Neural Processing Letters

, Volume 21, Issue 2, pp 81–93 | Cite as

Data Fusion in Radial Basis Function Networks for Spatial Regression



Conventional radial basis function (RBF) networks for spatial regression assume independent and identical distribution and ignore spatial information. In contrast to input fusion, we push spatial information further into RBF networks by fusing output from hidden and output layers. Three case studies demonstrate the advantage of hidden fusion over others and indicate the optimal value is around 1 for the coefficient used in hidden fusion, which links the output from the hidden layer for each site with their neighbors.


data fusion spatial autocorrelation spatial regression radial basis function network 



Radial Basis Function


Mean Squared Error


independent and identical distribution


Input Fusion


Hidden Fusion


Output Fusion


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  1. 1.
    Ambroise, C., Govaert, G. 1998Convergence of an EM-type algorithm for spatial clusteringPattern Recognition Letters19919927Google Scholar
  2. 2.
    Anselin, L. 1988Spatial Econometrics: Methods and ModelsKluwer Academic PublishersDordrechtGoogle Scholar
  3. 3.
    Bishop, C. M. 1995Neural Networks for Pattern RecognitionOxford University PressDordrechtGoogle Scholar
  4. 4.
    De Carvalho, A., Brizzotti, M. M. 2001Combining RBF networks trained by different clustering techniquesNeural Processing Letters14227240Google Scholar
  5. 5.
    Cressie, N. A. 1993Statistics for Spatial DataWileyNew Yorkrevised editionGoogle Scholar
  6. 6.
    Ester, M., Kriegel, H. P. and Sander, J.: Spatial data mining: A database approach, In:Proceedings of 5th Symposium on Spatial Databases: 47–66, 1997.Google Scholar
  7. 7.
    Geman, S., Geman, D. 1984Stochastic relaxation, gibbs distributions and the bayesian restoration of imagesIEEE Transactions on Pattern Analysis and Machine Intelligence6721741Google Scholar
  8. 8.
    Gilardi, N., Bengio, S. 2000Local machine learning models for spatial data analysisJournal of Geographic Information and Decision Analysis41128Google Scholar
  9. 9.
    Gilley, O. W., Pace, R. K. 1996On the harrison and rubinfeld dataJournal of Environmental Economics and Management31403405Google Scholar
  10. 10.
    Harrison, D., Rubinfeld, D. L. 1978Hedonic prices and the demand for clean airJournal of Environmental Economics and Management581102Google Scholar
  11. 11.
    Hartman, E. J., Keller, J. D., Kowalski, J. M. 1990Layered neural networks with gaussian hidden units as universal approximationsNeural Computation2210215Google Scholar
  12. 12.
    Hastie, T., Tibshirani, R., Friedman, J. 2001Elements of Statistical Learning: Data MiningInference and PredictionSpringer-Verlag, BerlinGoogle Scholar
  13. 13.
    Hermes, L. and Buhmann, J. M.: Contextual classification by entropy-based polygonization, In: Proceedings of the IEEE Conference on Computer Vision and Pattern recognition 2001, 442–447.Google Scholar
  14. 14.
    Jain, A. K., Farrokhnia, F. 1991Unsupervised texture segmentation using gabor filtersPattern Recognition2411671186Google Scholar
  15. 15.
    Jhung, Y., Swain, P. H. 1996Bayesian contextural classification based on modified m-estimates and markov random fieldsIEEE Transactions on Geoscience and Remote Sensing346775Google Scholar
  16. 16.
    Kaufman, L, Rousseeuw, P.J. 1990Finding Groups in Data: An Introduction to Cluster AnalysisWileyNew YorkGoogle Scholar
  17. 17.
    Legendre, P.: Constrained clustering, In: Developments in Numerical Ecology, 289–307, NATO ASI Series G 14, (1987).Google Scholar
  18. 18.
    LeSage, J. P.: MATLAB Toolbox for Spatial Econometrics, http: //, 1999.Google Scholar
  19. 19.
    Leung, H., Hennessey, G., Drosopoulos, A. 2000Signal detection using the radial basis function coupled map lattice, IEEE Transactions on Neural Networks1111331151Google Scholar
  20. 20.
    Oliver, M. A., Webster, R. 1989A geostatistical basis for spatial weighting in multivariate classificationMathematical Geology211535Google Scholar
  21. 21.
    Pace, R. K., Barry, R. 1997Quick computation of spatial autoregressive estimatorsGeographical Analysis29232247Google Scholar
  22. 22.
    Poggio, T., Girosi, F. 1990Networks for approximation and learningProceedings of the IEEE7814811497Google Scholar
  23. 23.
    Powell, M. J. D.: Radial basis functions for multivariable interpolation: A review, In: Algorithms for Approximation, 143–167, Clarendon Press, Oxford, (1987).Google Scholar
  24. 24.
    Shekhar, S. and Chawla, S.: Spatial Databases: A Tour, Prentice-Hall, 2002.Google Scholar
  25. 25.
    Silverman, B. W. 1986Density Estimation for Statistics and Data AnalysisChapman & HallLondonGoogle Scholar
  26. 26.
    Solberg, A. H., Taxt, T., Jain, A. K. 1996A markov random field model for classification of multisource satellite imageryIEEE Transactions on Geoscience and Remote Sensing34100113Google Scholar
  27. 27.
    Yan, Y. 1999Understanding speech recognition using correlation-generated neural network targetsIEEE Transactions on Speech and Audio Processing7350352Google Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Department of ComputerScienceNational University of SingaporeSingapore

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