Robust Estimation of Gaussian Mixture Models Using Anomaly Scores and Bayesian Information Criterion for Missing Value Imputation

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 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.

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.

EM algorithm is a very popular iteration-based method to estimate the parameters of Gaussian Mixture Model from a large observation set. However, in most cases, EM algorithm is not guaranteed to converge to the global optimum. Instead, it stops at some local optimums, which can be much worse than the global optimum. Therefore, it is usually required to run multiple procedures of EM algorithm with different initial configurations and return the best solution. To improve the efficiency of this scheme, we propose a new method which can estimate an upper bound on the logarithm likelihood of the local optimum, based on the current configuration after the latest EM iteration. This is accomplished by first deriving some region bounding the possible locations of local optimum, followed by some upper bound estimation on the maximum likelihood. With this estimation, we can terminate an EM algorithm procedure if the estimated local optimum is definitely worse than the best solution seen so far. Extensive experiments show that our method can effectively and efficiently accelerate conventional multiple restart EM algorithm.

This paper describes a local and distributed expectation maximization algorithm for learning parameters of Gaussian mixture models (GMM) in large peer-to-peer (P2P) environments. The algorithm can be used for a variety of well-known data mining tasks in distributed environments such as clustering, anomaly detection, target tracking, and density estimation to name a few, necessary for many emerging P2P applications in bioinformatics, webmining and sensor networks. Centralizing all or some of the data to build global models is impractical in such P2P environments because of the large number of data sources, the asynchronous nature of the P2P networks, and dynamic nature of the data/network. The proposed algorithm takes a two-step approach. In the monitoring phase, the algorithm checks if the model `quality' is acceptable by using an efficient local algorithm. This is then used as a feedback loop to sample data from the network and rebuild the GMM when it is outdated. We present thorough experimental results to verify our theoretical claims.

In this paper we apply generative modeling to Gaussian Mixture Models (GMM) as a solution to high dimensional novel event detection and analysis on radio frequency (RF) power generators. A family of Denoising and β-Disentangling Variational Autoencoders β-DVAE) is used to encode datasets into lower and more salient dimensions. The reduced feature sets are modeled using GMMs where model parameters are learned using the Expectation Maximization (EM) algorithm. The data is obtained from two different generator models operating under normal as well as known not-normal conditions. This approach is also tested on standard classification sets. Robust testing is reported to achieve a target class accuracy of 98.16% for the target RF generator. Additionally, the GMM parameters are decoded by the generative model to the original data space for calculating per-variable fitness in order to provide an initial novel event analysis for engineers and technicians. This per-variable fitness is very critical for determining each variable's contribution to a novelty so that engineers can perform informed trouble shooting and maintenance of the RF generators.

There has been growing interest in performing signal processing tasks directly on compressive measurements, e.g. low-dimensional linear measurements of signals taken with Gaussian random vectors. In this paper, we present a highly efficient algorithm to recover the covariance matrix of high-dimensional data from compressive measurements. We show that, as the number of data samples increases, the eigenvectors (principal components) of the empirical covariance matrix of a simple matrix-vector multiplication of the compressive measurements converge to the true principal components of the original data. Also, we investigate the perturbation of eigenvalues of the covariance matrix under random projection of the data to find conditions for approximate recovery of them. Furthermore, we introduce an important application of our proposed method for efficient estimation of the parameters of Gaussian Mixture Models from compressive measurements. We present experimental results demonstrating the performance and efficiency of our proposed algorithms.

In this paper we seek a Gaussian mixture model (GMM) of an n-variate probability density function. Usually the parameters of GMMs are determined in the original n-dimensional space by optimizing a maximum likelihood (ML) criterion. A practical deficiency of this method of fitting GMMs is its poor performance when dealing with high-dimensional data since a large sample size is needed to match the accuracy that is possible in low dimensions. We propose a method for fitting the GMM based on the projection pursuit strategy. This GMM is highly constrained and hence its ability to model structure in subspaces is enhanced, compared to a direct ML fitting of a GMM in high dimensions. Our method is closely related to recently developed independent factor analysis (IFA) mixture models. The comparisons with ML fitting of GMM in n-dimensions and IFA mixtures show that the proposed method is an attractive choice for fitting GMMs using small sizes of training sets.

Condition assessment of cables by pattern recognition of vehicle-induced cable tension ratio

Salient object detection in a real-time environment demands high accuracy with less computation time. It is a changeling task to investigate a saliency model which improves saliency detection accuracy as well as reduces computation time. In this paper, we propose a hybrid model that improves detection accuracy with low computational time. The input image is simplified and clustered at multiple scales using SLIC and K-means clustering algorithms respectively. Gaussian Mixture Model (GMM) is developed on various color components of the digital image at multiple scales. The parameters of GMM are learnt using Expectation Maximization (EM) algorithm. The spatial variance of each color component is determined using GMM parameters and hence object position is estimated. Further, spatial variance of color components and object position is exploited to compute saliency map at a scale level. Afterwards, all the saliency maps generated across various scales are linearly combined to produce the final saliency map. The performance of the proposed model is compared in terms of Precision, Recall, F-Measure, Area under the Curve (AUC), Receiver Operating Characteristics (ROC) and Mean Absolute Error (MAE). Extensive experiments on six publicly available datasets viz. MSRA10K, DUT-OMRON, ECSSD, PASCAL-S, SED2, and THUR15K show that the proposed model outperforms or comparable against 11 state-of-the-art methods of the last decade. The key features of the proposed method are object completeness and efficiency in terms of computational time.

In this paper, the Gaussian mixture model (GMM) is introduced to implement channel multipath clustering. The GMM incorporates the covariance structure and the mean information of the channel multipaths, thus it can effectively reveal the similarity of the channel multipaths. First, the expectation-maximization (EM) algorithm is utilized to search for the posterior estimation of the GMM parameters. Then, the variational Bayesian (VB) algorithm is employed to optimize the GMM parameters to enhance the searching ability of EM and further to determine the optimal number of Gaussian distributions without resorting to cross-validation. Finally, a compact index (CI) is proposed to validate the clustering results reasonably. Thanks to the proposed CI, it is possible to find a close relationship among the GMM clustering mechanism, the multipath propagation characteristics and the CI evaluation index. Experiments with synthetic data and outdoor-to-indoor (O2I) channel data are presented to demonstrate the effectiveness of the proposed method.

We present a novel approach for prediction of satellite image frame that uses a fuzzy rule based framework. The input-output membership functions for the premise and consequent parts of the rules are derived using a Gaussian Mixture Model (GMM). The weights of the fuzzy rules are represented as the prior probabilities of the respective Gaussian components. For obtaining the predictive fuzzy model, the GMM parameters are estimated via EM algorithm using a spatiotemporal representation of image sequence or video clips. Minimum Description Length (MDL) criterion is used to obtain a suitable predictive fuzzy model. The resulting model is successfully applied on a sequence of satellite images of tropical cyclone, Nargis, that made landfall in Myanmar on May 2, 2008. The quality of the predicted image is assessed using two criteria. The proposed approach is found to predict image frame successfully.

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.

Intelligent vehicles need reliable information about the environment in order to operate with total safety. In this paper we propose a flexible multi-module architecture for a multi-target detection and tracking system (MTDTS) complemented with a Bayesian object classification layer based on finite Gaussian mixture models (GMM). The GMM parameters are estimated by an expectation maximization (EM) algorithm, hence finite-component models were generated based on feature-vectors extracted from object's classes during the training stage. Using the joint mixture Gaussian pdf modelled for each class, a Bayesian approach is used to distinct the object's categories (persons, tree-trunks/posts, and cars) in a semi-structured outdoor environment based on data from a laser range finder (LRF). Experiments using real data scan confirm the robustness of the proposed architecture. This paper investigates a particular problem: detection, tracking and classification of objects in cybercars-like outdoor environments

적외선 영상은 야간에 표적의 탐지가 가능하여 보완과 감시분야에 활용도가 높다. 그러나 가시광선 영상에 비하여 해상도가 낮고 잡음의 영향이 크다는 단점이 있다. 본 논문에서는 적외선 영상의 표적을 분할하는 방법을 연구한다. 표적을 포함하는 다수의 관심영역(Region of Interest)을 다단계 분할 방법을 이용하여 추출하고 관심영역을 입력영상으로 다단계 분할방법을 다시 적용하여 표적을 분할한다. 다단계 분할 방법의 각 단계는 가우시안 혼합모델의 파라미터를 초기화 하고 추정하는 k-means 클러스터링(Clustering)과 EM(Expectation-Maximization) 알고리즘과 추정된 사후확률을 이용하여 각 화소의 클러스터를 결정하는 단계로 구성된다. 본 논문에서 추출된 관심영역을 선택하고 통합하는 방법을 제안한다. 관심영역의 통합은 근접한 모든 관심영역의 윈도우를 포함하도록 이루어진다. 실험에서는 야간의 보행자로부터 획득한 적외선 영상에 제안된 방법을 적용하고 다른 분할 방법과 비교하여 제안한 방법이 우수함을 보인다. Infrared (IR) imaging is capable of detecting targets that are not visible at night, thus it has been widely used for the security and defense system. However, the quality of the IR image is often degraded by low resolution and noise corruption. This paper addresses target segmentation with the IR image. Multiple regions of interest (ROI) are extracted by the multi-level segmentation and targets are segmented from the individual ROI. Each level of the multi-level segmentation is composed of a k-means clustering algorithm an expectation-maximization (EM) algorithm, and a decision process. The k-means clustering algorithm initializes the parameters of the Gaussian mixture model (GMM) and the EM algorithm iteratively estimates those parameters. Each pixel is assigned to one of clusters during the decision. This paper proposes the selection and the merging of the extracted ROIs. ROI regions are selectively merged in order to include the overlapped ROI windows. In the experiments, the proposed method is tested on an IR image capturing two pedestrians at night. The performance is compared with conventional methods showing that the proposed method outperforms others.