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BookDOI

Modern Multivariate Statistical Techniques

TL;DR: The identity matrices have different dimensions — In the top row of each matrix, the identity matrix has dimension r and in the bottom row it has dimension s.
Abstract: CHAPTER 3 Page 46, line –15: (K × J)-matrix. Page 47, Equation (3.5): −EF should be −EF . Page 49, line –6: R should be <. Page 53, line –7: “see Exercise 3.4” is not relevant here. Page 53, Equation (3.43): Last term on rhs should be ∂yJ ∂xK . Page 60, Equation (3.98): σ should be σ. Page 61, line 8: (3.106) should be (3.105). Pages 61, 62, Equations (3.109), (3.110), and (3.111): The identity matrices have different dimensions — In the top row of each matrix, the identity matrix has dimension r and in the bottom row it has dimension s. Page 62, line 1: “r-vector” should be “(r + s)-vector.” Page 62, Equation (3.111): ΣXY should be ΣY X . Page 64, Equation (3.127): |Σ| should be |Σ|. Page 62, Equation (3.133): I(2.2) should be I(2,2). Page 65, line 8 (2nd line of property 2): Wr should be Wp. Page 65, property 4: Restate as follows. Let X = (X1, · · · ,Xn) , where Xi ∼ Nr(0,Σ), i = 1, 2, . . . , n, are independently and identically distributed (iid). Let A be a symmetric (n×n)-matrix with ν = rank(A), and let a be a fixed r-vector. Let y = Xa. Then, X AX ∼ Wr(ν,Σ) iff yAy ∼ σ aχ 2 ν , where σ 2 a = a Σa. Page 66, Equation (3.143): last term on rhs, +n should be −n2 . Page 67, line 3: Should read tr(TT ) = ∑r i=1 t 2 ii + ∑ i>j t 2 ij . Page 67, line –6: Should read “idempotent with rank n − 1.” Page 67, line –3: bX should be X b. Page 67, Equation (3.148): n should be n − 1.
Citations
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Book
11 Apr 2019
TL;DR: This book provides a self-contained introduction to the area of high-dimensional statistics, aimed at the first-year graduate level, and includes chapters that are focused on core methodology and theory - including tail bounds, concentration inequalities, uniform laws and empirical process, and random matrices.
Abstract: Recent years have witnessed an explosion in the volume and variety of data collected in all scientific disciplines and industrial settings. Such massive data sets present a number of challenges to researchers in statistics and machine learning. This book provides a self-contained introduction to the area of high-dimensional statistics, aimed at the first-year graduate level. It includes chapters that are focused on core methodology and theory - including tail bounds, concentration inequalities, uniform laws and empirical process, and random matrices - as well as chapters devoted to in-depth exploration of particular model classes - including sparse linear models, matrix models with rank constraints, graphical models, and various types of non-parametric models. With hundreds of worked examples and exercises, this text is intended both for courses and for self-study by graduate students and researchers in statistics, machine learning, and related fields who must understand, apply, and adapt modern statistical methods suited to large-scale data.

748 citations

Journal ArticleDOI
12 Apr 2016-eLife
TL;DR: A new dimensionality reduction technique, demixed principal component analysis (dPCA), that decomposes population activity into a few components and exposes the dependence of the neural representation on task parameters such as stimuli, decisions, or rewards is demonstrated.
Abstract: Neurons in higher cortical areas, such as the prefrontal cortex, are often tuned to a variety of sensory and motor variables, and are therefore said to display mixed selectivity. This complexity of single neuron responses can obscure what information these areas represent and how it is represented. Here we demonstrate the advantages of a new dimensionality reduction technique, demixed principal component analysis (dPCA), that decomposes population activity into a few components. In addition to systematically capturing the majority of the variance of the data, dPCA also exposes the dependence of the neural representation on task parameters such as stimuli, decisions, or rewards. To illustrate our method we reanalyze population data from four datasets comprising different species, different cortical areas and different experimental tasks. In each case, dPCA provides a concise way of visualizing the data that summarizes the task-dependent features of the population response in a single figure.

443 citations

Journal ArticleDOI
TL;DR: This Review highlights important technical prerequisites as well as recent developments in metabolomics and metabolomics data analysis with special emphasis on their utility in biomarker identification and qualification, aswell as targeted metabolomics by employing high-throughput mass spectrometry.
Abstract: Metabolomics is a truly interdisciplinary field of science, which combines analytical chemistry, platform technology, mass spectrometry, and NMR spectroscopy with sophisticated data analysis. Applied to biomarker discovery, it includes aspects of pathobiochemistry, systems biology/medicine, and molecular diagnostics and requires bioinformatics and multivariate statistics. While successfully established in the screening of inborn errors in neonates, metabolomics is now widely used in the characterization and diagnostic research of an ever increasing number of diseases. In this Review we highlight important technical prerequisites as well as recent developments in metabolomics and metabolomics data analysis with special emphasis on their utility in biomarker identification and qualification, as well as targeted metabolomics by employing high-throughput mass spectrometry.

312 citations

Journal ArticleDOI
TL;DR: The aim in this paper is to improve the general understanding of how PCA and DA process and display differential sensing data, which should lead to the ability to better interpret the final results.
Abstract: Statistical analysis techniques such as principal component analysis (PCA) and discriminant analysis (DA) have become an integral part of data analysis for differential sensing. These multivariate statistical tools, while extremely versatile and useful, are sometimes used as “black boxes”. Our aim in this paper is to improve the general understanding of how PCA and DA process and display differential sensing data, which should lead to the ability to better interpret the final results. With various sets of model data, we explore several topics, such as how to choose an appropriate number of hosts for an array, selectivity compared to cross-reactivity, when to add hosts, how to obtain the best visually representative plot of a data set, and when arrays are not necessary. We also include items at the end of the paper as general recommendations which readers can follow when using PCA or DA in a practical application. Through this paper we hope to present these statistical analysis methods in a manner such that chemists gain further insight into approaches that optimize the discriminatory power of their arrays.

269 citations

Journal ArticleDOI
TL;DR: A group-lasso type penalty is applied that treats each row of the matrix of the regression coefficients as a group and shows that this penalty satisfies certain desirable invariance properties of the reduced-rank regression coefficient matrix.
Abstract: The reduced-rank regression is an effective method in predicting multiple response variables from the same set of predictor variables. It reduces the number of model parameters and takes advantage of interrelations between the response variables and hence improves predictive accuracy. We propose to select relevant variables for reduced-rank regression by using a sparsity-inducing penalty. We apply a group-lasso type penalty that treats each row of the matrix of the regression coefficients as a group and show that this penalty satisfies certain desirable invariance properties. We develop two numerical algorithms to solve the penalized regression problem and establish the asymptotic consistency of the proposed method. In particular, the manifold structure of the reduced-rank regression coefficient matrix is considered and studied in our theoretical analysis. In our simulation study and real data analysis, the new method is compared with several existing variable selection methods for multivariate regression...

230 citations


Cites methods from "Modern Multivariate Statistical Tec..."

  • ...It is well known that RRR contains many classical multivariate regression models as special cases, including principal component and factor analysis, canonical correlation analysis, linear discriminant analysis, and correspondence analysis; see chapter 6 of Izenman (2008)....

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References
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Book
01 Jan 2008
TL;DR: This paper presents a meta-modelling framework called CART, which automates the very labor-intensive and therefore time-heavy and expensive process of Classification and Regression Trees (CART) that is currently used in statistical inference.
Abstract: Statistical Learning as a Regression Problem.- Regression Splines and Regression Smoothers.- Classification and Regression Trees (CART).- Bagging.- Random Forests.- Boosting.- Support Vector Machines.- Broader Implications and a Bit of Craft Lore.

375 citations

Journal ArticleDOI
TL;DR: This article discusses methodology for multidimensional scaling (MDS) and its implementation in two software systems, GGvis and XGvis, and shows applications to the mapping of computer usage data, to the dimension reduction of marketing segmentation data,to the layout of mathematical graphs and social networks, and finally to the spatial reconstruction of molecules.
Abstract: We discuss methodology for multidimensional scaling (MDS) and its implementation in two software systems, GGvis and XGvis. MDS is a visualization technique for proximity data, that is, data in the form of N × N dissimilarity matrices. MDS constructs maps (“configurations,” “embeddings”) in IRk by interpreting the dissimilarities as distances. Two frequent sources of dissimilarities are high-dimensional data and graphs. When the dissimilarities are distances between high-dimensional objects, MDS acts as a (often nonlinear) dimension-reduction technique. When the dissimilarities are shortest-path distances in a graph, MDS acts as a graph layout technique. MDS has found recent attention in machine learning motivated by image databases (“Isomap”). MDS is also of interest in view of the popularity of “kernelizing” approaches inspired by Support Vector Machines (SVMs; “kernel PCA”).This article discusses the following general topics: (1) the stability and multiplicity of MDS solutions; (2) the analysis of struc...

298 citations

Journal ArticleDOI
TL;DR: These uncertainties will be addressed by the following interactive techniques: algorithm animation, random restarts, and manual editing of configurations, (a) interactive control over parameters that determine the criterion and its minimization, and (c) diagnostics for pinning down artifactual point configurations.
Abstract: These uncertainties will be addressed by the following interactive techniques: (a) algorithm animation, random restarts, and manual editing of configurations, (b) interactive control over parameters that determine the criterion and its minimization, (c) diagnostics for pinning down artifactual point configurations, and (d) restricting MDS to subsets of objects and subsets of pairs of objects. A system, called "XGvis", which implments these techniques, is freely available with the "XGobi" distribution. XGobi is a multivariate data visualization system that is used here for visualizing point configurations.

87 citations