Multi-Variate Splines with Non-Degenerate Partitions.
Scientific interim rept.,
WASHINGTON UNIV ST LOUIS MO CONTROL SYSTEMS SCIENCE AND ENGINEERING
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Any set of hyperplanes partitions E sup N into a set of polyhedra. A multivariate spline of degree n is a polynomial of total degree n on each polyhedron with all partial derivatives of order n-1 being continuous everywhere. An especially simple canonical form is presented for splines with respect to nondegenerate if a set of hyperplanes has nonempty intersection then the corresponding set of normal vectors is linearly independent partitions. Use of the canonical form, for fitting data, involves linear regression for fixed partitions and nonlinear regression for varying partitions. The canonical form gives rise to an ill-condition linear regression problem. However, some preliminary numerical experience in low dimensions indicates that the ill-conditioning is overcome with the use of singular value decomposition. Author
- Statistics and Probability