Radial Basis Function Network

Introduction

This program demonstrates some function approximation capabilities of a Radial Basis Function Network.

The user supplies a set of training points which represent some "sample" points for some arbitrary curve. Next, the user specifies the number of equally spaced gaussian centers and the variance for the network. Using the training samples, the weights multiplying each of the gaussian basis functions arecalculated using the pseudo-inverse (yielding the minimum least-squares solution). The resulting network is then used to approximate the function between the given "sample" points.

Credits

This Java applet (© 1996 - Jesse W. Hong, Massachusetts Institute of Technology) was integrated as is into this web page with the agreement of the author. Please send comments about this applet directly to jesse@mit.edu.

Instructions

• Use the first mouse button to place training points in the upper section. Try to space them approximately equally, and place more points than number of centers being used.
• Adjust the number of centers and the standard deviation (width of Gaussian) for the Gaussians used in the approximation.
• Click on "Redraw" to redraw the basis functions and the training points. After any adjustment to the number of centers or variance, click on this button to show the new basis functions.
• Click on "Go!" to train the network.
• The "Reset" button erases all the training points and lets you start again.
• Status messages are shown at the bottom of the applet.

Comments

After learning, the scaled plots of each of the gaussians is shown in green in the upper graph. The resulting approximation of the curve which is the sum of all the scaled gaussians is shown in red.

Try to make sure that your data points are equally spaced. If they are not and you have a lot of centers, then in between some of the data points, the fitted curve may go off screen. Just add a new point in these regions and click on "Go!" again.

Play around with the number of centers and the standard deviation and see how the smoothness and accuracy of the approximation is affected. E.g. make the standard deviation 0.25, place 10 equally spaced data points, click on "Go!", then make the standard deviation 1.0, and click on "Go!" again. You can also play with the number of data points, etc.

Questions

1. Put down 32 data points so that the data looks like a noisy version of some function. Take the width of the Gaussians as 1.5.
   a) As a first step, try to do an interpolation, that is, use 32 Gaussians. Type 'go' and look at the result. Then reduce the number of centers to 16, then to 8, and finally to 4. What is the result?
   b) Repeat the same sequence but with a width of 1.0 and 0.5. What is the result?
2. Put down about 30 data points in two clusters, one for low x-values and another one for high x-values. Take 10 centers.
   a) Do a fit with standard deviation 1.5, then with 1.0. Is the result reasonable?
   b) Now add one extra data point in the middle between the two clusters. Is the result getting better? Try to optimize parameters (number of centers and standard deviation).