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SPRINGER BRIEFS IN STATISTICS JSS RESEARCH SERIES IN STATISTICS Toshio Sakata E ditor Applied Matrix and Tensor Variate Data Analysis 123 SpringerBriefs in Statistics JSS Research Series in Statistics Editor-in-chief Naoto Kunitomo Akimichi Takemura Series editors Genshiro Kitagawa Tomoyuki Higuchi Nakahiro Yoshida Yutaka Kano Toshimitsu Hamasaki Shigeyuki Matsui Manabu Iwasaki ThecurrentresearchofstatisticsinJapanhasexpandedinseveraldirectionsinline with recent trends in academic activities in the area of statistics and statistical sciences over the globe. The core of these research activities in statistics in Japan has been the Japan Statistical Society (JSS). This society, the oldest and largest academicorganization for statistics inJapan, was founded in1931by ahandful of pioneerstatisticiansandeconomistsandnowhasahistoryofabout80years.Many distinguished scholars have been members, including the influential statistician Hirotugu Akaike, who was a past president of JSS, and the notable mathematician Kiyosi Itô, who was an earlier member of the Institute of Statistical Mathematics (ISM), which has been a closely related organization since the establishment of ISM. The society has two academic journals: the Journal of the Japan Statistical Society (English Series) and the Journal of the Japan Statistical Society (Japanese Series). The membership of JSS consists of researchers, teachers, and professional statisticians in many different fields including mathematics, statistics, engineering, medical sciences, government statistics, economics, business, psychology, education, and many other natural, biological, and social sciences. The JSS Series of Statistics aims to publish recent results of current research activities in the areas of statistics and statistical sciences in Japan that otherwise would not be available in English; they are complementary to the two JSS academic journals, both English and Japanese. Because the scope of a research paper in academic journals inevitably has become narrowly focused and condensed in recent years, this series is intended to fill the gap between academic research activities and the form of a single academic paper. The series will be of great interest to a wide audience of researchers, teachers, professional statisticians, and graduate students in many countries who are interested in statistics and statistical sciences, in statistical theory, and in various areas of statistical applications. More information about this series at http://www.springer.com/series/13497 Toshio Sakata Editor Applied Matrix and Tensor Variate Data Analysis 123 Editor ToshioSakata Faculty of Design Kyushu University Fukuoka Japan ISSN 2191-544X ISSN 2191-5458 (electronic) SpringerBriefs inStatistics ISSN 2364-0057 ISSN 2364-0065 (electronic) JSSResearch Series in Statistics ISBN978-4-431-55386-1 ISBN978-4-431-55387-8 (eBook) DOI 10.1007/978-4-431-55387-8 LibraryofCongressControlNumber:2015959581 ©TheAuthor(s)2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained hereinorforanyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerJapanKK To the late Prof. Akio Kudo, Prof. Akimichi Okuma and Prof. Takashi Yanagawa, who taught me mathematical statistics in my younger days and guided me through my research life. Preface Statisticians encounter many types of complex data, and among them matrix and tensor data are currently popular data forms that play a vital role in various anal- yses. Matrix data constitute an essential data form of multivariate analysis and tensordataarecomprisedofamulti-waydatumwithwhichstatisticiansarefamiliar in analyzing contingency tables. The difference between matrix and tensor data formsandcontingencytablesisthattheentriesinthetensordataarerealvaluesand that theentriesinthecontingency tables areintegers. Furthermore, tensor data can treat complex data in more flexible ways. In this expository book we present both foundations and applications of matrix and tensor data analysis in six chapters. Chapter 1 deals with the principal component analysis (PCA) for 3-way tensor data, popularly referred to as 3WPCA. The author of this chapter, Prof. Adachi of Osaka University, has conducted extensive research in this field and published papers in respected journals like Psychometrika and Computational Statistics. His research emphasizes hierarchical relationships among Tucker2, Tucker3, Parafac, andthecommonPCA.Heillustratesthetheorybyapplyingittopsychologicaldata. Chapter 2 discusses nonnegative matrix factorization (NMF) and its application toaudiosignaldata. Afterthebasics,includingalgorithms, ofNMFareexplained, remarkable applications to audio signal data are given. The author of this chapter, Dr. Kameoka, is a promising award winning young researcher who works at NTT Communication Science Laboratories and is an adjunct associate professor at the Graduate School of Information Science and Technology of the University of Tokyo. Chapter3introducesanapplicationoftensorPCAtoimagedata.Theauthorof this chapter, Associate Prof. Inoue of Kyushu University, is a specialist of tensor principalcomponentsanalysistoimagedataandhaswrittenmanypapersrelatedto these topics in journals like Lecture Note of Computer Sciences and others. He formulated a generalized tensor PCA (GTPCA) and derived multi-linear PCA (MPCA), robust MPCA, simultaneous low-rank approximation of tensors (SLART), and robust SLART from GTPCA. Applications for image analysis are vii viii Preface presented in this chapter such as outlier removal, image compression, and face recognition. Chapter 4 focuses on sparse coding theory of image data. The author of this chapter,Prof. MurataofWaseda University, isoneofthefoundersofIndependent Components Analysis in Japan and has published books about the topic. He has developed unified treatments of several principles in matrix factorizations. HeshowsthatthedifferencebetweenPCA,independentcomponentanalysis(ICA), nonnegativematrixfactorization(NMF),andsparserepresentationsareunderstood as different constraints when a data matrix is decomposed into a product of two matrices. Applications to image processing are also given. Chapter5centersontheapplicationoftensornormaldistributiontogeneticdata. Theauthorofthischapter,Dr.Akdemir,ResearchAssociateatCornellUniversity, haspublishedapioneeringpaperabouttensornormaldistributionintheJournalof AlgebraicStatistics.Heexpoundedtheestimationofmeanandcovarianceofarray (tensor)undernormaldistributionswithmissingdata.Applicationstoreal-lifedata involving the estimation of genotype and environment interaction and effects on possible correlated traits are detailed. Chapter 6 introduces one-sided tests for the mean matrix of a matrix normal distribution. The late Prof. Kudo of Kyushu University, who was a pioneer of multivariate one-sided tests in Japan, taught the authors of this chapter, Associate Prof.IwasaofKumamotoUniversityandProf.SakataofKyushuUniversity.Both authors have published several papers about one-sided tests in statistical journals such as the Journal of Multivariate Analysis and the Journal of Institute of StatisticalMathematics.Theydevelopedaone-sidedtestformeanmatrixofmatrix normal distributions and constructed a class of similar test statistics that includes similartests previouslyobtainedbySasabuchi,Hu,andBanerjee.Theseworksare positioned as the starting point to one-sided tests for the mean tensor of a tensor normal distribution. All the chapters in this book distinctly develop interesting application fields for matrix and tensor data. Finally, the authors would like to express their sincere appreciation to Prof. Akimichi Takemura for his encouragment to write this book, and to Japan Statistical Society for giving us this opportunity to publish it. Contents 1 Three-Way Principal Component Analysis with Its Applications to Psychology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Kohei Adachi 2 Non-negative Matrix Factorization and Its Variants for Audio Signal Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Hirokazu Kameoka 3 Generalized Tensor PCA and Its Applications to Image Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Kohei Inoue 4 Matrix Factorization for Image Processing . . . . . . . . . . . . . . . . . . . 73 Noboru Murata 5 Array Normal Model and Incomplete Array Variate Observations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Deniz Akdemir 6 One-Sided Tests for Matrix Variate Normal Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Manabu Iwasa and Toshio Sakata ix Contributors Kohei Adachi Graduate School of Human Sciences, Osaka University, Suita, Osaka, Japan Deniz Akdemir Department of Plant Breeding and Genetics, Cornell University, Ithaca, NY, USA Kohei Inoue Kyushu University, Minami-ku, Fukuoka, Japan Manabu Iwasa Kumamoto University, Kumamoto, Japan Hirokazu Kameoka The University of Tokyo, Tokyo, Japan; Nippon Telegraph and Telephone Corporation, Atsugi, Kanagawa, Japan Noboru Murata Faculty of Science and Engineering, Waseda University, Shinjuku-ku, Tokyo, Japan Toshio Sakata Department of Human Science of Kyushu University, Fukuka, Japan xi