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Multivariate Data Analysis on Matrix Manifolds

(with Manopt)

  • Textbook
  • © 2021

Overview

Authors:
  1. Nickolay Trendafilov

    1. Dept of Mathematics and Statistics, Open University, Milton Keynes, UK

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  2. Michele Gallo

    1. Dept of Human & Social Sciences, Univ degli Studi di Napoli "L'Orientale, Naples, Italy

    View author publications

    You can also search for this author in PubMed   Google Scholar

  • Integrates multivariate data analysis with Riemannian geometry
  • Provides a unified treatment of several MDA techniques
  • Incorporates new tools and technology into current theory of MDA
  • Includes Manpot codes which can be directly used to solve a number of problems or be used as templates to create new codes

Part of the book series: Springer Series in the Data Sciences (SSDS)

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Table of contents (11 chapters)

  1. Front Matter

    Pages i-xx
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  2. Introduction

    • Nickolay Trendafilov, Michele Gallo
    Pages 1-8

  3. Matrix analysis and differentiation

    • Nickolay Trendafilov, Michele Gallo
    Pages 9-44

  4. Matrix manifolds in MDA

    • Nickolay Trendafilov, Michele Gallo
    Pages 45-88

  5. Principal component analysis (PCA)

    • Nickolay Trendafilov, Michele Gallo
    Pages 89-139

  6. Factor analysis (FA)

    • Nickolay Trendafilov, Michele Gallo
    Pages 141-186

  7. Procrustes analysis (PA)

    • Nickolay Trendafilov, Michele Gallo
    Pages 187-228

  8. Linear discriminant analysis (LDA)

    • Nickolay Trendafilov, Michele Gallo
    Pages 229-268

  9. Cannonical correlation analysis (CCA)

    • Nickolay Trendafilov, Michele Gallo
    Pages 269-288

  10. Common principal components (CPC)

    • Nickolay Trendafilov, Michele Gallo
    Pages 289-324

  11. Metric multidimensional scaling (MDS) and related methods

    • Nickolay Trendafilov, Michele Gallo
    Pages 325-371

  12. Data analysis on simplexes

    • Nickolay Trendafilov, Michele Gallo
    Pages 373-402

  13. Back Matter

    Pages 403-450
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Keywords

About this book

This graduate-level textbook aims to give a unified presentation and solution of several commonly used techniques for multivariate data analysis (MDA). Unlike similar texts, it treats the MDA problems as optimization problems on matrix manifolds defined by the MDA model parameters, allowing them to be solved using (free) optimization software Manopt. The book includes numerous in-text examples as well as Manopt codes and software guides, which can be applied directly or used as templates for solving similar and new problems. The first two chapters provide an overview and essential background for studying MDA, giving basic information and notations. Next, it considers several sets of matrices routinely used in MDA as parameter spaces, along with their basic topological properties. A brief introduction to matrix (Riemannian) manifolds and optimization methods on them with Manopt complete the MDA prerequisite. The remaining chapters study individual MDA techniques in depth. The number ofexercises complement the main text with additional information and occasionally involve open and/or challenging research questions. Suitable fields include computational statistics, data analysis, data mining and data science, as well as theoretical computer science, machine learning and optimization. It is assumed that the readers have some familiarity with MDA and some experience with matrix analysis, computing, and optimization. 

Authors and Affiliations

  • Dept of Mathematics and Statistics, Open University, Milton Keynes, UK

    Nickolay Trendafilov

  • Dept of Human & Social Sciences, Univ degli Studi di Napoli "L'Orientale, Naples, Italy

    Michele Gallo

About the authors

Nickolay T. Trendafilov is Reader of Computational Statistics in the School of Mathematics and Statistics, Open University, UK. He received MSc and PhD in the Department of Mathematics and Informatics, University of Sofia “St. Kl. Ohridski”, and then joined the Laboratory of Computational Stochastic, Bulgarian Academy of Sciences. He held research and visiting positions in a number of universities in Belgium, Italy, Japan and USA. His interests are in the computational aspects of multivariate data analysis and interpretation. Other activities include elected memberships in the International Statistical Institute (ISI), the Royal Statistical Society's (RSS) Computing Section, and the Board of Directors, European Regional Section of the International Association for Statistical Computing (IASC). 

Michele Gallo is Professor in the Department of Human and Social Sciences at the University of Naples – L’Orientale. He received his PhD degree in Total Quality Management from the University of Naples – Federico II, in 2000. His current research interest is in Multivariate Data Analysis, Compositional and Ordinal Data, Rasch Analysis. He has published more than 90 research articles. He is Associate-Editor of the journal Computational Statistics. 


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