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A Brief Survey of Modern Optimization for Statisticians

INTERNATIONAL STATISTICAL REVIEW 2014年第1期82卷 46-70页

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

Modern computational statistics is turning more and more to high-dimensional optimization to handle the deluge of big data. Once a model is formulated, its parameters can be estimated by optimization. Because model parsimony is important, models routinely include non-differentiable penalty terms such as the lasso. This sober reality complicates minimization and maximization. Our broad survey stresses a few important principles in algorithm design. Rather than view these principles in isolation, it is more productive to mix and match them. A few well-chosen examples illustrate this point. algorithm derivation is also emphasized, and theory is downplayed, particularly the abstractions of the convex calculus. Thus, our survey should be useful and accessible to a broad audience.

关键词： acceleration augmented Lagrangian penalization mm algorithm Newton's method Block relaxation

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Coordinate descent algorithm for covariance graphical lasso

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STATISTICS AND COMPUTING 2014年第4期24卷 521-529页

作者： Wang, Hao Univ S Carolina Dept Stat Columbia SC 29208 USA

Bien and Tibshirani (Biometrika, 98(4):807-820, 2011) have proposed a covariance graphical lasso method that applies a lasso penalty on the elements of the covariance matrix. This method is definitely useful because it not only produces sparse and positive definite estimates of the covariance matrix but also discovers marginal independence structures by generating exact zeros in the estimated covariance matrix. However, the objective function is not convex, making the optimization challenging. Bien and Tibshirani (Biometrika, 98(4):807-820, 2011) described a majorize-minimize approach to optimize it. We develop a new optimization method based on coordinate descent. We discuss the convergence property of the algorithm. Through simulation experiments, we show that the new algorithm has a number of advantages over the majorize-minimize approach, including its simplicity, computing speed and numerical stability. Finally, we show that the cyclic version of the coordinate descent algorithm is more efficient than the greedy version.

关键词： Coordinate descent Covariance graphical lasso Covariance matrix estimation L-1 penalty mm algorithm Marginal independence Regularization Shrinkage Sparsity

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Solution path clustering with adaptive concave penalty

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ELECTRONIC JOURNAL OF STATISTICS 2014年第1期8卷 1569-1603页

作者： Marchetti, Yuliya Zhou, Qing Univ Calif Los Angeles Dept Stat Los Angeles CA 90095 USA

Fast accumulation of large amounts of complex data has created a need for more sophisticated statistical methodologies to discover interesting patterns and better extract information from these data. The large scale of the data often results in challenging high-dimensional estimation problems where only a minority of the data shows specific grouping patterns. To address these emerging challenges, we develop a new clustering methodology that introduces the idea of a regularization path into unsupervised learning. A regularization path for a clustering problem is created by varying the degree of sparsity constraint that is imposed on the differences between objects via the minimax concave penalty with adaptive tuning parameters. Instead of providing a single solution represented by a cluster assignment for each object, the method produces a short sequence of solutions that determines not only the cluster assignment but also a corresponding number of clusters for each solution. The optimization of the penalized loss function is carried out through an mm algorithm with block coordinate descent. The advantages of this clustering algorithm compared to other existing methods are as follows: it does not require the input of the number of clusters;it is capable of simultaneously separating irrelevant or noisy observations that show no grouping pattern, which can greatly improve data interpretation;it is a general methodology that can be applied to many clustering problems. We test this method on various simulated datasets and on gene expression data, where it shows better or competitive performance compared against several clustering methods.

关键词： Clustering sparsity concave regularization co-ordinate descent mm algorithm

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Estimation for General Birth-Death Processes

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JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION 2014年第506期109卷 730-747页

作者： Crawford, Forrest W. Minin, Vladimir N. Suchard, Marc A. Yale Univ Sch Publ Hlth Dept Biostat New Haven CT 06510 USA Univ Washington Dept Stat Seattle WA 98195 USA Univ Calif Los Angeles Dept Biomath Dept Biostat Los Angeles CA 90095 USA Univ Calif Los Angeles Dept Human Genet Los Angeles CA 90095 USA

Birth-death processes (BDPs) are continuous-time Markov chains that track the number of "particles" in a system over time. While widely used in population biology, genetics, and ecology, statistical inference of the instantaneous particle birth and death rates remains largely limited to restrictive linear BDPs in which per-particle birth and death rates are constant. Researchers often observe the number of particles at discrete times, necessitating data augmentation procedures such as expectation-maximization (EM) to find maximum likelihood estimates (MLEs). For BDPs on finite state-spaces, there are powerful matrix methods for computing the conditional expectations needed for the E-step of the EM algorithm. For BDPs on infinite state-spaces, closed-form solutions for the E-step are available for some linear models, but most previous work has resorted to time-consuming simulation. Remarkably, we show that the E-step conditional expectations can be expressed as convolutions of computable transition probabilities for any general BDP with arbitrary rates. This important observation, along with a convenient continued fraction representation of the Laplace transforms of the transition probabilities, allows for novel and efficient computation of the conditional expectations for all BDPs, eliminating the need for truncation of the state-space or costly simulation. We use this insight to derive EM algorithms that yield maximum likelihood estimation for general BDPs characterized by various rate models, including generalized linear models (GLM). We show that our Laplace convolution technique outperforms competing methods when they are available and demonstrate a technique to accelerate EM algorithm convergence. We validate our approach using synthetic data and then apply our methods to cancer cell growth and estimation of mutation parameters in microsatellite evolution.

关键词： Continuous-time Markov chain EM algorithm Maximum likelihood estimation Microsatellite evolution mm algorithm

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Biomarker Detection in Association Studies: Modeling SNPs Simultaneously via Logistic ANOVA

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JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION 2014年第508期109卷 1355-1367页

作者： Jung, Yoonsuh Huang, Jianhua Z. Hu, Jianhua Univ Waikato Dept Stat Hamilton 3240 New Zealand Texas A&M Univ Dept Stat College Stn TX 77843 USA Captial Univ Econ & Business ISEM Beijing Peoples R China Univ Texas MD Anderson Canc Ctr Dept Biostat Houston TX 77030 USA

In genome-wide association studies, the primary task is to detect biomarkers in the form of single nucleotide polymorphisms (SNPs) that have nontrivial associations with a disease phenotype and some other important clinical/environmental factors. However, the extremely large number of SNPs compared to the sample size inhibits application of classical methods such as the multiple logistic regression. Currently, the most commonly used approach is still to analyze one SNP at a time. In this article, we propose to consider the genotypes of the SNPs simultaneously via a logistic analysis of variance (ANOVA) model, which expresses the logit transformed mean of SNP genotypes as the summation of the SNP effects, effects of the disease phenotype and/or other clinical variables, and the interaction effects. We use a reduced-rank representation of the interaction-effect matrix for dimensionality reduction, and employ the L-1-penalty in a penalized likelihood framework to filter out the SNPs that have no associations. We develop a majorization-minimization algorithm for computational implementation. In addition, we propose a modified BIC criterion to select the penalty parameters and determine the rank number. The proposed method is applied to a multiple sclerosis dataset and simulated datasets and shows promise in biomarker detection.

关键词： BIC GWAS L-1-penalty mm algorithm Penalized Bernoulli likelihood Simultaneous modeling of SNPs

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Nested coordinate descent algorithms for empirical likelihood

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JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION 2014年第9期84卷 1917-1930页

作者： Tang, Cheng Yong Wu, Tong Tong Univ Colorado Business Sch Denver CO 80204 USA Univ Maryland Dept Epidemiol & Biostat College Pk MD 20742 USA

Empirical likelihood (EL) as a nonparametric approach has been demonstrated to have many desirable merits. While it has intensive development in methodological research, its practical application is less explored due to the requirements of intensive optimizations. Effective and stable algorithms therefore are highly desired for practical implementation of EL. This paper bears the effort to narrow the gap between methodological research and practical application of EL. We try to tackle the computation problems, which are considered difficult by practitioners, by introducing a nested coordinate descent algorithm and one modified version to EL. Coordinate descent as a class of convenient and robust algorithms has been shown in the existing literature to be effective in optimizations. We show that the nested coordinate descent algorithms can be conveniently and stably applied in general EL problems. The combination of nested coordinate descent with the mm algorithm further simplifies the computation. The nested coordinate descent algorithms are a natural and perfect match with inferences based on profile estimation and variable selection in high-dimensional data. Extensive examples are conducted to demonstrate the performance of the nested coordinate descent algorithms in the context of EL.

关键词： confidence regions cyclic coordinate descent empirical likelihood likelihood ratio mm algorithm profiled empirical likelihood 62G05 65K10

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ASYMPTOTIC INFERENCE FOR HIDDEN PROCESS REGRESSION MODELS

ASYMPTOTIC INFERENCE FOR HIDDEN PROCESS REGRESSION MODELS

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IEEE Workshop on Statistical Signal Processing (SSP)

作者： Nguyen, Hien D. McLachlan, Geoffrey J. Univ Queensland Sch Math & Phys Brisbane Qld 4072 Australia

ISBN: (纸本)9781479949755

Hidden process regression (HPR) is a relatively new solution to fitting time series data that undergo a regime change. Current research in HPR has concentrated entirely on its prediction capacity, and modifications for time series classification and clustering, rather than inferential problems such as confidence interval construction. In this article, we investigate results under which maximum likelihood estimates for HPR are consistent and asymptotically normal. Proceeding from these results, confidence intervals of interesting quantities are derived. In the process, we give a monotonically converging modification to the currently used maximum likelihood estimation algorithm for HPR. As a demonstration of our approach, the algorithm and intervals are applied to a climate science example.

关键词： Hidden process regression Mixture-of-experts Confidence intervals mm algorithm

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Cluster and propensity based approximation of a network

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BMC SYSTEMS BIOLOGY 2013年第1期7卷 21页

作者： Ranola, John Michael Langfelder, Peter Lange, Kenneth Horvath, Steve Univ Calif Los Angeles Los Angeles CA 90095 USA

Background: The models in this article generalize current models for both correlation networks and multigraph networks. Correlation networks are widely applied in genomics research. In contrast to general networks, it is straightforward to test the statistical significance of an edge in a correlation network. It is also easy to decompose the underlying correlation matrix and generate informative network statistics such as the module eigenvector. However, correlation networks only capture the connections between numeric variables. An open question is whether one can find suitable decompositions of the similarity measures employed in constructing general networks. Multigraph networks are attractive because they support likelihood based inference. Unfortunately, it is unclear how to adjust current statistical methods to detect the clusters inherent in many data sets. Results: Here we present an intuitive and parsimonious parametrization of a general similarity measure such as a network adjacency matrix. The cluster and propensity based approximation (CPBA) of a network not only generalizes correlation network methods but also multigraph methods. In particular, it gives rise to a novel and more realistic multigraph model that accounts for clustering and provides likelihood based tests for assessing the significance of an edge after controlling for clustering. We present a novel Majorization-Minimization (mm) algorithm for estimating the parameters of the CPBA. To illustrate the practical utility of the CPBA of a network, we apply it to gene expression data and to a bi-partite network model for diseases and disease genes from the Online Mendelian Inheritance in Man (OMIM). Conclusions: The CPBA of a network is theoretically appealing since a) it generalizes correlation and multigraph network methods, b) it improves likelihood based significance tests for edge counts, c) it directly models higher-order relationships between clusters, and d) it suggests novel clustering al

关键词： Network decomposition Model-based clustering mm algorithm Propensity Network conformity

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EM vs mm: A case study

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COMPUTATIONAL STATISTICS & DATA ANALYSIS 2012年第12期56卷 3909-3920页

作者： Zhou, Hua Zhang, Yiwen N Carolina State Univ Dept Stat Raleigh NC 27695 USA

The celebrated expectation-maximization (EM) algorithm is one of the most widely used optimization methods in statistics. In recent years it has been realized that EM algorithm is a special case of the more general minorization-maximization (mm) principle. Both algorithms create a surrogate function in the first (E or M) step that is maximized in the second M step. This two step process always drives the objective function uphill and is iterated until the parameters converge. The two algorithms differ in the way the surrogate function is constructed. The expectation step of the EM algorithm relies on calculating conditional expectations, while the minorization step of the mm algorithm builds on crafty use of inequalities. For many problems. EM and mm derivations yield the same algorithm. This expository note walks through the construction of both algorithms for estimating the parameters of the Dirichlet-Multinomial distribution. This particular case is of interest because EM and mm derivations lead to two different algorithms with completely distinct operating characteristics. The EM algorithm converges quickly but involves solving a nontrivial maximization problem in the M step. In contrast the mm updates are extremely simple but converge slowly. An EM-mm hybrid algorithm is derived which shows faster convergence than the mm algorithm in certain parameter regimes. The local convergence rates of the three algorithms are studied theoretically from the unifying mm point of view and also compared on numerical examples. (C) 2012 Elsevier B.V. All rights reserved.

关键词： Convergence rate Dirichlet-multinomial distribution EM algorithm mm algorithm

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Reconstructing DNA copy number by joint segmentation of multiple sequences

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BMC BIOINFORMATICS 2012年第1期13卷 1-15页

作者： Zhang, Zhongyang Lange, Kenneth Sabatti, Chiara Univ Calif Los Angeles Dept Stat Los Angeles CA USA Univ Calif Los Angeles Dept Human Genet Biomath & Stat Los Angeles CA USA Stanford Univ Dept Hlth Res & Policy & Stat Stanford CA 94305 USA

Background: Variations in DNA copy number carry information on the modalities of genome evolution and mis-regulation of DNA replication in cancer cells. Their study can help localize tumor suppressor genes, distinguish different populations of cancerous cells, and identify genomic variations responsible for disease phenotypes. A number of different high throughput technologies can be used to identify copy number variable sites, and the literature documents multiple effective algorithms. We focus here on the specific problem of detecting regions where variation in copy number is relatively common in the sample at hand. This problem encompasses the cases of copy number polymorphisms, related samples, technical replicates, and cancerous sub-populations from the same individual. Results: We present a segmentation method named generalized fused lasso (GFL) to reconstruct copy number variant regions. GFL is based on penalized estimation and is capable of processing multiple signals jointly. Our approach is computationally very attractive and leads to sensitivity and specificity levels comparable to those of state-of-the-art specialized methodologies. We illustrate its applicability with simulated and real data sets. Conclusions: The flexibility of our framework makes it applicable to data obtained with a wide range of technology. Its versatility and speed make GFL particularly useful in the initial screening stages of large data sets.

关键词： Copy number variant Copy number polymorphism Fused lasso Group fused lasso mm algorithm

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