A simulation study to compare robust clustering methods based on mixtures

The following mixture model-based clustering methods are compared in a simulation study with one-dimensional data, fixed number of clusters and a focus on outliers and uniform “noise”: an ML-estimator (MLE) for Gaussian mixtures, an MLE for a mixture of Gaussians and a uniform distribution (interpreted as “noise component” to catch outliers), an MLE for a mixture of Gaussian distributions where a uniform distribution over the range of the data is fixed (Fraley and Raftery in Comput J 41:578–588, 1998), a pseudo-MLE for a Gaussian mixture with improper fixed constant over the real line to catch “noise” (RIMLE; Hennig in Ann Stat 32(4): 1313–1340, 2004), and MLEs for mixtures of t-distributions with and without estimation of the degrees of freedom (McLachlan and Peel in Stat Comput 10(4):339–348, 2000). The RIMLE (using a method to choose the fixed constant first proposed in Coretto, The noise component in model-based clustering. Ph.D thesis, Department of Statistical Science, University College London, 2008) is the best method in some, and acceptable in all, simulation setups, and can therefore be recommended.     

Model-based clustering of high-dimensional data: Variable selection versus facet determination

Variable selection is an important problem for cluster analysis of high-dimensional data. It is also a difficult one. The difficulty originates not only from the lack of class information but also the fact that high-dimensional data are often multifaceted and can be meaningfully clustered in multiple ways. In such a case the effort to find one subset of attributes that presumably gives the “best” clustering may be misguided. It makes more sense to identify various facets of a data set (each being based on a subset of attributes), cluster the data along each one, and present the results to the domain experts for appraisal and selection. In this paper, we propose a generalization of the Gaussian mixture models and demonstrate its ability to automatically identify natural facets of data and cluster data along each of those facets simultaneously. We present empirical results to show that facet determination usually leads to better clustering results than variable selection.     

A LASSO-penalized BIC for mixture model selection

The efficacy of family-based approaches to mixture model-based clustering and classification depends on the selection of parsimonious models. Current wisdom suggests the Bayesian information criterion (BIC) for mixture model selection. However, the BIC has well-known limitations, including a tendency to overestimate the number of components as well as a proclivity for underestimating, often drastically, the number of components in higher dimensions. While the former problem might be soluble by merging components, the latter is impossible to mitigate in clustering and classification applications. In this paper, a LASSO-penalized BIC (LPBIC) is introduced to overcome this problem. This approach is illustrated based on applications of extensions of mixtures of factor analyzers, where the LPBIC is used to select both the number of components and the number of latent factors. The LPBIC is shown to match or outperform the BIC in several situations.     

From here to infinity: sparse finite versus Dirichlet process mixtures in model-based clustering

In model-based clustering mixture models are used to group data points into clusters. A useful concept introduced for Gaussian mixtures by Malsiner Walli et al. (Stat Comput 26:303–324, 2016) are sparse finite mixtures, where the prior distribution on the weight distribution of a mixture with K components is chosen in such a way that a priori the number of clusters in the data is random and is allowed to be smaller than K with high probability. The number of clusters is then inferred a posteriori from the data. The present paper makes the following contributions in the context of sparse finite mixture modelling. First, it is illustrated that the concept of sparse finite mixture is very generic and easily extended to cluster various types of non-Gaussian data, in particular discrete data and continuous multivariate data arising from non-Gaussian clusters. Second, sparse finite mixtures are compared to Dirichlet process mixtures with respect to their ability to identify the number of clusters. For both model classes, a random hyper prior is considered for the parameters determining the weight distribution. By suitable matching of these priors, it is shown that the choice of this hyper prior is far more influential on the cluster solution than whether a sparse finite mixture or a Dirichlet process mixture is taken into consideration.

     

Fuzzy clustering using multiple Gaussian kernels with optimized-parameters

In this paper, we propose a new kernel-based fuzzy clustering algorithm which tries to find the best clustering results using optimal parameters of each kernel in each cluster. It is known that data with nonlinear relationships can be separated using one of the kernel-based fuzzy clustering methods. Two common fuzzy clustering approaches are: clustering with a single kernel and clustering with multiple kernels. While clustering with a single kernel doesn’t work well with “multiple-density” clusters, multiple kernel-based fuzzy clustering tries to find an optimal linear weighted combination of kernels with initial fixed (not necessarily the best) parameters. Our algorithm is an extension of the single kernel-based fuzzy c-means and the multiple kernel-based fuzzy clustering algorithms. In this algorithm, there is no need to give “good” parameters of each kernel and no need to give an initial “good” number of kernels. Every cluster will be characterized by a Gaussian kernel with optimal parameters. In order to show its effective clustering performance, we have compared it to other similar clustering algorithms using different databases and different clustering validity measures.     

Evolutionary Monte Carlo Methods for Clustering

The problem of clustering a group of observations according to some objective function (e.g., K-means clustering, variable selection) or a density (e.g., posterior from a Dirichlet process mixture model prior) can be cast in the framework of Monte Carlo sampling for cluster indicators. We propose a new method called the evolutionary Monte Carlo clustering (EMCC) algorithm, in which three new “crossover moves,” based on swapping and reshuffling sub cluster intersections, are proposed. We apply the EMCC algorithm to several clustering problems including Bernoulli clustering, biological sequence motif clustering, BIC based variable selection, and mixture of normals clustering. We compare EMCC's performance both as a sampler and as a stochastic optimizer with Gibbs sampling, “split-merge” Metropolis–Hastings algorithms, K-means clustering, and the MCLUST algorithm.     

Criteria for Capital Investment: An Approach Through Decision Theory

The arbitrary character of such concepts as “Discounted Present Worth”, which are often advocated as methods of assessing and comparing investment opportunities, is discussed in Part 1 of this paper. It is also pointed out that, by their nature, these concepts are not particularly well adapted to situations in which “risk” is an important factor. In order to develop a method in which a logical approach to risk can be adopted, an understanding of the basic problem of Decision Making under Uncertainty is required. An introduction to this subject forms Part 2 of this paper. In Part 3 the problem of Plant Investment under Risk is considered, and a rational approach to this is developed. The importance of the “portfolio” concept in such problems is particularly demonstrated.     

Identifying Mixtures of Mixtures Using Bayesian Estimation

The use of a finite mixture of normal distributions in model-based clustering allows us to capture non-Gaussian data clusters. However, identifying the clusters from the normal components is challenging and in general either achieved by imposing constraints on the model or by using post-processing procedures. Within the Bayesian framework, we propose a different approach based on sparse finite mixtures to achieve identifiability. We specify a hierarchical prior, where the hyperparameters are carefully selected such that they are reflective of the cluster structure aimed at. In addition, this prior allows us to estimate the model using standard MCMC sampling methods. In combination with a post-processing approach which resolves the label switching issue and results in an identified model, our approach allows us to simultaneously (1) determine the number of clusters, (2) flexibly approximate the cluster distributions in a semiparametric way using finite mixtures of normals and (3) identify cluster-specific parameters and classify observations. The proposed approach is illustrated in two simulation studies and on benchmark datasets. Supplementary materials for this article are available online.     

混料切片试验设计

混料试验设计在众多领域中都有广泛的应用,有时试验者不仅仅需要考虑各混料成分所占比例对响应变量的影响,同时还关心其它被称为过程变量的因素.在实际中,对于这类问题通常使用的设计方案是混料设计和因子设计的组合设计.这种组合设计在过程变量的不同水平组合下,使用的是相同的设计阵,因此空间填充性较差.基于混料球体堆积设计,文章提出了一类新的混料设计,称之为混料切片设计,它的整体设计和所有子设计(过程变量的每一水平组合对应的混料设计)都具有很好的空间填充性,从而比组合设计有更好的模型稳健性.基于同余子群的陪集分解方法,针对过程变量水平组合数的不同情况提出了相应的简单快速的构造算法,文章最后的数值例子解释了算法的可行性和设计的有效性.     

Penalized multinomial mixture logit model

Normal distribution based discriminant methods have been used for the classification of new entities into different groups based on a discriminant rule constructed from the learning set. In practice if the groups are not homogeneous, then mixture discriminant analysis of Hastie and Tibshirani (J R Stat Soc Ser B 58(1):155–176, 1996) is a useful approach, assuming that the distribution of the feature vectors is a mixture of multivariate normals. In this paper a new logistic regression model for heterogenous group structure of the learning set is proposed based on penalized multinomial mixture logit models. This approach is shown through simulation studies to be more effective. The results were compared with the standard mixture discriminant analysis approach using the probability of misclassification criterion. This comparison showed a slight reduction in the average probability of misclassification using this penalized multinomial mixture logit model as compared to the classical discriminant rules. It also showed better results when applied to practical life data problems producing smaller errors.     

Sum-based weighted belief base merging: From commensurable to incommensurable framework

Different methods have been proposed for merging multiple and potentially conflicting information. The merging process based on the so-called “Sum” operation offers a natural method for merging commensurable prioritized belief bases. Their popularity is due to the fact that they satisfy the majority property and they adopt a non-cautious attitude in deriving plausible conclusions.This paper analyzes the sum-based merging operator when sources to merge are incommensurable, namely when they do not share the same meaning of uncertainty scales. We first show that the obtained merging operator can be equivalently characterized either in terms of an infinite set of compatible scales, or by a well-known Pareto ordering on a set of propositional logic interpretations. We also study some restrictions on compatible scales based on different commensurability hypothesis.Moreover, this paper provides a postulate-based analysis of our merging operators. We show that when prioritized bases to merge are not commensurable, the majority property is no longer satisfied. We provide conditions to recovering it. We also analyze the fairness postulate, which represents the unique postulate unsatisfied when belief bases to merge are commensurable and we propose a new postulate of consensuality. This postulate states that the result of the merging process must be consensual. It obtains the consent of all parties by integrating a piece of belief of each base.Finally, in the incommensurable case, we show that the fairness and consensuality postulates are satisfied when all compatible scales are considered. However, we provide an impossibility theorem stating that there is no way to satisfy fairness and consensuality postulates if only one canonical compatible scale is considered.     

High breakdown mixture discriminant analysis

Robust S-estimation is proposed for multivariate Gaussian mixture models generalizing the work of Hastie and Tibshirani (J. Roy. Statist. Soc. Ser. B 58 (1996) 155). In the case of Gaussian Mixture models, the unknown location and scale parameters are estimated by the EM algorithm. In the presence of outliers, the maximum likelihood estimators of the unknown parameters are affected, resulting in the misclassification of the observations. The robust S-estimators of the unknown parameters replace the non-robust estimators from M-step of the EM algorithm. The results were compared with the standard mixture discriminant analysis approach using the probability of misclassification criterion. This comparison showed a slight reduction in the average probability of misclassification using robust S-estimators as compared to the standard maximum likelihood estimators.     

A global algorithm to estimate the expectations of the components of an observed univariate mixture

This paper deals with the unsupervised classification of univariate observations. Given a set of observations originating from a K-component mixture, we focus on the estimation of the component expectations. We propose an algorithm based on the minimization of the “K-product” (KP) criterion we introduced in a previous work. We show that the global minimum of this criterion can be reached by first solving a linear system then calculating the roots of some polynomial of order K. The KP global minimum provides a first raw estimate of the component expectations, then a nearest-neighbour classification enables to refine this estimation. Our method’s relevance is finally illustrated through simulations of various mixtures. When the mixture components do not strongly overlap, the KP algorithm provides better estimates than the Expectation-Maximization algorithm.     

Statistical estimation of a mixture of Gaussian distributions

The paper is devoted to the problem of statistical estimation of a multivariate distribution density, which is a discrete mixture of Gaussian distributions. A heuristic approach is considered, based on the use of the EM algorithm and nonparametric density estimation with a sequential increase in the number of components of the mixture. Criteria for testing of model adequacy are discussed.     

Bayesian nonparametric regression with varying residual density

We consider the problem of robust Bayesian inference on the mean regression function allowing the residual density to change flexibly with predictors. The proposed class of models is based on a Gaussian process (GP) prior for the mean regression function and mixtures of Gaussians for the collection of residual densities indexed by predictors. Initially considering the homoscedastic case, we propose priors for the residual density based on probit stick-breaking mixtures. We provide sufficient conditions to ensure strong posterior consistency in estimating the regression function, generalizing existing theory focused on parametric residual distributions. The homoscedastic priors are generalized to allow residual densities to change nonparametrically with predictors through incorporating GP in the stick-breaking components. This leads to a robust Bayesian regression procedure that automatically down-weights outliers and influential observations in a locally adaptive manner. The methods are illustrated using simulated and real data applications.     

On Testing for the Number of Components in a Mixed Poisson Model

Poisson mixtures are usually used to describe overdispersed data. Finite Poisson mixtures are used in many practical situations where often it is of interest to determine the number of components in the mixture. Identifying how many components comprise a mixture remains a difficult problem. The likelihood ratio test (LRT) is a general statistical procedure to use. Unfortunately, a number of specific problems arise and the classical theory fails to hold. In this paper a new procedure is proposed that is based on testing whether a new component can be added to a finite Poisson mixture which eventually leads to the number of components in the mixture. It is a sequential testing procedure based on the well known LRT that utilises a resampling technique to construct the distribution of the test statistic. The application of the procedure to real data reveals some interesting features of the distribution of the test statistic.     

Integer Fuzzy Programming Approach in Bi-objective Selective Maintenance Allocation Problem

An attempt has been made to obtain a compromise allocation based on maximization of individual reliabilities of repairable and replaceable components with in the subsystems, using the information of failed and operational components and a non linear cost function with fixed budget. A solution algorithm of fuzzy programming technique is used to solve the Bi-Objective Selective Maintenance Allocation Problem (BSMAP). Also, the problem has been solved by two other suggested methods; “Weighted Criterion Technique” and “Desirability Function Technique”. A numerical example is also presented to illustrate the computational details.     

Bayesian optimization of pump operations in water distribution systems

Bayesian optimization has become a widely used tool in the optimization and machine learning communities. It is suitable to problems as simulation/optimization and/or with an objective function computationally expensive to evaluate. Bayesian optimization is based on a surrogate probabilistic model of the objective whose mean and variance are sequentially updated using the observations and an “acquisition” function based on the model, which sets the next observation at the most “promising” point. The most used surrogate model is the Gaussian Process which is the basis of well-known Kriging algorithms. In this paper, the authors consider the pump scheduling optimization problem in a Water Distribution Network with both ON/OFF and variable speed pumps. In a global optimization model, accounting for time patterns of demand and energy price allows significant cost savings. Nonlinearities, and binary decisions in the case of ON/OFF pumps, make pump scheduling optimization computationally challenging, even for small Water Distribution Networks. The well-known EPANET simulator is used to compute the energy cost associated to a pump schedule and to verify that hydraulic constraints are not violated and demand is met. Two Bayesian Optimization approaches are proposed in this paper, where the surrogate model is based on a Gaussian Process and a Random Forest, respectively. Both approaches are tested with different acquisition functions on a set of test functions, a benchmark Water Distribution Network from the literature and a large-scale real-life Water Distribution Network in Milan, Italy.     

Bayesian Ising Graphical Model for Variable Selection

In this article, we propose a new Bayesian variable selection (BVS) approach via the graphical model and the Ising model, which we refer to as the “Bayesian Ising graphical model” (BIGM). The BIGM is developed by showing that the BVS problem based on the linear regression model can be considered as a complete graph and described by an Ising model with random interactions. There are several advantages of our BIGM: it is easy to (i) employ the single-site updating and cluster updating algorithm, both of which are suitable for problems with small sample sizes and a larger number of variables, (ii) extend this approach to nonparametric regression models, and (iii) incorporate graphical prior information. In our BIGM, the interactions are determined by the linear model coefficients, so we systematically study the performance of different scale normal mixture priors for the model coefficients by adopting the global-local shrinkage strategy. Our results indicate that the best prior for the model coefficients in terms of variable selection should place substantial weight on small, nonzero shrinkage. The methods are illustrated with simulated and real data. Supplementary materials for this article are available online.