Identification and estimation of superposed Neyman–Scott spatial cluster processes

This paper proposes an estimation method for superposed spatial point patterns of Neyman–Scott cluster processes of different distance scales and cluster sizes. Unlike the ordinary single Neyman–Scott model, the superposed process of Neyman–Scott models is not identified solely by the second-order moment property of the process. To solve the identification problem, we use the nearest neighbor distance property in addition to the second-order moment property. In the present procedure, we combine an inhomogeneous Poisson likelihood based on the Palm intensity with another likelihood function based on the nearest neighbor property. The derivative of the nearest neighbor distance function is regarded as the intensity function of the rotation invariant inhomogeneous Poisson point process. The present estimation procedure is applied to two sets of ecological location data.     

Clustering via finite nonparametric ICA mixture models

We propose a novel extension of nonparametric multivariate finite mixture models by dropping the standard conditional independence assumption and incorporating the independent component analysis (ICA) structure instead. This innovation extends nonparametric mixture model estimation methods to situations in which conditional independence, a necessary assumption for the unique identifiability of the parameters in such models, is clearly violated. We formulate an objective function in terms of penalized smoothed Kullback–Leibler distance and introduce the nonlinear smoothed majorization-minimization independent component analysis algorithm for optimizing this function and estimating the model parameters. Our algorithm does not require any labeled observations a priori; it may be used for fully unsupervised clustering problems in a multivariate setting. We have implemented a practical version of this algorithm, which utilizes the FastICA algorithm, in the R package icamix. We illustrate this new methodology using several applications in unsupervised learning and image processing.

     

Asymptotic properties of the minimum contrast estimators for projections of inhomogeneous space-time shot-noise Cox processes

We consider a flexible class of space-time point process models—inhomogeneous shot-noise Cox point processes. They are suitable for modelling clustering phenomena, e.g. in epidemiology, seismology, etc. The particular structure of the model enables the use of projections to the spatial and temporal domain. They are used to formulate a stepwise estimation method to estimate different parts of the model separately. In the first step, the Poisson likelihood approach is used to estimate the inhomogeneity parameters. In the second and third steps, the minimum contrast estimation based on K-functions of the projected processes is used to estimate the interaction parameters. We study the asymptotic properties of the resulting estimators and formulate a set of conditions sufficient for establishing consistency and asymptotic normality of the estimators under the increasing domain asymptotics.     

Multivariate Survival Models with a Mixture of Positive Stable Frailties

In this paper, we describe models for dependent multivariate survival data using finite mixtures of positive stable frailty distributions. We investigate the cross-ratio function as a local measure of association. We estimate the parameters in the stable mixture together with the parameters of the (conditional) proportional hazards model in a Bayesian framework using Markov chain Monte Carlo algorithms. We illustrate the methodology using data on kidney infections.     

AMlet,RAMlet, and GAMlet: Automatic Nonlinear Fitting of Additive Models,Robust and Generalized,With Wavelets

A simple and yet powerful method is presented to estimate nonlinearly and nonparametrically the components of additive models using wavelets. The estimator enjoys the good statistical and computational properties of the Waveshrink scatterplot smoother and it can be efficiently computed using the block coordinate relaxation optimization technique. A rule for the automatic selection of the smoothing parameters, suitable for data mining of large datasets, is derived. The wavelet-based method is then extended to estimate generalized additive models. A primal-dual log-barrier interior point algorithm is proposed to solve the corresponding convex programming problem. Based on an asymptotic analysis, a rule for selecting the smoothing parameters is derived, enabling the estimator to be fully automated in practice. We illustrate the finite sample property with a Gaussian and a Poisson simulation.     

New algorithms for statistical modeling of inhomogeneous Poisson ensembles

New algorithms for statistical modeling of inhomogeneous Poisson ensembles are proposed. They are based on a special method for modeling discrete random variables. The corresponding modification of the well-known maximum cross section method is developed.     

Classification et mélanges de processusClustering and mixtures of processes

We propose a clustering method based on the estimation of mixtures of probability distributions, the new point being that the statistical units are described by probability distributions. The components of the mixtures are Dirichlet processes, normalized weighted Gamma processes, and Kraft processes. Mixtures obtained by applying some algorithms to the finite dimensional distributions of the components converge to the desired mixture as the dimension increases, since the components are mutually singular due to a theorem of Kakutani. The desired clusters are then the support of these components. To cite this article: R. Emilion, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 189–193.     

Computing the noncentral gamma distribution, its inverse and the noncentrality parameter

The noncentral gamma distribution can be viewed as a generalization of the noncentral chi-squared distribution and it can be expressed as a mixture of a Poisson density function with a incomplete gamma function. The noncentral gamma distribution is not available in free conventional statistical programs. This paper aimed to propose an algorithm for the noncentral gamma by combining the method originally proposed by Benton and Krishnamoorthy (Comput Stat Data Anal 43(2):249–267, 2003) for the noncentral distributions with the method of inversion of the distribution function with respect to the noncentrality parameter using Newton–Raphson. The algorithms are available in pseudocode and implemented as R functions. To evaluate the accuracy and speed of computation of the algorithms implemented in R, results of the distribution function, density function, quantiles and noncentrality parameter of the noncentral incomplete gamma and its particular case, the noncentral chi-squared, were obtained for the arguments settings used by Benton and Krishnamoorthy (Comput Stat Data Anal 43(2):249–267, 2003) and Chen (J Stat Comput Simul 75(10):813–829, 2005). The implemented routines performed well and, in general, were as accurate than other approximations. The R package denoted ncg is available to download on the CRAN-R package repository http://cran.r-project.org/.     

A Haar-Fisz Algorithm for Poisson Intensity Estimation

This article introduces a new method for the estimation of the intensity of an inhomogeneous one-dimensional Poisson process. The Haar-Fisz transformation transforms a vector of binned Poisson counts to approximate normality with variance one. Hence we can use any suitable Gaussian wavelet shrinkage method to estimate the Poisson intensity. Since the Haar-Fisz operator does not commute with the shift operator we can dramatically improve accuracy by always cycle spinning before the Haar-Fisz transform as well as optionally after. Extensive simulations show that our approach usually significantly outperformed state-of-the-art competitors but was occasionally comparable. Our method is fast, simple, automatic, and easy to code. Our technique is applied to the estimation of the intensity of earthquakes in northern California. We show that our technique gives visually similar results to the current state-of-the-art.     

Insurance claims modulated by a hidden Brownian marked point process

Aimed at better modeling insurance claims in an economic environment driven by business cycles, a new Markov-modulated Poisson process model is proposed, and an algorithm is derived to estimate the hidden Markov process by using the observed information. Our method differs from existing ones in the following ways: the new hidden process can model more efficiently the cyclic state of the economic environment; our theory is based on a variation of the law of large numbers and is easy to understand; the Fourier expansion-based parameter estimation algorithm is flexible and can be more easily implemented than other algorithms. Simulation results not only demonstrate the practicality of our model and algorithm, but also show the efficiency and robustness of the estimation algorithm.     

“Poisson” models of random fields with applications in transport theory

A comparative analysis of various models of exponentially correlated random fields associated with Poisson point ensembles is given. Algorithms for the modeling of radiative transfer in random media of this type are considered. An asymptotic estimate for the particle passage probability is constructed assuming that the flow of trajectory intersections with domains of constant random density is Poisson distributed.     

Asymptotic equivalence for nonparametric regression with non-regular errors

Asymptotic equivalence in Le Cam’s sense for nonparametric regression experiments is extended to the case of non-regular error densities, which have jump discontinuities at their endpoints. We prove asymptotic equivalence of such regression models and the observation of two independent Poisson point processes which contain the target curve as the support boundary of its intensity function. The intensity of the point processes is of order of the sample size n and involves the jump sizes as well as the design density. The statistical model significantly differs from regression problems with Gaussian or regular errors, which are known to be asymptotically equivalent to Gaussian white noise models.     

Semiparametric mixtures of regressions with single-index for model based clustering

In this article, we propose two classes of semiparametric mixture regression models with single-index for model based clustering. Unlike many semiparametric/nonparametric mixture regression models that can only be applied to low dimensional predictors, the new semiparametric models can easily incorporate high dimensional predictors into the nonparametric components. The proposed models are very general, and many of the recently proposed semiparametric/nonparametric mixture regression models are indeed special cases of the new models. Backfitting estimates and the corresponding modified EM algorithms are proposed to achieve optimal convergence rates for both parametric and nonparametric parts. We establish the identifiability results of the proposed two models and investigate the asymptotic properties of the proposed estimation procedures. Simulation studies are conducted to demonstrate the finite sample performance of the proposed models. Two real data applications using the new models reveal some interesting findings.

     

Stochastic EM algorithms for parametric and semiparametric mixture models for right-censored lifetime data

Mixture models in reliability bring a useful compromise between parametric and nonparametric models, when several failure modes are suspected. The classical methods for estimation in mixture models rarely handle the additional difficulty coming from the fact that lifetime data are often censored, in a deterministic or random way. We present in this paper several iterative methods based on EM and Stochastic EM methodologies, that allow us to estimate parametric or semiparametric mixture models for randomly right censored lifetime data, provided they are identifiable. We consider different levels of completion for the (incomplete) observed data, and provide genuine or EM-like algorithms for several situations. In particular, we show that simulating the missing data coming from the mixture allows to plug a standard R package for survival data analysis in an EM algorithm’s M-step. Moreover, in censored semiparametric situations, a stochastic step is the only practical solution allowing computation of nonparametric estimates of the unknown survival function. The effectiveness of the new proposed algorithms are demonstrated in simulation studies and an actual dataset example from aeronautic industry.     

Adaptive Rejection Metropolis Simulated Annealing for Detecting Global Maximum Regions

A finite mixture model has been used to fit the data from heterogeneous populations to many applications. An Expectation Maximization (EM) algorithm is the most popular method to estimate parameters in a finite mixture model. A Bayesian approach is another method for fitting a mixture model. However, the EM algorithm often converges to the local maximum regions, and it is sensitive to the choice of starting points. In the Bayesian approach, the Markov Chain Monte Carlo (MCMC) sometimes converges to the local mode and is difficult to move to another mode. Hence, in this paper we propose a new method to improve the limitation of EM algorithm so that the EM can estimate the parameters at the global maximum region and to develop a more effective Bayesian approach so that the MCMC chain moves from one mode to another more easily in the mixture model. Our approach is developed by using both simulated annealing (SA) and adaptive rejection metropolis sampling (ARMS). Although SA is a well-known approach for detecting distinct modes, the limitation of SA is the difficulty in choosing sequences of proper proposal distributions for a target distribution. Since ARMS uses a piecewise linear envelope function for a proposal distribution, we incorporate ARMS into an SA approach so that we can start a more proper proposal distribution and detect separate modes. As a result, we can detect the maximum region and estimate parameters for this global region. We refer to this approach as ARMS annealing. By putting together ARMS annealing with the EM algorithm and with the Bayesian approach, respectively, we have proposed two approaches: an EM-ARMS annealing algorithm and a Bayesian-ARMS annealing approach. We compare our two approaches with traditional EM algorithm alone and Bayesian approach alone using simulation, showing that our two approaches are comparable to each other but perform better than EM algorithm alone and Bayesian approach alone. Our two approaches detect the global maximum region well and estimate the parameters in this region. We demonstrate the advantage of our approaches using an example of the mixture of two Poisson regression models. This mixture model is used to analyze a survey data on the number of charitable donations.     

Geometric ergodicity of the Gibbs sampler for the Poisson change-point model

Poisson change-point models have been widely used for modelling inhomogeneous time-series of count data. There are a number of methods available for estimating the parameters in these models using iterative techniques such as MCMC. Many of these techniques share the common problem that there does not seem to be a definitive way of knowing the number of iterations required to obtain sufficient convergence. In this paper, we show that the Gibbs sampler of the Poisson change-point model is geometrically ergodic. Establishing geometric ergodicity is crucial from a practical point of view as it implies the existence of a Markov chain central limit theorem, which can be used to obtain standard error estimates. We prove that the transition kernel is a trace-class operator, which implies geometric ergodicity of the sampler. We then provide a useful application of the sampler to a model for the quarterly driver fatality counts for the state of Victoria, Australia.     

Tweedie’s Compound Poisson Model With Grouped Elastic Net

Tweedie’s compound Poisson model is a popular method to model data with probability mass at zero and nonnegative, highly right-skewed distribution. Motivated by wide applications of the Tweedie model in various fields such as actuarial science, we investigate the grouped elastic net method for the Tweedie model in the context of the generalized linear model. To efficiently compute the estimation coefficients, we devise a two-layer algorithm that embeds the blockwise majorization descent method into an iteratively reweighted least square strategy. Integrated with the strong rule, the proposed algorithm is implemented in an easy-to-use R package HDtweedie, and is shown to compute the whole solution path very efficiently. Simulations are conducted to study the variable selection and model fitting performance of various lasso methods for the Tweedie model. The modeling applications in risk segmentation of insurance business are illustrated by analysis of an auto insurance claim dataset. Supplementary materials for this article are available online.     

Variational Approximation for Mixtures of Linear Mixed Models

Mixtures of linear mixed models (MLMMs) are useful for clustering grouped data and can be estimated by likelihood maximization through the Expectation–Maximization algorithm. A suitable number of components is then determined conventionally by comparing different mixture models using penalized log-likelihood criteria such as Bayesian information criterion. We propose fitting MLMMs with variational methods, which can perform parameter estimation and model selection simultaneously. We describe a variational approximation for MLMMs where the variational lower bound is in closed form, allowing for fast evaluation and develop a novel variational greedy algorithm for model selection and learning of the mixture components. This approach handles algorithm initialization and returns a plausible number of mixture components automatically. In cases of weak identifiability of certain model parameters, we use hierarchical centering to reparameterize the model and show empirically that there is a gain in efficiency in variational algorithms similar to that in Markov chain Monte Carlo (MCMC) algorithms. Related to this, we prove that the approximate rate of convergence of variational algorithms by Gaussian approximation is equal to that of the corresponding Gibbs sampler, which suggests that reparameterizations can lead to improved convergence in variational algorithms just as in MCMC algorithms. Supplementary materials for the article are available online.     

基于方向多尺度变换和统计建模的纹理分类方法

纹理是图像分析和识别中经常使用的关键特征, 而小波变换则是图像纹理表示和分类中的常用工具. 然而, 基于小波变换的纹理分类方法常常忽略了小波低频子带信息, 并且无法提取图像纹理的块状奇异信息. 本文提出小波子带系数的局部能量直方图建模方法、轮廓波特征的Poisson 混合模型建模方法和基于轮廓波子带系数聚类的特征提取方法, 并将其应用于图像纹理分类上. 基于局部能量直方图的纹理分类方法解决了小波低频子带的建模难题, 基于Poisson 混合模型的纹理分类方法则首次将Poisson 混合模型用于轮廓子带特征的建模, 而基于轮廓波域聚类的纹理分类方法是一种快速的分类方法. 实验结果显示, 本文所提出的三类方法都超过了当前典型的纹理分类方法.     

Linear latent variable models: the lava-package

An R package for specifying and estimating linear latent variable models is presented. The philosophy of the implementation is to separate the model specification from the actual data, which leads to a dynamic and easy way of modeling complex hierarchical structures. Several advanced features are implemented including robust standard errors for clustered correlated data, multigroup analyses, non-linear parameter constraints, inference with incomplete data, maximum likelihood estimation with censored and binary observations, and instrumental variable estimators. In addition an extensive simulation interface covering a broad range of non-linear generalized structural equation models is described. The model and software are demonstrated in data of measurements of the serotonin transporter in the human brain.