This example aims to assess whether a lump in a breast could be malignant (cancerous) or benign (non-cancerous) from digitized images of a fine-needle aspiration biopsy.
The breast cancer database was obtained from the University of Wisconsin Hospitals, Madison, from Dr. William H. Wolberg.
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The variable to be predicted can have two values (malignant or benignant tumor). Therefore, this is a binary classification project.
The goal here is to model the probability of a malignant tumor, conditioned on the fine needle aspiration test features.
The breast_cancer.csv file contains the data for this application. Target variables can only have two values in a classification model: 0 (false) or 1 (true). The number of instances (rows) in the data set is 683, and the number of variables (columns) is 10.
The number of input variables, or attributes for each sample, is 9. All input variables are numeric-valued and represent measurements from digitized images of a fine-needle aspiration biopsy. The number of target variables is 1 and represents the absence or presence of cancer in an individual. The following list summarizes the variables information:
Finally, the use of all instances is set. Note that each instance contains the input and target variables of a different patient. The data set is divided into training, validation, and testing subsets. 60% of the instances will be assigned for training, 20% for generalization, and 20% for testing.
Once the data set has been set, we are ready to perform a few related analytics. With that, we check the provided information and make sure that the data has good quality.
We can calculate the data statistics and draw a table with the minimums, maximums, means, and standard deviations of all variables in the data set. The next table depicts the values.
All variables range from 1 to 10. On the other hand, note that the mean of all variables is less than 5. Note that the input variable with the smallest standard deviation is "mitoses".
Also, we can calculate the distributions for all variables. The following figure shows a pie chart with the numbers of malignant (positives) and benign (negatives) tumors in the data set.
As we can see, malignant tumors represent 35% of the samples, and benign tumors represent 65% of the samples approximately.
The inputs-targets correlations might indicate to us what factors most influence a tumor to be malignant or benign.
Here, the most correlated variables with malignant tumors are bare nuclei, cell shape uniformity, and cell size uniformity.
The second step is to set a neural network to represent the classification function. For this class of applications, the neural network is composed of:
The scaling layer contains the statistics on the inputs calculated from the data file and the method for scaling the input variables. Here the minimum-maximum method has been set. Nevertheless, the mean-standard deviation method would produce very similar results.
Two perceptron layers with a hidden logistic layer and a logistic output layer are used. Note that, since the logistic function ranges from 0 to 1, the outputs from that layer can be interpreted as probabilities. The neural network must have nine inputs since there are eight input variables and one output since there is one target variable. As an initial guess, we use three neurons in the hidden layer.
The probabilistic layer only contains the method for interpreting the outputs as probabilities. Indeed, as the sum of all outputs from a probabilistic layer must be 1, that two methods would always yield one here since there is only one output. Moreover, as the output layer's activation function is the logistic, that output can already be interpreted as a probability of class membership.
The following figure is a graphical representation of this neural network for breast cancer diagnosis.
The fourth step is to set the training strategy, which is composed of two terms:
The loss index is the weighted squared error with L2 regularization. This is the default loss index for binary classification applications.
The learning problem can be stated as finding a neural network that minimizes the loss index. That is, a neural network that fits the data set (error term) and does not oscillate (regularization term).
The optimization algorithm that we use is the quasi-Newton method. This is also the standard optimization algorithm for this type of problem.
The following chart shows how the error decreases with the iterations during the training process. The final training and selection errors are training error = 0.054 WSE and selection error = 0.072 WSE, respectively.
The objective of model selection is to find the network architecture with the best generalization properties, that is, that which minimizes the error on the selected instances of the data set.
More specifically, we want to find a neural network with a selection error of less than 0.072 WSE, which is the value that we have achieved so far.
Order selection algorithms train several network architectures with a different number of neurons and select that with the smallest selection error.
The incremental order method starts with a small number of neurons and increases the complexity at each iteration. The following chart shows the training error (blue) and the selection error (orange) as a function of the number of neurons.
The figure below shows the final architecture for the neural network.
The objective of the testing analysis is to validate the generalization performance of the trained neural network. To validate a classification technique, we need to compare the values provided by this technique to the observed values. We can use the ROC curve as it is the standard testing method for binary classification projects.
The following table contains the elements of the confusion matrix. This matrix contains the true positives, false positives, false negatives, and true negatives for the variable diagnose.
|Predicted positive||Predicted negative|
The binary classification tests are parameters for measuring the performance of a classification problem with two classes:
Once the neural network's generalization performance has been tested, the neural network can be saved for future use in the so-called model deployment mode.
We can diagnose new patients by calculating the neural network outputs. For that, we need to know the input variables for them. An example is the following:
The mathematical expression represented by the neural network is written below. It takes the inputs clump_thickness, cell_size_uniformity, cell_shape_uniformity, marginal_adhesion, single_epithelial_cell_size, bare_nuclei, bland_chromatin, normal_nucleoli and mitoses to produce the output diagnose. For classification problems, the information is propagated in a feed-forward fashion through the scaling, perceptron, and probabilistic layers.
scaled_clump_thickness = (clump_thickness-4.44217)/2.82076; scaled_cell_size_uniformity = (cell_size_uniformity-3.15081)/3.06514; scaled_cell_shape_uniformity = (cell_shape_uniformity-3.21523)/2.98858; scaled_marginal_adhesion = (marginal_adhesion-2.83016)/2.86456; scaled_single_epithelial_cell_size = (single_epithelial_cell_size-3.23426)/2.22309; scaled_bare_nuclei = (bare_nuclei-3.54466)/3.64386; scaled_bland_chromatin = (bland_chromatin-3.4451)/2.4497; scaled_normal_nucleoli = (normal_nucleoli-2.86969)/3.05267; scaled_mitoses = (mitoses-1.60322)/1.73267; y_1_1 = Logistic (-1.35621+ (scaled_clump_thickness*-2.54409)+ (scaled_cell_size_uniformity*-5.01572) + (scaled_cell_shape_uniformity*-3.39576)+ (scaled_marginal_adhesion*-0.278873)+ (scaled_single_epithelial_cell_size*-2.61646) + (scaled_bare_nuclei*-5.51018)+ (scaled_bland_chromatin*-0.979982)+ (scaled_normal_nucleoli*-1.71412)+ (scaled_mitoses*0.410197)); non_probabilistic_diagnose = Logistic (3.94959+ (y_1_1*-9.14654)); diagnose = Probability(non_probabilistic_diagnose);
The above expression can be exported anywhere, for instance, to a dedicated diagnosis software used by doctors.
You can watch the step by step tutorial video below to help you complete this Machine Learning example for free using the easy-to-use machine learning software Neural Designer.