Bard Y. Nonlinear Parameter Estimation

New York: Academic Press, 1974. — 341 p.This book is intended primarily for use by the scientist or engineer who is concerned with fitting mathematical models to numerical data, and for use in courses on data analysis which deal with that subject. Such fitting is frequently done by the method of least squares, with no regard paid to previous knowledge concerning the values of the parameters (coefficients), nor to the statistical nature of the measurement errors.In Chapters II — IV we show how the problem can be formulated so as to take all these factors into account. In Chapters V-VI we discuss the computational methods used to solve the problem, once its formulation has been completed. Chapter VII is devoted to the question of what conclusions can be drawn, after the estimates have been computed, concerning the validity of the estimates, or of the model which has been fitted. In Chapter VIII we discuss the important special case of models which are stated in the form of differential equations. Other special problems are treated in Chapter IX. Finally, in Chapter X we suggest methods for planning the experiments in such a way that the data will shed the greatest possible light on the model and its parameters. We cannot stress too strongly the point that if data are to be gathered for the purpose of establishing a mathematical model, then the experiments should be designed with this purpose in mind. Hence the importance of Chapter X.A practical, rather than theoretical point of view has been taken through-out this book. We describe computational algorithms which have performed well on a variety of problems, even if their convergence has not been proven, and even if they have failed on some other problems. We have as yet no foolproof, efficient methods for solving nonlinear problems; hence we cannot afford to throw away useful tools just because they are not perfect.
The presentation uses matrix algebra and probability theory on a very elementary level. Reviews of the needed concepts and proofs of some important theorems will be found in the appendixes. Some supplementary material has been included in the form of problems at the ends of chapters. Problems requiring actual computation have not been included; the reader is likely to have his own data to compute with, and additional data may be found in many of the cited references. Several numerical problems have, however, been worked out in great detail in separate sections at the ends of Chapters V-IX for the purpose of illustrating (he methods discussed in those chapters.