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Support Vector Machines

In this assignment we will be using support vector machines to separate data points in a binary classification setup. We will be using the breast cancer dataset later on in the assignment.

About the Breast Cancer Dataset: The dataset contains 569 samples. Each feature vector is 30-dimensional and each target label is either 0 (meaning benign) or 1 (meaning malignant). Each point has the following features (read left to right, top to bottom):
radius_mean texture_mean perimeter_mean area_mean smoothness_mean compactness_mean
concavity_mean concave points_mean symmetry_mean fractal_dimension_mean radius_se texture_se
perimeter_se area_se smoothness_se compactness_se concavity_se concave points_se
symmetry_se fractal_dimension_se radius_worst texture_worst perimeter_worst area_worst
smoothness_worst compactness_worst concavity_worst concave points_worst symmetry_worst fractal_dimension_worst

A single data point might have the following feature vector:

[17.99, 10.38, 122.8, 1001, 0.1184, 0.2776, 0.3001, 0.1471, 0.2419, 0.07871, 1.095, 0.9053, 8.589, 153.4, 0.006399, 0.04904, 0.05373, 0.01587, 0.03003, 0.006193, 25.38, 17.33, 184.6, 2019, 0.1622, 0.6656, 0.7119, 0.2654, 0.4601, 0.1189]

which corresponds tha malignant diagnosis (the target is 0).

Section 1

We start by exploring the effect of using different kernels using a simple dataset. Next we look at how we train maximum margin classifiers using either hard or soft margins and then we apply support vector machines on a larger data set.

Section 1.1

Lets draw the decision boundary and margins of linear kernel support vector machine (SVM) of some data.

You can use _plot_linear_kernel() for this

  1. generate some data with sklearn.datasets.make_blobs. Make your blobs consist of 40 samples and 2 centers. 
    X, t = make_blobs(...)
  2. Create an instance of sklearn.svm.SVC and select linear as the kernel type. Choose the regularization parameter C=1000 to avoid regularization. 
    clf = scm.SVC(...)
  3. Plot the boundary using tools.plot_svm_margin. 
    plot_svm_margin(...)

Turn in your plot as 1_1_1.png in the PDF document.

For a very boring example of only two points, this plot looks like this:

Boring linear

Section 1.2

This question should be answered in your PDF document

  1. How many support vectors are there for each class in your example?
  2. What is the shape of the decision boundary?

Section 1.3

Implement a support vector machine with a radial basis function (rbf) using scikit learn and plot the outcome using the function plot_svm_margin. Use a very high value of C as before.

You should plot three different figures using plt.subplot as we did for example in Assignment 00.

You can use _compare_gamma() for this

These three plots will be used to compare the results you get for different values of the gamma parameter. Compare:

  1. Default value of gamma
  2. Low value gamma = 0.2
  3. High value gamma = 2

You will again use the sklearn.svm.SVC and the same data blobs as before.

To achieve this plot you can slightly tweak the tool.plot_svm_margin as you desire.

For the very boring case of only 4 data points you should get results similar to the following

Simple gamma

Present your plot as 1_3_1.png in your PDF document.

Section 1.4

This question should be answered in your PDF documnet

  1. How many support vectors are there for each class for each value of gamma?
  2. What is the shape of the decision boundary for each value of gamma?
  3. What difference does the gamma parameter make and why?

Section 1.5

Now using a linear basis function again as the kernel, compare different values of C: 1000, 0.5, 0.3, 0.05, 0.0001

Again turn in a single plot with all those cases using plt.subplot. You can use _compare_C for this.

For the very boring case of 4 points the plots should look something like this

Simple C compare

Turn in your plot as 1_5_1.png in your PDF document.

Section 1.6

This question should be answered in your PDF document

  1. How many support vectors are there for each class for each case of C?
  2. How many of those support vectors are within the margins?.
  3. Are any support vectors misclassified? If so, why?

Section 2

Lets try applying SVMs to larger datasets We will apply SVMs to the breast cancer dataset. You can access the dataset via:

(X_train, t_train), (X_test, t_test) = tools.load_cancer()

Apply an SVM with a linear kernel and a sigmoidal kernel and calculate the accuracy, precision and recall for each classifier that you design and implement.

Section 2.1

Create a function train_test_SVM(svc, X_train, t_train, X_test, t_test) that trains the SVM (svc) on [X_train, t_train] and returns the accuracy, precision and recall on the test set [X_test, t_test].

If we have a prediction y and the targets t_test, we can use the functions accuracy_score(t_test, y), precision_score(t_test, y) and recall_score(t_test, y).

Example inputs and outputs:

(X_train, t_train), (X_test, t_test) = load_cancer()
svc = svm.SVC(C=1000)

train_test_SVM(svc, X_train, t_train, X_test, t_test)

Output:

(0.9181286549707602, 0.9801980198019802, 0.8918918918918919)

Section 2.2

This question should be answered in your PDF document

Compare the results of your train_test_SVM function between linear, radial basis and polynomial kernel functions.

Which method seems to be the best for the task?

Independent section

This is an open ended independent question. You can choose to compare visually different parameters on the cancer dataset, different types of models, create your own data, etc.