Spline Models for Observational Data.

TLDR

Spline models provide a theoretical framework for estimating smooth functions from noisy observations across diverse data distributions such as Gaussian, Poisson, and binomial. The book aims to develop theory and practice for estimating functions from noisy functional data, investigate optimal experimental design, and extend methods to distributed parameter system identification. It presents spline‑based estimation techniques that incorporate side conditions and prior information to solve ill‑posed inverse problems, while addressing experimental design and extensions to distributed parameter systems. The authors establish convergence properties, data‑driven smoothing parameter selection, confidence intervals, and efficient numerical methods for these spline estimation problems.

Abstract

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.