Gaussian Mixture Probability Density Function Research Articles

A just-in-time modeling approach for multimode soft sensor based on Gaussian mixture variational autoencoder

Industrial data are often high-dimensional, nonlinear and multiple-modal. This paper develops a soft sensor model based on Gaussian mixture Variational Autoencoder (GMVAE) under the just-in-time learning (JITL) framework. To extract latent representations with multimode characteristics, GMVAE as a deep neural network model is utilized by considering Gaussian mixture models (GMM) in the latent space. After training the GMVAE model, each latent (or feature) variable can be described through a Gaussian mixture distribution. Subsequently, when a new sample arrives, a mixture symmetric Kullback-Leibler (MSKL) divergence is utilized to measure its similarity with historical data samples. MSKL divergence can measure similarity between two Gaussian mixture probability density functions. Based on the MSKL divergence, weighted input and output historical data are obtained, and then a local model is established. The effectiveness of the proposed soft sensor modeling method is validated through a numerical example along with simulation on the Tennessee Eastman benchmark process.

Information-Theoretic Joint Probabilistic Data Association Filter

This article proposes a novel information-theoretic joint probabilistic data association filter for tracking unknown number of targets. The proposed information-theoretic joint probabilistic data association algorithm is obtained by the minimization of a weighted reverse Kullback–Leibler divergence to approximate the posterior Gaussian mixture probability density function. Theoretical analysis of mean performance and error covariance performance with ideal detection probability is presented to provide insights of the proposed approach. Extensive empirical simulations are undertaken to validate the performance of the proposed multitarget tracking algorithm.

A sea-sky-line detection method based on Gaussian mixture models and image texture features

This article presents a sea-sky-line detection algorithm in a sea-sky environment for unmanned surface vehicles. Obstacle detection is a vital branch for unmanned surface vehicles on the ocean. Because of the specificity and complexity of the marine navigation environment, we first apply semantic segmentation for marine images. The complete marine scene is divided into sky area, middle mixture area, and seawater area before sea-sky-line detection. Segmenting the marine environment is beneficial for narrowing the obstacle search area, accelerating the rate of obstacle detection, and improving detection accuracy. Therefore, a fast, robust, and accurate sea-sky image segmentation method is urgently required. Therefore, we present a method that lies in a probabilistic graphical model for segmenting marine images. The Gaussian mixture model is introduced as the probability distribution model for the marine image. The sky, middle mixture, and seawater areas are generated by three Gaussian models. The expectation–maximization algorithm is utilized to maximize the log-likelihood function, and the parameters of the Gaussian mixture probability density function that recover the marine image distribution are available after several iterations. Furthermore, to solve the problem of incorrect convergence direction caused by unsatisfactory initialization conditions, the gray level co-occurrence matrix is referenced to initialize the Gaussian components. The coarse segmentation results rely on the gray level co-occurrence matrix and are used to calculate the prior initialization parameters of Gaussian components and obtain the prior distribution information of marine images, which mitigates the harmful influence of poor initialization. The algorithm is tested on a data set consisting of the marine obstacle detection dataset (MODD) public data set and our collected images. The results on this data set demonstrate that the proposed method is more robust and that a superior initialization condition can effectively accelerate the convergence velocity of the iterative process for Gaussian components.

Error propagation in traffic modeling: solutions for the mesoscopic LWR model

ABSTRACTTraffic flow models are a valuable tool to design road networks, assess the effectiveness of traffic controls and provide real-time information for traffic management centers and users. When integrating data, they become the cornerstone for traffic monitoring and forecasting. However, these models are based on a simplified representation of the traffic, subject to uncertainties and with parameters likely to be refined. The paper proposes the formulation of a new traffic flow modeling framework that accounts for error propagation. It is based on the mesoscopic approach of the Lighthill-Whitham-Richards model, with traffic variables represented as Gaussian-Mixture probability density functions. Sensitivity for the error propagation model is analyzed regarding: the network topology, traffic flow parameters and parameterization of the error model. Applications for traffic indicators associated to their errors/uncertainties are illustrated. Novel indicators and decision-support tools can be built for advanced traffic modeling solutions based on the proposed model.

SAR Image Despeckling Based on a Mixture of Gaussian Distributions with Local Parameters and Multiscale Edge Detection in Lapped Transform Domain

A new Lapped transform domain SAR image despeckling algorithm using a two-state Gaussian mixture probability density function that uses local parameters for the mixture model, is proposed. The use of lapped orthogonal transform (LOT) is motivated by its low computational complexity and robustness to oversmoothing. It is shown that the dyadic rearranged LOT coefficients of logarithmically transformed SAR images can be well approximated using two-state Gaussian mixture distribution compared to Laplacian, Gamma, generalized Gaussian and Cauchy distributions, based on the Kolmogorov–Smirnov (KS) goodness of fit test. The LOT coefficients of speckle noise are modeled using zero mean Gaussian distributions. A maximum a posteriori (MAP) estimator within Bayesian framework is developed using this proposed prior distribution and is used to restore the noisy LOT coefficients. The parameters of mixture distribution are estimated using the expectation-maximization algorithm. This paper presents a new technique to identify LOT modulus maxima which allows us to classify LOT coefficients into the edge and non edge coefficients. The classified edge coefficients are kept unmodified by the proposed algorithm whereas the noise-free estimates of non-edge coefficients are obtained using Bayesian MAP estimator developed using two state Gaussian mixture distribution with local parameters. Finally the proposed technique is combined with the cycle spinning scheme to further improve the despeckling performance. Experimental results show that the proposed method very efficiently reduces speckle in homogeneous regions while preserving more edge structures compared to some recent well known methods.

Directionlet-based method using the Gaussian mixture prior to SAR image despeckling

In this article, a new denoising algorithm is proposed based on the directionlet transform and the maximum a posteriori (MAP) estimation. The detailed directionlet coefficients of the logarithmically transformed noise-free image are considered to be Gaussian mixture probability density functions (PDFs) with zero means, and the speckle noise in the directionlet domain is modelled as additive noise with a Gaussian distribution. Then, we develop a Bayesian MAP estimator using these assumed prior distributions. Because the estimator that is the solution of the MAP equation is a function of the parameters of the assumed mixture PDF models, the expectation-maximization (EM) algorithm is also utilized to estimate the parameters, including weight factors and variances. Finally, the noise-free SAR image is restored from the estimated coefficients yielded by the MAP estimator. Experimental results show that the directionlet-based MAP method can be successfully applied to images and real synthetic aperture radar images to denoise speckle.

Likelihood function modeling of particle filter in presence of non-stationary non-gaussian measurement noise

A generalized likelihood function model of a sampling importance resampling (SIR) particle filter (PF) has been derived for state estimation of a nonlinear system in the presence of non-stationary, non-Gaussian white measurement noise. The measurement noise is modeled by Gaussian mixture probability density function and the noise parameters are estimated by maximizing the log likelihood function of the noise model. This model is then included in the likelihood function of the SIR particle filter (PF) at each time step for online state estimation of the system. The performance of the proposed algorithm has been evaluated by estimating the states of (i) a non-linear system in the presence of non-stationary Rayleigh distributed noise and (ii) a radar tracking system in the presence of glint noise. The simulation results show that the proposed modified SIR PF offers best performance among the considered algorithms for these examples.

Using Probability Density Functions to Derive Consistent Closure Relationships among Higher-Order Moments

Parameterizations of turbulence often predict several lower-order moments and make closure assumptions for higher-order moments. In principle, the low- and high-order moments share the same probability density function (PDF). One closure assumption, then, is the shape of this family of PDFs. When the higher-order moments involve both velocity and thermodynamic scalars, often the PDF shape has been assumed to be a double or triple delta function. This is equivalent to assuming a mass-flux model with no subplume variability. However, PDF families other than delta functions can be assumed. This is because the assumed PDF methodology is fairly general. This paper proposes closures for several third- and fourth-order moments. To derive the closures, the moments are assumed to be consistent with a particular PDF family, namely, a mixture of two trivariate Gaussians. (This PDF is also called a double Gaussian or binormal PDF by some authors.) Separately from the PDF assumption, the paper also proposes a simplified relationship between scalar and velocity skewnesses. This PDF family and skewness relationship are simple enough to yield simple, analytic closure formulas relating the moments. If certain conditions hold, this set of moments is specifically realizable. By this it is meant that the set of moments corresponds to a real Gaussian-mixture PDF, one that is normalized and nonnegative everywhere. This paper compares the new closure formulas with both large eddy simulations (LESs) and closures based on double and triple delta PDFs. This paper does not implement the closures in a single-column model and test them interactively. Rather, the comparisons are diagnostic; that is, low-order moments are extracted from the LES and treated as givens that are input into the closures. This isolates errors in the closures from errors in a single-column model. The test cases are three atmospheric boundary layers: a trade wind cumulus layer, a stratocumulus layer, and a clear convective case. The new closures have shortcomings, but nevertheless are superior to the double or triple delta closures in most of the cases tested.

Specifics of Hidden Markov Model Modifications for Large Vocabulary Continuous Speech Recognition

Specifics of hidden Markov model-based speech recognition are investigated. Influence of modeling simple and context-dependent phones, using simple Gaussian, two and three-component Gaussian mixture probability density functions for modeling feature distribution, and incorporating language model are discussed. Word recognition rates and model complexity criteria are used for evaluating suitability of these modifications for practical applications. Development of large vocabulary continuous speech recognition system using HTK toolkit and WSJCAM0 English speech corpus is described. Results of experimental investigations are presented.

Parameter estimation for autoregressive Gaussian-mixture processes: the EMAX algorithm

The problem of estimating parameters of discrete-time non-Gaussian autoregressive (AR) processes is addressed. The subclass of such processes considered is restricted to those whose driving noise samples are statistically independent and identically distributed according to a Gaussian-mixture probability density function (pdf). Because the likelihood function for this problem is typically unbounded in the vicinity of undesirable, degenerate parameter estimates, the maximum likelihood approach is not fruitful. Hence, an alternative approach is taken whereby a finite local maximum of the likelihood surface is sought. This approach, which is termed the quasimaximum likelihood (QML) approach, is used to obtain estimates of the AR parameters as well as the means, variances, and weighting coefficients that define the Gaussian-mixture pdf. A technique for generating solutions to the QML problem is derived using a generalized version of the expectation-maximization principle. This technique, which is referred to as the EMAX algorithm, is applied in four illustrative examples; its performance is compared directly with that of previously proposed algorithms based on the same data model and that of conventional least-squares techniques.

Bound for Minkowski metric or quadratic metric applied to VQ codeword search

A bound for a Minkowski metric based on L/sub p/ distortion measure is proposed and evaluated as a means to reduce the computation in vector quantisation. This bound provides a better criterion than the absolute error inequality (AEI) elimination rule on the Euclidean distortion measure. For the Minkowski metric of order n, this bound contributes the elimination criterion from the L/sub 1/ metric to L/sub n/ metric. This bound can also be an extended quadratic metric which can be a hidden Markov model (HMM) with a Gaussian mixture probability density function (PDF). In speech recognition, the HMM with the Gaussian mixture VQ codebook PDF has been shown to be a promising method.

A mixture modeling approach to text‐independent speaker ID

Automatic speaker identification (ASI) systems generally fall into two classes: text dependent and text independent. High recognition rates for short utterances (<1 s) are more common for text‐dependent systems since the process benefits from the a priori knowledge of the underlying acoustic‐phonetic stream. Without this added information, text‐independent ASI usually requires long test utterances for averaging out the unknown phonetic variations. In this paper a new technique is introduced to bridge the gap between text‐dependent and text‐independent ASI, which allows for high recognition rates using short utterances in a text‐independent ASI system. Speakers are parametrically represented by a Gaussian mixture probability density function, where the parameters are maximum likelihood estimates obtained from a form of the iterative estimate‐maximize (EM) algorithm [G. J. McLachlan, Mixture Models (Dekker, New York, 1988)]. The components in the mixture model can be considered to represent “hidden” acousti...