Description
1. **Support Vector Machines with Synthetic Data**
, 50
points. ¶
For this problem, we will generate synthetic data for a nonlinear binary classification problem and partition it into
training, validation and test sets. Our goal is to understand the behavior of SVMs with Radial-Basis Function
(RBF) kernels with different values of C and γ.
In [1]: # DO NOT EDIT THIS FUNCTION; IF YOU WANT TO PLAY AROUND WITH DATA GENERATION,
# MAKE A COPY OF THIS FUNCTION AND THEN EDIT
#
import numpy as np
from sklearn.datasets import make_moons
from sklearn.model_selection import train_test_split
import matplotlib.pyplot as plt
from matplotlib.colors import ListedColormap
def generate_data(n_samples, tst_frac=0.2, val_frac=0.2):
# Generate a non-linear data set
X, y = make_moons(n_samples=n_samples, noise=0.25, random_state=42)
# Take a small subset of the data and make it VERY noisy; that is, generate
outliers
m = 30
np.random.seed(30) # Deliberately use a different seed
ind = np.random.permutation(n_samples)[:m]
X[ind, :] += np.random.multivariate_normal([0, 0], np.eye(2), (m, ))
y[ind] = 1 – y[ind]
# Plot this data
cmap = ListedColormap([‘#b30065’, ‘#178000′])
plt.scatter(X[:, 0], X[:, 1], c=y, cmap=cmap, edgecolors=’k’)
# First, we use train_test_split to partition (X, y) into training and test
sets
X_trn, X_tst, y_trn, y_tst = train_test_split(X, y, test_size=tst_frac,
random_state=42)
# Next, we use train_test_split to further partition (X_trn, y_trn) into tra
ining and validation sets
X_trn, X_val, y_trn, y_val = train_test_split(X_trn, y_trn, test_size=val_fr
ac,
random_state=42)
return (X_trn, y_trn), (X_val, y_val), (X_tst, y_tst)
In [2]: #
# DO NOT EDIT THIS FUNCTION; IF YOU WANT TO PLAY AROUND WITH VISUALIZATION,
# MAKE A COPY OF THIS FUNCTION AND THEN EDIT
#
def visualize(models, param, X, y):
# Initialize plotting
if len(models) % 3 == 0:
nrows = len(models) // 3
else:
nrows = len(models) // 3 + 1
fig, axes = plt.subplots(nrows=nrows, ncols=3, figsize=(15, 5.0 * nrows))
cmap = ListedColormap([‘#b30065’, ‘#178000’])
# Create a mesh
xMin, xMax = X[:, 0].min() – 1, X[:, 0].max() + 1
yMin, yMax = X[:, 1].min() – 1, X[:, 1].max() + 1
xMesh, yMesh = np.meshgrid(np.arange(xMin, xMax, 0.01),
np.arange(yMin, yMax, 0.01))
for i, (p, clf) in enumerate(models.items()):
# if i > 0:
# break
r, c = np.divmod(i, 3)
ax = axes[r, c]
# Plot contours
zMesh = clf.decision_function(np.c_[xMesh.ravel(), yMesh.ravel()])
zMesh = zMesh.reshape(xMesh.shape)
ax.contourf(xMesh, yMesh, zMesh, cmap=plt.cm.PiYG, alpha=0.6)
if (param == ‘C’ and p > 0.0) or (param == ‘gamma’):
ax.contour(xMesh, yMesh, zMesh, colors=’k’, levels=[-1, 0, 1],
alpha=0.5, linestyles=[‘–‘, ‘-‘, ‘–‘])
# Plot data
ax.scatter(X[:, 0], X[:, 1], c=y, cmap=cmap, edgecolors=’k’)
ax.set_title(‘{0} = {1}’.format(param, p))
In [3]: # Generate the data
n_samples = 300 # Total size of data set
(X_trn, y_trn), (X_val, y_val), (X_tst, y_tst) = generate_data(n_samples)
a. (25 points) The effect of the regularization parameter,
Complete the Python code snippet below that takes the generated synthetic 2-d data as input and learns nonlinear SVMs. Use scikit-learn’s SVC (https://scikit-learn.org/stable/modules/generated/sklearn.svm.SVC.html)
function to learn SVM models with radial-basis kernels for fixed and various choices of
. The value of is fixed to , where is the data dimension and
is the standard deviation of the data set . SVC can automatically use these setting for if you pass the
argument gamma = ‘scale’ (see documentation for more details).
Plot: For each classifier, compute both the training error and the validation error. Plot them together, making
sure to label the axes and each curve clearly.
Discussion: How do the training error and the validation error change with ? Based on the visualization of the
models and their resulting classifiers, how does changing change the models? Explain in terms of minimizing
the SVM’s objective function , where is the hinge loss for each training
example .
Final Model Selection: Use the validation set to select the best the classifier corresponding to the best value,
. Report the accuracy on the test set for this selected best SVM model. Note: You should report a single
number, your final test set accuracy on the model corresponding to $C{best}$_.
C
γ
C ∈ {10 , ⋯, 1, ⋯ }
−3 10
−2 10
5 γ γ =
1
d⋅σX
d
σX X γ
C
C
w + C ℓ(w ∣ , )
1
2 w′ Σn
i=1 xi yi ℓ
(xi, yi)
Cbest
In [4]: # Learn support vector classifiers with a radial-basis function kernel with
# fixed gamma = 1 / (n_features * X.std()) and different values of C
C_range = np.arange(-3.0, 6.0, 1.0)
C_values = np.power(10.0, C_range)
models = dict()
trnErr = dict()
valErr = dict()
for C in C_values:
#
#
# Insert your code here to learn SVM models
#
#
visualize(models, ‘C’, X_trn, y_trn)
#
#
# Insert your code here to perform model selection
#
#
b. (25 points) The effect of the RBF kernel parameter,
Complete the Python code snippet below that takes the generated synthetic 2-d data as input and learns various
non-linear SVMs. Use scikit-learn’s SVC (https://scikitlearn.org/stable/modules/generated/sklearn.svm.SVC.html) function to learn SVM models with radial-basis
kernels for fixed and various choices of . The value of is fixed to
.
Plot: For each classifier, compute both the training error and the validation error. Plot them together, making
sure to label the axes and each curve clearly.
Discussion: How do the training error and the validation error change with ? Based on the visualization of the
models and their resulting classifiers, how does changing change the models? Explain in terms of the
functional form of the RBF kernel,
Final Model Selection: Use the validation set to select the best the classifier corresponding to the best value,
. Report the accuracy on the test set for this selected best SVM model. Note: You should report a single
number, your final test set accuracy on the model corresponding to $\gamma{best}$_.
γ
C γ ∈ {10 , 1, 10, }
−2 10
−1 10
2 10
3 C
C = 10
γ
γ
κ(x, z) = exp(−γ ⋅ ∥x − z∥ )
2
γbest
File “
visualize(models, ‘C’, X_trn, y_trn)
^
IndentationError: expected an indented block
In [ ]: # Learn support vector classifiers with a radial-basis function kernel with
# fixed C = 10.0 and different values of gamma
gamma_range = np.arange(-2.0, 4.0, 1.0)
gamma_values = np.power(10.0, gamma_range)
models = dict()
trnErr = dict()
valErr = dict()
for G in gamma_values:
#
#
# Insert your code here to learn SVM models
#
#
visualize(models, ‘gamma’, X_trn, y_trn)
#
#
# Insert your code here to perform model selection
#
#
2. **Breast Cancer Diagnosis with Support Vector
Machines**
, 25 points.
For this problem, we will use the Wisconsin Breast Cancer
(https://archive.ics.uci.edu/ml/datasets/Breast+Cancer+Wisconsin+(Diagnostic)) data set, which has already
been pre-processed and partitioned into training, validation and test sets. Numpy’s loadtxt
(https://docs.scipy.org/doc/numpy-1.13.0/reference/generated/numpy.loadtxt.html) command can be used to load
CSV files.
In [ ]: # Load the Breast Cancer Diagnosis data set; download the files from eLearning
# CSV files can be read easily using np.loadtxt()
#
# Insert your code here.
#
Use scikit-learn’s SVC (https://scikit-learn.org/stable/modules/generated/sklearn.svm.SVC.html) function to learn
SVM models with radial-basis kernels for each combination of and
. Print the tables corresponding to the training and validation errors.
Final Model Selection: Use the validation set to select the best the classifier corresponding to the best
parameter values, and . Report the accuracy on the test set for this selected best SVM model. Note:
You should report a single number, your final test set accuracy on the model corresponding to $C{best}
\gamma{best}$.
C ∈ {10 , , 1, , ⋯ }
−2 10
−1 10
1 10
4
γ ∈ {10 , , 1, 10, } −3 10
−2 10
−1 10
2
Cbest γbest
and
In [ ]: #
#
# Insert your code here to perform model selection
#
#
3. **Breast Cancer Diagnosis with -Nearest
Neighbors**
, 25 points.
k
Use scikit-learn’s k-nearest neighbor (https://scikitlearn.org/stable/modules/generated/sklearn.neighbors.KNeighborsClassifier.html) classifier to learn models for
Breast Cancer Diagnosis with , with the kd-tree algorithm.
Plot: For each classifier, compute both the training error and the validation error. Plot them together, making
sure to label the axes and each curve clearly.
Final Model Selection: Use the validation set to select the best the classifier corresponding to the best
parameter value, . Report the accuracy on the test set for this selected best kNN model. Note: You should
report a single number, your final test set accuracy on the model corresponding to $k{best}$_.
k ∈ {1, 5, 11, 15, 21}
kbest
In [ ]: #
#
# Insert your code here to perform model selection
#
#
Discussion: Which of these two approaches, SVMs or kNN, would you prefer for this classification task?
Explain.