Canonical Correlation Analysis and Graphical Modeling for Huaman
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Canonical Correlation a Tutorial
Canonical Correlation a Tutorial Magnus Borga January 12, 2001 Contents 1 About this tutorial 1 2 Introduction 2 3 Definition 2 4 Calculating canonical correlations 3 5 Relating topics 3 5.1 The difference between CCA and ordinary correlation analysis . 3 5.2Relationtomutualinformation................... 4 5.3 Relation to other linear subspace methods ............. 4 5.4RelationtoSNR........................... 5 5.4.1 Equalnoiseenergies.................... 5 5.4.2 Correlation between a signal and the corrupted signal . 6 A Explanations 6 A.1Anoteoncorrelationandcovariancematrices........... 6 A.2Affinetransformations....................... 6 A.3Apieceofinformationtheory.................... 7 A.4 Principal component analysis . ................... 9 A.5 Partial least squares ......................... 9 A.6 Multivariate linear regression . ................... 9 A.7Signaltonoiseratio......................... 10 1 About this tutorial This is a printable version of a tutorial in HTML format. The tutorial may be modified at any time as will this version. The latest version of this tutorial is available at http://people.imt.liu.se/˜magnus/cca/. 1 2 Introduction Canonical correlation analysis (CCA) is a way of measuring the linear relationship between two multidimensional variables. It finds two bases, one for each variable, that are optimal with respect to correlations and, at the same time, it finds the corresponding correlations. In other words, it finds the two bases in which the correlation matrix between the variables is diagonal and the correlations on the diagonal are maximized. The dimensionality of these new bases is equal to or less than the smallest dimensionality of the two variables. An important property of canonical correlations is that they are invariant with respect to affine transformations of the variables. This is the most important differ- ence between CCA and ordinary correlation analysis which highly depend on the basis in which the variables are described. -
A Tutorial on Canonical Correlation Analysis
Finding the needle in high-dimensional haystack: A tutorial on canonical correlation analysis Hao-Ting Wang1,*, Jonathan Smallwood1, Janaina Mourao-Miranda2, Cedric Huchuan Xia3, Theodore D. Satterthwaite3, Danielle S. Bassett4,5,6,7, Danilo Bzdok,8,9,10,* 1Department of Psychology, University of York, Heslington, York, YO10 5DD, United Kingdome 2Centre for Medical Image Computing, Department of Computer Science, University College London, London, United Kingdom; MaX Planck University College London Centre for Computational Psychiatry and Ageing Research, University College London, London, United Kingdom 3Department of Psychiatry, Perelman School of Medicine, University of Pennsylvania, Philadelphia, PA, 19104, USA 4Department of Bioengineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA 5Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA 6Department of Neurology, Perelman School of Medicine, University of Pennsylvania, Philadelphia, PA 19104 USA 7Department of Physics & Astronomy, School of Arts & Sciences, University of Pennsylvania, Philadelphia, PA 19104 USA 8 Department of Psychiatry, Psychotherapy and Psychosomatics, RWTH Aachen University, Germany 9JARA-BRAIN, Jülich-Aachen Research Alliance, Germany 10Parietal Team, INRIA, Neurospin, Bat 145, CEA Saclay, 91191, Gif-sur-Yvette, France * Correspondence should be addressed to: Hao-Ting Wang [email protected] Department of Psychology, University of York, Heslington, York, YO10 5DD, United Kingdome Danilo Bzdok [email protected] Department of Psychiatry, Psychotherapy and Psychosomatics, RWTH Aachen University, Germany 1 1 ABSTRACT Since the beginning of the 21st century, the size, breadth, and granularity of data in biology and medicine has grown rapidly. In the eXample of neuroscience, studies with thousands of subjects are becoming more common, which provide eXtensive phenotyping on the behavioral, neural, and genomic level with hundreds of variables. -
Graphical Models for Discrete and Continuous Data Arxiv:1609.05551V3
Graphical Models for Discrete and Continuous Data Rui Zhuang Department of Biostatistics, University of Washington and Noah Simon Department of Biostatistics, University of Washington and Johannes Lederer Department of Mathematics, Ruhr-University Bochum June 18, 2019 Abstract We introduce a general framework for undirected graphical models. It generalizes Gaussian graphical models to a wide range of continuous, discrete, and combinations of different types of data. The models in the framework, called exponential trace models, are amenable to estimation based on maximum likelihood. We introduce a sampling-based approximation algorithm for computing the maximum likelihood estimator, and we apply this pipeline to learn simultaneous neural activities from spike data. Keywords: Non-Gaussian Data, Graphical Models, Maximum Likelihood Estimation arXiv:1609.05551v3 [math.ST] 15 Jun 2019 1 Introduction Gaussian graphical models (Drton & Maathuis 2016, Lauritzen 1996, Wainwright & Jordan 2008) describe the dependence structures in normally distributed random vectors. These models have become increasingly popular in the sciences, because their representation of the dependencies is lucid and can be readily estimated. For a brief overview, consider a 1 random vector X 2 Rp that follows a centered normal distribution with density 1 −x>Σ−1x=2 fΣ(x) = e (1) (2π)p=2pjΣj with respect to Lebesgue measure, where the population covariance Σ 2 Rp×p is a symmetric and positive definite matrix. Gaussian graphical models associate these densities with a graph (V; E) that has vertex set V := f1; : : : ; pg and edge set E := f(i; j): i; j 2 −1 f1; : : : ; pg; i 6= j; Σij 6= 0g: The graph encodes the dependence structure of X in the sense that any two entries Xi;Xj; i 6= j; are conditionally independent given all other entries if and only if (i; j) 2= E. -
Graphical Models
Graphical Models Michael I. Jordan Computer Science Division and Department of Statistics University of California, Berkeley 94720 Abstract Statistical applications in fields such as bioinformatics, information retrieval, speech processing, im- age processing and communications often involve large-scale models in which thousands or millions of random variables are linked in complex ways. Graphical models provide a general methodology for approaching these problems, and indeed many of the models developed by researchers in these applied fields are instances of the general graphical model formalism. We review some of the basic ideas underlying graphical models, including the algorithmic ideas that allow graphical models to be deployed in large-scale data analysis problems. We also present examples of graphical models in bioinformatics, error-control coding and language processing. Keywords: Probabilistic graphical models; junction tree algorithm; sum-product algorithm; Markov chain Monte Carlo; variational inference; bioinformatics; error-control coding. 1. Introduction The fields of Statistics and Computer Science have generally followed separate paths for the past several decades, which each field providing useful services to the other, but with the core concerns of the two fields rarely appearing to intersect. In recent years, however, it has become increasingly evident that the long-term goals of the two fields are closely aligned. Statisticians are increasingly concerned with the computational aspects, both theoretical and practical, of models and inference procedures. Computer scientists are increasingly concerned with systems that interact with the external world and interpret uncertain data in terms of underlying probabilistic models. One area in which these trends are most evident is that of probabilistic graphical models. -
A Call for Conducting Multivariate Mixed Analyses
Journal of Educational Issues ISSN 2377-2263 2016, Vol. 2, No. 2 A Call for Conducting Multivariate Mixed Analyses Anthony J. Onwuegbuzie (Corresponding author) Department of Educational Leadership, Box 2119 Sam Houston State University, Huntsville, Texas 77341-2119, USA E-mail: [email protected] Received: April 13, 2016 Accepted: May 25, 2016 Published: June 9, 2016 doi:10.5296/jei.v2i2.9316 URL: http://dx.doi.org/10.5296/jei.v2i2.9316 Abstract Several authors have written methodological works that provide an introductory- and/or intermediate-level guide to conducting mixed analyses. Although these works have been useful for beginning and emergent mixed researchers, with very few exceptions, works are lacking that describe and illustrate advanced-level mixed analysis approaches. Thus, the purpose of this article is to introduce the concept of multivariate mixed analyses, which represents a complex form of advanced mixed analyses. These analyses characterize a class of mixed analyses wherein at least one of the quantitative analyses and at least one of the qualitative analyses both involve the simultaneous analysis of multiple variables. The notion of multivariate mixed analyses previously has not been discussed in the literature, illustrating the significance and innovation of the article. Keywords: Mixed methods data analysis, Mixed analysis, Mixed analyses, Advanced mixed analyses, Multivariate mixed analyses 1. Crossover Nature of Mixed Analyses At least 13 decision criteria are available to researchers during the data analysis stage of mixed research studies (Onwuegbuzie & Combs, 2010). Of these criteria, the criterion that is the most underdeveloped is the crossover nature of mixed analyses. Yet, this form of mixed analysis represents a pivotal decision because it determines the level of integration and complexity of quantitative and qualitative analyses in mixed research studies. -
Graphical Modelling of Multivariate Spatial Point Processes
Graphical modelling of multivariate spatial point processes Matthias Eckardt Department of Computer Science, Humboldt Universit¨at zu Berlin, Berlin, Germany July 26, 2016 Abstract This paper proposes a novel graphical model, termed the spatial dependence graph model, which captures the global dependence structure of different events that occur randomly in space. In the spatial dependence graph model, the edge set is identified by using the conditional partial spectral coherence. Thereby, nodes are related to the components of a multivariate spatial point process and edges express orthogo- nality relation between the single components. This paper introduces an efficient approach towards pattern analysis of highly structured and high dimensional spatial point processes. Unlike all previous methods, our new model permits the simultane- ous analysis of all multivariate conditional interrelations. The potential of our new technique to investigate multivariate structural relations is illustrated using data on forest stands in Lansing Woods as well as monthly data on crimes committed in the City of London. Keywords: Crime data, Dependence structure; Graphical model; Lansing Woods, Multi- variate spatial point pattern arXiv:1607.07083v1 [stat.ME] 24 Jul 2016 1 1 Introduction The analysis of spatial point patterns is a rapidly developing field and of particular interest to many disciplines. Here, a main concern is to explore the structures and relations gener- ated by a countable set of randomly occurring points in some bounded planar observation window. Generally, these randomly occurring points could be of one type (univariate) or of two and more types (multivariate). In this paper, we consider the latter type of spatial point patterns. -
Decomposable Principal Component Analysis Ami Wiesel, Member, IEEE, and Alfred O
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 11, NOVEMBER 2009 4369 Decomposable Principal Component Analysis Ami Wiesel, Member, IEEE, and Alfred O. Hero, Fellow, IEEE Abstract—In this paper, we consider principal component anal- computer networks [6], [12], [15]. In particular, [9] and [18] ysis (PCA) in decomposable Gaussian graphical models. We exploit proposed to approximate the global PCA using a sequence of the prior information in these models in order to distribute PCA conditional local PCA solutions. Alternatively, an approximate computation. For this purpose, we reformulate the PCA problem in the sparse inverse covariance (concentration) domain and ad- solution which allows a tradeoff between performance and com- dress the global eigenvalue problem by solving a sequence of local munication requirements was proposed in [12] using eigenvalue eigenvalue problems in each of the cliques of the decomposable perturbation theory. graph. We illustrate our methodology in the context of decentral- DPCA is an efficient implementation of distributed PCA ized anomaly detection in the Abilene backbone network. Based on the topology of the network, we propose an approximate statistical based on a prior graphical model. Unlike the above references graphical model and distribute the computation of PCA. it does not try to approximate PCA, but yields an exact solution Index Terms—Anomaly detection, graphical models, principal up to on any given error tolerance. DPCA assumes additional component analysis. prior knowledge in the form of a graphical model that is not taken into account by previous distributed PCA methods. Such models are very common in signal processing and communi- I. INTRODUCTION cations. -
A Probabilistic Interpretation of Canonical Correlation Analysis
A Probabilistic Interpretation of Canonical Correlation Analysis Francis R. Bach Michael I. Jordan Computer Science Division Computer Science Division University of California and Department of Statistics Berkeley, CA 94114, USA University of California [email protected] Berkeley, CA 94114, USA [email protected] April 21, 2005 Technical Report 688 Department of Statistics University of California, Berkeley Abstract We give a probabilistic interpretation of canonical correlation (CCA) analysis as a latent variable model for two Gaussian random vectors. Our interpretation is similar to the proba- bilistic interpretation of principal component analysis (Tipping and Bishop, 1999, Roweis, 1998). In addition, we cast Fisher linear discriminant analysis (LDA) within the CCA framework. 1 Introduction Data analysis tools such as principal component analysis (PCA), linear discriminant analysis (LDA) and canonical correlation analysis (CCA) are widely used for purposes such as dimensionality re- duction or visualization (Hotelling, 1936, Anderson, 1984, Hastie et al., 2001). In this paper, we provide a probabilistic interpretation of CCA and LDA. Such a probabilistic interpretation deepens the understanding of CCA and LDA as model-based methods, enables the use of local CCA mod- els as components of a larger probabilistic model, and suggests generalizations to members of the exponential family other than the Gaussian distribution. In Section 2, we review the probabilistic interpretation of PCA, while in Section 3, we present the probabilistic interpretation of CCA and LDA, with proofs presented in Section 4. In Section 5, we provide a CCA-based probabilistic interpretation of LDA. 1 z x Figure 1: Graphical model for factor analysis. 2 Review: probabilistic interpretation of PCA Tipping and Bishop (1999) have shown that PCA can be seen as the maximum likelihood solution of a factor analysis model with isotropic covariance matrix. -
Cross-Modal Subspace Clustering Via Deep Canonical Correlation Analysis
The Thirty-Fourth AAAI Conference on Artificial Intelligence (AAAI-20) Cross-Modal Subspace Clustering via Deep Canonical Correlation Analysis Quanxue Gao,1 Huanhuan Lian,1∗ Qianqian Wang,1† Gan Sun2 1State Key Laboratory of Integrated Services Networks, Xidian University, Xi’an 710071, China. 2State Key Laboratory of Robotics, Shenyang Institute of Automation, Chinese Academy of Sciences, China. [email protected], [email protected], [email protected], [email protected] Abstract Different Different Self- S modalities subspaces expression 1 For cross-modal subspace clustering, the key point is how to ZZS exploit the correlation information between cross-modal data. Deep 111 X Encoding Z However, most hierarchical and structural correlation infor- 1 1 S S ZZS 2 mation among cross-modal data cannot be well exploited due 222 to its high-dimensional non-linear property. To tackle this problem, in this paper, we propose an unsupervised frame- Z X 2 work named Cross-Modal Subspace Clustering via Deep 2 Canonical Correlation Analysis (CMSC-DCCA), which in- Different Similar corporates the correlation constraint with a self-expressive S modalities subspaces 1 ZZS layer to make full use of information among the inter-modal 111 Deep data and the intra-modal data. More specifically, the proposed Encoding X Z A better model consists of three components: 1) deep canonical corre- 1 1 S ZZS S lation analysis (Deep CCA) model; 2) self-expressive layer; Maximize 2222 3) Deep CCA decoders. The Deep CCA model consists of Correlation convolutional encoders and correlation constraint. Convolu- X Z tional encoders are used to obtain the latent representations 2 2 of cross-modal data, while adding the correlation constraint for the latent representations can make full use of the infor- Figure 1: The illustration shows the influence of correla- mation of the inter-modal data. -
Sequential Mean Field Variational Analysis
Computer Vision and Image Understanding 101 (2006) 87–99 www.elsevier.com/locate/cviu Sequential mean field variational analysis of structured deformable shapes Gang Hua *, Ying Wu Department of Electrical and Computer Engineering, 2145 Sheridan Road, Northwestern University, Evanston, IL 60208, USA Received 15 March 2004; accepted 25 July 2005 Available online 3 October 2005 Abstract A novel approach is proposed to analyzing and tracking the motion of structured deformable shapes, which consist of multiple cor- related deformable subparts. Since this problem is high dimensional in nature, existing methods are plagued either by the inability to capture the detailed local deformation or by the enormous complexity induced by the curse of dimensionality. Taking advantage of the structured constraints of the different deformable subparts, we propose a new statistical representation, i.e., the Markov network, to structured deformable shapes. Then, the deformation of the structured deformable shapes is modelled by a dynamic Markov network which is proven to be very efficient in overcoming the challenges induced by the high dimensionality. Probabilistic variational analysis of this dynamic Markov model reveals a set of fixed point equations, i.e., the sequential mean field equations, which manifest the interac- tions among the motion posteriors of different deformable subparts. Therefore, we achieve an efficient solution to such a high-dimen- sional motion analysis problem. Combined with a Monte Carlo strategy, the new algorithm, namely sequential mean field Monte Carlo, achieves very efficient Bayesian inference of the structured deformation with close-to-linear complexity. Extensive experiments on tracking human lips and human faces demonstrate the effectiveness and efficiency of the proposed method. -
The Geometry of Kernel Canonical Correlation Analysis
Max–Planck–Institut für biologische Kybernetik Max Planck Institute for Biological Cybernetics Technical Report No. 108 The Geometry Of Kernel Canonical Correlation Analysis Malte Kuss1 and Thore Graepel2 May 2003 1 Max Planck Institute for Biological Cybernetics, Dept. Schölkopf, Spemannstrasse 38, 72076 Tübingen, Germany, email: [email protected] 2 Microsoft Research Ltd, Roger Needham Building, 7 J J Thomson Avenue, Cambridge CB3 0FB, U.K, email: [email protected] This report is available in PDF–format via anonymous ftp at ftp://ftp.kyb.tuebingen.mpg.de/pub/mpi-memos/pdf/TR-108.pdf. The complete series of Technical Reports is documented at: http://www.kyb.tuebingen.mpg.de/techreports.html The Geometry Of Kernel Canonical Correlation Analysis Malte Kuss, Thore Graepel Abstract. Canonical correlation analysis (CCA) is a classical multivariate method concerned with describing linear dependencies between sets of variables. After a short exposition of the linear sample CCA problem and its analytical solution, the article proceeds with a detailed characterization of its geometry. Projection operators are used to illustrate the relations between canonical vectors and variates. The article then addresses the problem of CCA between spaces spanned by objects mapped into kernel feature spaces. An exact solution for this kernel canonical correlation (KCCA) problem is derived from a geometric point of view. It shows that the expansion coefficients of the canonical vectors in their respective feature space can be found by linear CCA in the basis induced by kernel principal component analysis. The effect of mappings into higher dimensional feature spaces is considered critically since it simplifies the CCA problem in general. -
Lecture 7: Analysis of Factors and Canonical Correlations
Lecture 7: Analysis of Factors and Canonical Correlations M˚ansThulin Department of Mathematics, Uppsala University [email protected] Multivariate Methods • 6/5 2011 1/33 Outline I Factor analysis I Basic idea: factor the covariance matrix I Methods for factoring I Canonical correlation analysis I Basic idea: correlations between sets I Finding the canonical correlations 2/33 Factor analysis Assume that we have a data set with many variables and that it is reasonable to believe that all these, to some extent, depend on a few underlying but unobservable factors. The purpose of factor analysis is to find dependencies on such factors and to use this to reduce the dimensionality of the data set. In particular, the covariance matrix is described by the factors. 3/33 Factor analysis: an early example C. Spearman (1904), General Intelligence, Objectively Determined and Measured, The American Journal of Psychology. Children's performance in mathematics (X1), French (X2) and English (X3) was measured. Correlation matrix: 2 1 0:67 0:64 3 R = 4 1 0:67 5 1 Assume the following model: X1 = λ1f + 1; X2 = λ2f + 2; X3 = λ3f + 3 where f is an underlying "common factor" ("general ability"), λ1, λ2, λ3 are \factor loadings" and 1, 2, 3 are random disturbance terms. 4/33 Factor analysis: an early example Model: Xi = λi f + i ; i = 1; 2; 3 with the unobservable factor f = \General ability" The variation of i consists of two parts: I a part that represents the extent to which an individual's matehematics ability, say, differs from her general ability I a "measurement error" due to the experimental setup, since examination is only an approximate measure of her ability in the subject The relative sizes of λi f and i tell us to which extent variation between individuals can be described by the factor.