Classification of Paper Quality Using Neural Networks:Radial Basis Function Network

A radial basis function (RBF) network will be used on the cardboard paper data. RBF networks are often not suitable for use in high-dimensional problems like this one, which has 15 input dimensions. It is, however, possible to choose to use only a subset of the inputs. Choosing a subset of inputs can be seen as a simple form of feature extraction. To simplify the problem, only three inputs will be used and only two of the paper types will be classified. Due to the random nature of the initialization and training processes, the result will vary each time the commands are evaluated.

Load the Mathematica application package Neural Networks and the data.

Select input data from sensors 6, 7, and 8 and class data from classes 4 through 6.

For classification, it is advantageous to add a sigmoidal nonlinearity at the output of the RBF network. This constrains the output to the range 0 to 1. Also, a better-conditioned training problem is often obtained if the linear submodel is excluded.

Initialize the RBF network with six neurons, no linear submodel, and a Sigmoid output nonlinearity.

Train the RBF network for 20 iterations.

The trained RBF network can now be used to classify input vectors by applying the network to them.

Classify the 24th paper data sample from the validation data.

The output of this RBF network is a real-valued function that takes values between 0 and 1. A crisp classification can be obtained in various ways. A simple way is to set values larger than 0.5 to 1 and smaller than 0.5 to 0 in the following manner. (Here, True rather than 1 indicates the sample's class.)

The classification can also be displayed with a bar chart in which the correctly classified data is on the diagonal and the incorrectly classified samples are off the diagonal. A crisp classification can be obtained by changing the output nonlinearity from the smooth step sigmoid to the discrete step by entering the option OutputNonlinearity -> UnitStep.

Plot the classification result on the validation data.

The classification result is not particularly impressive, but due to the randomness in the initialization, you may reevaluate the example, and the result might become better.

Use NetPlot to look at the classification performance improvement during training for each class.

Plot the progress of the classifier on the validation data.

You can repeat the example, selecting different components of the input values and different classes for the output. Because the radial basis functions have local support where they are nonzero, the training often gets trapped in poor local minima. Quite frequently, one of the classes will be totally misclassified. You will see this if you reevaluate the example with a different initialization of the RBF network. Correct classification is obtained when there is one basis function at each class center. If any of the class centers has not managed to attract a basis function during the training, then this class will not be correctly classified.