This Article 
 Bibliographic References 
 Add to: 
Multivariate Classification through Adaptive Delaunay-Based C0 Spline Approximation
April 1995 (vol. 17 no. 4)
pp. 403-417

Abstract—This paper introduces a new method for adaptively building a multivariate C0 spline approximation from scattered samples of an unknown function. The central feature of the method is a means for adaptively tesselating an approximation space to form a multidimensional mesh over which the spline fitting then occurs. The mesh used is a Delaunay tesselation of the approximation space whose vertices lie at a subset of the scattered sample locations. The specific subset of sample locations used is adaptively determined by repeated overfitting and simplification of the resulting spline approximation.

Overfitting and simplification is an attractive paradigm for high-dimensional approximation problems because it provides a means for forming an approximation that is complex only in regions where the scattered sample data provide sufficient evidence of complexity in the underlying unknown function. Overfitting and simplification is effectively exploited in this new approach as the function representation used is not subject to certain recursive dependencies.

The properties of the new technique are demonstrated in the context of an easily visualized bivariate classification problem. The technique is then applied to a 10-dimensional clinical ECG classification problem, and the results are compared to those obtained with a perceptron based neural network.

[1] R.E. Bellman,Adaptive Control Processes: A Guided Tour, p. 94, Princeton Univ. Press, N.J., 1961.
[2] R.P. Lippmann, "An Introduction to Computing with Neural Nets," IEEE Acoustics, Speech, and Signal Processing Magazine, vol. 4, pp. 4-22, Apr. 1987.
[3] R.S. Scalero and N. Tepedelenlioglu,“A fast new algorithm for training feedforward neural networks,” IEEE Trans. Signal Processing, vol. 40, no. 1, pp. 202-210, Jan. 1992.
[4] E.D. Karnin,“A simple procedure for pruning back-propagation trained neural networks,” IEEE Trans. Neural Networks, vol. 1, no. 2, pp. 239-242, June 1990.
[5] S.E. Fahlman and C. Lebiere,“The cascade correlation learning architecture,” Carnegie Mellon Univ. technical report CMU-CS-90-100, Feb. 1990.
[6] J.H. Friedman,“Multivariate adaptive regression splines,” The Annals of Statistics, vol. 19, no. 1, pp. 1-141, 1991.
[7] D.F. Specht, “A General Regression Neural Network,” IEEE Trans. Neural Networks, vol. 2, pp. 568-576, Nov. 1991.
[8] S. Qian,Y.C. Lee,R.D. Jones,C.W. Barnes,, and K. Lee,“Function approximation with an orthogonal basis set,” IEEE INNS Int’l Joint Conf. Neural Networks, vol. 3, pp. 605-619, 1990.
[9] O.J. Murphy,“An information theoretic design and training algorithm for neural networks,” IEEE Trans. Circuits and Systems, vol. 38, no. 12, pp. 1542-1547, Dec. 1991.
[10] A. Sankar,Neural Tree Networks, Computational Methods of Signal Recoveryand Recognition, Richard J. Mammone, ed., Chap. 12, pp. 327-367,Wiley, New York, 1992.
[11] L. Breiman,J.H. Friedman,R.A. Olshen,, and C.J. Stone,Classification and Regression Trees, Wadsworth, Belmont, Calif., 1984.
[12] D.F. Watson,“Computing the N-dimensional Delaunay tesselation with application tovoronoi polytopes,” The Computer J., vol. 24, no. 2, pp. 167-172, 1981.
[13] D. Cubanski,“Ambulatory ECG arrhythmia classification using adaptive multivariatefunctional approximation,” PhD dissertation, Worcester Polytechnic Inst., Worcester, Mass., 1993.
[14] K.J. Bathe and T. Sussman,“The gradient of the finite element variational indicator with respect tonodal point coordinates,” Int’l J. Numerical Methods in Eng., vol. 17, pp. 1717-1734, 1981.
[15] D. Cubanski,D. Cyganski,E.M. Antman,, and C.L. Feldman,“A neural network system for automatic detection of atrial fibrillation,” Amer. Coll. Cardiology, 42nd Ann. Scientific Sess., JACC, vol. 21, no. 2,182A, Feb. 1993.

Index Terms:
Pattern classification, approximation, neural networks, non-linear optimization, splines, ECG.
David Cubanski, David Cyganski, "Multivariate Classification through Adaptive Delaunay-Based C0 Spline Approximation," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 17, no. 4, pp. 403-417, April 1995, doi:10.1109/34.385978
Usage of this product signifies your acceptance of the Terms of Use.