SPRINGER BRIEFS IN STATISTICS JSS RESEARCH SERIES IN STATISTICS Masanori Sawa Masatake Hirao Sanpei Kageyama Euclidean Design Theory 123 SpringerBriefs in Statistics JSS Research Series in Statistics Editors-in-Chief Naoto Kunitomo, Graduate School of Economics, Meiji University, Bunkyo-ku, Tokyo, Japan Akimichi Takemura, The Center for Data Science Education and Research, Shiga University, Bunkyo-ku, Tokyo, Japan Series Editors Genshiro Kitagawa, Meiji Institute for Advanced Study of Mathematical Sciences, Nakano-ku, Tokyo, Japan Tomoyuki Higuchi, The Institute of Statistical Mathematics, Tachikawa, Tokyo, Japan Toshimitsu Hamasaki, Office of Biostatistics and Data Management, National Cerebral and Cardiovascular Center, Suita, Osaka, Japan Shigeyuki Matsui, Graduate School of Medicine, Nagoya University, Nagoya, Aichi, Japan Manabu Iwasaki, School of Data Science, Yokohama City University, Yokohama, Tokyo, Japan Yasuhiro Omori, Graduate School of Economics, The University of Tokyo, Bunkyo-ku, Tokyo, Japan Masafumi Akahira, Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki, Japan Takahiro Hoshino, Department of Economics, Keio University, Tokyo, Japan Masanobu Taniguchi, Department of Mathematical Sciences/School, Waseda University/Science & Engineering, Shinjuku-ku, Japan 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, educa- tion, and many other natural, biological, and social sciences. The JSS Series of Statisticsaimstopublishrecent results ofcurrentresearchactivities intheareas 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 andJapanese.Becausethescopeofaresearchpaperinacademicjournalsinevitably hasbecomenarrowlyfocusedandcondensedinrecentyears,thisseriesisintended 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 countrieswhoareinterestedinstatisticsandstatisticalsciences,instatisticaltheory, and in various areas of statistical applications. More information about this series at http://www.springer.com/series/13497 Masanori Sawa Masatake Hirao (cid:129) (cid:129) Sanpei Kageyama Euclidean Design Theory 123 Masanori Sawa Masatake Hirao Graduate Schoolof System Informatics Schoolof Information andScience Kobe University Technology Kobe,Hyogo,Japan AichiPrefectural University Nagakute,Aichi, Japan SanpeiKageyama Research Centerfor Mathematics andScience Education Tokyo University of Science Tokyo,Japan Hiroshima University Hiroshima, Japan ISSN 2191-544X ISSN 2191-5458 (electronic) SpringerBriefs inStatistics ISSN 2364-0057 ISSN 2364-0065 (electronic) JSSResearch Series in Statistics ISBN978-981-13-8074-7 ISBN978-981-13-8075-4 (eBook) https://doi.org/10.1007/978-981-13-8075-4 ©TheAuthor(s),underexclusivelicensetoSpringerNatureSingaporePteLtd.2019 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseof illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. 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The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Preface Research in the area of discrete optimal designs has been steadily and rapidly growing,especiallyduringpastseveraldecades.Thenumberofpublicationsavail- ableinthisareaisinseveralhundreds.Theoptimalityproblemshavebeenformulated in various models arising in the experimental designs and substantial progress has been made toward solving these problems. In the meantime, the theory of mainly continuousoptimaldesignshasbeenreviewedandreorganizedinacomprehensive surveybyPukelsheim’sbookOptimalDesignofExperiments(1993;JohnWiley)of over400pages,whichalsotriestogiveaunifiedoptimality theoryofembracinga widevarietyofdesignproblems.FurtherdevelopmentscanbefoundinabookTopics inOptimalDesign(2002;Springer)byLiski,Mandal,Shah,andSinhawhocovera widerangeoftopicsinbothdiscreteandcontinuousoptimaldesigns. We want to deal with the construction of optimal experimental designs pure-mathematically and systematically under a new framework. The aim of the present book is to show the modern first treatment of experimental designs for giving a comprehensive introduction to the interrelationship of the theory of opti- mal designs with the theory of cubature formulas in numerical analysis. It also provides the reader with original new ideas for constructing optimal designs, though this is not a full-length treatment of the subject. ThebookopenswiththebasicsonreproducingkernelHilbertspace,andbuilds up to more advanced topics including, bounds for the number of cubature formula points, equivalence theorems for statistical optimalities, and the Sobolev theorem for the cubature formula. It ends with a generalization for the abovementioned classical results in a functional analytic manner. Since papers on optimal designs are published in a variety of journals, and because of the extensive role of these designs in design of experiments and other areas we believe it is imperative to gather these results and present them in varied form to suit diverse interests. This book is an instance of such an attempt. AstheContentsshow,thematerialiscoveredinfivechapters.Chapter1provides a brief summary of basic ideas and facts concerning kernel functions which are closelyrelatedtothetheoriesofcubatureformulainnumericalanalysisaswellasof Euclidean design which is a special point configuration in the Euclidean space. v vi Preface Chapter2istoshowthatcubatureformulascanbeusedforfindingoptimaldesignsof experimentasastatisticalapplication.Inthissense,therelationshipbetweencubature formulas and Euclidean designs is discussed. Chapter 3 is devoted to organically combining optimal experimental designs and Euclidean designs in algebraic com- binatorics.Chapter4discussesanddescribestwoadvancedmethodsofconstructing optimalEuclideandesigns,onebasedonorbitsofreflectiongroupsandanotherbased oncombinatorial orstatistical subjects suchascombinatorialt-designs andorthog- onalarrays.Theclimaxofthisbook,Chap.5,introducestheconceptofgeneralized cubature formula and lays the foundation of Euclidean Design Theory, which not only produces a novel framework for understanding optimal designs, based on the theoryofcubatureformulasinanalysisandsphericalorEuclideandesignsincom- binatorics,butalsofindssomeapplicationstodesignofexperiments. Thisbookisespeciallyintendedforreaderswhoareinterestedinrecentadvances in the construction theory of cubature formulas, Euclidean designs, and optimal experimental designs. Moreover, it is recommended to research workers who seek rich interactions between optimal experimental designs and various mathematical subjects including spherical design theory in combinatorics, embedding theory of Banach spaces in functional analysis, and cubature theory in numerical analysis. Anovelcommunicatingplatformisfinallyprovidedfor“designtheorists”inawide varietyofresearchfields.Ofcourse,sincewearealsoaimingatanaudiencewitha wide range of backgrounds, including postgraduate students in statistics or combi- natorics orboth,we have assumeda reasonable knowledge of linear algebra,anal- ysis, finite field theory, but very little else. Number theory and some essential statisticaloranalyticalconceptsaredevelopedasneeded.Ifyouwouldglanceattitles inthereferences,youmightconceivehowtodealwithtopicsontheseproblems. Itishopedthatthebookwillalsobeusefulasasecondaryandprimaryreference for statisticians and mathematicians doing research on the design of experiments, and also for the experimenters in diverse fields of applications. Acknowledgments Masanori Sawa sincerely appreciates his wife Ikumi and three children, Kojiro,Kenji,andShiori,whohavebeenpatientlywatchinghimindulginginwritingthepresent bookduringholidaysorevenvacations.HewouldalsoliketothankhismotherKazuko,hislate fatherSeiji,andthefamilyofuncle,Koji,Sumiko,Takujifortheirwarmsupportssofar.Masatake Hirao is eagerly thankful to his family for their support and encouragement, especially his wife Tomoko and wonderful child Rintaro. Finally, we would like to thank Prof. M. Iwasaki, YokohamaCityUniversity,whoisoneofSerieseditors,JSSResearchSeriesinStatistics,forhis warmencouragementsofpreparingourdraftfortheSeries. Kobe, Japan Masanori Sawa Nagakute, Japan Masatake Hirao Hiroshima, Japan Sanpei Kageyama May 2019 Contents 1 Kernel Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Kernels and Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Kernels and Compact Formulas. . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Kernels and Dimension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Further Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 Cubature Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.1 Cubature Formula and Elementary Construction Methods . . . . . . . 20 2.2 Existence Theorems and Lower Bounds. . . . . . . . . . . . . . . . . . . . 23 2.3 Euclidean Design and Spherical Symmetry . . . . . . . . . . . . . . . . . 31 2.4 Further Remarks and Open Questions . . . . . . . . . . . . . . . . . . . . . 38 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3 Optimal Euclidean Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.1 Regression and Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 Optimal Euclidean Design and Characterization Theorems . . . . . . 51 3.3 Realization of the Kiefer Characterization Theorem . . . . . . . . . . . 55 3.4 Further Remarks and Open Questions . . . . . . . . . . . . . . . . . . . . . 58 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4 Constructions of Optimal Euclidean Design . . . . . . . . . . . . . . . . . . . 63 4.1 Finite Reflection Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.1.1 Group Ad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.1.2 Group Bd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.1.3 Group D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 d 4.2 Invariant Polynomial and the Sobolev Theorem . . . . . . . . . . . . . . 73 4.2.1 Group A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 d 4.2.2 Group Bd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.2.3 Group Dd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 vii viii Contents 4.3 Corner Vector Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.3.1 Group Ad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.3.2 Group Bd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.3.3 Group Dd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.4 Combinatorial Thinning Methods . . . . . . . . . . . . . . . . . . . . . . . . 94 4.5 Further Remarks and Open Questions . . . . . . . . . . . . . . . . . . . . . 98 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5 Euclidean Design Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.1 Generalized Cubature Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.2 Generalized Tchakaloff Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.3 Generalized Fisher Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.4 Generalized LP Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.5 Generalized Sobolev Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.6 Conclusion and Further Implications . . . . . . . . . . . . . . . . . . . . . . 121 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 131 Chapter 1 Kernel Functions The present chapter provides a brief summary of basic ideas and facts concerning kernelfunctions,whicharecloselyrelatedtothetheoriesofcubatureformulabeing acertainclassofintegrationformulasinnumericalanalysis,aswellasofEuclidean design,whichisaspecialpointconfigurationintheEuclideanspace. ThefirstthreesectionsreviewsomegeneralelementaryfactssuchastheAronszajn theoremandtheRieszrepresentationtheorem,afterwhichtheemphasisgradually shifts to technical details, including explicit computations of kernel functions and dimensions of finite-dimensional normed vector spaces. The final section covers related topics such as the Seidel’nikov inequality in discrete geometry [14] and a novelconnectionbetweenkernelmachinesandcubatureformulas[5]. Theaimofthischapteristomatchthebreadthoftheoryofkernelfunctionsand that of cubature formulas. Each section includes results about cubature formulas without detailed explanations on terminologies; such details will be explained in the subsequent chapters. The prerequisite for reading this chapter is a knowledge of functional analysis at the standard undergraduate level, but the reader who has not learned it can still read this chapter by accepting some advanced materials in sphericalharmonicsandorthogonalpolynomials. Most of the materials concerning kernel functions, which appear in this chap- ter without proofs, can be found in [13]. For an extensive treatment of spherical harmonicsandorthogonalpolynomials,wereferthereaderto[16, 17]. 1.1 KernelsandPolynomials At first, the definition of kernel function is described. Throughout let Ω be a set, usually,ofrealvectors,ormoregenerallyatopologicalspace. Definition1.1 (Kernel function) A function K :Ω ×Ω →R is called a kernel (function)onΩ if ©TheAuthor(s),underexclusivelicensetoSpringerNatureSingaporePteLtd.2019 1 M.Sawaetal.,EuclideanDesignTheory,JSSResearchSeriesinStatistics, https://doi.org/10.1007/978-981-13-8075-4_1