Session 9A, Paper 1

1. On Automatic Mapping of Environmental Data Using Adaptive General Regression Neural Network Mikhail Kanevski and Vadim Timonin GISRUK 2010, UCL, London [email_address] , [email_address] , www.unil.ch/igar

3. SIMPLE PROBLEM: AUTOMATIC INTERPOLATION (from measurements to maps) Interpolator Automatic

6. GRNN is a modification of Nadaraya-Watson nonparametric regressor (GRNN is a winner of the SIC2004 – Spatial Interpolation Competition organised by EU JRC, Ispra)

7. Regression = conditional mean where is a conditional distribution of Z given x.

8. where joint pdf can be estimated using Parzen-Rozenblatt kernel density estimator ( K(.) is a kernel ): Conditional pdf is defined by:

9. Therefore the regression can be represented as follows:

10. There are different valid kernels. For an isotropic Gaussian kernel

11. In a more general setting of adaptive/anisotropic kernel we have:

12. General Regression Neural Network INPUTS INTEGRATION LAYER IMAGE LAYER OUTPUT GRNN estimate using measurements Z k :

14. GRNN: influence of bandwidth True function Too large, oversmoothing Too small, overfitting Optimal

16. Case study – precipitation mapping Swiss DEM and Precipitation Monitoring Network

17. Data (raw and shuffled) and corresponding training curves

18. The same is valid for Adaptive GRNN: variables (features, inputs) which are irrelevant are “filtered out” automatically by large corresponding bandwidths.

19. An example with added artificial coordinate 4135 191 7474 6949 420 4D (3D+Noise) 192 7601 7011 419 3D σ Znoise σ z σ y σ x Sigma values (metres) Cross-Validation error Model

20. 3D and 4D modelling results

21. Quality of model? Analysis of the residuals using… GRNN! CV error = 92.8; sigma=inf

22. GRNN Mapping & uncertainties (illustrative example)

23. Geokernels.org

24. The research was partly supported by Swiss NSF grants N 200021-126505 and N 200020-121835 www.unil.ch/igar 2009 Thank you for your attention! 2004 2008