viscosities, indicate a highly dynamic response to ice mass loss.

Q-Markov Covariance equivalent realizations for unstable and marginally stable systems

An observer-based q-Markov Covariance equivalent realization (QMC) formulation is presented in the paper. It is shown that by inserting an observer in the input–output relationship associated with the dynamical system, a QMC approach can be developed that is applicable to systems with marginally stable or unstable equilibrium points. A system of equations governing all linear state-space realizations, along with the corresponding observers, is derived from matching a set of Markov and Covariance parameters. The solution to this equation system is shown to parameterize all state-space realizations that match the pre-specified correlation functions. Four numerical examples are used to show the utility of the proposed approach. The numerical results show that the observer-based QMC presented in the paper can identify linear SISO and MIMO systems with stable, marginally stable, and unstable characteristics.

Generalized kernel regularized least squares estimator with parametric error covariance

A two-step estimator of a nonparametric regression function via Kernel regularized least squares (KRLS) with parametric error covariance is proposed. The KRLS, not considering any information in the error covariance, is improved by incorporating a parametric error covariance, allowing for both heteroskedasticity and autocorrelation, in estimating the regression function. A two step procedure is used, where in the first step, a parametric error covariance is estimated by using KRLS residuals and in the second step, a transformed model using the error covariance is estimated by KRLS. Theoretical results including bias, variance, and asymptotics are derived. Simulation results show that the proposed estimator outperforms the KRLS in both heteroskedastic errors and autocorrelated errors cases. An empirical example is illustrated with estimating an airline cost function under a random effects model with heteroskedastic and correlated errors. The derivatives are evaluated, and the average partial effects of the inputs are determined in the application.

Finite mixtures of matrix variate Poisson-log normal distributions for three-way count data.

Three-way data structures, characterized by three entities, the units, the variables and the occasions, are frequent in biological studies. In RNA sequencing, three-way data structures are obtained when high-throughput transcriptome sequencing data are collected for n genes across p conditions at r occasions. Matrix variate distributions offer a natural way to model three-way data and mixtures of matrix variate distributions can be used to cluster three-way data. Clustering of gene expression data is carried out as means of discovering gene co-expression networks. In this work, a mixture of matrix variate Poisson-log normal distributions is proposed for clustering read counts from RNA sequencing. By considering the matrix variate structure, full information on the conditions and occasions of the RNA sequencing dataset is simultaneously considered, and the number of covariance parameters to be estimated is reduced. We propose three different frameworks for parameter estimation: a Markov chain Monte Carlo-based approach, a variational Gaussian approximation-based approach, and a hybrid approach. Various information criteria are used for model selection. The models are applied to both real and simulated data, and we demonstrate that the proposed approaches can recover the underlying cluster structure in both cases. In simulation studies where the true model parameters are known, our proposed approach shows good parameter recovery. The GitHub R package for this work is available at https://github.com/anjalisilva/mixMVPLN and is released under the open source MIT license.

A new framework for ground motion selection for structural seismic assessment

Performance-based structural seismic assessment generally requires the selection of ground motions wherein the target acceleration response spectrum (Sa) should be matched in a desired period range. The pros and cons of the commonly used uniform hazard spectrum (UHS), conditional mean spectrum (CMS) and conditional spectrum (CS) are thoroughly examined. Aiming at combining the merits and overcoming the challenges of these methods, a new framework is proposed to construct a series of conditional target spectra for selecting ground motions. The response spectrum is expressed as the product of peak ground acceleration (PGA) and the normalized spectra (β = Sa/PGA) which can be smoothed by four spectral shape parameters. The proposed framework provides the expected median and covariance of the spectral shape parameters conditioned on specified design values of PGA and PGV/PGA (peak ground velocity/PGA). Efficient step-by-step procedures are developed for the application of the proposed alternative approach, and a sample application is illustrated and compared with the CS approach. It is demonstrated that the proposed methodology is capable of incorporating conventional probabilistic seismic hazard analysis (PSHA), considering spectral shape and spectral variability of realistic ground motions while avoiding the choice of the conditioning period. Site class effects can also be incorporated into the proposed approach.

Modelling intra-annual tree stem growth with a distributional regression approach for Gaussian process responses

Abstract High-resolution circumference dendrometers measure the irreversible growth and the reversible shrinking and swelling due to the water content of a tree stem. We propose a novel statistical method to decompose these measurements into a permanent and a temporary component, while explaining differences between the trees and years by covariates. Our model embeds Gaussian processes with parametric mean and covariance functions as response structures in a distributional regression framework with structured additive predictors. We discuss different mean and covariance functions, connections with other model classes, Markov chain Monte Carlo inference, and the efficiency of our sampling scheme.

Development of a sensitivity analysis framework for aquatic biogeochemical models using machine learning

Our evolving understanding of ecosystem functioning along with the advent of computational power have paved the way for the development of complex mathematical models that explicitly represent the functional diversity of biotic communities and multiple biogeochemical cycles. The ever-growing demand for more complex models underscores the importance of robust sensitivity analysis (SA) to elucidate the impact of the uncertainty on model inputs and to untangle the parameter covariance patterns that ultimately lead to the emergence of equifinality problems. In this study, we propose a novel multi-pronged SA framework that integrates advanced statistical and machine learning (ML) techniques. Principal component analysis (PCA) is first applied to dissect the wide array of predictive outputs and identify modes of variability in time and/or space. Classification and Regression Tree (CART) analysis is then used to identify a set of splitting decisions connecting threshold values of key state variables and model parameters with different ranges of predictive outputs with management interest. Self-Organizing Maps (SOM) are implemented as a final step to unravel any non-linear associations between model parameters and responses. As a proof-of-concept, we used a complex aquatic biogeochemical model developed for the Bay of Quinte, a eutrophic embayment in Lake Ontario, to examine competition patterns and structural shifts among multiple functional phytoplankton (diatoms, N-fixing cyanobacteria, and Microcystis) and zooplankton (herbivores and omnivores) groups. Our sensitivity analysis framework showed that the parameters representing the dependence of growth and metabolic processes on temperature are particularly influential to recreate plankton community dynamics during the cold period of the year, whereas the interplay among the interspecific resource competition, strength of the prey-predator interactions, and phosphorus availability mainly regulate their dynamics during the growing season. The growth strategies of diatoms, their nutritional quality that determines the assimilation efficiency by zooplankton, along with the ambient nutrient availability determine our capacity to reproduce patterns of cyanobacteria dominance and faithfully depict the severity of harmful algal blooms. Finally, our study discusses the benefits of a broader use of the ML-based SA framework to unravel influential parametric interactions in modulating the behaviors of complex mathematical models.

Fixed-Domain Posterior Contraction Rates for Spatial Gaussian Process Model with Nugget

Spatial Gaussian process regression models typically contain finite dimensional covariance parameters that need to be estimated from the data. We study the Bayesian estimation of covariance parameters including the nugget parameter in a general class of stationary covariance functions under fixed-domain asymptotics, which is theoretically challenging due to the increasingly strong dependence among spatial observations. We propose a novel adaptation of the Schwartz’s consistency theorem for showing posterior contraction rates of the covariance parameters including the nugget. We derive a new polynomial evidence lower bound, and propose consistent higher-order quadratic variation estimators that satisfy concentration inequalities with exponentially small tails. Our Bayesian fixed-domain asymptotics theory leads to explicit posterior contraction rates for the microergodic and nugget parameters in the isotropic Matérn covariance function under a general stratified sampling design. We verify our theory and the Bayesian predictive performance in simulation studies and an application to sea surface temperature data. Supplementary materials for this article are available online.

Alcohol or Drug Self-Help Use Among Adults in the United States: Age, Period, and Cohort Effects Between 2002 and 2018.

The online version contains supplementary material available at 10.1007/s11469-023-01012-2.

Genetic parameters of fear-related behaviors in Nigerian indigenous turkey poults: A pilot study

Accurate estimates of genetic parameters will enable turkey breeders to determine the effect of selection on fear-related behaviors in breeding programs. Currently, no published reports exist on genetic parameters of fear-related behaviors in Nigerian indigenous turkey (NIT). The objective of this pilot study was to estimate genetic parameters of fear-related behaviors in NIT poults to know the relative contribution of genes and environment to these traits. Three behavioral tests were used as measures of fear response in randomly selected 200 NIT poults generated from mating of 10 sires and 20 dams. Emergence, inversion and open field tests were conducted at 9, 12, and 14 days of age, respectively. Fear tests were conducted in early life to measure innate fear and not learned fear. Heritability, genetic and phenotypic correlations were estimated for latency to emerge from a dark box (EMG), duration of wing flapping during inversion (INV), and behavioral activities performed during open field test such as preening frequency (PN), preening duration (PD), wall pecking duration (DPW), number of times a bird rested on its chest (NRC), duration of movement (DM), total number of floor squares explored (BE) and percentage of floor squares explored (PBC). Variance and covariance parameters of the fear-related behaviors were estimated using restricted maximum likelihood method. Heritability estimates ranged from very low to moderate. The lowest heritability (0.004) was estimated for NRC performed during the open field test while the highest heritability (0.364) was estimated for EMG. All genetic correlations between all pairs of fear-related behaviors were low except between INV and NRC (0.8132); INV and DM (0.7326), and between INV and PN (0.7115). Positive phenotypic correlations were found between INV and other fear-related behaviors while negative phenotypic correlations were found between NRC and the other fear-related behaviors. The highest phenotypic correlation was found between INV and BE (0.9985) while the lowest was between EMG and NRC (-0.0001). The very low to moderate heritability estimated for most of the fear-related behaviors indicates little genetic contribution to fear-related behaviors in NIT, suggesting that fearfulness can be best reduced in these birds by improving the rearing environment. The high genetic correlation between INV and NRC; DPW and PN; INV and DM as well as between INV and PN may indicate that the genes that contribute to these pairs of fear-related behaviors are co-inherited.

Assessing the accuracy of exponentially weighted moving average models for Value-at-Risk and Expected Shortfall of crypto portfolios

A plethora of academic papers on generalized autoregressive conditional heteroscedasticity (GARCH) models for bitcoin and other cryptocurrencies have been published in academic journals. Yet few, if indeed any, of these are employed by practitioners. Previous academic studies produce results that are fragmented, confusing and conflicting, so there is no commercial incentive to drive an expensive implementation of complex multivariate GARCH models, which anyway would commonly require more data for calibration than are available in the history of most cryptocurrencies, at least at the daily frequency. Consequently, this paper assesses the forecasting accuracy of simple parametric RiskMetrics type volatility and covariance models, with a focus on ad hoc parameter choice instead of a data-intensive calibration procedure. We provide extensive backtests of hourly and daily Value-at-Risk (VaR) and Expected Shortfall (ES) forecasts that are regarded as best practice in the industry and commonly used for regulatory approval. Our results demonstrate that much simpler models in the exponentially weighted moving average (EWMA) class are just as accurate as GARCH models for VaR and ES forecasting, provided they capture an asymmetric volatility response and a heavy-tailed returns distribution. Moreover, on ranking each model's variance and covariance forecasts using average scores generated from proper univariate and multivariate scoring rules, there is no evidence of superior performance of variance and covariance forecasts generated by GARCH models, using either daily or hourly data.

Uncertainty quantification in operational modal analysis of time-varying structures based on time-dependent autoregressive moving average model

The identified modal parameters in Operational Modal Analysis (OMA) are always subject to statistical uncertainties from many sources and the uncertainty is also important to assess the quality of the identified results. This study proposes an uncertainty quantification method in OMA for time-varying structures based on the Functional Series Time-dependent AutoRegressive Moving Average (FS-TARMA) model. A multi-stage pseudo-linear optimization scheme is first adopted to obtain the approximate Maximum Likelihood Estimates (MLEs) of FS-TARMA model parameters and the covariance is computed as the inverse Hessian matrix of the Negative Log-Likelihood Function (NLLF) at the MLEs. Finally, the approximate closed-form covariance of modal parameters is derived by the first-order sensitivity method. The proposed method is verified by two numerical examples and an experimental beam. Furthermore, the ability of the uncertainty to extinguish the spurious modes caused by colored noise excitation and the noise modes caused by measurement noise or inappropriate model structure is also demonstrated in these examples.

An Efficient q-Markov Covariance Equivalent Realization Approach to System Identification

An efficient q-Markov covariance equivalent realization (QMC) formulation is presented in this paper. This efficient solution matches all Markov and Covariance parameters of a finite-dimensional system using a finite amount of data. The frequency responses of that system is also matched by the efficient QMC solution. The existence condition of this efficient QMC solution and its parametrization is given in this paper. Then, a generalized algorithm for system identification using the efficient QMC is given. The example is set up by a cantilever beam. Results show that the efficient QMC fully captures Markov and covariance parameters and frequency responses of the cantilever beam.

Covariances and parameter confidence intervals from light-element R-matrix evaluations

R-matrix parameter covariances for light elements with A 16 are calculated within the Wigner-Eisenbud multichannel unitary R-matrix theory. We review the theoretical foundation, numerical approach, and determination of the parameter covariances within this approach. We derive the relation between the parameter variance, as the diagonal elements of the covariance matrix, and the parameter confidence interval based upon the chi-squared distribution. Cross section covariances computed from the parameter confidence intervals are calculated for several compound systems and discussed.

Covariance analysis of 235U(nth,f) independent and cumulative fission yields: Propositions for JEFF4

The study of fission yields has a major impact on the characterization and understanding of the fission process and its applications. Fission yield evaluations represent the synthesis of experimental and theoretical knowledge in order to perform the best estimation of independent fission yields. Today, the lack of correlations between the different fission observables induces several inconsistencies in the evaluations. Different works proposed to estimate the correlations of the independent fission yields satisfying the consistency to the chain yield evaluations. Nevertheless, none of them introduces a prior correlations of the independent and chain evaluations in the evaluation process. Covariance matrix of fission yields depends on the evaluation method used according to the kinds of existing measurements. The consistency is deeply entangled to the statistical agreement between each dataset considering the covariance of measurements. Moreover, covariance of model parameters does not represent the only contribution to the evaluation covariance matrix. Thus, a new evaluation process is crucial to provide a complete and coherent evaluation file. The LEPh Laboratory of CEA Cadarache is developing this program for the future version of the JEFF-library.