Mixture Of Normal Distributions Research Articles

Modelling and analysing the daily temperature of several cities using mixture Gaussian distributions

Modelling and analysing the daily temperature of several cities using mixture Gaussian distributions

In Situ Damage Monitoring of CFRPs by Electromagnetic Tomography With the Compatible Multitemplate Supervised Descent Method

In this paper, an in-situ damage detection framework for carbon fiber-reinforced plastics (CFRPs) is proposed. The methodology involves the design of an electromagnetic tomography (EMT) sensor and developing an absolute image reconstruction method for electromagnetic tomography. A linear EMT sensor is designed for in-situ structural health monitoring of CFRPs tensile load test specimens. A compatible multi-template supervised descent method (cmt-SDM) based on a normal mixture distribution model of pixels is proposed for EMT image reconstruction. The cmt-SDM was developed to overcome the ill-posed problem in absolute image reconstruction. Simulation and experimental results show that the method proposed in this paper can be employed to monitor and evaluate the local damage degree of CFRPs specimens under quasi-static tensile loading. In addition, when the local fiber-breaking ratio reaches 40%, the damage location can be accurately reflected by the imaging results. Compared with the traditional SDM method, the method proposed in this paper shows smaller estimation error and delivers more accurate reconstructed images.

Domain Generalization Via Encoding and Resampling in a Unified Latent Space

Domain generalization aims to generalize a network trained on multiple domains to unknown yet related domains. Operating under the assumption that invariant information generalizes well to unknown domains, previous work has aimed to minimize the discrepancies amongst distributions across given domains. However, without prior regularization of feature distributions, the network in practice overfits the invariant information in the given domains. Moreover, if there are insufficient samples in given domains, then domain generalizability is limited, as diverse domain variations are not captured. To address these two drawbacks, we propose to explicitly map features in known and unknown domains onto latent space in a fixed Gaussian mixture distribution by variational coding. As a result, features in different classes follow Gaussian distributions with different mean values. The predefined latent space narrows discrepancies between known and unknown domains and effectively separates samples into different classes. Moreover, we propose to perturb sample features with gradients from the distribution regularized loss. This perturbation generates samples beyond but near the latent space of prior distributions, which has a profound impact on domain variations. Experiments and visualizations demonstrate the effectiveness of our proposed method.

Joint Design of Transmit Sequence and Receive Filter Based on Riemannian Manifold of Gaussian Mixture Distribution for MIMO Radar

To improve target detection performance in non-Gaussian backgrounds, the joint design of transmit sequence and receive filter for multiple-input-multiple-output (MIMO) radar is studied. By approximating the probability density function of observed non-Gaussian data with the Gaussian mixture model, a Riemannian manifold of Gaussian mixture distribution is developed to depict the complicated background first. Then, maximizing the geometric distance on manifolds, which is converted by maximizing the discrimination between the target and clutter, is proposed as the criterion for the joint design of transmit sequence and receive filter. Thereby, under the constant-modulus constraint, the joint design problem can be transformed into an optimization problem. However, the proposed optimization problem is non-convex and constrained. To solve this problem, a Riemannian optimization framework is provided. By taking the advantage of the underlying geometric and algebraic structure of the constraint space, the original constrained optimization problem in Euclidean space can be transformed into the unconstraint optimization problem over Riemannian product manifolds. Moreover, to obtain the global optimal solution, the Riemannian gradient of the geometric distance cost is derived for the conjugate gradient algorithm. Experiments demonstrate that the proposed method shows advantages in detection performance compared with competitive methods.

Crowd Localization from Gaussian Mixture Scoped Knowledge and Scoped Teacher.

Crowd localization is to predict each instance head position in crowd scenarios. Since the distance of pedestrians being to the camera are variant, there exists tremendous gaps among scales of instances within an image, which is called the intrinsic scale shift. The core reason of intrinsic scale shift being one of the most essential issues in crowd localization is that it is ubiquitous in crowd scenes and makes scale distribution chaotic. To this end, the paper concentrates on access to tackle the chaos of the scale distribution incurred by intrinsic scale shift.We propose Gaussian Mixture Scope (GMS) to regularize the chaotic scale distribution. Concretely, the GMS utilizes a Gaussian mixture distribution to adapt to scale distribution and decouples the mixture model into sub-normal distributions to regularize the chaos within the sub-distributions. Then, an alignment is introduced to regularize the chaos among sub-distributions. However, despite that GMS is effective in regularizing the data distribution, it amounts to dislodging the hard samples in training set, which incurs overfitting. We assert that it is blamed on the block of transferring the latent knowledge exploited by GMS from data to model. Therefore, a Scoped Teacher playing a role of bridge in knowledge transform is proposed. What' s more, the consistency regularization is also introduced to implement knowledge transform. To that effect, the further constraints are deployed on Scoped Teacher to derive feature consistence between teacher and student end. With proposed GMS and Scoped Teacher implemented on four mainstream datasets of crowd localization, the extensive experiments demonstrate the superiority of our work. Moreover, comparing with existing crowd locators, our work achieves state-of-the-art via F1-measure comprehensively on four datasets.

Multivariate Mixture Model for Small Area Estimation of Poverty Indicators

Abstract When disaggregation of national estimates in several domains or areas is required, direct survey estimators, which use only the domain-specific survey data, are usually design-unbiased even under complex survey designs (at least approximately) and require no model assumptions. Nevertheless, they are appropriate only for domains or areas with sufficiently large sample size. For example, when estimating poverty in a domain with a small sample size (small area), the volatility of a direct estimator might make that area seems like very poor in one period and very rich in the next one. Small area (or indirect) estimators have been developed in order to avoid such undesired instability. Small area estimators borrow strength from the other areas so as to improve the precision and therefore obtain much more stable estimators. However, the usual model-based assumptions, which include some kind of area homogeneity, may not hold in real applications. A more flexible model based on multivariate mixtures of normal distributions that generalises the usual nested error linear regression model is proposed for estimation of general parameters in small areas. This flexibility makes the model adaptable to more general situations, where there may be areas with a different behaviour from the other ones, making the model less restrictive (hence, more close to nonparametric) and more robust to outlying areas. An expectation-maximisation (E-M) method is designed for fitting the proposed mixture model. Under the proposed mixture model, two different new predictors of general small area indicators are proposed. A parametric bootstrap method is used to estimate the mean squared errors of the proposed predictors. Small sample properties of the new predictors and of the bootstrap procedure are analysed by simulation studies and the new methodology is illustrated with an application to poverty mapping in Palestine.

Quality measures for fully automatic CT histogram-based fat estimation on a corpse sample

In a previous article a new algorithm for fully automatic ‘CT histogram based Fat Estimation and quasi-Segmentation’ (CFES) was validated on synthetic data, on a special CT phantom, and tested on one corpse. Usage of said data in FE-modelling for temperature-based death time estimation is the investigation’s number one long-term goal. The article presents CFES’s results on a human corpse sample of size R = 32, evaluating three different performance measures: the τ-value, measuring the ability to differentiate fat from muscle, the anatomical fat-muscle misclassification rate D, and the weighted distance S between the empirical and the theoretical grey-scale value histogram. CFES-performance on the sample was: D = 3.6% for weight exponent α = 1, slightly higher for α ≥ 2 and much higher for α ≤ 0. Investigating τ, S and D on the sample revealed some unexpected results: While large values of τ imply small D-values, rising S implies falling D and there is a positive linear relationship between τ and S. The latter two findings seem to be counter-intuitive. Our Monte Carlo analysis detected a general umbrella type relation between τ and S, which seems to stem from a pivotal problem in fitting Normal mixture distributions.

Robust mixture regression modeling based on the normal mean-variance mixture distributions

Mixture regression models (MRMs) are widely used to capture the heterogeneity of relationships between the response variable and one or more predictors coming from several non-homogeneous groups. Since the conventional MRMs are quite sensitive to departures from normality caused by extra skewness and possible heavy tails, various extensions built on more flexible distributions have been put forward in the last decade. The class of normal mean-variance mixture (NMVM) distributions that arise from scaling both the mean and variance of a normal random variable with a common mixing distribution encompasses many prominent (symmetric or asymmetrical) distributions as special cases. A unified approach to robustifying MRMs is proposed by considering the class of NMVM distributions for component errors. An expectation conditional maximization either (ECME) algorithm, which incorporates membership indicators and the latent scaling variables as the missing data, is developed for carrying out maximum likelihood (ML) estimation of model parameters. Four simulation studies are conducted to examine the finite-sample property of ML estimators and the robustness of the proposed model against outliers for contaminated and noisy data. The usefulness and superiority of our methodology are demonstrated through applications to two real datasets.

Maximizing Realism: Mapping Plastic Particles at the Ocean Surface Using Mixtures of Normal Distributions.

Current methods of characterizing plastic debris use arbitrary, predetermined categorizations and assume that the properties of particles are independent. Here we introduce Gaussian mixture models (GMM), a technique suitable for describing non-normal multivariate distributions, as a method to identify mutually exclusive subsets of floating macroplastic and microplastic particles (latent class analysis) based on statistically defensible categories. Length, width, height and polymer type of 6,942 particles and items from the Atlantic Ocean were measured using infrared spectroscopy and image analysis. GMM revealed six underlying normal distributions based on length and width; two within each of the lines, films, and fragments categories. These classes differed significantly in polymer types. The results further showed that smaller films and fragments had a higher correlation between length and width, indicating that they were about the same size in two dimensions. In contrast, larger films and fragments showed low correlations of height with length and width. This demonstrates that larger particles show greater variability in shape and thus plastic fragmentation is associated with particle rounding. These results offer important opportunities for refinement of risk assessment and for modeling the fragmentation and distribution of plastic in the ocean. They further illustrate that GMM is a useful method to map ocean plastics, with advantages over approaches that use arbitrary categorizations and assume size independence or normal distributions.

UTILITY INDIFFERENCE PRICING AND MEAN-VARIANCE APPROACH WHEN THE DEGREE OF RISK AVERSION IS SMALL

We study the properties of the risk-sensitive value measure and mean-variance approach studied by Hodoshima et al. [2] when the degree of risk aversion is small. We show that the two value measures work similarly in normal mixture distributions as well as any distribution when the degree of risk aversion is small. Therefore, the advantage of the risk-sensitive value measure disappears compared to mean- variance approach in any distribution when the degree of risk aversion is small. We obtain the result by approximating the risk-sensitive value measure by Taylor series expansion.

Utility of apparent diffusion coefficient histogram analysis in differentiating benign and malignant palate lesions

ObjectivesTo explore the ability of conventional MRI and histogram analysis of apparent diffusion coefficient (ADC) for differentiating between malignant and benign palatal lesions. Materials and MethodsA retrospective analysis was performed on MRI images of 86 patients with palatal lesions confirmed by histopathology between January 2015 and December 2018. Each lesion was evaluated based on the conventional MRI characteristics, including size, location, morphology, inner texture, enhancement, capsule, bone destruction, and nerve invasion. ADC histogram analysis was fitted to a bimodal Gaussian distribution mixture curve. Nine histogram analysis parameters were extracted from ADC maps and were compared between malignant and benign palatal lesions. Statistical analysis was performed to assess the differential performance of each parameter individually and combined. ResultsOn conventional MRI, the capsule structure and nerve invasion were useful characteristics for the differential diagnosis of palatal lesions. Histogram analysis showed significant differences between the groups in terms of mean ADC values of the lower curve and overall curve of the bimodal histogram, as well as ADC10 and ADC50. The optimal diagnostic threshold of ADC value was ADC50 = 1.17 × 10−3 mm2/s (area under curve [AUC] = 0.934, sensitivity = 93.0 %, specificity = 86.2 %). The combination of conventional MRI and ADC values yielded the best predictive performance, with an AUC of 0.949, and an increase in sensitivity from 80.7 % to 96.5 % and specificity from 58.6 % to 82.8 %, compared to conventional MRI. ConclusionsHistogram analysis of DWI combined with conventional MRI allows accurate differentiation between benign and malignant palatal lesions.

Features of the Application of the Detection Algorithm Based on the Background Subtraction Method in the Scanning Optoelectronic Surveillance System

The article considers the approach developed by the authors to adapt the detection algorithm based on the background subtraction method to work in a scanning optoelectronic surveillance system. This algorithm is the basis of the developed image processing method applied to the problem of detecting moving objects by an optoelectronic thermal imaging surveillance system. At the same time, this method provides a solution to the tasks of protecting a critical object with an optoelectronic surveillance system, which must function both in stationary and scanning mode. The use of an improved mixture of Gaussian distributions to build a background model makes it possible to adapt to periodic changes in pixel brightness from dynamic background elements of the observed scene. However, the application of this detection algorithm in a scanning optoelectronic system requires the solution of the following problematic issues: the exclusion of false highlighting of the areas of the processed image in the foreground when transferring the sensor field of view; the elimination of errors in the initial values of the parameters of a mixture of Gaussian distributions. The application of the approach proposed by the authors to solving these issues is implemented in a specialized optoelectronic surveillance system and allows achieving the same quality of the detector based on the background subtraction method in scanning mode as in stationary mode.

A Decoder-Free Variational Deep Embedding for Unsupervised Clustering.

In deep clustering frameworks, autoencoder (AE)- or variational AE-based clustering approaches are the most popular and competitive ones that encourage the model to obtain suitable representations and avoid the tendency for degenerate solutions simultaneously. However, for the clustering task, the decoder for reconstructing the original input is usually useless when the model is finished training. The encoder-decoder architecture limits the depth of the encoder so that the learning capacity is reduced severely. In this article, we propose a decoder-free variational deep embedding for unsupervised clustering (DFVC). It is well known that minimizing reconstruction error amounts to maximizing a lower bound on the mutual information (MI) between the input and its representation. That provides a theoretical guarantee for us to discard the bloated decoder. Inspired by contrastive self-supervised learning, we can directly calculate or estimate the MI of the continuous variables. Specifically, we investigate unsupervised representation learning by simultaneously considering the MI estimation of continuous representations and the MI computation of categorical representations. By introducing the data augmentation technique, we incorporate the original input, the augmented input, and their high-level representations into the MI estimation framework to learn more discriminative representations. Instead of matching to a simple standard normal distribution adversarially, we use end-to-end learning to constrain the latent space to be cluster-friendly by applying the Gaussian mixture distribution as the prior. Extensive experiments on challenging data sets show that our model achieves higher performance over a wide range of state-of-the-art clustering approaches.

Prognostics of rotating machines through generalized Gaussian hidden Markov models

Nowadays, the industrial scenario is driven by the need of costs and time reduction. In this contest, system failure prediction plays a pivotal role in order to program maintenance operations only in the last stages of the real operating life, avoiding unnecessary machine downtime. In the last decade, Hidden Markov Models have been widely exploited for machinery prognostic purposes. The probabilistic dependency between the measured observations and the real damaging stage of the system has usually been described as a mixture of Gaussian distributions. This paper aims to generalize the probabilistic function as a mixture of generalized Gaussian distributions in order to consider possible distribution variations during the different states. In this direction, this work proposes an algorithm for the estimation of the model parameters exploiting the observations measured on the real system. The prognostic effectiveness of the resulting model has been demonstrated through the analysis of several run-to-failure datasets concerning both rolling element bearings and more complex systems.

A Revision of the Traditional Analysis Method of Allometry to Allow Extension of the Normality-Borne Complexity of Error Structure: Examining the Adequacy of a Normal-Mixture Distribution-Driven Error Term.

Huxley's model of simple allometry provides a parsimonious scheme for examining scaling relationships in scientific research, resource management, and species conservation endeavors. Factors including biological error, analysis method, sample size, and overall data quality can undermine the reliability of a fit of Huxley's model. Customary amendments enhance the complexity of the power function-conveyed systematic term while keeping the usual normality-borne error structure. The resulting protocols bear multiple-parameter complex allometry forms that could pose interpretative shortcomings and parameter estimation difficulties, and even being empirically pertinent, they could potentially bear overfitting. A subsequent heavy-tailed Q-Q normal spread often remains undetected since the adequacy of a normally distributed error term remains unexplored. Previously, we promoted the advantages of keeping Huxley's model-driven systematic part while switching to a logistically distributed error term to improve fit quality. Here, we analyzed eelgrass leaf biomass and area data exhibiting a marked size-related heterogeneity, perhaps explaining a lack of systematization at data gathering. Overdispersion precluded adequacy of the logistically adapted protocol, thereby suggesting processing data through a median absolute deviation scheme aimed to remove unduly replicates. Nevertheless, achieving regularity to Huxley's power function-like trend required the removal of many replicates, thereby questioning the integrity of a data cleaning approach. But, we managed to adapt the complexity of the error term to reliably identify Huxley's model-like systematic part masked by variability in data. Achieving this relied on an error term conforming to a normal mixture distribution which successfully managed overdispersion in data. Compared to normal-complex allometry and data cleaning composites present arrangement delivered a coherent Q-Q normal mixture spread and a remarkable reproducibility strength of derived proxies. By keeping the analysis within Huxley's original theory, the present approach enables substantiating nondestructive allometric proxies aimed at eelgrass conservation. The viewpoint endorsed here could also make data cleaning unnecessary.

Sparse Online Variational Bayesian Regression

This work considers variational Bayesian inference as an inexpensive and scalable alternative to a fully Bayesian approach in the context of sparsity-promoting priors. In particular, the priors considered arise from scale mixtures of Normal distributions with a generalized inverse Gaussian mixing distribution. This includes the variational Bayesian LASSO as an inexpensive and scalable alternative to the Bayesian LASSO introduced in [65]. It also includes a family of priors which more strongly promote sparsity. For linear models the method requires only the iterative solution of deterministic least squares problems. Furthermore, for p unknown covariates the method can be implemented exactly online with a cost of $O(p^3)$ in computation and $O(p^2)$ in memory per iteration -- in other words, the cost per iteration is independent of n, and in principle infinite data can be considered. For large $p$ an approximation is able to achieve promising results for a cost of $O(p)$ per iteration, in both computation and memory. Strategies for hyper-parameter tuning are also considered. The method is implemented for real and simulated data. It is shown that the performance in terms of variable selection and uncertainty quantification of the variational Bayesian LASSO can be comparable to the Bayesian LASSO for problems which are tractable with that method, and for a fraction of the cost. The present method comfortably handles $n = 65536$, $p = 131073$ on a laptop in less than 30 minutes, and $n = 10^5$, $p = 2.1 \times 10^6$ overnight.

Wiener models robust identification of multi-rate process with time-varying delay using expectation-maximization algorithm

Multi-rate systems exist widely in the field of computer process control. Research on the multi-rate issue of linear systems is relatively mature and is gradually turning to multi-rate issues of nonlinear systems. Concurrently, irregular process disturbances (impulse noise, outliers, non-zero mean noise) and varying transmission delay problems are often overlooked or considered alone, which is lacking generality. Thus, the main objective of this paper is to formulate and solve the identification problem of multi-rate nonlinear Wiener models with time-varying delay and irregular process disturbances simultaneously. The probability graph model of the Wiener process is constructed. Under the expectation–maximization (EM) framework, the Scale outlier model and Location outlier model (both based on Gaussian mixture distribution) are separately introduced to model the contaminated output data. The time-varying delay at each sampling instant is assumed as a uniform distribution. An auxiliary model is applied to acquire the unmeasured middle variable. Then, a robust EM identification algorithm is developed and its convergence is analyzed. The validity of the developed approach is illustrated through two numerical examples and a simulation of the continuous stirred tank reactor.

Multi-mode non-Gaussian variational autoencoder network with missing sources for anomaly detection of complex electromechanical equipment

Anomaly detection is crucial to the safety of complex electromechanical equipment. With the rapid accumulation of industrial data, intelligent methods without human intervention have become the mainstream of anomaly detection. Among them, variational autoencoder (VAE) performs well in anomaly detection with missing fault samples due to the self-supervised learning paradigm. However, the data from electromechanical equipment is usually non-Gaussian, making it difficult for the standard VAE based on Gaussian distribution to recognize the abnormal states. To solve the above problems, we proposed multi-mode non-Gaussian VAE (MNVAE) to detect anomalies from unknown distribution vibration signals without fault samples or prior knowledge. Firstly, the encoder maps the input to a Gaussian mixture distribution in latent space and samples a latent variable from it, after which the Householder Flow is applied to the latent variable to capture more abundant features. Finally, to describe the non-Gaussianity of the signal, Weibull distribution serves as the likelihood function of the reconstructed signal output from the decoder and as the basis for anomaly discrimination. In comparison to 6 related methods, our method yields the best results across various datasets. Through further experiments, the robustness of our method is proved and the proposed improvements are effective.

EGMM: An evidential version of the Gaussian mixture model for clustering

The Gaussian mixture model (GMM) provides a simple yet principled framework for clustering, with properties suitable for statistical inference. In this paper, we propose a new model-based clustering algorithm, called EGMM (evidential GMM), in the theoretical framework of belief functions to better characterize cluster-membership uncertainty. With a mass function representing the cluster membership of each object, the evidential Gaussian mixture distribution composed of the components over the powerset of the desired clusters is proposed to model the entire dataset. The parameters in EGMM are estimated by a specially designed Expectation–Maximization (EM) algorithm. A validity index allowing automatic determination of the proper number of clusters is also provided. The proposed EGMM is as simple as the classical GMM, but can generate a more informative evidential partition for the considered dataset. The synthetic and real dataset experiments show that the proposed EGMM performs better than other representative clustering algorithms. Besides, its superiority is also demonstrated by an application to multi-modal brain image segmentation.

Unsupervised hierarchical methodology of maritime traffic pattern extraction for knowledge discovery

Owing to the space–air–ground integrated networks (SAGIN), seaborne shipping has attracted increasing interest in the research on the motion behavior knowledge extraction and navigation pattern mining problems in the era of maritime big data for improving maritime traffic safety management. This study aims to develop a novel unsupervised methodology for feature extraction and knowledge discovery based on automatic identification system (AIS) data, allowing for seamless knowledge transfer to support trajectory data mining. The unsupervised hierarchical methodology is constructed from three parts: trajectory compression, trajectory similarity measure, and trajectory clustering. In the first part, an adaptive Douglas–Peucker with speed (ADPS) algorithm is created to preserve critical features, obtain useful information, and simplify trajectory information. Then, dynamic time warping (DTW) is utilized to measure the similarity between trajectories as the critical indicator in trajectory clustering. Finally, the improved spectral clustering with mapping (ISCM) is presented to extract vessel traffic behavior characteristics and mine movement patterns for enhancing marine safety and situational awareness. Comprehensive experiments are conducted and implemented in the Chengshan Jiao Promontory in China to verify the feasibility and effectiveness of the novel methodology. Experimental results show that the proposed methodology can effectively compress the trajectories, determine the number of clusters in advance, guarantee the clustering accuracy, and extract useful navigation knowledge while significantly reducing the computational cost. The clustering results are further explored and follow the Gaussian mixture distribution, which can help provide new discriminant criteria for trajectory clustering.