Inequality-Constrained Multivariate Smoothing Splines with Application to the Estimation of Posterior Probabilities

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Abstract
We consider the problem of estimating a smooth function of several variables given discrete, scattered, noisy observations of its values, and given that it satisfies a family of linear inequality constraints. Constraints such as positivity over some region are included. The model is z i = f(y 1(i), …, yd + ε (i = 1, …, n), where the ε i 's are independent zero mean random variables, f is assumed to be smooth, with smoothness defined in terms of square integrability of certain derivatives, and f is known to satisfy a given set of linear inequality constraints. It is proposed that f be estimated as the minimizer, in an appropriate function space, of subject to f satisfying the constraints if they are a finite family, or satisfying a finite approximating set of constraints if they are not a finite family, where J m(f) is the thin-plate penalty functional defined by J m(f) = Σα1 + … + αd = m (m!/α1! … αd!) ∫ … ∫ [δ m f/δyα 11 … δ yα d d]2 dy 1 … dy d. More generally, the results apply to the model where the Li 's are bounded linear functionals on an appropriate Hilbert space, and the inequality constraints are of the form with the Nj also bounded linear functionals. Data and constraints involving function values, integrals, and some derivatives can be included in (2) and (3). Following Kimeldorf and Wahba (1971), we characterize the minimizer of (1) when the constraints are finite in number, and we provide a representation for the minimizer that can be computed using available nonlinear programming algorithms. When the data as well as the constraints involve function values, the minimizer is shown to be a thin-plate spline. The method of generalized cross-validation for constrained problems can be used here for estimating a good value of the smoothing parameter λ, and an efficient computational algorithm for it is developed in this article. As an example the method is applied to the estimation of posterior probabilities in the classification problem. Numerical results and figures for both synthetic and experimental data are given.

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