Multivariate Gaussian Research Articles

부유선별공정 모니터링을 위한 다변수 가우시안 혼합모델기반 색상 분석

Froth flotation is a type of mineral processing that selectively separates useful minerals from theBRfroth produced by pulp aeration. The physical features of the froth contain information to understandBRthe status of the process of froth flotation. In particular, by obtaining the color feature of the froth, itBRis possible to predict not only the type of useful mineral but also the grade of the mineral. Therefore,BRit is very important to obtain high-precision color analysis results in froth flotation monitoring. In thisBRpaper, we propose a color analysis method based on multivariate Gaussian mixture models to guaranteeBRthe cyclicity with a period of 360 degrees in hue component of color. In order to verify the proposedBRmethod, we present the color analysis results of images which contain around 0º (360º ) of hue andBRimages obtained by froth flotation. Through the experimental results, it shows that the color analysisBRresults obtained by the proposed method are more precise.BR

Decentralized Groupwise Expectation Propagation Detector for Uplink Massive MU-MIMO Systems

With the proliferation of emerging Internet of Things (IoT) applications, massive multiuser multiple-input–multiple-output (MU-MIMO) is a promising technology to support the massive connectivity requirement with limited spectral resources. The existing detection algorithms are mainly designed based on a centralized baseband processing architecture, which requires extremely high raw baseband data rates transferred between base-station antennas and the central processing unit (CPU). Decentralized baseband processing (DBP) architecture has recently been proposed to alleviate high interconnect data rates and chip input/output bandwidth bottlenecks. Since the information is not fully shared among each antenna cluster, conventional decentralized detectors suffer from significant performance loss, especially for systems with high overload ratios and/or spatially correlated channels. In this work, we first propose a decentralized groupwise detection paradigm for the star architecture by dividing users into multiple user groups, which can be effectively exploited by factor graph-based message passing algorithms, such as the expectation propagation (EP) algorithm. Then, an efficient message fusion rule is devised based on the product principle of the multivariate Gaussian probability density function at the CPU. To reduce the computational complexity, an approximate groupwise EP (AGW-EP) method is proposed by judiciously selecting reliable constellation vectors during the symbol belief calculation phase. In addition, we extend the proposed methods to the daisy-chain architecture, which requires constant interconnect data rates. Simulation results demonstrate that the proposed groupwise paradigm greatly enhances the performance of the conventional EP detector. Moreover, the proposed decentralized groupwise EP and AGW-EP detection schemes outperform their counterparts, particularly in correlated MIMO channels.

Failure probability estimation through high-dimensional elliptical distribution modeling with multiple importance sampling

This paper addresses the challenge of performing importance sampling in high-dimensional space (several hundred inputs) in order to estimate the failure probability of a physical system subject to randomness. It is assumed that the failure domain defined in the input space can possibly include multiple failure regions. A new approach is developed to construct auxiliary importance sampling densities sequentially for each failure region identified as part of the failure domain. The search for failure regions is achieved through optimization. A stochastic decomposition of the elliptically distributed inputs is exploited in the structure of the auxiliary densities, which are expressed as the product of a parametric conditional distribution for the radial component, and a parametric von Mises–Fisher distribution for the directional vector. The failure probability is then estimated by multiple importance sampling with a mixture of the densities. To demonstrate the efficiency of the proposed method in high-dimensional space, several numerical examples are considered involving the multivariate Gaussian and Student distributions, which are commonly used elliptical distributions for input modeling. In comparison with other simulation methods, the numerical cost of the proposed approach is found to be quite low when the gradient of the performance function defining the failure domain is available.

A multivariate model to estimate environmental load on an offshore structure

Offshore structures such as oil platforms are subjected to significant environmental loads caused by wind, waves, and current. The complexity of offshore environment requires robust and reliable models to capture dependencies among environmental variables. A vine copula is a powerful tool that can be used to construct multivariate models by decomposing the complex structure into a series of simple pair copulas. Previous studies have shown that simple and symmetric copulas can be used as building blocks to construct vine copula models. In this study, symmetric and asymmetric copula functions are considered building blocks to capture all possible dependency structures. The c-vine model is then used to estimate the total environmental load on an offshore structure. Estimated loads are compared with those using the traditional independent variable approach and a multi-Gaussian distribution function-based method. The results reveal that both symmetric and asymmetric copula functions can be fitted to build c-vine copula models for the trivariate case. C-vine copulas, constructed using asymmetric copulas, provide a better estimation of the total environmental load than the independent and multivariate Gaussian methods. The result of this study is useful in probabilistic structural analysis of offshore structures for design and resilience analysis.

Sample-path large deviations for stochastic evolutions driven by the square of a Gaussian process.

Recently, a number of physical models have emerged described by a random process with increments given by a quadratic form of a fast Gaussian process. We find that the rate function which describes sample-path large deviations for such a process can be computed from the large domain size asymptotic of a certain Fredholm determinant. The latter can be evaluated analytically using a theorem of Widom which generalizes the celebrated Szegő-Kac formula to the multidimensional case. This provides a large class of random dynamical systems with timescale separation for which an explicit sample-path large-deviation functional can be found. Inspired by problems in hydrodynamics and atmosphere dynamics, we construct a simple example with a single slow degree of freedom driven by the square of a fast multivariate Gaussian process and analyze its large-deviation functional using our general results. Even though the noiseless limit of this example has a single fixed point, the corresponding large-deviation effective potential has multiple fixed points. In other words, it is the addition of noise that leads to metastability. We use the explicit answers for the rate function to construct instanton trajectories connecting the metastable states.

Survey of Distances between the Most Popular Distributions

We present a number of upper and lower bounds for the total variation distances between the most popular probability distributions. In particular, some estimates of the total variation distances in the cases of multivariate Gaussian distributions, Poisson distributions, binomial distributions, between a binomial and a Poisson distribution, and also in the case of negative binomial distributions are given. Next, the estimations of Lévy–Prohorov distance in terms of Wasserstein metrics are discussed, and Fréchet, Wasserstein and Hellinger distances for multivariate Gaussian distributions are evaluated. Some novel context-sensitive distances are introduced and a number of bounds mimicking the classical results from the information theory are proved.

MR-BOIL: Causal inference in one-sample Mendelian randomization for binary outcome with integrated likelihood method.

Mendelian randomization is a statistical method for inferring the causal relationship between exposures and outcomes using an economics-derived instrumental variable approach. The research results are relatively complete when both exposures and outcomes are continuous variables. However, due to the noncollapsing nature of the logistic model, the existing methods inherited from the linear model for exploring binary outcome cannot take the effect of confounding factors into account, which leads to biased estimate of the causal effect. In this article, we propose an integrated likelihood method MR-BOIL to investigate causal relationships for binary outcomes by treating confounders as latent variables in one-sample Mendelian randomization. Under the assumption of a joint normal distribution of the confounders, we use expectation maximization algorithm to estimate the causal effect. Extensive simulations demonstrate that the estimator of MR-BOIL is asymptotically unbiased and that our method improves statistical power without inflating type I error rate. We then apply this method to analyze the data from Atherosclerosis Risk in Communications Study. The results show that MR-BOIL can better identify plausible causal relationships with high reliability, compared with the unreliable results of existing methods. MR-BOIL is implemented in R and the corresponding R code is provided for free download.

CMS Spread Options Pricing under the CHH Model

Based on the Chen, Hsieh, and Huang (2017) interest rate model, this research explores the analytical approach for pricing CMS spread options. We first derive a complex joint density for two swap rates composed of sequential forward rates and approximate the joint density by bivariate normals. After applying the methods of Pearson (1995) and Li, Deng, and Zhou (2008), we obtain two analytical pricing models and examine their accuracy using numerical analysis. Finally, we empirically show the predictive power of the implied volatility of CMS options for future economic states.

Gaussian Markov Random Fields and totally positive matrices

The present paper focuses on the study of the conditions under which the covariance matrix of a multivariate Gaussian distribution is totally positive, paying particular attention to multivariate Gaussian distributions that are Gaussian Markov Random Fields. More specifically, it is proven that, if the graph over which the Gaussian Markov Random Field is defined consists of path graphs and the covariances between adjacent variables on the graph are non-negative, then there always exists a reordering of the variables that renders the resulting covariance matrix totally positive. Moreover, this reordering is identified and some cases for which the conditions for the covariance matrix of a multivariate Gaussian distribution to be totally positive are necessary and sufficient are provided.

General construction and classes of explicit L1-optimal couplings

The main scope of this paper is to give some explicit classes of examples of L1-optimal couplings. Optimal transportation w.r.t. the Kantorovich metric ℓ1 (resp. the Wasserstein metric W1) between two absolutely continuous measures is known since the basic papers of Kantorovich and Rubinstein (Dokl. Akad. Nauk SSSR 115 (1957) 1058–1061) and Sudakov (Proc. Steklov Inst. Math. 141 (1979) 1–178) to occur on rays induced by a decomposition of the basic space (and more generally to higher dimensional decompositions in the case of general measures) induced by the corresponding dual potentials. Several papers have given this kind of structural result and established existence and uniqueness of solutions in varying generality. Since the dual problems pose typically too strong challenges to be solved in explicit form, these structural results have so far been applied for the solution of few particular instances. First, we give a self-contained review of some basic optimal coupling results and we propose and investigate in particular some basic principles for the construction of L1-optimal couplings given by a reduction principle and some usable forms of the decomposition method. This reduction principle, together with symmetry properties of the reduced measures, gives a hint to the decomposition of the space into sectors and via the non crossing property of optimal transport leads to the choice of transportation rays. The optimality of the induced transports is then a consequence of the characterization results of optimal couplings. Then, we apply these principles to determine in explicit form L1-optimal couplings for several classes of examples of elliptical distributions. In particular, we give for the first time a general construction of L1-optimal couplings between two bivariate Gaussian distributions. We also discuss optimality of special constructions like shifts and scalings, and provide an extended class of dual functionals allowing for the closed-form computation of the ℓ1-metric or of accurate lower bounds of it in a variety of examples.

Objective Bayesian Approach for Binary Hypothesis Testing of Multivariate Gaussian Observations

This paper proposes an objective Bayesian detector for the binary hypothesis testing problem in which the observation vectors are IID and follow multivariate Gaussian distribution under both the null and alternative hypotheses. The mean vector and covariance matrix under each hypothesis are not known, they also do not have special structure other than the covariance matrices are partially ordered in the positive definite cone with their difference positive semi-definite. An example of such a problem is the detection of Gaussian random signal in Gaussian noise. Non-informative priors are imposed on the unknown parameters for computing the marginals and evaluating the Bayes factor test statistic, where the prior for the covariance matrix is applied to the Cholesky factorization of the precision matrix. For the null hypothesis, we propose conjugate priors for the unknowns. For the alternative hypothesis, we propose uniform prior on the mean vector, and a class of objective priors that encompasses the Jeffreys, Independence Jeffreys, Geisser and Cornfield, Haar Measure and Reference priors for the precision matrix. The proposed priors enable closed-form expressions for the marginals and lead to an objective Bayes factor having the posteriors mainly governed by the data. The proposed detector is analyzed for justifying the priors used, deriving the theoretical moments, and approximating the false alarm and detection probability densities by the shifted Gamma distributions. The developed detector exhibits improvement in detection performance over the energy detector (ED) and the generalized likelihood ratio test (GLRT).

Uncertainty modelling and dynamic risk assessment for long-sequence AIS trajectory based on multivariate Gaussian Process

A long-sequence multi-step prediction method based on multivariate Gaussian hypothesis and Gaussian process is proposed to model the uncertainty in the future ship path. This is a necessary step to predict the area where the ship is likely to be located at each future moment and to perform a dynamic risk assessment. Through data fusion, the uncertainty of the prediction is reduced, and more accurate support can be achieved for risk assessment. Firstly, from the current trajectory, the initial uncertainty intervals for the future trajectory are predicted based on the Gaussian process. Then, from the historical data, a reference trajectory set suitable for predicting the future path is generated based on a feature extracting process, named the reference trajectory prediction model in this paper, and the uncertainty intervals are also predicted. After that, the two parts are fused for a more accurate prediction to calculate the dynamic collision probability. The Gaussian process and a Laplacian Eigenmaps-Self-Organizing Maps model are adopted for fast batch processing. The experimental results demonstrate that the proposed model can combine the advantages of both and achieve a more accurate dynamic risk assessment.

Online Streaming Features Causal Discovery Algorithm Based on Partial Rank Correlation

Aimedat the problem of dynamic causal discovery in the era of artificial intelligence, this article combines partial rank correlation coefficients and streaming features in the field of Bayesian network structure learning and proposes a new online streaming feature causal discovery algorithm based on partial rank correlation named the partial rank casual discovery streaming feature based algorithm. This algorithm is not only suitable for Bayesian causal structure learning in dynamic feature spaces generated by sequential streams of features but can also effectively process multivariate linear Gaussian and nonlinear non-Gaussian data. We present three main contributions. First, for arbitrarily distributed datasets, which can be generated by the additive noise model, we proved that the partial rank correlation coefficient can be used as the criterion for the conditional independence test and explored the distribution of corresponding statistics. Second, the PCSDSF algorithm redefined the relevance based on partial rank statistic prospects and then redefined conditional dependence or independence. This method can significantly reduce the number of conditional independence tests and achieves a good time performance. Finally, theoretical analysis and several experiments proved the reliability of the algorithm. A simulation showed that on average, the PCSDSF algorithm outperforms existing algorithms in terms of both the accuracy and time performance.

SwinVRNN: A Data‐Driven Ensemble Forecasting Model via Learned Distribution Perturbation

AbstractThe data‐driven approaches for medium‐range weather forecasting are recently shown to be extraordinarily promising for ensemble forecasting due to their fast inference speed compared to the traditional numerical weather prediction models. However, their forecast accuracy can hardly match the state‐of‐the‐art operational ECMWF Integrated Forecasting System (IFS) model. Previous data‐driven approaches perform ensemble forecasting using some simple perturbation methods, like the initial condition perturbation and the Monte Carlo dropout. However, their ensemble performance is often limited arguably by the sub‐optimal ways of applying perturbation. We propose a Swin Transformer‐based Variational Recurrent Neural Network (SwinVRNN), which is a stochastic weather forecasting model combining a SwinRNN predictor with a perturbation module. SwinRNN is designed as a Swin Transformer‐based recurrent neural network, which predicts the future states deterministically. Furthermore, to model the stochasticity in the prediction, we design a perturbation module following the Variational Auto‐Encoder paradigm to learn the multivariate Gaussian distributions of a time‐variant stochastic latent variable from the data. Ensemble forecasting can be easily performed by perturbing the model features leveraging the noise sampled from the learned distribution. We also compare four categories of perturbation methods for ensemble forecasting, that is, fixed distribution perturbation, learned distribution perturbation, MC dropout, and multi model ensemble. Comparisons on the WeatherBench data set show that the learned distribution perturbation method using our SwinVRNN model achieves remarkably improved forecasting accuracy and reasonable ensemble spread due to the joint optimization of the two targets. More notably, SwinVRNN surpasses operational IFS on the surface variables of the 2‐m temperature and the 6‐hourly total precipitation at all lead times up to 5 days (Code is available at https://github.com/tpys/wwprediction).

A Deep Learning Framework for Classifying and Evaluating Yoga Exercises

Yoga is a type of physical activity that combines physical postures with breathing techniques. Cardiovascular disease is the leading cause of illness and mortality. Yoga's potential involvement in cardiac recovery can be incorporated into cardiovascular rehabilitation. Classifying and evaluating yoga exercises using a deep learning framework plays a significant role in postoperative recovery. Home based rehabilitation often suffers from a lack of patient adherence to prescribed exercise routines, resulting in longer treatment times and higher healthcare expenses. This study proposes a framework for classifying and evaluating the quality of yoga exercises. The ConvNet followed by the deep belief network model is used to create quality scores for input movements to evaluate the quality of yoga asanas. Additionally, various machine learning algorithms are deployed to identify the best performing algorithm. The proposed framework is tested using the COCO dataset and the publicly available dataset from KAGGLE. The model is implemented using a three stage pipeline. In the first stage, an estimator and detector model is used for feature extraction. In the second stage, restricted Boltzmann machines are used for dimensionality reduction in two steps: The forward pass and the backward pass. Finally, the ConvNet followed by the deep belief network is used to classify yoga postures. The quaternion data is treated as a multivariate Gaussian variable to create Markov random fields. To quantify yoga asanas, the Markov random fields network is triggered, and the resulting quaternion point is compared to standard pose data, increasing the likelihood that the yoga position is correct. The importance of our study is that it allows participants to watch their movement in real-time and to obtain a numerical estimate of the accuracy of their yoga stance. Using this method, we were able to achieve a precision of 99.99%, which was previously unattainable on this dataset. Our model is more robust than previous models in terms of performance.

Partial Identification of Latent Correlations with Ordinal Data

The polychoric correlation is a popular measure of association for ordinal data. It estimates a latent correlation, i.e., the correlation of a latent vector. This vector is assumed to be bivariate normal, an assumption that cannot always be justified. When bivariate normality does not hold, the polychoric correlation will not necessarily approximate the true latent correlation, even when the observed variables have many categories. We calculate the sets of possible values of the latent correlation when latent bivariate normality is not necessarily true, but at least the latent marginals are known. The resulting sets are called partial identification sets, and are shown to shrink to the true latent correlation as the number of categories increase. Moreover, we investigate partial identification under the additional assumption that the latent copula is symmetric, and calculate the partial identification set when one variable is ordinal and another is continuous. We show that little can be said about latent correlations, unless we have impractically many categories or we know a great deal about the distribution of the latent vector. An open-source R package is available for applying our results.

Multivariate Gaussian processes: definitions, examples and applications

Gaussian processes occupy one of the leading places in modern statistics and probability theory due to their importance and a wealth of strong results. The common use of Gaussian processes is in connection with problems related to estimation, detection, and many statistical or machine learning models. In this paper, we propose a precise definition of multivariate Gaussian processes based on Gaussian measures on vector-valued function spaces, and provide an existence proof. In addition, several fundamental properties of multivariate Gaussian processes, such as stationarity and independence, are introduced. We further derive two special cases of multivariate Gaussian processes, including multivariate Gaussian white noise and multivariate Brownian motion, and present a brief introduction to multivariate Gaussian process regression as a useful statistical learning method for multi-output prediction problems.

Cost polarization by dequantizing for JPEG steganography

In this article, we study a recently proposed method for improving empirical security of steganography in JPEG images in which the sender starts with an additive embedding scheme with symmetrical costs of ± 1 changes and then decreases the cost of one of these changes based on an image obtained by applying a deblocking (JPEG dequantization) algorithm to the cover JPEG. This approach provides rather significant gains in security at negligible embedding complexity overhead for a wide range of quality factors and across various embedding schemes. Challenging the original explanation of the inventors of this idea, which is based on interpreting the dequantized image as an estimate of the precover (uncompressed) image, we provide alternative arguments. The key observation and the main reason why this approach works is how the polarizations of individual DCT coefficients work together. By using a MiPOD model of content complexity of the uncompressed cover image, we show that the cost polarization technique decreases the chances of “bad” combinations of embedding changes that would likely be introduced by the original scheme with symmetric costs. This statement is quantified by computing the likelihood of the stego image w.r.t. the multivariate Gaussian precover distribution in DCT domain. Furthermore, it is shown that the cost polarization decreases spatial discontinuities between blocks (blockiness) in the stego image and enforces desirable correlations of embedding changes across blocks. To further prove the point, it is shown that in a source that adheres to the precover model, a simple Wiener filter can serve equally well as a deep-learning based deblocker

Influence of process-based, stochastic and deterministic methods for representing heterogeneity in fluvial geothermal systems

Focus is on comparing stochastic, process-based and deterministic methods for modelling heterogeneity in hydraulic properties of fluvial geothermal reservoirs. Models are considered a generalized representation of a fluvial sequence in the upper part of the Gassum Formation in northern Denmark. Two ensemble realizations of process-based and stochastic heterogeneity were simulated using the state-of-the-art numerical modelling software, Delft Advanced Research Terra Simulator (DARTS), to assess differences on a statistically relevant sample size. Simulator settings were optimized to achieve two orders of magnitude improvement in simulation time. Our general findings show that the stochastic and process-based methods produce nearly identical results in terms of predicted breakthrough time and production temperature decline for high net-to-gross ratios (N/G). Simple homogenous and layered models overestimate breakthrough and underestimate temperature decline. More complex representation of facies in process-based models show smaller variance in results and stay within the limits of ensemble runs compared to simpler facies representation. Results indicate that a multivariate Gaussian based stochastic representation of heterogeneity provides comparable thermal response to a process-based model in fluvial systems of similar quality.

Winning a Tournament According to Bradley-Terry Probability Model

We analyze the chances of winning a tournament under the assumption that the probabilities of winning individual matches follow Bradley-Terry model [2]. We present an exact solution and show a few examples of its use. The examples are from California volleyball tournaments, the round of sixteen in the World Cup and the Champions League, the group stage of the Association of Tennis Professionals tournament, and the volleyball SuperLega in Italy.
 The computational complexity of the solution grows exponentially fastwith the number of teams and we seek approximations via multivariate Gaussian laws.