Reconstruction of Highway Vehicle Paths Using a Two-Stage Model

The expectation maximisation (EM) algorithm is an iterative maximum likelihood procedure often used for estimating the parameters of a mixture model. Theoretically, increases in the likelihood function are guaranteed as the algorithm iteratively improves upon previously derived parameter estimates. The algorithm is considered to converge when all parameter estimates become stable and no further improvements can be made to the likelihood value. However, to reduce computational time, it is often common practice for the algorithm to be stopped before complete convergence using heuristic approaches. In this paper, we consider various stopping criteria and evaluate their effect on fitting Gaussian mixture models (GMMs) to patient length of stay (LOS) data. Although the GMM can be successfully fitted to positively skewed data such as LOS, the fitting procedure often requires many iterations of the EM algorithm. To our knowledge, no previous study has evaluated the effect of different stopping criteria on fitting GMMs to skewed distributions. Hence, the aim of this paper is to evaluate the effect of various stopping criteria in order to select and justify their use within a patient spell classification methodology. Results illustrate that criteria based on the difference in the likelihood value and on the GMM parameters may not always be a good indicator for stopping the algorithm. In fact we show that the values of the difference in the variance parameters should be used instead, as these parameters are the last to stabilise. In addition, we also specify threshold values for the other stopping criteria.

One effective iterative method for estimating parameters in latent variable probabilistic models is the Expectation-Maximization (EM) algorithm, which finds extensive use in classification issues. A versatile probabilistic framework for modeling data from multiple Gaussian distributions, Gaussian Mixture Models (GMMs) employ EM for parameter estimation, which is explored in this article. Problems with class membership estimation (including means, covariances, and mixing coefficients) are the primary focus of this study. Starting with initial parameter estimations, estimating posterior probabilities in the E-step, and repeatedly updating model parameters in the M-step until convergence, the technique entails applying the EM algorithm on both synthetic and real-world datasets. In order to measure performance, we compare it to K-means and other baseline classifiers using measures like classification accuracy, Adjusted Rand Index (ARI), and log-likelihood progression. When data is distributed according to a multimodal Gaussian distribution, the findings show that EM-GMM provides better parameter estimation and better classification accuracy. Initialization procedures have a substantial effect on performance, as convergence study further shows. In addition to reiterating the EM algorithm's usefulness for probabilistic classification, this paper also describes its advantages and disadvantages and offers suggestions for improving its performance in settings with high-dimensional data and noise.

This paper addresses the problem of initialization of the expectation-maximization (EM) algorithm for maximum likelihood estimation of Gaussian mixture models. In order to avoid local maxima of the likelihood function, a genetic algorithm (GA) which searches for best initial conditions of the EM algorithm is proposed. In the GA, a chromosome represents a set of initial conditions, in which initial mean vectors of mixture components are feature vectors chosen from the training set. The chromosome also encodes variances of initial spherical covariance matrices of mixture components. To evaluate each chromosome in the GA we run the EM algorithm until convergence and use the obtained log likelihood as the fitness. In computational experiments our approach was applied to clustering problem and tested on two datasets from the image processing domain. The results indicate that our method outperforms the standard multiple restart EM algorithm and is at least comparable to the state-of-the art random swap EM method.

The traditional Expectation-Maximization (EM) training of Gaussian Mixture Model (GMM) is essentially a batch mode procedure which requires the multiple data samples with the sufficient size to update the model parameters. This severely limits the deployment and adaptation of GMM in many real-time online systems since the newly observed data samples are expected to be incorporated into the system upon available via retraining the model. This paper presents a new online incremental EM training procedure of GMM, which aims to perform the EM training incrementally and so can adapt GMM online sample by sample. The proposed method is extended on two kinds of EM algorithms for GMM, namely, Split-and-Merge EM and the traditional EM. Experiments on both the synthetic data and a speech processing task show the advantages and efficiency of the new method.

In a distribution power network, the load model has no certain pattern or predicted behaviour due to large range of data and changes in energy consumption for end-user consumers. Thus, a powerful analysis based on probabilistic structure is required. For this paper Gaussian Mixture Model (GMM) has been used. GMM is a powerful probability model that allows different types of load distributions to be presented as a combination of several Gaussian distributions. The parameters of GMM is unknown for large random data such as real load data and these parameters can be identified by Expectation Maximization (EM) algorithm. This paper presents a method to evaluate probabilistic load data concerning the time-evolution of any type of distribution load for any duration of time. The proposed method is explained through generated load data of 100 residential houses for duration of one year.

The expectation-maximization (EM) algorithm is a popular approach for parameter estimation of finite mixture model (FMM). A drawback of this approach is that the number of components of the finite mixture model is not known in advance, nevertheless, it is a key issue for EM algorithms. In this paper, a penalized minimum matching distance-guided EM algorithm is discussed. Under the framework of Greedy EM, a fast and accurate algorithm for estimating the number of components of the Gaussian mixture model (GMM) is proposed. The performance of this algorithm is validated via simulative experiments of univariate and bivariate Gaussian mixture models.

Model-based clustering approaches concern the paradigm of exploratory data analysis relying on the finite mixture model to automatically find a latent structure governing observed data. They are one of the most popular and successful approaches in cluster analysis. The mixture density estimation is generally performed by maximizing the observed-data log-likelihood by using the expectation-maximization (EM) algorithm. However, it is well-known that the EM algorithm initialization is crucial. In addition, the standard EM algorithm requires the number of clusters to be known a priori. Some solutions have been provided in [31], [12] for model-based clustering with Gaussian mixture models for multivariate data. In this paper we focus on model-based curve clustering approaches, when the data are curves rather than vectorial data, based on regression mixtures. We propose a new robust EM algorithm for clustering curves. We extend the model-based clustering approach presented in [31] for Gaussian mixture models, to the case of curve clustering by regression mixtures, including polynomial regression mixtures as well as spline or B-spline regressions mixtures. Our approach both handles the problem of initialization and the one of choosing the optimal number of clusters as the EM learning proceeds, rather than in a twofold scheme. This is achieved by optimizing a penalized log-likelihood criterion. A simulation study confirms the potential benefit of the proposed algorithm in terms of robustness regarding initialization and funding the actual number of clusters.

In recent year, stock investor in Indonesia increased rapidly, so it is required to do analysis about the stock that helps the investor in their investment plan. Clustering is beneficial to select the appropriate stock for investors. Unfortunately, stock prices keep varying from time to time. Consequently, it is not an easy work to select the stock for investment. In addition, stock price time series data are high dimensional data that influenced by many factors. In this study, high dimensional data are obtained by the time frame of each factor. Therefore, it is important to use a suitable technique to cluster high dimensional data. This paper presents High Dimensional Data Clustering (HDDC), a model-based clustering based on Gaussian Mixture Model, using the Expectation-Maximization (EM) algorithm. HDDC via EM algorithm gives a more robust result, and it possible to make an additional assumption. Moreover, this paper combines a high-dimensional clustering technique HDDC via EM algorithm and the most popular feature extraction technique Principal Component Analysis (PCA). This paper comparing methods of clustering technique HDDC and the combination between HDDC and PCA to know the most effective method which gives better result in clustering high-dimensional time series data. The 155 data features are reduced to 7 principal components using PCA analysis. Despite PCA has increased the time efficiency of building the model, clustering technique HDDC via EM algorithm enables to handle the high-dimensional data better than the combination with PCA.

The accurate modeling of uncertain variables, such as wind power and load uncertainty, is an essential foundation for the planning operation and scheduling decisions in power systems. Currently, Gaussian mixture model (GMM) and Dirichlet process mixture model (DPMM) are the primary techniques used to characterize uncertainties in power systems, which are commonly solved by expectation-maximization (EM) algorithm. However, for the massive data of uncertain variables, the algorithm encounters challenges in accurately obtaining GMM and DPMM with a lower time consumption. To address this issue, we propose a method for GMM uncertainty modeling in power systems considering the mutual assistance of latent variables. Specifically, the GMM of uncertain variables is first constructed, and the conditional probability is employed to characterize the mutual assistance of latent variables. Then, an improved EM algorithm is developed to obtain the optimal GMM parameters, in which the expectation step (E-step) and maximization step (M-step) of the algorithm are revised using the conditional probability. Importantly, the closed-form solutions for GMM parameters are rederived in the revised E-step and M-step. Finally, the proposed uncertainty modeling method is compared with the traditional GMM and DPMM on actual wind power and load data from Australia. The proposed method performs the efficiency and accuracy in characterizing uncertainty.

In the last decades, Gaussian Mixture Models (GMMs) have attracted considerable interest in data mining and pattern recognition. A GMM-based clustering algorithm models a dataset with a mixture of multiple Gaussian components and estimates the model parameters using the Expectation-Maximization (EM) algorithm. Recently, a new Locally Consistent GMM (LCGMM) has been proposed to improve the clustering performance by exploiting the local manifold structure of the data using a p nearest neighbor graph. In addition to the underlying manifold structure, many other forms of prior knowledge may guide the clustering process and improve the performance. In this paper, we introduce a Semi-Supervised LCGMM (Semi-LCGMM), where the prior knowledge is provided in the form of class labels of partial data. In particular, the new Semi-LCGMM incorporates the prior knowledge into the maximum likelihood function of the original LCGMM, and the model parameters are estimated using the EM algorithm. It is worth noting that, in our algorithm, each class may be modeled by multiple Gaussian components while in the unsupervised setting each class is modeled by a single Gaussian component. Our algorithm has shown promising results in many different applications, including clustering breast cancer data, heart disease data, handwritten digit images, human face images, and image segmentation.

Chapter 12 - Bayesian Learning: Inference and the EM Algorithm

This paper deals with a collision avoidance method for UAVs based on potential field. We derive potential vectors by using a Gaussian distribution function to design an avoidance trajectory and use a Gaussian Mixture Model (GMM) to represent two-dimensional complex shaped obstacles. In addition, we apply the Expectation-Maximization (EM) algorithm to modify the potential field in order to update the information using measured data on the obstacles. For this purpose, after briefly introducing the potential field, we construct a GMM with the EM algorithm and conduct guidance simulations using a point-mass UAV model in order to demonstrate the performance of the algorithm. Finally, we discuss the implications of the current approach and future research direction.

Segmentation of clusters by template rotation expectation maximization

Gaussian-input Gaussian mixture model for representing density maps and atomic models.

This article proposes the distribution similarity measure–based Gaussian mixtures model for the contact-state (CS) modelling in force-guided robotic assembly processes of flexible rubber parts. The wrench (Cartesian force and torque) signals of the manipulated object are captured for different states of the given assembly process. The distribution similarity measure–based Gaussian mixtures model CS modelling scheme is employed in modelling the captured wrench signals for different CSs. The proposed distribution similarity measure–based Gaussian mixtures model CS modelling scheme uses the Gaussian mixtures model in modelling the captured signals. The parameters of the Gaussian mixtures models are computed using expectation maximisation. The optimal number of Gaussian mixtures model components for each CS model is determined by considering the classification success rate as an index for the similarity measure between the distribution of the captured signals and the developed models. The optimal number of Gaussian mixtures model components corresponds to the highest classification success rate; hence, object elasticity variation would be accommodated by properly choosing the optimal number of Gaussian mixtures model components. The performance of the proposed distribution similarity measure–based Gaussian mixtures model CS modelling strategy is evaluated by a test stand composed of a KUKA lightweight robot doing peg-in-hole assembly processes for flexible rubber objects. Two rubber objects with different elasticity are considered for two experiments; in the first experiment, an elastic peg of 30 Shore A hardness is considered and that of the second experiment has hardness of 6 Shore A which is even softer than the one used in experiment 1. Employing the proposed distribution similarity measure–based Gaussian mixtures model CS modelling strategy excellent classification success rate was obtained for both experiments. However, more Gaussian mixtures model components are required for the softer one that gives a strong impression of the non-stationarity behaviour increment for softer materials. Comparison is performed with the available CS modelling schemes and the distribution similarity measure–based Gaussian mixtures model is shown to provide the best classification success rate performance with a reduced computational time.