view:25918 Last Update: 2023-2-6
Semi-reproducing kernel Hilbert spaces, splines and increment kriging
A reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions on an index set E, say, for which function evaluation is continuous in the Hilbert norm. One of the classic applications of such spaces is to show that the solution of a certain penalised least squares problem is given by a smoothing spline. However, the formulation of this problem in an RKHS setting involves several arbitrary choices. In this paper, we propose the use of a semi-RKHS (SRKHS), which provides a more natural setting for the smoothing spline solution. In addition, a systematic study is made of the properties of an SRKHS. It is well known that there is a one-to-one correspondence between an RKHS and positive semi-definite (psd) function on E, which in turn can be viewed as the covariance function of a stochastic process on E. In this paper, we extend this result to show that there is a one-to-one correspondence between an SRKHS and conditionally positive semi-definite (cpsd) function on E, which in turn defines the covariance behaviour of certain increments on E. Further, it is shown how optimal smoothing in the functional setting corresponds to optimal prediction in the stochastic process setting.