Nothing Special   »   [go: up one dir, main page]

skip to main content
article

A tutorial on support vector regression

Published: 01 August 2004 Publication History

Abstract

In this tutorial we give an overview of the basic ideas underlying Support Vector (SV) machines for function estimation. Furthermore, we include a summary of currently used algorithms for training SV machines, covering both the quadratic (or convex) programming part and advanced methods for dealing with large datasets. Finally, we mention some modifications and extensions that have been applied to the standard SV algorithm, and discuss the aspect of regularization from a SV perspective.

References

[1]
Aizerman M.A., Braverman É.M., and Rozonoér L.I. 1964. Theoretical foundations of the potential function method in pattern recognition learning. Automation and Remote Control 25: 821-837.
[2]
Aronszajn N. 1950. Theory of reproducing kernels. Transactions of the American Mathematical Society 68: 337-404.
[3]
Bazaraa M.S., Sherali H.D., and Shetty C.M. 1993. Nonlinear Programming: Theory and Algorithms, 2nd edition, Wiley.
[4]
Bellman R.E. 1961. Adaptive Control Processes. Princeton University Press, Princeton, NJ.
[5]
Bennett K. 1999. Combining support vector and mathematical programming methods for induction. In: Schölkopf B., Burges C.J.C., and Smola A.J., (Eds.), Advances in Kernel Methods--SV Learning, MIT Press, Cambridge, MA, pp. 307-326.
[6]
Bennett K.P. and Mangasarian O.L. 1992. Robust linear programming discrimination of two linearly inseparable sets. Optimization Methods and Software 1: 23-34.
[7]
Berg C., Christensen J.P.R., and Ressel P. 1984. Harmonic Analysis on Semigroups. Springer, New York.
[8]
Bertsekas D.P. 1995. Nonlinear Programming. Athena Scientific, Belmont, MA.
[9]
Bishop C.M. 1995. Neural Networks for Pattern Recognition. Clarendon Press, Oxford.
[10]
Blanz V., Schölkopf B., Bülthoff H., Burges C., Vapnik V., and Vetter T. 1996. Comparison of view-based object recognition algorithms using realistic 3D models. In: von der Malsburg C., von Seelen W., Vorbrüggen J.C., and Sendhoff B. (Eds.), Artificial Neural Networks ICANN'96, Berlin. Springer Lecture Notes in Computer Science, Vol. 1112, pp. 251-256.
[11]
Bochner S. 1959. Lectures on Fourier integral. Princeton Univ. Press, Princeton, New Jersey.
[12]
Boser B.E., Guyon I.M., and Vapnik V.N. 1992. A training algorithm for optimal margin classifiers. In: Haussler D. (Ed.), Proceedings of the Annual Conference on Computational Learning Theory. ACM Press, Pittsburgh, PA, pp. 144-152.
[13]
Bradley P.S., Fayyad U.M., and Mangasarian O.L. 1998. Data mining: Overview and optimization opportunities. Technical Report 98-01, University of Wisconsin, Computer Sciences Department, Madison, January. INFORMS Journal on Computing, to appear.
[14]
Bradley P.S. and Mangasarian O.L. 1998. Feature selection via concave minimization and support vector machines. In: Shavlik J. (Ed.), Proceedings of the International Conference on Machine Learning, Morgan Kaufmann Publishers, San Francisco, California, pp. 82-90. ftp://ftp.cs.wisc.edu/math-prog/tech-reports/98- 03.ps.Z.
[15]
Bunch J.R. and Kaufman L. 1977. Some stable methods for calculating inertia and solving symmetric linear systems. Mathematics of Computation 31: 163-179.
[16]
Bunch J.R. and Kaufman L. 1980. A computational method for the indefinite quadratic programming problem. Linear Algebra and Its Applications, pp. 341-370, December.
[17]
Bunch J.R., Kaufman L., and Parlett B. 1976. Decomposition of a symmetric matrix. Numerische Mathematik 27: 95-109.
[18]
Burges C.J.C. 1996. Simplified support vector decision rules. In L. Saitta (Ed.), Proceedings of the International Conference on Machine Learning, Morgan Kaufmann Publishers, San Mateo, CA, pp. 71-77.
[19]
Burges C.J.C. 1998. A tutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery 2(2): 121- 167.
[20]
Burges C.J.C. 1999. Geometry and invariance in kernel based methods. In Schölkopf B., Burges C.J.C., and Smola A.J., (Eds.), Advances in Kernel Methods--Support Vector Learning, MIT Press, Cambridge, MA, pp. 89-116.
[21]
Burges C.J.C. and Schölkopf B.1997. Improving the accuracy and speed of support vector learning machines. In Mozer M.C., Jordan M.I., and Petsche T., (Eds.), Advances in Neural Information Processing Systems 9, MIT Press, Cambridge, MA, pp. 375-381.
[22]
Chalimourda A., Schölkopf B., and Smola A.J. 2004. Experimentally optimal v in support vector regression for different noise models and parameter settings. Neural Networks 17(1): 127-141.
[23]
Chang C.-C., Hsu C.-W., and Lin C.-J. 1999. The analysis of decomposition methods for support vector machines. In Proceeding of IJCAI99, SVM Workshop.
[24]
Chang C.C. and Lin C.J. 2001. Training v-support vector classifiers: Theory and algorithms. Neural Computation 13(9): 2119- 2147.
[25]
Chen S., Donoho D., and Saunders M. 1999. Atomic decomposition by basis pursuit. Siam Journal of Scientific Computing 20(1): 33-61.
[26]
Cherkassky V. and Mulier F. 1998. Learning from Data. John Wiley and Sons, New York.
[27]
Cortes C. and Vapnik V. 1995. Support vector networks. Machine Learning 20: 273-297.
[28]
Cox D. and O'Sullivan F. 1990. A symptotic analysis of penalized likelihood and related estimators. Annals of Statistics 18: 1676-1695.
[29]
CPLEX Optimization Inc. Using the CPLEX callable library. Manual, 1994.
[30]
Cristianini N. and Shawe-Taylor J. 2000. An Introduction to Support Vector Machines. Cambridge University Press, Cambridge, UK.
[31]
Cristianini N., Campbell C., and Shawe-Taylor J. 1998. Multiplicative updatings for support vector learning. Neuro COLT Technical Report NC-TR-98-016, Royal Holloway College.
[32]
Dantzig G.B. 1962. Linear Programming and Extensions. Princeton Univ. Press, Princeton, NJ.
[33]
Devroye L., Györfi L., and Lugosi G. 1996. A Probabilistic Theory of Pattern Recognition. Number 31 in Applications of mathematics. Springer, New York.
[34]
Drucker H., Burges C.J.C., Kaufman L., Smola A., and Vapnik V. 1997. Support vector regression machines. In: Mozer M.C., Jordan M.I., and Petsche T. (Eds.), Advances in Neural Information Processing Systems 9, MIT Press, Cambridge, MA, pp. 155-161.
[35]
Efron B. 1982. The jacknife, the bootstrap, and other resampling plans. SIAM, Philadelphia.
[36]
Efron B. and Tibshirani R.J. 1994. An Introduction to the Bootstrap. Chapman and Hall, New York.
[37]
El-Bakry A., Tapia R., Tsuchiya R., and Zhang Y. 1996. On the formulation and theory of the Newton interior-point method for nonlinear programming. J. Optimization Theory and Applications 89: 507- 541.
[38]
Fletcher R. 1989. Practical Methods of Optimization. John Wiley and Sons, New York.
[39]
Girosi F. 1998. An equivalence between sparse approximation and support vector machines. Neural Computation 10(6): 1455-1480.
[40]
Girosi F., Jones M., and Poggio T. 1993. Priors, stabilizers and basis functions: From regularization to radial, tensor and additive splines. A.I. Memo No. 1430, Artificial Intelligence Laboratory, Massachusetts Institute of Technology.
[41]
Guyon I., Boser B., and Vapnik V. 1993. Automatic capacity tuning of very large VC-dimension classifiers. In: Hanson S.J., Cowan J.D., and Giles C.L. (Eds.), Advances in Neural Information Processing Systems 5. Morgan Kaufmann Publishers, pp. 147-155.
[42]
Härdle W. 1990. Applied nonparametric regression, volume 19 of Econometric Society Monographs. Cambridge University Press.
[43]
Hastie T.J. and Tibshirani R.J. 1990. Generalized Additive Models, volume 43 of Monographs on Statistics and Applied Probability. Chapman and Hall, London.
[44]
Haykin S. 1998. Neural Networks: A Comprehensive Foundation. 2nd edition. Macmillan, New York.
[45]
Hearst M.A., Schölkopf B., Dumais S., Osuna E., and Platt J. 1998. Trends and controversies--support vector machines. IEEE Intelligent Systems 13: 18-28.
[46]
Herbrich R. 2002. Learning Kernel Classifiers: Theory and Algorithms. MIT Press.
[47]
Huber P.J. 1972. Robust statistics: A review. Annals of Statistics 43: 1041.
[48]
Huber P.J. 1981. Robust Statistics. John Wiley and Sons, New York.
[49]
IBM Corporation. 1992. IBM optimization subroutine library guide and reference. IBM Systems Journal, 31, SC23-0519.
[50]
Jaakkola T.S. and Haussler D. 1999. Probabilistic kernel regression models. In: Proceedings of the 1999 Conference on AI and Statistics.
[51]
Joachims T. 1999. Making large-scale SVM learning practical. In: Schölkopf B., Burges C.J.C., and Smola A.J. (Eds.), Advances in Kernel Methods--Support Vector Learning, MIT Press, Cambridge, MA, pp. 169-184.
[52]
Karush W. 1939. Minima of functions of several variables with inequalities as side constraints. Master's thesis, Dept. of Mathematics, Univ. of Chicago.
[53]
Kaufman L. 1999. Solving the quadratic programming problem arising in support vector classification. In: Schölkopf B., Burges C.J.C., and Smola A.J. (Eds.), Advances in Kernel Methods--Support Vector Learning, MIT Press, Cambridge, MA, pp. 147-168
[54]
Keerthi S.S., Shevade S.K., Bhattacharyya C., and Murthy K.R.K. 1999. Improvements to Platt's SMO algorithm for SVM classifier design. Technical Report CD-99-14, Dept. of Mechanical and Production Engineering, Natl. Univ. Singapore, Singapore.
[55]
Keerthi S.S., Shevade S.K., Bhattacharyya C., and Murty K.R.K. 2001. Improvements to platt's SMO algorithm for SVM classifier design. Neural Computation 13: 637-649.
[56]
Kimeldorf G.S. and Wahba G. 1970. A correspondence between Bayesian estimation on stochastic processes and smoothing by splines. Annals of Mathematical Statistics 41: 495-502.
[57]
Kimeldorf G.S. and Wahba G. 1971. Some results on Tchebycheffian spline functions. J. Math. Anal. Applic. 33: 82-95.
[58]
Kowalczyk A. 2000. Maximal margin perceptron. In: Smola A.J., Bartlett P.L., Schölkopf B., and Schuurmans D. (Eds.), Advances in Large Margin Classifiers, MIT Press, Cambridge, MA, pp. 75- 113.
[59]
Kuhn H.W. and Tucker A.W. 1951. Nonlinear programming. In: Proc. 2nd Berkeley Symposium on Mathematical Statistics and Probabilistics, Berkeley. University of California Press, pp. 481-492.
[60]
Lee Y.J. and Mangasarian O.L. 2001. SSVM: A smooth support vector machine for classification. Computational optimization and Applications 20(1): 5-22.
[61]
Li M. and Vitányi P. 1993. An introduction to Kolmogorov Complexity and its applications. Texts and Monographs in Computer Science. Springer, New York.
[62]
Lin C.J. 2001. On the convergence of the decomposition method for support vector machines. IEEE Transactions on Neural Networks 12(6): 1288-1298.
[63]
Lustig I.J., Marsten R.E., and Shanno D.F. 1990. On implementing Mehrotra's predictor-corrector interior point method for linear programming. Princeton Technical Report SOR 90-03., Dept. of Civil Engineering and Operations Research, Princeton University.
[64]
Lustig I.J., Marsten R.E., and Shanno D.F. 1992. On implementing Mehrotra's predictor-corrector interior point method for linear programming. SIAM Journal on Optimization 2(3): 435- 449.
[65]
MacKay D.J.C. 1991. Bayesian Methods for Adaptive Models. PhD thesis, Computation and Neural Systems, California Institute of Technology, Pasadena, CA.
[66]
Mangasarian O.L. 1965. Linear and nonlinear separation of patterns by linear programming. Operations Research 13: 444-452.
[67]
Mangasarian O.L. 1968. Multi-surface method of pattern separation. IEEE Transactions on Information Theory IT-14: 801-807.
[68]
Mangasarian O.L. 1969. Nonlinear Programming. McGraw-Hill, New York.
[69]
Mattera D. and Haykin S. 1999. Support vector machines for dynamic reconstruction of a chaotic system. In: Schölkopf B., Burges C.J.C., and Smola A.J. (Eds.), Advances in Kernel Methods-- Support Vector Learning, MIT Press, Cambridge, MA, pp. 211- 242.
[70]
McCormick G.P. 1983. Nonlinear Programming: Theory, Algorithms, and Applications. John Wiley and Sons, New York.
[71]
Megiddo N. 1989. Progressin Mathematical Programming, chapter Pathways to the optimal set in linear programming, Springer, New York, NY, pp. 131-158.
[72]
Mehrotra S. and Sun J. 1992. On the implementation of a (primal-dual) interior point method. SIAM Journal on Optimization 2(4): 575- 601.
[73]
Mercer J. 1909. Functions of positive and negative type and their connection with the theory of integral equations. Philosophical Transactions of the Royal Society, London A 209: 415-446.
[74]
Micchelli C.A. 1986. Algebraic aspects of interpolation. Proceedings of Symposia in Applied Mathematics 36: 81-102.
[75]
Morozov V.A. 1984. Methods for Solving Incorrectly Posed Problems. Springer.
[76]
Müller K.-R., Smola A., Rätsch G., Schölkopf B., Kohlmorgen J., and Vapnik V. 1997. Predicting time series with support vector machines. In: Gerstner W., Germond A., Hasler M., and Nicoud J.-D. (Eds.), Artificial Neural Networks ICANN'97, Berlin. Springer Lecture Notes in Computer Science Vol. 1327 pp. 999-1004.
[77]
Murtagh B.A. and Saunders M.A. 1983. MINOS 5.1 user's guide. Technical Report SOL 83-20R, Stanford University, CA, USA, Revised 1987.
[78]
Neal R. 1996. Bayesian Learning in Neural Networks. Springer.
[79]
Nilsson N.J. 1965. Learning machines: Foundations of Trainable Pattern Classifying Systems. McGraw-Hill.
[80]
Nyquist. H. 1928. Certain topics in telegraph transmission theory. Trans. A.I.E.E., pp. 617-644.
[81]
Osuna E., Freund R., and Girosi F. 1997. An improved training algorithm for support vector machines. In Principe J., Gile L., Morgan N., and Wilson E. (Eds.), Neural Networks for Signal Processing VII--Proceedings of the 1997 IEEE Workshop, pp. 276-285, New York, IEEE.
[82]
Osuna E. and Girosi F. 1999. Reducing the run-time complexity in support vector regression. In: Schölkopf B., Burges C.J.C., and Smola A. J. (Eds.), Advances in Kernel Methods--Support Vector Learning, pp. 271-284, Cambridge, MA, MIT Press.
[83]
Ovari Z. 2000. Kernels, eigenvalues and support vector machines. Honours thesis, Australian National University, Canberra.
[84]
Platt J. 1999. Fast training of support vector machines using sequential minimal optimization. In: Schölkopf B., Burges C.J.C., and Smola A.J. (Eds.) Advances in Kernel Methods--Support Vector Learning, pp. 185-208, Cambridge, MA, MIT Press.
[85]
Poggio T. 1975. On optimal nonlinear associative recall. Biological Cybernetics, 19: 201-209.
[86]
Rasmussen C. 1996. Evaluation of Gaussian Processes and Other Methods for Non-Linear Regression. PhD thesis, Department of Computer Science, University of Toronto, ftp://ftp.cs.toronto.edu/pub/carl/thesis.ps.gz.
[87]
Rissanen J. 1978. Modeling by shortest data description. Automatica, 14: 465-471.
[88]
Saitoh S. 1988. Theory of Reproducing Kernels and its Applications. Longman Scientific & Technical, Harlow, England.
[89]
Saunders C., Stitson M.O., Weston J., Bottou L., Schölkopf B., and Smola A. 1998. Support vector machine--reference manual. Technical Report CSD-TR-98-03, Department of Computer Science, Royal Holloway, University of London, Egham, UK. SVM available at http://svm.dcs.rhbnc.ac.uk/.
[90]
Schoenberg I. 1942. Positive definite functions on spheres. Duke Math. J., 9: 96-108.
[91]
Schölkopf B. 1997. Support Vector Learning. R. Oldenbourg Verlag, München. Doktorarbeit, TU Berlin. Download: http://www.kernel-machines.org.
[92]
Schölkopf B., Burges C., and Vapnik V. 1995. Extracting support data for a given task. In: Fayyad U.M. and Uthurusamy R. (Eds.), Proceedings, First International Conference on Knowledge Discovery & Data Mining, Menlo Park, AAAI Press.
[93]
Schölkopf B., Burges C., and Vapnik V. 1996. Incorporating invariances in support vector learning machines. In: von der Malsburg C., von Seelen W., Vorbrüggen J. C., and Sendhoff B. (Eds.), Artificial Neural Networks ICANN'96, pp. 47-52, Berlin, Springer Lecture Notes in Computer Science, Vol. 1112.
[94]
Schölkopf B., Burges C.J.C., and Smola A.J. 1999a. (Eds.) Advances in Kernel Methods--Support Vector Learning. MIT Press, Cambridge, MA.
[95]
Schölkopf B., Herbrich R., Smola A.J., and Williamson R.C. 2001. A generalized representer theorem. Technical Report 2000-81, NeuroCOLT, 2000. To appear in Proceedings of the Annual Conference on Learning Theory, Springer (2001).
[96]
Schölkopf B., Mika S., Burges C., Knirsch P., Müller K.-R., Rätsch G., and Smola A. 1999b. Input space vs. feature space in kernel-based methods. IEEE Transactions on Neural Networks, 10(5): 1000- 1017.
[97]
Schölkopf B., Platt J., Shawe-Taylor J., Smola A.J., and Williamson R.C. 2001. Estimating the support of a high-dimensional distribution. Neural Computation, 13(7): 1443-1471.
[98]
Schölkopf B., Simard P., Smola A., and Vapnik V. 1998a. Prior knowledge in support vector kernels. In: Jordan M.I., Kearns M.J., and Solla S.A. (Eds.) Advances in Neural Information Processing Systems 10, MIT Press. Cambridge, MA, pp. 640-646.
[99]
Schölkopf B., Smola A., and Müller K.-R. 1998b. Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation, 10: 1299-1319.
[100]
Schölkopf B., Smola A., Williamson R.C., and Bartlett P.L. 2000. New support vector algorithms. Neural Computation, 12: 1207-1245.
[101]
Schölkopf B. and Smola A.J. 2002. Learning with Kernels. MIT Press.
[102]
Schölkopf B., Sung K., Burges C., Girosi F., Niyogi P., Poggio T., and Vapnik V. 1997. Comparing support vector machines with Gaussian kernels to radial basis function classifiers. IEEE Transactions on Signal Processing, 45: 2758-2765.
[103]
Shannon C.E. 1948. A mathematical theory of communication. Bell System Technical Journal, 27: 379-423, 623-656.
[104]
Shawe-Taylor J., Bartlett P.L., Williamson R.C., and Anthony M. 1998. Structural risk minimization over data-dependent hierarchies. IEEE Transactions on Information Theory, 44(5): 1926- 1940.
[105]
Smola A., Murata N., Schölkopf B., and Müller K.-R. 1998a. A symptotically optimal choice of ¿-loss for support vector machines. In: Niklasson L., Bodén M., and Ziemke T. (Eds.) Proceedings of the International Conference on Artificial Neural Networks, Perspectives in Neural Computing, pp. 105-110, Berlin, Springer.
[106]
Smola A., Schölkopf B., and Müller K.-R. 1998b. The connection between regularization operators and support vector kernels. Neural Networks, 11: 637-649.
[107]
Smola A., Schölkopf B., and Müller K.-R. 1998c. General cost functions for support vector regression. In: Downs T., Frean M., and Gallagher M. (Eds.) Proc. of the Ninth Australian Conf. on Neural Networks, pp. 79-83, Brisbane, Australia. University of Queensland.
[108]
Smola A., Schölkopf B., and Rätsch G. 1999. Linear programs for automatic accuracy control in regression. In: Ninth International Conference on Artificial Neural Networks, Conference Publications No. 470, pp. 575-580, London. IEE.
[109]
Smola. A.J. 1996. Regression estimation with support vector learning machines. Diplomarbeit, Technische Universität München.
[110]
Smola A.J. 1998. Learning with Kernels. PhD thesis, Technische Universität Berlin. GMD Research Series No. 25.
[111]
Smola A.J., Elisseeff A., Schölkopf B., and Williamson R.C. 2000. Entropy numbers for convex combinations and MLPs. In Smola A.J., Bartlett P.L., Schölkopf B., and Schuurmans D. (Eds.) Advances in Large Margin Classifiers, MIT Press, Cambridge, MA, pp. 369-387.
[112]
Smola A.J., Óvári Z.L., and Williamson R.C. 2001. Regularization with dot-product kernels. In: Leen T.K., Dietterich T.G., and Tresp V. (Eds.) Advances in Neural Information Processing Systems 13, MIT Press, pp. 308-314.
[113]
Smola A.J. and Schölkopf B. 1998a. On a kernel-based method for pattern recognition, regression, approximation and operator inversion. Algorithmica, 22: 211-231.
[114]
Smola A.J. and Schölkopf B. 1998b. A tutorial on support vector regression. NeuroCOLT Technical Report NC-TR-98-030, Royal Holloway College, University of London, UK.
[115]
Smola A.J. and Schölkopf B. 2000. Sparse greedy matrix approximation for machine learning. In: Langley P. (Ed.), Proceedings of the International Conference on Machine Learning, Morgan Kaufmann Publishers, San Francisco, pp. 911-918.
[116]
Stitson M., Gammerman A., Vapnik V., Vovk V., Watkins C., and Weston J. 1999. Support vector regression with ANOVA decomposition kernels. In: Schölkopf B., Burges C.J.C., and Smola A.J. (Eds.), Advances in Kernel Methods--Support Vector Learning, MIT Press Cambridge, MA, pp. 285-292.
[117]
Stone C.J. 1985. Additive regression and other nonparametric models. Annals of Statistics, 13: 689-705.
[118]
Stone M. 1974. Cross-validatory choice and assessment of statistical predictors (with discussion). Journal of the Royal Statistical Society, B36: 111-147.
[119]
Street W.N. and Mangasarian O.L. 1995. Improved generalization via tolerant training. Technical Report MP-TR-95-11, University of Wisconsin, Madison.
[120]
Tikhonov A.N. and Arsenin V.Y. 1977. Solution of Ill-posed problems. V. H. Winston and Sons.
[121]
Tipping M.E. 2000. The relevance vector machine. In: Solla S.A., Leen T.K., and Müller K.-R. (Eds.), Advances in Neural Information Processing Systems 12, MIT Press, Cambridge, MA, pp. 652-658.
[122]
Vanderbei R.J. 1994. LOQO: An interior point code for quadratic programming. TR SOR-94-15, Statistics and Operations Research, Princeton Univ., NJ.
[123]
Vanderbei R.J. 1997. LOQO user's manual--version 3.10. Technical Report SOR-97-08, Princeton University, Statistics and Operations Research, Code available at http://www.princeton.edu/ ~rvdb/.
[124]
Vapnik V. 1995. The Nature of Statistical Learning Theory. Springer, New York.
[125]
Vapnik V. 1998. Statistical Learning Theory. John Wiley and Sons, New York.
[126]
Vapnik. V. 1999. Three remarks on the support vector method of function estimation. In: Schölkopf B., Burges C.J.C., and Smola A.J. (Eds.), Advances in Kernel Methods--Support Vector Learning, MIT Press, Cambridge, MA, pp. 25-42.
[127]
Vapnik V. and Chervonenkis A. 1964. A note on one class of perceptrons. Automation and Remote Control, 25.
[128]
Vapnik V. and Chervonenkis A. 1974. Theory of Pattern Recognition {in Russian}. Nauka, Moscow. (German Translation: Wapnik W. & Tscherwonenkis A., Theorie der Zeichenerkennung, Akademie-Verlag, Berlin, 1979).
[129]
Vapnik V., Golowich S., and Smola A. 1997. Support vector method for function approximation, regression estimation, and signal processing. In: Mozer M.C., Jordan M.I., and Petsche T. (Eds.) Advances in Neural Information Processing Systems 9, MA, MIT Press, Cambridge. pp. 281-287.
[130]
Vapnik V. and Lerner A. 1963. Pattern recognition using generalized portrait method. Automation and Remote Control, 24: 774-780.
[131]
Vapnik V.N. 1982. Estimation of Dependences Based on Empirical Data. Springer, Berlin.
[132]
Vapnik V.N. and Chervonenkis A.Y. 1971. On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability and its Applications, 16(2): 264-281.
[133]
Wahba G. 1980. Spline bases, regularization, and generalized cross-validation for solving approximation problems with large quantities of noisy data. In: Ward J. and Cheney E. (Eds.), Proceedings of the International Conference on Approximation theory in honour of George Lorenz, Academic Press, Austin, TX, pp. 8-10.
[134]
Wahba G. 1990. Spline Models for Observational Data, volume 59 of CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia.
[135]
Wahba G. 1999. Support vector machines, reproducing kernel Hilbert spaces and the randomized GACV. In: Schölkopf B., Burges C.J.C., and Smola A.J. (Eds.), Advances in Kernel Methods-- Support Vector Learning, MIT Press, Cambridge, MA. pp. 69-88.
[136]
Weston J., Gammerman A., Stitson M., Vapnik V., Vovk V., and Watkins C. 1999. Support vector density estimation. In: Schölkopf B., Burges C.J.C., and Smola A.J. (Eds.) Advances in Kernel Methods--Support Vector Learning, MIT Press, Cambridge, MA. pp. 293-306.
[137]
Williams C.K.I. 1998. Prediction with Gaussian processes: From linear regression to linear prediction and beyond. In: Jordan M.I. (Ed.), Learning and Inference in Graphical Models, Kluwer Academic, pp. 599-621.
[138]
Williamson R.C., Smola A.J., and Schölkopf B. 1998. Generalization performance of regularization networks and support vector machines via entropy numbers of compact operators. Technical Report 19, NeuroCOLT, http://www.neurocolt.com. Published in IEEE Transactions on Information Theory, 47(6): 2516-2532 (2001).
[139]
Yuille A. and Grzywacz N. 1988. The motion coherence theory. In: Proceedings of the International Conference on Computer Vision, IEEE Computer Society Press, Washington, DC, pp. 344- 354.

Cited By

View all
  • (2025)Online motion accuracy compensation of industrial servomechanisms using machine learning approachesRobotics and Computer-Integrated Manufacturing10.1016/j.rcim.2024.10283891:COnline publication date: 1-Feb-2025
  • (2024)Applying machine learning techniques in the form of ensemble and hybrid models to appraise hardness properties of high-performance concreteJournal of Intelligent & Fuzzy Systems: Applications in Engineering and Technology10.3233/JIFS-23440946:1(903-921)Online publication date: 1-Jan-2024
  • (2024)An Interdisciplinary Survey on Origin-destination Flows Modeling: Theory and TechniquesACM Computing Surveys10.1145/368205857:1(1-49)Online publication date: 26-Jul-2024
  • Show More Cited By

Recommendations

Comments

Please enable JavaScript to view thecomments powered by Disqus.

Information & Contributors

Information

Published In

cover image Statistics and Computing
Statistics and Computing  Volume 14, Issue 3
August 2004
106 pages

Publisher

Kluwer Academic Publishers

United States

Publication History

Published: 01 August 2004

Author Tags

  1. machine learning
  2. regression estimation
  3. support vector machines

Qualifiers

  • Article

Contributors

Other Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

  • Downloads (Last 12 months)0
  • Downloads (Last 6 weeks)0
Reflects downloads up to 22 Nov 2024

Other Metrics

Citations

Cited By

View all
  • (2025)Online motion accuracy compensation of industrial servomechanisms using machine learning approachesRobotics and Computer-Integrated Manufacturing10.1016/j.rcim.2024.10283891:COnline publication date: 1-Feb-2025
  • (2024)Applying machine learning techniques in the form of ensemble and hybrid models to appraise hardness properties of high-performance concreteJournal of Intelligent & Fuzzy Systems: Applications in Engineering and Technology10.3233/JIFS-23440946:1(903-921)Online publication date: 1-Jan-2024
  • (2024)An Interdisciplinary Survey on Origin-destination Flows Modeling: Theory and TechniquesACM Computing Surveys10.1145/368205857:1(1-49)Online publication date: 26-Jul-2024
  • (2024)Research on Stock Volatility Prediction Using Attention Mechanism Enhanced GRU Ensemble ModelProceedings of the 2024 International Conference on Generative Artificial Intelligence and Information Security10.1145/3665348.3665359(54-58)Online publication date: 10-May-2024
  • (2024)Zero-Shot Learning to Enable Error Awareness in Data-Driven HRIProceedings of the 2024 ACM/IEEE International Conference on Human-Robot Interaction10.1145/3610977.3634940(592-601)Online publication date: 11-Mar-2024
  • (2024)MetaKernel: Learning Variational Random Features With Limited LabelsIEEE Transactions on Pattern Analysis and Machine Intelligence10.1109/TPAMI.2022.315493046:3(1464-1478)Online publication date: 1-Mar-2024
  • (2024)PIGAT: Physics-Informed Graph Attention Transformer for Air Traffic State PredictionIEEE Transactions on Intelligent Transportation Systems10.1109/TITS.2024.338612825:9(12561-12577)Online publication date: 1-Sep-2024
  • (2024)ENCFIS: An Exclusionary Neural Complex Fuzzy Inference System for Robust Regression LearningIEEE Transactions on Fuzzy Systems10.1109/TFUZZ.2023.333052832:3(1539-1552)Online publication date: 1-Mar-2024
  • (2024)Detecting Facial Action Units From Global-Local Fine-Grained ExpressionsIEEE Transactions on Circuits and Systems for Video Technology10.1109/TCSVT.2023.328890334:2(983-994)Online publication date: 1-Feb-2024
  • (2024)Annotate Smarter, not Harder: Using Active Learning to Reduce Emotional Annotation EffortIEEE Transactions on Affective Computing10.1109/TAFFC.2023.332956315:3(1213-1227)Online publication date: 1-Jul-2024
  • Show More Cited By

View Options

View options

Media

Figures

Other

Tables

Share

Share

Share this Publication link

Share on social media