Clustering Discrete Data Through the Multinomial Mixture Model

In this work, the multinomial mixture model is studied, through a maximum likelihood approach. The convergence of the maximum likelihood estimator to a set with characteristics of interest is shown. A method to select the number of mixture components is developed based on the form of the maximum likelihood estimator. A simulation study is then carried out to verify its behavior. Finally, two applications on real data of multinomial mixtures are presented.     

Principal Volatility Component Analysis

Many empirical time series such as asset returns and traffic data exhibit the characteristic of time-varying conditional covariances, known as volatility or conditional heteroscedasticity. Modeling multivariate volatility, however, encounters several difficulties, including the curse of dimensionality. Dimension reduction can be useful and is often necessary. The goal of this article is to extend the idea of principal component analysis to principal volatility component (PVC) analysis. We define a cumulative generalized kurtosis matrix to summarize the volatility dependence of multivariate time series. Spectral analysis of this generalized kurtosis matrix is used to define PVCs. We consider a sample estimate of the generalized kurtosis matrix and propose test statistics for detecting linear combinations that do not have conditional heteroscedasticity. For application, we applied the proposed analysis to weekly log returns of seven exchange rates against U.S. dollar from 2000 to 2011 and found a linear combination among the exchange rates that has no conditional heteroscedasticity.     

A Variable Selection Criterion for Two Sets of Principal Component Scores in Principal Canonical Correlation Analysis

Canonical correlation analysis (CCA) is often used to analyze the correlation between two random vectors. However, sometimes interpretation of CCA results may be hard. In an attempt to address these difficulties, principal canonical correlation analysis (PCCA) was proposed. PCCA is CCA between two sets of principal component (PC) scores. We consider the problem of selecting useful PC scores in CCA. A variable selection criterion for one set of PC scores has been proposed by Ogura (2010), here, we propose a variable selection criterion for two sets of PC scores in PCCA. Furthermore, we demonstrate the effectiveness of this criterion.     

Principal curves revisited

A principal curve (Hastie and Stuetzle, 1989) is a smooth curve passing through the middle of a distribution or data cloud, and is a generalization of linear principal components. We give an alternative definition of a principal curve, based on a mixture model. Estimation is carried out through an EM algorithm. Some comparisons are made to the Hastie-Stuetzle definition.     

Semiparametric estimation of the cumulative incidence function under competing risks and left-truncated sampling

In this article, we propose semiparametric methods to estimate the cumulative incidence function of two dependent competing risks for left-truncated and right-censored data. The proposed method is based on work by Huang and Wang (1995). We extend previous model by allowing for a general parametric truncation distribution and a third competing risk before recruitment. Based on work by Vardi (1989), several iterative algorithms are proposed to obtain the semiparametric estimates of cumulative incidence functions. The asymptotic properties of the semiparametric estimators are derived. Simulation results show that a semiparametric approach assuming the parametric truncation distribution is correctly specified produces estimates with smaller mean squared error than those obtained in a fully nonparametric model.     

Sensitivity Coefficient in Principal Component Analysis: Robust Case

In the classical principal component analysis (PCA), the empirical influence function for the sensitivity coefficient ρ is used to detect influential observations on the subspace spanned by the dominants principal components. In this article, we derive the influence function of ρ in the case where the reweighted minimum covariance determinant (MCD¹) is used as estimator of multivariate location and scatter. Our aim is to confirm the reliability in terms of robustness of the MCD¹ via the approach based on the influence function of the sensitivity coefficient.     

An EM and a Stochastic Version of the EM Algorithm for Nonparametric Hidden Semi-Markov Models

The Hidden semi-Markov models (HSMMs) were introduced to overcome the constraint of a geometric sojourn time distribution for the different hidden states in the classical hidden Markov models. Several variations of HSMMs were proposed that model the sojourn times by a parametric or a nonparametric family of distributions. In this article, we concentrate our interest on the nonparametric case where the duration distributions are attached to transitions and not to states as in most of the published papers in HSMMs. Therefore, it is worth noticing that here we treat the underlying hidden semi-Markov chain in its general probabilistic structure. In that case, Barbu and Limnios (2008 Barbu , V. , Limnios , N. ( 2008 ). Semi-Markov Chains and Hidden Semi-Markov Models Toward Applications: Their Use in Reliability and DNA Analysis . New York : Springer . [Google Scholar]) proposed an Expectation–Maximization (EM) algorithm in order to estimate the semi-Markov kernel and the emission probabilities that characterize the dynamics of the model. In this article, we consider an improved version of Barbu and Limnios' EM algorithm which is faster than the original one. Moreover, we propose a stochastic version of the EM algorithm that achieves comparable estimates with the EM algorithm in less execution time. Some numerical examples are provided which illustrate the efficient performance of the proposed algorithms.     

Packages for Estimating Finite Mixtures: A Review

We review five packages for estimating finite mixtures, BINOMIX, C.A. MAN, MIX, and the maximum likelihood routines of BMDP and STATA. The focus of the review is on numerical issues rather than matters such as user interface because the success or failure of an algorithm to yield a mixture model is likely to be the most important issue facing a researcher. The problem of suitable initial values is discussed throughout.     

主成分与因子分析中指标同趋势化方法探讨

样本主成分和样本因子分析法已成为一种最主要的综合评价方法之一,指标变量的同趋势化是运用该方法的重要步骤。文章总结了主成分与因子分析中指标同趋势化的具体方法,论述了这些方法对综合评价的影响,并指出了这些方法的适用条件。     

Semiparametric mixture models and repeated measures: the multinomial cut point model

Summary.  Suppose that we have m repeated measures on each subject, and we model the observation vectors with a finite mixture model.  We further assume that the repeated measures are conditionally independent. We present methods to estimate the shape of the component distributions along with various features of the component distributions such as the medians, means and variances. We make no distributional assumptions on the components; indeed, we allow different shapes for different components.     

主成分分析在我国区域经济梯度评价中的应用

本文利用数据统计和SPSS计算机软件进行主成分分析,对我国30个省份及地区的经济梯度状况进行评分、划类和排序,比较客观地阐述了我国区域经济大致的发展状况和未来发展前景,并提出了相关的政策建议,以期为各地经济的健康发展提供更好的思路。     

Influence Analysis in Principal Component Analysis Through Power-Series Expansions

ABSTRACT

In influence analysis several problems arise in the field of Principal Components when applying different sample versions. Among these are the difficulty of determining a certain correspondence between the eigenvalues before and after the deletion of observations, the choice of the sign of the eigenvectors and the computational problem derived from the resolution of a great number of eigenproblems. In this article, such problems are discussed from the joint influence point of view and a solution is proposed by using approximations. Furthermore, the influence on a new parameter of interest is introduced: the proportion of variance explained by a set of principal components.     

Analysis of nonlinear state space model with dependent measurement noises

This paper presents a nonlinear state space model with considering a first-order autoregressive model for measurement noises. A recursive method using Taylor series based approximations for filtering, prediction and smoothing problem of hidden states from the noisy observations is designed. Also, an expectation-maximization algorithm for calculating the maximum likelihood estimators of parameters is presented. The closed form solutions are obtained for estimating of the hidden states and the unknown parameters. Finally, the performance of the designed methods are verified in a simulation study.     

Likelihood ratio tests based on subglobal optimization: A power comparison in exponential mixture models

The paper compares several versions of the likelihood ratio test for exponential homogeneity against mixtures of two exponentials. They are based on different implementations of the likelihood maximization algorithm. We show that global maximization of the likelihood is not appropriate to obtain a good power of the LR test. A simple starting strategy for the EM algorithm, which under the null hypothesis often fails to find the global maximum, results in a rather powerful test. On the other hand, a multiple starting strategy that comes close to global maximization under both the null and the alternative hypotheses leads to inferior power.     

Inference and optimal censoring schemes for progressively censored Birnbaum–Saunders distribution

The aim of this paper is twofold. First we discuss the maximum likelihood estimators of the unknown parameters of a two-parameter Birnbaum–Saunders distribution when the data are progressively Type-II censored. The maximum likelihood estimators are obtained using the EM algorithm by exploiting the property that the Birnbaum–Saunders distribution can be expressed as an equal mixture of an inverse Gaussian distribution and its reciprocal. From the proposed EM algorithm, the observed information matrix can be obtained quite easily, which can be used to construct the asymptotic confidence intervals. We perform the analysis of two real and one simulated data sets for illustrative purposes, and the performances are quite satisfactory. We further propose the use of different criteria to compare two different sampling schemes, and then find the optimal sampling scheme for a given criterion. It is observed that finding the optimal censoring scheme is a discrete optimization problem, and it is quite a computer intensive process. We examine one sub-optimal censoring scheme by restricting the choice of censoring schemes to one-step censoring schemes as suggested by Balakrishnan (2007), which can be obtained quite easily. We compare the performances of the sub-optimal censoring schemes with the optimal ones, and observe that the loss of information is quite insignificant.     

Nonlinear semiparametric AR(1) model with skew-symmetric innovations

In this paper, we expand a first-order nonlinear autoregressive (AR) model with skew normal innovations. A semiparametric method is proposed to estimate a nonlinear part of model by using the conditional least squares method for parametric estimation and the nonparametric kernel approach for the AR adjustment estimation. Then computational techniques for parameter estimation are carried out by the maximum likelihood (ML) approach using Expectation-Maximization (EM) type optimization and the explicit iterative form for the ML estimators are obtained. Furthermore, in a simulation study and a real application, the accuracy of the proposed methods is verified.     

Sparse Functional Principal Component Analysis via Regularized Basis Expansions and Its Application

This article introduces principal component analysis for multidimensional sparse functional data, utilizing Gaussian basis functions. Our multidimensional model is estimated by maximizing a penalized log-likelihood function, while previous mixed-type models were estimated by maximum likelihood methods for one-dimensional data. The penalized estimation performs well for our multidimensional model, while maximum likelihood methods yield unstable parameter estimates and some of the parameter estimates are infinite. Numerical experiments are conducted to investigate the effectiveness of our method for some types of missing data. The proposed method is applied to handwriting data, which consist of the XY coordinates values in handwritings.     

Fitting mixtures of linear regressions

In most applications, the parameters of a mixture of linear regression models are estimated by maximum likelihood using the expectation maximization (EM) algorithm. In this article, we propose the comparison of three algorithms to compute maximum likelihood estimates of the parameters of these models: the EM algorithm, the classification EM algorithm and the stochastic EM algorithm. The comparison of the three procedures was done through a simulation study of the performance (computational effort, statistical properties of estimators and goodness of fit) of these approaches on simulated data sets.

Simulation results show that the choice of the approach depends essentially on the configuration of the true regression lines and the initialization of the algorithms.     

Bias correction in logistic regression with missing categorical covariates

Logistic regression plays an important role in many fields. In practice, we often encounter missing covariates in different applied sectors, particularly in biomedical sciences. Ibrahim (1990) proposed a method to handle missing covariates in generalized linear model (GLM) setup. It is well known that logistic regression estimates using small or medium sized missing data are biased. Considering the missing data that are missing at random, in this paper we have reduced the bias by two methods; first we have derived a closed form bias expression using Cox and Snell (1968), and second we have used likelihood based modification similar to Firth (1993). Here we have analytically shown that the Firth type likelihood modification in Ibrahim led to the second order bias reduction. The proposed methods are simple to apply on an existing method, need no analytical work, with the exception of a little change in the optimization function. We have carried out extensive simulation studies comparing the methods, and our simulation results are also supported by a real world data.     

Recovering extra-binomial variation

A hierarchical logit-normal model for analysis of binary data with extra-binomial variation is examined. A method of approximate maximum likelihood estimation of the parameters is proposed. The method uses the EM algorithm and approximations to facilitate its implementation are derived. Approximate standard errors of the estimates are provided and a numerical example is used to illustrate the method.