Spline Models for Observational Data

by Grace Wahba

This book serves well as an introduction into the more theoretical aspects of the use of spline models. It develops a theory and practice for the estimation of functions from noisy data on functionals. The simplest example is the estimation of a smooth curve, given noisy observations on a finite number of its values. Convergence properties, data based smoothing parameter selection, confidence intervals, and numerical methods are established which are appropriate to a number of problems within this framework. Methods for including side conditions and other prior information in solving ill posed inverse problems are provided. Data which involves samples of random variables with Gaussian, Poisson, binomial, and other distributions are treated in a unified optimization context. Experimental design questions, i.e., which functionals should be observed, are studied in a general context. Extensions to distributed parameter system identification problems are made by considering implicitly defined functionals.

Genres: Mathematics

180 pages, Paperback

First published September 1, 1990

Reviews

[Heather Terran · Rating 2 out of 5]

Wahba is brilliant, and this book is full of fantastic information. Unfortunately, as a brilliant mathematician, she is also incapable of communicating with mortals. This book is "concise", which in mathematician-ese means "cannot be comprehended by any except people who are already experts in the field". So, basically, if you already know what's in this book, then you can read it. Otherwise, it's an exercise in excruciation. Rewarding, if you can get through it, but agony in every step.

(And just to be clear, it's not like I'm a mathematical novice here -- I've been studying statistics at one level or another for more than a dozen years here. I have a PhD in machine learning, which means, essentially, computational statistics. I'm cool with Bayesian statistics, functional analysis, topology, etc. I read abstract algebra and analysis for fun. And this book is still like chewing on broken glass for me.)

This book covers a lot of good information about smoothing splines and reproducing kernel Hilbert spaces (RKHSs), including polynomial splines, thin plate splines, and Kriging estimators. It's all terribly useful stuff, and she does build them up from the ground.

Unfortunately, she commits essentially all of the classic sins of a lazy mathematician. She skips almost every step, she offers no explanations for most of the steps that she does bother to include, she doesn't motivate most of her development, she changes notation without warning, she fails to define terms, she gives nothing even vaguely resembling an overview or "big picture", she doesn't clearly state results and theorems, she implies that axioms are consequences and vice versa, she gives no examples or intuition, etc. etc. It's essentially a textbook on how not to write mathematics.

I would have given this book 1 star and filed it under "to be forcefully thrown", but it does have enough good information that I can't quite bring myself to do so. Still, if you can get this information from any other source, I would highly recommend that you do so. I hear that Gu has a good text in this area. That's on my to-read list...