COS 702 Spring 2012 Assignment 1. Radial Basis Functions University of Southern Mississippi Tyler Reese

Transcription

1 COS 702 Spring 2012 Assignment 1 Radial Basis Functions University of Southern Mississippi Tyler Reese

2 The Problem COS 702, Assignment 1: Consider the following test function (Franke s function) f(x, y) = 4 3 e 1 4 ((9x 2)2 +(9y 2) 2) e 1 49 (9x+1) (9y+1) e 1 4 ((9x 7)2 +(9y 3) 2) 1 5 e ((9x 4)2 +(9y 7) 2 ) Reconstruct the above function using three basis functions in the region [0,1] 2 : Inverse MQ (1/ r 2 c 2 + 1) where c is the shape parameter r 5 Gaussian e cr2 Choose 225 points as the center of the radial basis function (RBF) and another 100 random points as the test points using Halton quasi-random points random points uniform mesh points (15x15 mesh points) Maximum error = Max 1 k 100 f(x k, y k ) s(x k, y k ) RMSE = (f(x k, y k ) s(x k, y k )) k=1 Study the effect of shape parameter c and compare the results of using these basis functions and RBF centers. 1

3 Methods Overview Using the functions defined in the provided matlab files (DistanceMatrix.m, testfunction.m, hatonseq.m), six matlab scripts were developed to address this problem. The main procedure is to reconstruct a surface using radial basis functions (RBFs) is to use known data (i.e. the value of f(x,y)) to generate distance matrix with each element representing its cumulative distance in the x-y plane from the rest of the set. The elements from this distance matrix are then used to calculate the corresponding interpolation matrix with each element in the interpolation matrix equal to the value of the RBF evaluated at the value of the distance matrix. Appropriately setting the product of the interpolation matrix and a vector of unknown coefficients equal to the vector comprised of the values for f(x,y) leads to the solutions for the unknown coefficients. With these coefficients, the surface can be reconstructed over a set of known test points and the validity of the model confirmed using a similar procedure with the exception that now the coefficients are known and the values for f(x,y) are being calculated and compared to the known values. The Inverse MQ and the Gaussian RBFs both include the shape parameter, c. To study the effect of this shape parameter and to get an estimate of what the optimal value is under each scenario (Halton, Random, and Mesh), three of the scripts developed for this assignment evaluate the RMSE of the 100 random test points using values of the shape parameter ranging from 0.5 to 10.5 in increments of These scripts identify the value for the shape parameter that minimizes the RMSE and also generates a plot for each type of RBF to illustrate the effect that varying the shape parameter has on the RMSE. These scripts are based on the RBFeff.m script povided with modifications made to accomplished the desired goals and are named RBFeffHaltonFindC.m, RBFeffRandomFindC.m, and RBFeffMeshFindC.m. Once the values for the corresponding shape parameters were determined, they were incorporated into the scripts titled RBFeffHalton.m, RBFeffRandom.m, and RBFeffMesh.m which evaluate each RBF over a given set of data points and test points to evaluate the performance of each method. These scripts are again based on the method outlined in the RBFeff. script provided with modifications made to display the max error and RMSE based on 100 random test points and then plot the reconstructed surface and error mesh over a uniform mesh grid. 2

4 Results Finding the Shape Parameter Below in Figures 1, 2, 3, 4, 5, and 6 are the plots illustrating the effect of varying the value of the shape parameter on the RMSE. These plots employ a log scale on the vertical axis and help simultaneously indicate the strong effect the shape parameter has while allowing for a visual approximation for the optimum value of the shape parameter. Figure 1: Effect of Shape Parameter on RMSE for Inverse MQ RBFs on Halton Quasi-Random Points. Figure 2: Effect of Shape Parameter on RMSE for Gaussian RBFs on Halton Quasi-Random Points. 3

5 Figure 3: Effect of Shape Parameter on RMSE for Inverse MQ RBFs on Random Points. Figure 4: Effect of Shape Parameter on RMSE for Gaussian RBFs on Random Points. 4

6 Figure 5: Effect of Shape Parameter on RMSE for Inverse MQ RBFs on Uniform Mesh. Figure 6: Effect of Shape Parameter on RMSE forgaussian RBFs on Uniform Mesh. 5

7 While these plots are adequate for visually referencing the effect and optimum shape parameter, the scripts were developed to also display the shape parameter that corresponded to the minimum value of RMSE. The table below provides the values generated and abbreviated Halton as H., Random as R., Mesh as M., Inverse MQ as IMQ, and Gaussian as G. Table 1: Shape parameter and corresponding MSRE Conditions Shape Parameter RMSE H. IMQ x10 5 R. IMQ x10 4 M. IMQ x10 5 H. G x10 4 R. G x10 4 M. G x10 4 Reconstructing the Surface and Comparing the Values As was previously discussed, these shape parameters were then incorporated into the scripts tasked with yielding the main results for this assignment. The sequence of figures shown below illustrates the following for each of the Halton Quasi-Random, Random, and Uniform Mesh points: The first pair of plots indicates the points generated to function as the centers and build the interpolation matrix, and the follow three pairs of plots show the recreated surface and a mesh grid indicating the error for each grid point. These are on a relatively tight grid with increment size equal to Following the plots is a table providing a summary of the numerical comparison of the max error and RMSE for each set of RBFs based on the performance of the reconstruction over 100 random test points. 6

8 Figure 7: Halton Quasi-Random Points used to define the TestFunction. Figure 8: Resulting Reconstructed surface and error using Inverse MQ RBFs from Halton Quasi-Random Points. 7

9 Figure 9: Resulting Reconstructed surface and error using r 5 RBFs from Halton Quasi-Random Points. Figure 10: Resulting Reconstructed surface and error using Gaussian RBFs from Halton Quasi-Random Points. 8

10 Figure 11: Random Points used to define the TestFunction. Figure 12: Resulting Reconstructed surface and error using Inverse MQ RBFs from Random points. 9

11 Figure 13: Resulting Reconstructed surface and error using r 5 RBFs from Random points.. Figure 14: Resulting Reconstructed surface and error using Gaussian RBFs from Random points. 10

12 Figure 15: Uniform Mesh Points used to define the TestFunction. Figure 16: Resulting Reconstructed surface and error using Inverse MQ RBFs from Uniform Mesh points. 11

13 Figure 17: Resulting Reconstructed surface and error using r 5 RBFs from Uniform Mesh points. Figure 18: Resulting Reconstructed surface and error using Gaussian RBFs from Uniform Mesh points. 12

14 Table 2: RMSE and MaxError corresponding to each RBF under each set of different starting data points and evaluated over 100 random test points Conditions RMSE Max Error H. IMQ 8.693x R. IMQ 4.334x M. IMQ 7.859x H. r x R. r x M. r x H. G 2.173x R. G 9.584x M. G 4.254x Conclusions Given the values shown in Table 2, it can be seen that in every case, starting with random data points to build the subsequent distance matrix and interpolation matrix results in the largest RMSE as well as the largest Max Error regardless of the RBF utilized. It can also be seen that relative to each different set of starting points the Inverse MQ RBFs yielded smaller RMSE and Max Error than either the r 5 and the Gaussian RBFs with the only exception being that the r 5 for the Halton Quasi-Random points had a Max Error slightly lower than the Max Error resulting from the use of the Inverse MQ RBF. The largest Max Error occurred using the r 5 RBF on the Random starting points. In general, each set of RBFs performed relatively well, but if they were to be ranked according to the overall performance on this task from best to worst, the ranking would be the following: Inverse MQ, Gaussian, and then r 5. It should be noted that the values do fluctuate slightly from one execution of the script to the next as they are tested on a random set of 100 test points (though the same 100 test points are used for each evaluation of Halton, Random, and Mesh respectively). 13

15 References [1] C.S.Chen,Y.C.Hon,R.A.Schaback Scientific Computing with Radial Basis Functions. Department of Mathematics, University of Southern Mississippi, USA. [2] Gregory E. Fasshauer, Meshfree Approximation Methods with Matlab. World Scientific,