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Related titles. Carousel Previous Carousel Next. Kolmogorov, S. Jump to Page. Search inside document. Ripley This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press, First published Eighth printing Printed in the United Kingdom at the University Press, Cambridge A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication data Ripley, Brian D.
ISBN 0 1. Neural networks Computer science , 2. R56 The pattern recognition task. Overview of the remaining chapters. Parametric models. Logistic discrimination Predictive classification Alternative estimation procedures Performance assessment.
Computational learning approaches Shrinkage methods. Flexible Discriminants 4. Feed-forward Neural Networks 5. Belief Networks 8. Projection methods Statistical Sidelines A Maximum likelihood and MAP estimation. Hardware advances have made the concerns of pattern recognition of much wider applicability. There are many examples from everyday life Name the species of a flowering plant.
Grade bacon rashers from a visual image. Classify an X-ray image of a tumour as cancerous or benign. Decide to buy or sell a stock option. Give or refuse credit to a shopper. Neural networks have arisen from analogies with models of the way that humans might approach pattern recognition tasks, although they have developed a long way from the biological roots.
Great claims have been made for these procedures, and although few of these claims have withstood careful scrutiny, neural network methods have had great impact on pattern recognition practice.
A theoretical understanding of how they work is still under construction, and is attempted here by viewing neural networks within a statistical framework, together with methods developed in the field of machine learning. One of the aims of this book is to be a reference resource, so almost all the results used are proved and the remainder are given references to complete proofs. Another unusual feature of this book is that the methods are illustrated on examples, and those examples are either real ones or realistic abstractions.
Unlike the proofs, the examples are not optional! The formal pre-requisites to follow this book are rather few, espe- cially if no attempt is made to follow the proofs. A background in linear algebra is needed, including eigendecompositions. The singular value decomposition is used, but explained. A knowledge of calculus and its use in finding extrema such as local minima is needed, as well as the simplest notions of asymptotics Taylor series expansions and O n notation.
Graph theory is used in Chapter 8, but developed from scratch. Only a first course in probability and statistics is assumed, but considerable experience in manipulations will be needed to follow the derivations without writing out the intermediate steps. The glossary should help readers with non-technical backgrounds. A graduate-course knowledge of statistical concepts will be needed to appreciate fully the theoretical developments and proofs.
The sections on examples need a much less mathematical background; indeed a good overview of the state of the subject can be obtained by skimming the theoretical sections and concentrating on the examples. The theory and the insights it gives are important in understanding the relative merits of the methods, and it is often very much harder to show that an idea is unsound than to explain the idea. Several chapters have been used in graduate courses to statisticians and to engineers, computer scientists and physicists.
A core of material would be Sections 2. For example, statisticians should cover 2. Acknowledgements This book was originally planned as a joint work with Nils Lid Hjort University of Oslo , and his influence will be immediately apparent to those who have seen Hjort , a limited circulation report.
My own interest in neural networks was kindled by the invitation from Ole Barndorff-Nielsen and David Cox to give a short course at SemStat in , which resulted in Ripley I was introduced to the machine-learning literature and its distinctive goals by Donald Michie. I am grateful to Lionel Tarassenko and his co- authors for the cover picture of outlier detection in a mammogram fcom Tarassenko et al. Parts of this book have been used as source material for graduate lectures and seminar courses at Oxford, and I am grateful to my students and colleagues for feedback; present readers will appreciate the results of their insistence on more details in the mathematics.
The examples were computed within the statistical system S-Plus of MathSoft Inc, using software developed by the author and other contributors to the library of software for that system notably Trevor Hastie and Rob Tibshirani. It has been a pleasure to work with CUP staff on the design and production of this volume; especial thanks go to David Tranah, the editor for this project who also contributed many aspects of the design.
Random variables are usually denoted by capital letters; if X is a random variable then x denotes its value. E denotes expectation. A suffix denotes the random variable or distribution over which the averaging takes place. A is the indicator function of event A, one if A happens, otherwise zero.
Then the following holds. The error-reject curve plots pmc against pd for varying d. Proposition 2. Most of the rest of the theory presented here can be regarded as ways to estimate or approximate the posterior probabilities from the training set. Fukunaga calls it the Bayes error. If we disregard the doubt option a new feature vector x is allocated to the class k with smallest value of 5 x, 4?
If the classes are equally likely a priori then x is classified as coming from the nearest class, in the sense of having the smallest Mahalanobis distance to its mean. The error rate for the optimal rule can be computed explicitly in the two-class case. Note that the error rate is expressed in terms of the one- dimensional normal distribution even when the class distributions are p-dimensional normal. Then the optimal rule is to allocate to class if X The overall error rate is 0.
Suppose next that one can obtain two independent measurements X, and Xz from the object to be classified. How do the allocation rules and the error rates change?
The overall error rate has been reduced to 0. The inconveniences caused by a reject have to be judged against the consequences of a misclassification. In a serious application where the classifier is meant to work routinely on future examples one would typically try several d values on a training set of vectors with known classes, and obtain estimates of misclassification and doubt rates see Section 2.
Plotting misclassification rate against d is useful see Figure 3. This will lead to low error rates but on few classified vectors and a high doubt rate. There are no restrictions on the type of densities pi, The most popular special cases of the optimal rule are the normal distribution cases with common or different covariance matrices; see the example above and those discussed in Section 2.
Indeed, discriminant or classification analysis started with a sample version of 2. He derived the best linear rule in the two-class case but from a different perspective; see Section 3. However, these may be difficult to calculate. One technique is to simulate the missing features from p x x" and average p c x over the simulated values.
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Pattern recognition and neural networks I B. Pattern Recognition and Neural Networks bbmt. Pattern Recognition and Neural Networks. Pattern Recognition and Brian D. Ripley, University of Oxford.