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ESTIMATING A COmmON PERIOD FOR A SET OF IRREGULARLY SAMPLED FUNCTIONS WITH APPLICATIONS TO PERIODIC VARIABLE STAR DATA

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ANNALS OF APPLIED STATISTICS 2016年第1期10卷 165-197页

作者： Long, James P. Chi, Eric C. Baraniuk, Richard G. Texas A&M Univ Dept Stat College Stn TX 77843 USA N Carolina State Univ Dept Stat Raleigh NC 27607 USA Rice Univ Dept Elect & Comp Engn Houston TX 77005 USA

We consider the problem of estimating a common period for a set of functions sampled at irregular intervals. The motivating problem arises in astronomy, where the functions represent a star's observed brightness over time through different photometric filters. While current methods perform well when the brightness is sampled densely enough in at least one filter, they break down when no brightness function is densely sampled. In this paper we introduce two new methods for period estimation in this important latter case. The first, multiband generalized Lomb-Scargle (MGLS), extends the frequently used Lomb-Scargle method to naively combine information across filters. The second, penalized generalized Lomb-Scargle (PGLS), builds on MGLS by more intelligently borrowing strength across filters. Specifically, we incorporate constraints on the phases and amplitudes across the different functions using a nonconvex penalized likelihood function. We develop a fast algorithm to optimize the penalized likelihood that combines block coordinate descent with the majorization-minimization (mm) principle. We test and validate our methods on synthetic and real astronomy data. Both PGLS and MGLS improve period estimation accuracy over current methods based on using a single function;moreover, PGLS outperforms MGLS and other leading methods when the functions are sparsely sampled.

关键词： Astrostatistics penalized likelihood period estimation functional data mm algorithm block coordinate descent

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Variable selection for frailty transformation models with application to diabetic complications

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CANADIAN JOURNAL OF STATISTICS-REVUE CANADIENNE DE STATISTIQUE 2016年第3期44卷 375-394页

作者： Liu, Xu Song, Xinyuan Xie, Shangyu Zhou, Yong Shanghai Univ Finance & Econ Sch Stat & Management Shanghai Peoples R China Chinese Univ Hong Kong Shenzhen Res Inst Hong Kong Hong Kong Peoples R China Chinese Univ Hong Kong Dept Stat Hong Kong Hong Kong Peoples R China Univ Int Business & Econ RCAF Beijing Peoples R China Univ Int Business & Econ Sch Banking & Finance Beijing Peoples R China Chinese Acad Sci Acad Math & Syst Sci Beijing Peoples R China

This article focuses on variable selection in the context of gamma frailty transformation models for multivariate survival times. We propose a method based on a nonconcave penalty function to select relevant regression predictors and simultaneously estimate model parameters and nonparametric functions. Our procedure performs as well as the oracle one in which the true model is assumed to be known. We conduct simulation studies to demonstrate the performance of the proposed methodology. We apply our method to a case study based on the Hong Kong Diabetes Registry, and obtain new insights on the risk factors of cardiovascular-renal complications for type 2 diabetic patients. The Canadian Journal of Statistics 44: 375-394;2016 (c) 2016 Statistical Society of Canada

关键词： Diabetic complications Gamma frailty mm algorithm SCAD transformation models variable selection MSC 2010: Primary 62G08 secondary 62N01

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mm algorithms for geometric and signomial programming

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MATHEMATICAL PROGRAmmING 2014年第1-2期143卷 339-356页

作者： Lange, Kenneth Zhou, Hua Univ Calif Los Angeles Dept Biomath Los Angeles CA 90095 USA Univ Calif Los Angeles Dept Human Genet Los Angeles CA 90095 USA Univ Calif Los Angeles Dept Stat Los Angeles CA 90095 USA N Carolina State Univ Dept Stat Raleigh NC 27695 USA

This paper derives new algorithms for signomial programming, a generalization of geometric programming. The algorithms are based on a generic principle for optimization called the mm algorithm. In this setting, one can apply the geometric-arithmetic mean inequality and a supporting hyperplane inequality to create a surrogate function with parameters separated. Thus, unconstrained signomial programming reduces to a sequence of one-dimensional minimization problems. Simple examples demonstrate that the mm algorithm derived can converge to a boundary point or to one point of a continuum of minimum points. Conditions under which the minimum point is unique or occurs in the interior of parameter space are proved for geometric programming. Convergence to an interior point occurs at a linear rate. Finally, the mm framework easily accommodates equality and inequality constraints of signomial type. For the most important special case, constrained quadratic programming, the mm algorithm involves very simple updates.

关键词： Arithmetic-geometric mean inequality Geometric programming Global convergence mm algorithm Linearly constrained quadratic programming Parameter separation Penalty method Signomial programming

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Matrix completion discriminant analysis

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COMPUTATIONAL STATISTICS & DATA ANALYSIS 2015年 92卷 115-125页

作者： Wu, Tong Tong Lange, Kenneth Univ Rochester Dept Biostat & Computat Biol Rochester NY 14642 USA Univ Calif Los Angeles Dept Biomath Los Angeles CA 90095 USA Univ Calif Los Angeles Dept Human Genet Los Angeles CA 90095 USA Univ Calif Los Angeles Dept Stat Los Angeles CA 90095 USA

Matrix completion discriminant analysis (MCDA) is designed for semi-supervised learning where the rate of missingness is high and predictors vastly outnumber cases. MCDA operates by mapping class labels to the vertices of a regular simplex. With c classes, these vertices are arranged on the surface of the unit sphere in c - 1 dimensional Euclidean space. Because all pairs of vertices are equidistant, the classes are treated symmetrically. To assign unlabeled cases to classes, the data is entered into a large matrix (cases along rows and predictors along columns) that is augmented by vertex coordinates stored in the last c - 1 columns. Once the matrix is constructed, its missing entries can be filled in by niatrix completion. To carry out matrix completion, one minimizes a sum of squares plus a nuclear norm penalty. The simplest solution invokes an mm algorithm and singular value decomposition. Choice of the penalty tuning constant can be achieved by cross validation on randomly withheld case labels. Once the matrix is completed, an unlabeled case is assigned to the class vertex closest to the point deposited in its last c - 1 columns. A variety of examples drawn from the statistical literature demonstrate that MCDA is competitive on traditional problems and outperforms alternatives on large-scale problems. (C) 2015 Elsevier B.V. All rights reserved.

关键词： Classification Missing observations mm algorithm Semi-supervised learning Singular value decomposition

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Quantile regression methods with varying-coefficient models for censored data

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COMPUTATIONAL STATISTICS & DATA ANALYSIS 2015年 88卷 154-172页

作者： Xie, Shangyu Wan, Alan T. K. Zhou, Yong Univ Int Business & Econ RCAF Beijing Peoples R China Univ Int Business & Econ Sch Banking & Finance Beijing Peoples R China City Univ Hong Kong Dept Management Sci Kowloon Hong Kong Peoples R China Chinese Acad Sci Acad Math & Syst Sci Beijing Peoples R China Shanghai Univ Finance & Econ Sch Stat & Management Shanghai Peoples R China

Considerable intellectual progress has been made to the development of various semi-parametric varying-coefficient models over the past ten to fifteen years. An important advantage of these models is that they avoid much of the curse of dimensionality problem as the nonparametric functions are restricted only to some variables. More recently, varying-coefficient methods have been applied to quantile regression modeling, but all previous studies assume that the data are fully observed. The main purpose of this paper is to develop a varying-coefficient approach to the estimation of regression quantiles under random data censoring. We use a weighted inverse probability approach to account for censoring, and propose a majorize minimize type algorithm to optimize the non-smooth objective function. The asymptotic properties of the proposed estimator of the nonparametric functions are studied, and a resampling method is developed for obtaining the estimator of the sampling variance. An important aspect of our method is that it allows the censoring time to depend on the covariates. Additionally, we show that this varying-coefficient procedure can be further improved when implemented within a composite quantile regression framework. Composite quantile regression has recently gained considerable attention due to its ability to combine information across different quantile functions. We assess the finite sample properties of the proposed procedures in simulated studies. A real data application is also considered. (C) 2015 Elsevier B.V. All rights reserved.

关键词： Composite quantile regression Covariate-dependent censoring Inverse probability weighting mm algorithm Perturbation resampling

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Semiparametric Hierarchical Composite Quantile Regression

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COmmUNICATIONS IN STATISTICS-THEORY AND METHODS 2015年第5期44卷 996-1012页

作者： Chen, Yanliang Tang, Man-Lai Tian, Maozai Lanzhou Univ Finance & Econ Sch Stat Lanzhou Peoples R China Hang Seng Management Coll Dept Math & Stat Shatin Hong Kong Peoples R China Renmin Univ China Sch Stat Beijing Peoples R China

In biological, medical, and social sciences, multilevel structures are very common. Hierarchical models that take the dependencies among subjects within the same level are necessary. In this article, we introduce a semiparametric hierarchical composite quantile regression model for hierarchical data. This model (i) keeps the easy interpretability of the simple parametric model;(ii) retains some of the flexibility of the complex non parametric model;(iii) relaxes the assumptions that the noise variances and higher-order moments exist and are finite;and (iv) takes the dependencies among subjects within the same hierarchy into consideration. We establish the asymptotic properties of the proposed estimators. Our simulation results show that the proposed method is more efficient than the least-squares-based method for many non normally distributed errors. We illustrate our methodology with a real biometric data set.

关键词： Composite quantile regression Hierarchical data mm algorithm Semiparametric model

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An mm algorithm for Constrained Estimation in a Road Safety Measure Modeling

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COmmUNICATIONS IN STATISTICS-SIMULATION AND COMPUTATION 2010年第5期39卷 1057-1071页

作者： Mkhadri, Abdallah N'Guessan, Assi Hafidi, Bezza Univ Lille 1 Lab Paul Painleve Lille France Univ Lille 1 Ecole Polytech Lille France Univ Cadi Ayyad Dept Math Marrakech Morocco Univ Iub Zohr Agadir Morocco

We model the effect of a road safety measure on a set of target sites with a control area for each site, and we suppose that the accident data recorded at each site are classified in different mutually exclusive types. We adopt the before-after technique and we assume that at any one target site the total number of accidents recorded is multinomially distibuted between the periods and types of accidents. In this article, we propose a minorization-majorization (mm) algorithm for obtaining the constrained maximum likelihood estimates of the parameter vector. We compare it with a gradient projection-expectation maximization (GP-EM) algorithm, based on gradient projections. The performance of the algorithms is examined through a simulation study of road safety data.

关键词： Constrained maximum likelihood GP-EM algorithm mm algorithm Monte Carlo simulation Multinomial model Road safety measure Standard errors

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GenSVM: a generalized multiclass support vector machine

The Journal of Machine Learning Research

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The Journal of Machine Learning Research 2016年第1期17卷

作者： Kevin Murphy Bernhard Schölkopf Gerrit J. J. Van Den Burg Patrick J. F. Groenen Google MPI for Intelligent Systems Econometric Institute Erasmus University Rotterdam Rotterdam The Netherlands

Traditional extensions of the binary support vector machine (SVM) to multiclass problems are either heuristics or require solving a large dual optimization problem. Here, a generalized multiclass SVM is proposed called GenSVM. In this method classification boundaries for a K-class problem are constructed in a (K - 1)-dimensional space using a simplex encoding. Additionally, several different weightings of the misclassification errors are incorporated in the loss function, such that it generalizes three existing multiclass SVMs through a single optimization problem. An iterative majorization algorithm is derived that solves the optimization problem without the need of a dual formulation. This algorithm has the advantage that it can use warm starts during cross validation and during a grid search, which significantly speeds up the training phase. Rigorous numerical experiments compare linear GenSVM with seven existing multiclass SVMs on both small and large data sets. These comparisons show that the proposed method is competitive with existing methods in both predictive accuracy and training time, and that it significantly outperforms several existing methods on these criteria.

关键词： mm algorithm SVM classifier comparison iterative majorization multiclass classification support vector machines

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Estimating common principal components in high dimensions

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ADVANCES IN DATA ANALYSIS AND CLASSIFICATION 2014年第2期8卷 217-226页

作者： Browne, Ryan P. McNicholas, Paul D. Univ Guelph Dept Math & Stat Guelph ON N1G 2W1 Canada

We consider the problem of minimizing an objective function that depends on an orthonormal matrix. This situation is encountered, for example, when looking for common principal components. The Flury method is a popular approach but is not effective for higher dimensional problems. We obtain several simple majorization-minimization (mm) algorithms that provide solutions to this problem and are effective in higher dimensions. We use mixture model-based clustering applications to illustrate our mm algorithms. We then use simulated data to compare them with other approaches, with comparisons drawn with respect to convergence and computational time.

关键词： Clustering Common principal components GPCM Flury Minimization Maximization Mixture Mixture models Model-based clustering mm algorithm

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Quantile Regression via the EM algorithm

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COmmUNICATIONS IN STATISTICS-SIMULATION AND COMPUTATION 2014年第10期43卷 2162-2172页

作者： Zhou, Ying-Hui Ni, Zhong-Xin Li, Yong Shanghai Univ Sch Econ Shanghai 200444 Peoples R China Renmin Univ Hanqing Adv Inst Econ & Finance Beijing 100872 Peoples R China

The three-parameter asymmetric Laplace distribution (ALD) has received increasing attention in the field of quantile regression due to an important feature between its location and asymmetric parameters. On the basis of the representation of the ALD as a normal-variance-mean mixture with an exponential mixing distribution, this article develops EM and generalized EM algorithms, respectively, for computing regression quantiles of linear and nonlinear regression models. It is interesting to show that the proposed EM algorithm and the mm (Majorization-Minimization) algorithm for quantile regressions are really the same in terms of computation, since the updating formula of them are the same. This provides a good example that connects the EM and mm algorithms. Simulation studies show that the EM algorithm can successfully recover the true parameters in quantile regressions.

关键词： Normal-variance-mean mixture. Asymmetric Laplace distribution Generalized inverse Gaussian distribution mm algorithm Generalized EM algorithm

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