Pass/Fail Determination Using Examination Scores
In this chapter, we examine the k-fold cross-validation for small sample method (Method 1) by combining the resampling technique with k-fold cross-validation. By this breakthrough, we obtain the error rate means, M1 and M2, for the training and validation
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New Theory of Discriminant Analysis After R. Fisher Advanced Research by the Feature-Selection Method for Microarray Data
New Theory of Discriminant Analysis After R. Fisher
Shuichi Shinmura
New Theory of Discriminant Analysis After R. Fisher Advanced Research by the Feature-Selection Method for Microarray Data
123
Shuichi Shinmura Faculty of Economics Seikei University Musashinoshi, Tokyo Japan
ISBN 978-981-10-2163-3 DOI 10.1007/978-981-10-2164-0
ISBN 978-981-10-2164-0
(eBook)
Library of Congress Control Number: 2016947390 © Springer Science+Business Media Singapore 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part 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 dissimilar methodology now known or hereafter developed. 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 from the relevant protective laws and regulations and therefore free for general use. 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 herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer Science+Business Media Singapore Pte Ltd.
Preface
This book introduces the new theory of discriminant analysis based on mathematical programming (MP)-based optimal linear discriminant functions (OLDFs) (hereafter, “the Theory”) after R. Fisher. There are five serious problems of discriminant analysis in Sect. 1.1.2. I develop five OLDFs in Sect. 1.3. An OLDF based on a minimum number of misclassification (minimum NM, MNM) criterion using integer programing (IP-OLDF) reveals four relevant facts in Sect. 1.3.3. IP-OLDF tells us the relation between NM and LDF clearly in addition to a monotonic decrease of MNM. IP-OLDF and an OLDF using linear programing (LP-OLDF) are compared with Fisher’s LDF and a quadratic discriminant function (QDF) using Iris data in Chap. 2 and cephalo-pelvic disproportion (CPD) data in Chap. 3. However, because IP-OLDF may not find a true MNM if data do not satisfy the general position revealed by student data in Chap. 4 (Problem 1), I develop Revised IP-OLDF, Revised LP-OLDF, and Revised IPLP-OLDF that is a mixture model of Revised LP-OLDF and Revised IP-OLDF. Only Revised IP-OLDF can find true MNM corresponding to an interior point of optimal convex polyhedron (optimal CP, OCP) defined on the discriminant coefficient s
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