Multivariate Gaussian Assumption Research Articles

Density-driven free convection in heterogeneous aquifers with connectivity features

Free convection usually happens in variable-density groundwater flow systems, and it favors contaminant transport by enlarging length scales and shortening timescales compared to advection and diffusion/dispersion alone. Previous studies have demonstrated that heterogeneity with multi-variate Gaussian distribution for logarithmic permeabilities (log10k) plays an important role in the onset, growth, and/or decay of instability during the density-driven convective process. Nevertheless, the connectivity features (i.e., connected structures of extremely high/low-k values), which are common in natural aquifers, have received little attention. In this study, the classical problem of transient free convection has been modified and numerically simulated by Monte Carlo approach to investigate the effects of connectivity features in heterogeneity on the unstable convective processes. Results show that free convection is promoted by the connected high-k structures and retarded by the connected low-k structures during mainly the early-stage mass loading. The impacts of connectivity features tend to be amplified by higher variation in log10k distributions, and can be secondarily influenced by correlation lengths and anisotropy. Under the multi-variate Gaussian assumption, the existence of connected high-k structures leads to underestimation of density-driven instability, in which the risk differs based on the statistics of permeability fields, metrics of interest and timeframe. This study highlights the importance of understanding connectivity features in heterogeneous geological media when assessing density-dependent solute transport in groundwater systems.

Probability Density Functions with Correlations of Experimental and Evaluated 235U(nth,f) Fission Yields and Their Impacts on Reactivity Loss with Burnup

The study of fission yields has a major impact on the characterization and understanding of the fission process and its applications. For the latter, it is crucial to provide fission yields with their associated correlation matrix in order to estimate precisely the uncertainties of crucial quantities, such as the reactivity loss. In the last decade, different works have been proposed to estimate the correlations of the independent fission yields satisfying the consistency of the cumulative yield evaluations. In these previous works, only model parameters and conservation laws have driven the correlations. The novelty of the present work consists of new complete and consistent evaluations of 235U(nth,f) independent and cumulative yields and their correlation matrices, starting from experimental data. The characterization of the probability density functions of fission yields validates the multivariate Gaussian assumption and constitutes a major issue in the validation of the uncertainty propagation tools of the applications. In addition, the new 235U(nth,f) fission yield covariance data have resulted in a decrease in nuclear data related to uncertainty in associated burnup calculations.

Gaussian Anomaly Detection by Modeling the Distribution of Normal Data in Pretrained Deep Features

Anomaly detection (AD) in images is a fundamental computer vision problem and refers to identifying images that deviate significantly from normality. State-of-the-art AD algorithms commonly learn a model of normality from scratch using task-specific datasets in either semisupervised or self-supervised manner. We follow an alternative approach and model the distribution of normal data in deep feature representations learned from ImageNet via a multivariate Gaussian (MVG). This lightweight approach achieves a new state of the art in AD on the public MVTec AD dataset. In addition to the empirical benefits, we give a clear motivation for the seemingly simplistic approach via the ties between deep generative and discriminative modeling revealed recently. We further elucidate why ImageNet representations are discriminative in the transfer learning AD setting using the principal component analysis. Here, we find that the principal components containing little variance in normal data are the ones crucial for discriminating between normal and anomalous instances, giving an explanation for the unreasonable effectiveness of our approach. We also investigate setting the working point of our approach by selecting acceptable false-positive rate thresholds based on the MVG assumption and the resistance of our approach to unlabeled anomalies in the dataset. Finally, we investigate whether our approach is prone to exploiting spurious correlations using explainable AI techniques. Code is publicly available at https://github.com/ORippler/gaussian-ad-mvtec.

Quantifying Expected Uncertainty Reduction and Value of Information Using Ensemble-Variance Analysis

Summary Data-acquisition programs, such as surveillance and pilots, play an important role in minimizing subsurface risks and improving decision quality for reservoir management. For design optimization and investment justification of these programs, it is crucial to be able to quantify the expected uncertainty reduction and the value of information (VOI) attainable from a given design. This problem is challenging because the data from the acquisition program are uncertain at the time of the analysis. In this paper, a method called ensemble-variance analysis (EVA) is proposed. Derived from a multivariate Gaussian assumption between the observation data and the objective function, the EVA method quantifies the expected uncertainty reduction from covariance information that is estimated from an ensemble of simulations. The result of EVA can then be used with a decision tree to quantify the VOI of a given data-acquisition program. The proposed method has several novel features compared with existing methods. First, the EVA method directly considers the data/objective-function relationship. Therefore, it can handle nonlinear forward models and an arbitrary number of parameters. Second, for cases when the multivariate Gaussian assumption between the data and objective function does not hold, the EVA method still provides a lower bound on expected uncertainty reduction, which can be useful in providing a conservative estimate of the surveillance/pilot performance. Finally, EVA also provides an estimate of the shift in the mean of the objective-function distribution, which is crucial for VOI calculation. In this paper, the EVA work flow for expected-uncertainty-reduction quantification is described. The result from EVA is benchmarked with recently proposed rigorous sampling methods, and the capacity of the method for VOI quantification is demonstrated for a pilot-analysis problem using a field-scale reservoir model.

Graphical models via joint quantile regression with component selection

A graphical model is used for describing interrelationships among multiple variables. In many cases, the multivariate Gaussian assumption is made partly for its simplicity but the assumption is hardly met in actual applications. In order to avoid dependence on a rather strong assumption, we propose to infer the graphical model via joint quantile regression with component selection, since the components of quantile regression carry information to infer the conditional independence. We demonstrate the advantages of our approach using simulation studies and apply our method to an interesting real biological dataset, where the dependence structure is highly complex.

Optimality Claims for the FML Covariance Estimator with respect to Two Matrix Norms

In this correspondence we prove two interesting properties of the fast maximum likelihood (FML) covariance matrix estimator proposed in [1] under the assumption of zero-mean complex circular Gaussian training data sharing the same covariance matrix. The new properties represent optimality claims regardless of the statistical characterization of the data and, in particular, of the multivariate Gaussian assumption for the observables. The optimality is proved with respect to two cost functions involving either the Frobenius or the spectral norm of an Hermitian matrix.

Globular Clusters in the Milky Way and Dwarf Galaxies: A Distribution‐Free Statistical Comparison

It has been found that globular clusters (GCs) in dwarf galaxies and those in the Milky Way (MW) outer halo mostly have the same parent distributions, while GCs in the MW disk and inner halo have a different origin from those in dwarf galaxies. Thus, these dwarf galaxies did not play a crucial role in the formation of the Galactic disk or inner halo. In order to investigate this phenomenon in a more objective manner, a statistical comparison of the GCs of our Galaxy and those of neighboring dwarf galaxies has been carried out by a multivariate nonparametric method. For the various parameters of GCs in the MW and in dwarf galaxies, the multivariate Gaussian assumption fails, so a nonparametric method of comparison (instead of multivariate analysis of variance [MANOVA]) has been chosen. The test is performed on GCs of the MW disk, inner halo, and outer halo separately, with GCs from neighboring dwarf galaxies Canis Major, Fornax, and Sculptor, and the LMC dwarf irregular galaxy. The test is also performed for GCs from dwarf spheroidal galaxies in the neighborhood of M31: M33, NGC 147, NGC 185, and NGC 205.

Analysis of differentially coherent linear receivers over rician-faded CDMA channels

RicianAccurate performance analysis for linear receivers over frequency- and time-selective asynchronous code-division multiple-access Rician-fading channels is very useful and a general approach to this topic is very desirable. In this paper, by using a decision variable-based moment generating function approach, we provide a unified bit-error probability (BEP) analysis framework for different linear detectors with binary or quaternary differential phase-shift keying and postdetection combining over Rician-fading channels, taking into account the effects of the spreading code correlation, the system and fading-channel parameters, diversity combining, and branch correlation. To reduce the complexity of the exact BEP evaluation, we furthermore provide an approximate multivariate Gaussian assumption (MGA)-based method which entails a low complexity for BEP evaluation. Ideal and approximate linear minimum mean-squared error diversity receivers for correlated Rician-fading channels are proposed. Numerical results show that the phases of the line-of-sight (LOS) components of the desired user significantly affect the receiver performance over correlated multipath Rician channels, and this may be exploited to improve performance. Also, when the LOS components are affected by a significant Doppler shift, automatic frequency control is very useful in improving the receiver performance.

Variance–Covariance Matrix of the Experimental Variogram: Assessing Variogram Uncertainty

Assessment of the sampling variance of the experimental variogram is an important topic in geostatistics as it gives the uncertainty of the variogram estimates. This assessment, however, is repeatedly overlooked in most applications mainly, perhaps, because a general approach has not been implemented in the most commonly used software packages for variogram analysis. In this paper the authors propose a solution that can be implemented easily in a computer program, and which, subject to certain assumptions, is exact. These assumptions are not very restrictive: second-order stationarity (the process has a finite variance and the variogram has a sill) and, solely for the purpose of evaluating fourth-order moments, a Gaussian distribution for the random function. The approach described here gives the variance–covariance matrix of the experimental variogram, which takes into account not only the correlation among the experiemental values but also the multiple use of data in the variogram computation. Among other applications, standard errors may be attached to the variogram estimates and the variance–covariance matrix may be used for fitting a theoretical model by weighted, or by generalized, least squares. Confidence regions that hold a given confidence level for all the variogram lag estimates simultaneously have been calculated using the Bonferroni method for rectangular intervals, and using the multivariate Gaussian assumption for K-dimensional elliptical intervals (where K is the number of experimental variogram estimates). A general approach for incorporating the uncertainty of the experimental variogram into the uncertainty of the variogram model parameters is also shown. A case study with rainfall data is used to illustrate the proposed approach.