This book by the Professor of Mathematical Statistics in the University of Manchester is the first on the theory of stochastic processes to be published in this country. Apart from a number of specialized mono-graphs on the subject of stochastic processes which have appeared in recent years, particularly in Sweden and the U.S.S.R., the only general works available hitherto have been the books by Professor Paul Levy and Professor J Doob. Both of these latter are highly mathematical in nature and neither can be said to satisfy the need of the applied mathematician and statistician for a general introduction at a not too abstract mathematical level to the mathematical methods and statistical techniques in this important new branch of the theory of probability and statistics. ... Statisticians will be grateful to Professor Bartlett for this book. The subject of stochastic processes is in any case not an easy one, because the mathematical difficulties to which the techniques lead are in so many cases formidable. It follows that the whole subject bristles with heaven-sent opportunities for the pure mathematician to get his teeth into meaty problems. It also follows that Professor Bartlett's allotted task of providing a useful and readable exposition of the subject for statisticians and other potential users of these techniques was itself one of considerable difficulty. He has succeeded in doing just that.
Classical statistics is essentially a study of the stochastic aspects of static situations, with order in time or space largely irrelevant. This new branch of statistics, on the other hand, deals with situations which are changing in time and space, where the nature of the picture at any instant depends, for example, on what has gone before. Over recent years it has been developed in an ah hoc way by physicists studying statistical mechanics and the motion of fluids in turbulence; by mathematicians in games of chance; by electrical engineers in the communication of signals, the study of noise, and so on; and by statisticians studying population growth with births and deaths, sequential sampling, queues and renewals, and the correlation of time series. Professor Bartlett's book meets an important need in presenting a general treatment of the subject, showing the essential unity of the mathematical concepts and methods which pervade all these applications.
This book is the best which has yet been produced on the specialised topic. It is lucidly written by someone who clearly sees the statistical implications of the abstract theory and may be read without undue difficulty by any student who has two years of university mathematical training behind him. It will undoubtedly be found indispensable by anyone wishing to learn something of a subject which has engaged the attention of nearly all probabilists in the past decade.
This is not an easy book to review. It is a depository of a large amount of information about techniques of mathematical and statistical analysis, and its value can be properly appreciated only when a variety of practical problems have been investigated with its aid. Mere reading of the book, however painstaking and careful, cannot produce more than a rough assessment of its potential value. However, it is clear that Professor Bartlett has brought together in a compact and useful form an account of the recent advances in the theory and analysis of stochastic processes, which would otherwise be found only in scattered books and journals. To this accomplishment, he has added an illuminating commentary on the relation between apparently dissimilar approaches to the same problem.
This book, based on the author's lecture notes at the University of North Carolina, is the first of a proposed three-volume work on the theory and application of stochastic processes. It is an introductory work addressed to the applied mathematician and statistician and presents the elementary methods and statistical techniques involved in stochastic processes. ... Although Bartlett does not always conform to the commonly accepted notation of the theory, his volume is a model of clarity and organization. On the whole this book is to be highly recommended for the applied mathematician and statistician who like a sound but not too abstract treatment of the theory of stochastic processes. For research workers in the natural, physical, and social sciences, who have a strong mathematical background, this book provides a means of becoming acquainted with some applications of the theory to their respective fields.
This reviewer can only report that he found that he could read substantial parts of the book only with the greatest difficulty, and other substantial parts not at all. The fact that the reviewer was familiar with the subject matter often did not seem to help much; even some definitions were difficult to understand. Several distinguished probabilists have reported the same experience to the reviewer. It seems reason- able to conclude that this is not a book mathematicians can profitably use to obtain an understanding of the theory of stochastic processes. Perhaps applied mathematicians and statisticians, to whom the book is addressed, will think differently. The reviewer has often noticed divergences of opinion between the two groups as to what is clear and easy to understand.
A few years ago there was a very great need for general text-books on stochastic processes, as the literature of this extensive field was scattered throughout many scientific journals and specialized monographs. This situation has improved greatly thanks to the comprehensive treatises by Doob (Stochastic Processes, New York 1955) and by Blanc-Lapierre and Fortet (Théorie des fonctions aleatoires, Paris 1953), and now Bartlett's book comes as a valuable complement. Doob's treatise is exclusively theoretical, while Bartlett concentrates on applied work and statistical aspects, considering all sorts of applications whereas Blanc-Lapierre and Fortet specialize on physics. The title "Introduction" must not be taken to indicate an elementary text. The exposition is rather a broad yet penetrating orientation which displays the many different types of process, the richness of their applications, the diversity of mathematical methods in play, and the rapid development of theoretical and applied work. The emphasis on applications is a most attractive feature which reflects the insight and realism that the author has acquired by his numerous research contributions to various types of process. The treatment in a limited space has required a concentrated style, and the reader must have a working knowledge of calculus and applied statistics.
The question which recurred often to this reviewer while going through Bartlett's book is, "An introduction for whom?" According to the preface, the volume is addressed to applied mathematicians and statisticians, emphasis being on "methods and applications." The author's plan necessitates omission of much rigour and detail. A treatment of this nature could serve as a useful introduction to the subject for many readers (applied mathematicians and statisticians, as well as others) who want to obtain some idea of the applicability of the theory without spending time on a full mathematical treatment 'a la Doob'; however, it seems doubtful that the present volume will prove to be such a useful introduction for many of these readers. A main criticism of the book is that omission of some rigour and detail should not necessitate hazy (or complete omission of) definitions of words used technically in the text, and which make whole sections difficult (if not impossible) to decipher. ... Other aspects of the author's style are also disturbing or fatiguing.
Professor Bartlett has managed to incorporate a wide range of theory and applications in a single book of moderate size. Anyone with the requisite mathematical knowledge can gain an entry to the subject by reading the book as a whole. Others, already grounded in the fundamentals, may prefer to concentrate on particular chapters dealing with aspects of special interest to them. Owing to the large number of topics brought together for discussion much of the mathematical analysis is rather abbreviated in form, but as the book is written primarily for applied mathematicians and statisticians this hardly constitutes serious criticism. There can be no doubt that the book is a substantial contribution to the subject.
In this period of rapid expansion the field of stochastic processes naturally seems interesting and attractive but threateningly difficult to many people outside of it, so that expository treatments are eagerly sought and awaited. The book under review is, as its title indicates, just such an exposition, and it is entirely without precedent in completeness and variety. Today, anyone who wants to learn the subject (unless possibly his interest in it is exclusively that of the pure mathematician) will have to turn to this book both early and late in his study. Anyone who has a specific problem in the application of stochastic processes will turn to it for specific ideas and references ... the coverage of the book, except for certain mathematical areas at which it is not at all directed, is remarkable. The difficult problem of organization is met with thought and flexibility. High accuracy is maintained. In two respects, though, I am disappointed in the book. First, I had hoped to find more emphasis on the qualitative behaviour of stochastic processes; and, second, the writing seems trying and difficult.
It has long been hoped that Professor Bartlett would write a comprehensive account of stochastic processes, a subject to which he has made notable contributions in recent years. Unlike the treatise by J L Doob, the only other comprehensive work on this subject in English, this book is addressed to those more interested in practical applications of the theory than in its underlying mathematics. However, unless the reader already has specialized knowledge of at least some parts of this subject and its mathematical techniques he will find this introductory account difficult.
All statisticians, together with the present and future generations of statistical students, will welcome the reappearance of Professor M S Bartlett's well-known book as a substantially bound paper-back in the Cambridge University Press series.
The distinctive flavour of the first edition is retained. It is the combination of originality, generality, economy of argument and avoidance of mathematical detail that account for the book's attractions and difficulty. When the first edition appeared in 1955 there were, I think, no other books dealing with this material. Now there are a number of textbooks at all levels about the analytical solution of stochastic models, i.e. broadly Chapters 1-7. There is still much of interest on these topics in Professor Bartlett's book.
The first edition of Professor Bartlett's by now classic book appeared in 1955 and was reviewed in this Journal in the same year. The popularity of that book is evidenced by the fact that four reprints were produced during the period 1956-62. A paperback version appeared in 1960 and. was reviewed in this Journal in the subsequent year.
How has Bartlett's book fared in the eleven years? Is it too to be cast aside from the main stream of development? The answer is emphatically, No. The passage of time has only served to emphasize the value of this pioneering work: it has also served to make some parts of it more intelligible than they originally were, but it is still not easy reading. ... the rewards from perseverance are great and we can look forward to another eleven years' run of this important book.
The first edition of this book appeared in 1955. It was generally well received, except for the complaint voiced by some that it was too condensed for relatively easy reading. This complaint is apt to remain valid for the second edition, since the difference between the new and the old edition lies essentially in the addition of some new material.
This is the third edition, and the first paperback edition, of a book which originally appeared in 1955. At that time it was one of the pioneering text books in stochastic processes and was particularly notable for its bias towards applications, and the breadth and insight shown by the author. It is at least as true today that it lies towards the applied end of the spectrum, the subject of stochastic processes having become increasingly sophisticated and mathematical over the intervening years. ... this is not a book for mathematical purists but a useful source of applicable results and a reminder of the practical motivation of the subject.
... this is an unevenly patched version of the first edition. Those of us who cut our teeth on that first edition cannot help but regret that the third edition does not reflect the growth of the field in the last 25 years.
The author shows a healthy interest in finding practical realisations of the theoretical models; his own work on the critical population size which determines whether measles can maintain itself in a community without re-introduction from outside is of particular importance.
This book is an extension and summary of current work in the application of ecological models in which variance terms are explicitly retained. It is a difficult but valuable book. It presents the first explicit ecological discussion of Monte Carlo methods and the first interesting use of an electronic computer for model evaluation. Perhaps its prime value will be that statisticians or mathematicians who become interested in ecological problems (and for some fortunate and obscure reason the number of these people is increasing) can find here a clearly defined class of problems that is of biological interest and also of sufficient difficulty not to offend mathematical taste. To have such a book is a great advance.
This little volume would seem to be of greatest interest to workers in biometrics and mathematical biology; however, it is a good introduction to the techniques of stochastic model building in biology for anyone interested in applied probability theory. It should also serve as an introduction to works giving a more detailed treatment of similar topics; for example, the author's book (An Introduction to Stochastic Processes, 1955), the book of N T J Bailey (The Mathematical Theory of Epidemics, 1957), and the book of the reviewer (Elements of the Theory of Markov Processes and Their Applications, 1960). Bartlett's monograph should also be of interest to econometricians and mathematical economists concerned with the construction of models for economic planning and development, for models of population growth and ecology are important components of any general planning scheme.
This monograph would seem to be of greatest interest to students and research workers in biometrics and mathematical biology; however, it is a good introduction to the techniques of stochastic model building in biology for anyone interested in applied probability theory. The book is not suitable for self-study, unless the reader has some knowledge of the elements of the theory of stochastic processes and some acquaintance with the formulation of stochastic models. It would, however, be suitable for a lecture course or seminar devoted to stochastic population models. It should also serve as an introduction, and supplement, to works giving a more detailed treatment of similar topics; for example, the author's earlier book (An Introduction to Stochastic Processes, 1955).
This monograph is one of a series on applied probability and statistics devoted to recent developments in these areas. One must surely applaud this step since short accounts that can give fuller treatment than journal articles provide surely serve a vital function. Unfortunately, this contribution is unlikely to find acceptance either with mathematicians or biologists for somewhat different reasons, but essentially because the book gives every evidence of hasty writing.
Mathematically-sophisticated population ecologists will welcome the appearance in the Methuen Monograph series of a book that is completely theirs. Other ecologists, including the reviewer, to whom mathematics at the level of Fourier transforms is a bit obscure, will still find that the book is partially theirs. These latter ecologists can take heart from the reviewer's personal guarantee that at least some of the ecologists, whose contributions are quoted in this book, will likewise have no idea what Bartlett is saying about them mathematically. This is in no way a criticism of Bartlett: it is simply additional evidence, if such is needed, that in biology as in the physical sciences we must place more stress on mathematical education, to equip ourselves more adequately to handle the most powerful single tool in science.
... this is an excellent book, either for readers requiring an introduction to the application of stochastic models to certain population phenomena, or for those who wish to revise their knowledge of the specialized topics discussed.
Professor Bartlett's book ... represents precisely what is understood by the term 'applied probability.' By this is meant a healthy interplay between theoretical development and experimental investigation, the latter suggesting further modifications to the theory and so on. ... this book is the work of a scientist rather than of one over concerned with mathematical virtuosity. In these days when there are so many people who believe that statistics and probability theory are concerned solely with the proof of mathematical theorems, this book should serve as a tonic.
See J O Irwin, Review: Essays in Probability and Statistics by M S Bartlett, Journal of the Royal Statistical Society. Series A (General) 125 (3) (1962), 484-487 for a detailed review of these essays.
R Coleman, Review: Probability, Statistics and Time. A Collection of Essays by M S Bartlett, Journal of the Royal Statistical Society. Series A (General) 140 (1) (1977), 105. Rodney Coleman writes:-
My overwhelming first impression on reading these essays was of the vastness of Professor Bartlett's knowledge. Drawing on an enormous amount of reading and study, he writes about statistics, probability and stochastic processes, and their role in physics, biology, genetics and medicine.
The monograph extends and updates the author's earlier review of spatial processes, presented to a stochastic processes conference at Sheffield in 1972, and includes several additional examples. It is written in two parts (Theory and Examples) and the author expresses the hope that this division will not lead the mathematically inclined to ignore Part II or the applied brethren to skip Part I. Both would lose a lot by such an approach since the applications build upon the theoretical developments in a systematic manner.
This book of less than one hundred small-format pages offers a telescoped but well-arranged review of selected stochastic process models for the analysis of spatial pattern. The writing style frequently carries the informality of a lecture and is never offensive. Unlike Bartlett's widely read earlier book, 'Stochastic processes', the present volume is clearly meant not as a formal comprehensive treatise, but rather as a digest.
This is not a definitive text for a student, but rather a monograph for a researcher, providing a useful summary of some recent and continuing work.
With a few notable exceptions, biologists have tended to avoid anything but the most trivial uses of mathematics, partly because of a lack of mathematical training, partly because of a feeling that the complexities of living organisms cannot be reduced to a few simple equations. This may be strictly true, yet even if mathematical models are only approximate and highly simplified, they can often be revealing and helpful. This is now being much more appreciated. It is a sign of the times that a department which began as a "Lectureship in the Design of Experiment" has changed first to a "Unit of Biometry", and now a "Department of Biomathematics", with M S Bartlett as professor. In his inaugural lecture Professor Bartlett deals with mathematical work in ecology (models of competition and predation between species), epidemiology, population genetics (the spread of genes in a population as a result of mutation, selection and random fluctuations) and information theory (as in the untangling of the genetic code-only briefly referred to). Professor Bartlett has done some of his most important work in the study of stochastic processes in the spread of epidemics. He mentions that this shows the existence of a "threshold effect": the epidemic will only spread in a population above a certain critical size, of about a quarter of a million for measles. This has been observationally confirmed. The stochastic processes in population genetics present even more difficult mathematical problems than those in epidemiology. But progress is being made, steadily and gradually. Professor Bartlett mentions, for example, the problem of survival of a mutant gene, considered in essence first by Francis Galton but still of considerable relevance. He also has something to say about the difficult and controversial idea of "genetic load". In short, this lecture clearly shows the importance of biomathematics as a young discipline. It would be interesting to know what aspects will be most exciting in 30 years' time.
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