Skew-radial basis function expansions for empirical modeling

SIAM Journal on Scientific Computing

Volumn 31, Issue 6, 2010, Pages 4715-4743

  Jamshidi, Arta A ^a   Kirby, Michael J ^a  

^a Department of Environmental and Radiological Health Sciences   (United States)

Author keywords

Data fitting; Data interpolation; Function approximation; Manifold learning; Radial basis function; Skew radial basis function

Indexed keywords

DATA REDUCTION; IMAGE PROCESSING; INTERPOLATION; MATRIX ALGEBRA; NUMBER THEORY; RADIAL BASIS FUNCTION NETWORKS;

BASE FUNCTION; DATA FITTINGS; DATA INTERPOLATION; FUNCTIONS APPROXIMATIONS; MANIFOLD LEARNING; RADIAL BASE FUNCTION; RADIAL BASIS; SKEW-RADIAL BASE FUNCTION;

FUNCTIONS;

EID: 78650590323     PISSN: 10648275     EISSN: 10957197     Source Type: Journal    
DOI: 10.1137/08072293X     Document Type: Article

References (69)

1. E. J. Kansa and Y. C. Hon, eds., Computers and Mathematics with Applications, Vol. 43, Elsevier, New York, 2002, pp. 275-619.
2. R. B. Arellano-Valle and M. G. Genton, On fundamental skew-symmetric distributions, J. Multivariate Anal., 96 (2005), pp. 93-116.
3. R. B. Arnold and R. J. Beaver, The skew-Cauchy distribution, Statist. Probab. Lett., 49 (2000), pp. 285-290.
4. F. Azam and H. F. VanLandingham, An alternate radial basis function neural network model, 2000 IEEE Internat. Conf. Syst., Man, and Cybernet., 4 (2000), pp. 2679-2684.
5. A. Azzalini, A class of distributions which includes the normal ones, Scand. J. Statist., 12 (1985), pp. 171-178.
6. A. Azzalini, Further results on a class of distributions which includes the normal ones, Statistica (Bologna), 46 (1986), pp. 199-208.
7. A. Azzalini and A. Capitanio, Statistical applications of the multivariate skew-normal distribution, J. Roy. Statist. Soc., 61 (1999), pp. 579-602.
8. A. Azzalini and A. Capitanio, Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t distribution, J. Roy. Statist. Soc., 65 (2003), pp. 367- 389.
9. A. Azzalini and A. Dalla Valle, The multivariate skew-normal distribution, Biometrika, 83 (1996), pp. 715-726.
10. J. Behboodian, A. Jamalizadeh, and N. Balakrishnan, A new class of skew-Cauchy distributions, Statist. Probab. Lett., 76 (2006), pp. 1488-1493.
11. R. E. Bellman, Dynamic Programming, Princeton University Press, Princeton, NJ, 1957.
12. R. Bhatia and K. R. Parthasarathy, Positive definite functions and operator inequalities, Bull. Lond. Math. Soc., 32 (2000), pp. 214-228.
13. M. Bozzini, L. Lenarduzzi, M. Rossini, and R. Schaback, Interpolation by basis functions of different scales and shapes, Calcolo, 41 (2004), pp. 77-87.
14. M. D. Brancoa and D. K. Dey, A general class of multivariate skew-elliptical distributions, J. Multivariate Anal., 79 (2001), pp. 99-113.
15. D. S. Broomhead and M. Kirby, A new approach to dimensionality reduction: Theory and algorithms, SIAM J. Appl. Math., 60 (2000), pp. 2114-2142.
16. D. S. Broomhead and M. Kirby, The Whitney reduction network: A method for computing autoassociative graphs, Neural Comput., 13 (2001), pp. 2595-2616.
17. D. S. Broomhead and D. Lowe, Multivariable functional interpolation and adaptive networks, Complex Systems, 2 (1988), pp. 321-355.
18. G. Bugmann, Normalized Gaussian radial basis function networks, Neurocomputing, 20 (1998), pp. 97-110.
19. M. D. Buhmann, Radial Basis Functions, Cambridge University Press, Cambridge, 2003.
20. M. Casdagli, Nonlilnear prediction of chaotic time series, Phys. D, 35 (1989), pp. 335-356.
21. M. Casdagli, D. D. Jardins, S. Eubank, J. D. Farmer, J. Gibson, N. Hunter, and J. Theiler, Nonlinear modeling of chaotic time series: Theory and applications, in Applied Chaos, J.H. Kim and J. Stringer, eds., Wiley, New York, 1992, pp. 335-380.
22. T. A. Driscoll and B. Fornberg, Padé-based interpretation and correction of the Gibbs phenomenon, in The Gibbs Phenomenon in Various Representations and Applications, Sampling Publishing, Potsdam, NY, 2009.
23. K. A. Emanuel, An air-sea interaction theory for tropical cyclones. Part I: Steady state maintenance, J. Atmospheric Sci., 43 (1986), pp. 585-604.
24. K. A. Emanuel, The maximum intensity of hurricanes, J. Atmospheric Sci, 45 (1988), pp. 1143-1155.
25. R. Fletcher, Practical Methods of Optimization, 2nd ed., Wiley, New York, 1987.
26. B. Fornberg and N. Flyer, The Gibbs phenomenon for radial basis functions, in The Gibbs Phenomenon in Various Representations and Applications, Sampling Publishing, Potsdam, NY, 2009.
27. F. Girosi, M. Jones, and T. Poggio, Regularization theory and neural network architectures, Neural Comput., 7 (1995), pp. 219-269.
28. F. Girosi and T. Poggio, Networks and the Best Approximation Property, Biol. Cybernet., 63 (1998), pp. 169-176.
29. G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., The Johns Hopkins University Press, Baltimore, MD, 1996.
30. A. K. Gupta, F. C. Chang, and W. J. Huang, Some skew-symmetric models, Random Oper. Stoch. Equ., 10 (2002), pp. 133-140.
31. A. K. Gupta, G. González-Farías, and J. A. Domínguez-Molina, A multivariate skew normal distribution, J. Multivariate Anal., 89 (2004), pp. 181-190.
32. F. De Helguero, Sulla Rappresentazione Analitica Delle Curve Abnormali, in Atti del IV Congesso Internazionale dei Matematici, Vol. III, G. Castelnuovo, ed., Roma R. Accademia dei Lincei, Rome, Italy, 1909, pp. 287-299.
33. W. Huanga and Y. Chenb, Generalized skew-Cauchy distribution, Statist. Probab. Lett., 77 (2007), pp. 1137-1147.
34. A. A. Jamshidi, A New Spatio-temporal Resource Allocation Network (ST-RAN), M.Sc. thesis, Department of Mathematics, Colorado State University, Fort Collins, CO, 2004.
35. A. A. Jamshidi, Modeling Spatio-temporal Systems with Skew Radial Basis Functions: Theory, Algorithms and Applications, Ph.D. dissertation, Department of Mathematics, Colorado State University, Fort Collins, CO, 2008.
36. A. A. Jamshidi and M. J. Kirby, Examples of compactly supported functions for radial basis approximations, in Proceedings of the 2006 International Conference on Machine Learning; Models, Technologies and Applications, H. R. Arabnia, E. Kozerenko, and S. Shaumyan, eds., CSREA Press, Irvine, CA, 2006, pp. 155-160.
37. A. A. Jamshidi and M. J. Kirby, Towards a black box algorithm for nonlinear function approximation over high-dimensional domains, SIAM J. Sci. Comput., 29 (2007), pp. 941- 963.
38. A. A. Jamshidi and M. J. Kirby, Skew-radial basis functions for modeling edges and jumps, in Proceedings of the 8th IMA International Conference on Mathematics in Signal Processing, Cirencester, UK, 2008, pp. 51-54.
39. M. C. Jones and M. J. Faddy, A skew extension of the t distribution, with applications, J. Roy. Statist. Soc. Ser. B, 65 (2003), pp. 159-174.
40. R. D. Jones, Y.C. Lee, C. W. Barnes, G. W. Flake, K. Lee, P. S. Lewis, and S. Qian, Function approximation and time series prediction with neural networks, IJCNN Internat. Joint Conf. Neural Netw., 1 (1990), pp. 649-665.
41. J. H. Jung, A note on the Gibbs phenomenon with multiquadric radial basis functions, Appl. Numer. Math., 57 (2007), pp. 213-229.
42. N. B. Karayiannis and M. M. Randolph-Gips, On the construction and training of reformulated radial basis function neural networks, IEEE Trans. Neural Netw., 14 (2003), pp. 835-846.
43. E. Kleinschmidt, Grundlagen einer theorie der tropischen zyklonen (basis principles for a theory of tropical cyclones), Arch. Meteor. Geophys. Bioklimatol, A4 (1951), pp. 53-72.
44. C. C. Lee, C. C. Chung, J. R. Tsai, and C. I. Chang, Robust radial basis function neural networks, IEEE Trans. Syst., Man and Cybernet., Part B, 29 (1999), pp. 674-685.
45. P. V. Lee and S. Haykin, Regularized Radial Basis Function Networks Theory and Applications, Wiley, New York, 2001.
46. D. G. Luenberger, Linear and Nonlinear Programming, 2nd ed., Addison Wesley, Reading, MA, 1984.
47. G. Mclachlan and D. Peel, Finite Mixture Models, Wiley-Interscience, New York, 2000.
48. C. A. Micchelli, Interpolation of scattered data: Distance matrices and conditionally positive definite functions, Constr. Approx., 2 (1986), pp. 11-22.
49. J. Moody and C. Darken, Fast learning in networks of locally-tuned processing units, Neural Comput., 1 (1989), pp. 281-294.
50. S. Nadarajah and S. Kotz, Skewed distributions generated by the normal kernel, Statist. Probab. Lett., 65 (2003), pp. 269-277.
51. J. Park and I. W. Sandberg, Universal approximation using radial-basis-function networks, Neural Comput., 3 (1991), pp. 246-257.
52. J. Park and I. W. Sandberg, Approximation and radial-basis-function networks, Neural Comput., 5 (1993), pp. 305-316.
53. J. Persing and M. T. Montgomery, Hurricane superintensity, J. Atmospheric Sci., 60 (2003), pp. 2349-2371.
54. T. Poggio and F. Girosi, Networks for approximation and learning, Proc. IEEE, 78 (1990), pp. 1481-1496.
55. T. Poggio and F. Girosi, Regularization algorithm for learning that are equivalent to multilayer networks, Science, 247 (1990), pp. 978-982.
56. M. J. D. Powell, Radial basis functions for multivariable interpolation: A review, in Algorithms for Approximation, J. C. Mason and M. G. Cox, eds., Clarendon Press, Oxford, 1987, pp. 143-167.
57. Y. Qiao, Radial Basis Functions (RBFs) for Solving Color Conversion Problems, Ph.D. dissertation, Department of Mathematics, Colorado State University, Fort Collins, CO, 2008.
58. H. J. Rossberg, Positive definite probability densities and probability distributions, J. Math. Sci., 76 (1995), pp. 2181-2197.
59. S. Roweis and L. Saul, Nonlinear dimensionality reduction by locally linear embedding, Science, 290 (2000), pp. 2323-2326.
60. D. E. Rumelhart and J. L. McClelland, Parallel Distributed Processing, MIT Press, Cambridge, MA, 1986.
61. R. Schaback, Error estimates and condition numbers for radial basis function interpolation, Adv. Comput. Math., 3 (1995), pp. 251-264.
62. R. Schaback and H. Wendland, Characterization and construction of radial basis functions, in Multivariate Approximation and Applications, Eilat Proceedings, N. Dyn, D. Leviatan, D. Levin, and A. Pinkus, eds., Cambridge University Press, Cambridge, 2000, pp. 1-24.
63. B. Scholkopf, S. Kah-Kay, C. J. C. Burges, F. Girosi, P. Niyogi, T. Poggio, and V. Vapnik, Comparing support vector machines with Gaussian kernels to radial basis function classifiers, IEEE Trans. Signal Process., 45 (1997), pp. 2758-2765.
64. E. Tadmor and J. Tanner, Adaptive filters for piecewise smooth spectral data, IMA J. Numer. Anal., 25 (2005), pp. 635-647.
65. J. B. Tenenbaum, V. de Silva, and J. C. Langford, A global geometric framework for nonlinear dimensionality reduction, Science, 290 (2000), pp. 2319-2323.
66. M. Visani, C. Garcia, and J. M. Jolion, Normalized radial basis function networks and bilinear discriminant analysis for face recognition, IEEE Conf. Adv. Video Signal Based Surveillance, 15 (2005), pp. 342-347.
67. A. Wahed and M. M. Ali, The skew-logistic distribution, J. Statist. Res., 35 (2001), pp. 71-80.
68. H. Wendland, Scattered Data Approximation, Cambridge University Press, Cambridge, 2005.
69. P. J. Werbos, Beyond Regression: New Tools for Prediction and Analysis in the Behavioral Sciences, Ph.D. dissertation, Division of AppliedMathematics, Harvard University, Boston, MA, 1974.