def point_and_circle_clustering(eig_max=15):
"""
TO BE COMPLETED.
Used in question 2.8
"""
# Generate data and compute number of clusters
X, Y = point_and_circle(600)
num_classes = len(np.unique(Y))
"""
Choose parameters
"""
k = 0
var = 1.0 # exponential_euclidean's sigma^2
#chosen_eig_indices = [1, 2, 3] # indices of the ordered eigenvalues to pick
if k == 0: # compute epsilon
dists = sd.cdist(
X, X, 'euclidean'
) # dists[i, j] = euclidean distance between x_i and x_j
min_tree = min_span_tree(dists)
l = []
n1, m1 = min_tree.shape
for i in range(n1):
for j in range(m1):
if min_tree[i][j] == True:
l.append(dists[i][j])
#distance_threshold = sorted(l)[-1]
distance_threshold = sorted(l)[-1]
eps = np.exp(-(distance_threshold)**2.0 / (2 * var))
W = build_similarity_graph(X, var=var, eps=eps, k=k)
# build laplacian
else:
W = build_similarity_graph(X, var=var, k=k)
L_unn = build_laplacian(W, 'unn')
L_norm = build_laplacian(W, 'sym')
#eigenvalues,U = np.linalg.eig(L_unn)
#indexes = np.argsort(eigenvalues)
#eigenvalues = eigenvalues[indexes]
#U = U[:,indexes]
#chosen_eig_indices = choose_eigenvalues(eigenvalues, eig_max = eig_max)
chosen_eig_indices = [0, 1]
Y_unn = spectral_clustering(L_unn,
chosen_eig_indices,
num_classes=num_classes)
Y_norm = spectral_clustering(L_norm,
chosen_eig_indices,
num_classes=num_classes)
plot_clustering_result(X, Y, L_unn, Y_unn, Y_norm, 1)
def two_moons_clustering(eig_max=15):
"""
TO BE COMPLETED.
Used in question 2.7
"""
# Generate data and compute number of clusters
X, Y = two_moons(600)
num_classes = len(np.unique(Y))
"""
Choose parameters
"""
k = 0
var = 1.0 # exponential_euclidean's sigma^2
laplacian_normalization = 'unn'
# chosen_eig_indices = [0, 1, 2] # indices of the ordered eigenvalues to pick
if k == 0: # compute epsilon
dists = sd.cdist(
X, X, 'euclidean'
) # dists[i, j] = euclidean distance between x_i and x_j
min_tree = min_span_tree(dists)
l = []
n1, m1 = min_tree.shape
for i in range(n1):
for j in range(m1):
if min_tree[i][j] == True:
l.append(dists[i][j])
#distance_threshold = sorted(l)[-1]
distance_threshold = sorted(l)[-1]
eps = np.exp(-(distance_threshold)**2.0 / (2 * var))
# build laplacian
W = build_similarity_graph(X, var=var, eps=eps, k=k)
L = build_laplacian(W, laplacian_normalization)
# chose the eigenvalues
eigenvalues, U = np.linalg.eig(L)
indexes = np.argsort(eigenvalues)
eigenvalues = eigenvalues[indexes]
U = U[:, indexes]
chosen_eig_indices = choose_eigenvalues(eigenvalues, eig_max=eig_max)
Y_rec = spectral_clustering(L, chosen_eig_indices, num_classes=num_classes)
plot_clustering_result(X, Y, L, Y_rec, KMeans(num_classes).fit_predict(X))
def find_the_bend():
"""
TO BE COMPLETED
Used in question 2.3
:return:
"""
# the number of samples to generate
num_samples = 600
# Generate blobs and compute number of clusters
X, Y = blobs(num_samples, 4, 0.2)
num_classes = len(np.unique(Y))
"""
Choose parameters
"""
k = 0
var = 1 # exponential_euclidean's sigma^2
laplacian_normalization = 'unn' # either 'unn'normalized, 'sym'metric normalization or 'rw' random-walk normalization
# build laplacian
if k == 0:
dists = sd.cdist(X, X, metric="euclidean")
min_tree = min_span_tree(dists)
distance_threshold = dists[min_tree].max()
eps = np.exp(-distance_threshold**2.0 / (2 * var))
print(eps)
W = build_similarity_graph(X, var=var, k=k, eps=eps)
else:
W = build_similarity_graph(X, var=var, k=k)
L = build_laplacian(W, laplacian_normalization)
"""
compute first 15 eigenvalues and call choose_eigenvalues() to choose which ones to use.
"""
eigenvalues, vects = scipy.linalg.eig(L)
eigenvalues = sorted(eigenvalues.real)
# for ind,val in enumerate(eigenvalues[:15]):
# plt.scatter(ind, val)
# plt.xlabel("index of the eigenvalue")
# plt.ylabel("value of the eigenvalue")
# chosen_eig_indices = [0,1,2,3] # indices of the ordered eigenvalues to pick
"""
compute spectral clustering solution using a non-adaptive method first, and an adaptive one after (see handout)
Y_rec = (n x 1) cluster assignments [0,1,..., c-1]
"""
# run spectral clustering
# Y_rec = spectral_clustering(L, chosen_eig_indices, num_classes=num_classes)
Y_rec_adaptive = spectral_clustering_adaptive(L, num_classes=num_classes)
# plot_the_bend(X, Y, L, Y_rec, eigenvalues)
plot_the_bend(X, Y, L, Y_rec_adaptive, eigenvalues)
def two_blobs_clustering():
"""
TO BE COMPLETED
Clustering of two blobs. Used in questions 2.1 and 2.2
"""
# Get data and compute number of classes
X, Y = blobs(50, n_blobs=2, blob_var=0.15, surplus=0)
num_classes = len(np.unique(Y))
"""
Choose parameters
"""
k = 0
var = 1.0 # exponential_euclidean's sigma^2
laplacian_normalization = 'unn'
chosen_eig_indices = [0, 1,
2] # indices of the ordered eigenvalues to pick
if k == 0: # compute epsilon
dists = sd.cdist(
X, X, 'euclidean'
) # dists[i, j] = euclidean distance between x_i and x_j
min_tree = min_span_tree(dists)
l = []
n1, m1 = min_tree.shape
for i in range(n1):
for j in range(m1):
if min_tree[i][j] == True:
l.append(dists[i][j])
#distance_threshold = sorted(l)[-1]
distance_threshold = sorted(l)[-2]
eps = np.exp(-(distance_threshold)**2.0 / (2 * var))
#####
# build laplacian
W = build_similarity_graph(X, var=var, eps=eps, k=k)
plot_graph_matrix(X, Y, W)
L = build_laplacian(W, laplacian_normalization)
# run spectral clustering
Y_rec = spectral_clustering(L, chosen_eig_indices, num_classes=num_classes)
# Plot results
plot_clustering_result(X, Y, L, Y_rec, KMeans(num_classes).fit_predict(X))
def two_blobs_clustering():
"""
TO BE COMPLETED
Clustering of two blobs. Used in questions 2.1 and 2.2
"""
question = '2.2'
# Get data and compute number of classes
X, Y = blobs(600, n_blobs=2, blob_var=0.15, surplus=0)
num_classes = len(np.unique(Y))
n = X.shape[0]
"""
Choose parameters
"""
var = 1.0 # exponential_euclidean's sigma^2
laplacian_normalization = 'rw'
if question == '2.1':
# as the graph has to be connected in this question, we construct a epsilon-graph using a MST
dists = pairwise_distances(X).reshape(
(n, n)) # dists[i, j] = euclidean distance between x_i and x_j
min_tree = min_span_tree(dists)
distance_threshold = np.max(dists[min_tree])
eps = np.exp(-distance_threshold**2 / (2 * var))
# choice of eigenvectors to use
chosen_eig_indices = [1] # indices of the ordered eigenvalues to pick
# build similarity graph and laplacian
W = build_similarity_graph(X, var=var, eps=eps)
L = build_laplacian(W, laplacian_normalization)
elif question == '2.2':
# choice of eigenvectors to use
chosen_eig_indices = [0, 1]
# choice of k for the k-nn graph
k = 20
# build similarity graph and laplacian
W = build_similarity_graph(X, var=var, k=k)
L = build_laplacian(W, laplacian_normalization)
# run spectral clustering
Y_rec = spectral_clustering(L, chosen_eig_indices, num_classes=num_classes)
# Plot results
plot_clustering_result(X, Y, L, Y_rec, KMeans(num_classes).fit_predict(X))
def how_to_choose_epsilon():
"""
TO BE COMPLETED.
Consider the distance matrix with entries dist(x_i, x_j) (the euclidean distance between x_i and x_j)
representing a fully connected graph.
One way to choose the parameter epsilon to build a graph is to choose the maximum value of dist(x_i, x_j) where
(i,j) is an edge that is present in the minimal spanning tree of the fully connected graph. Then, the threshold
epsilon can be chosen as exp(-dist(x_i, x_j)**2.0/(2*sigma^2)).
"""
# the number of samples to generate
num_samples = 100
# the option necessary for worst_case_blob, try different values
gen_pam = 2.0 # to understand the meaning of the parameter, read worst_case_blob in generate_data.py
# get blob data
X, Y = worst_case_blob(num_samples, gen_pam)
# get two moons data
# X, Y = two_moons(num_samples)
n = X.shape[0]
"""
use the distance function and the min_span_tree function to build the minimal spanning tree min_tree
- var: the exponential_euclidean's sigma2 parameter
- dists: (n x n) matrix with euclidean distance between all possible couples of points
- min_tree: (n x n) indicator matrix for the edges in the minimal spanning tree
"""
var = 1.0
dists = pairwise_distances(X).reshape(
(n, n)) # dists[i, j] = euclidean distance between x_i and x_j
min_tree = min_span_tree(dists)
"""
set threshold epsilon to the max weight in min_tree
"""
distance_threshold = np.max(dists[min_tree])
eps = np.exp(-distance_threshold**2 / (2 * var))
"""
use the build_similarity_graph function to build the graph W
W: (n x n) dimensional matrix representing
the adjacency matrix of the graph
use plot_graph_matrix to plot the graph
"""
W = build_similarity_graph(X, var=var, eps=eps, k=0)
plot_graph_matrix(X, Y, W)
def two_moons_clustering():
"""
TO BE COMPLETED.
Used in question 2.7
"""
# Generate data and compute number of clusters
X, Y = two_moons(600)
num_classes = len(np.unique(Y))
"""
Choose parameters
"""
k = 0
var = 1.0 # exponential_euclidean's sigma^2
laplacian_normalization = 'unn'
chosen_eig_indices = [0, 1] # indices of the ordered eigenvalues to pick
# build laplacian
# build laplacian
if k == 0:
dists = sd.cdist(X, X, metric="euclidean")
min_tree = min_span_tree(dists)
distance_threshold = dists[min_tree].max()
eps = np.exp(-distance_threshold**2.0 / (2 * var))
print(eps)
W = build_similarity_graph(X, var=var, k=k, eps=eps)
else:
W = build_similarity_graph(X, var=var, k=k)
L = build_laplacian(W, laplacian_normalization)
# Y_rec = spectral_clustering(L, chosen_eig_indices, num_classes=num_classes)
#
# plot_clustering_result(X, Y, L, Y_rec, KMeans(num_classes).fit_predict(X))
Y_rec_adaptive = spectral_clustering_adaptive(L, num_classes=num_classes)
plot_clustering_result(X, Y, L, Y_rec_adaptive,
KMeans(num_classes).fit_predict(X))
def two_blobs_clustering():
"""
TO BE COMPLETED
Clustering of two blobs. Used in questions 2.1 and 2.2
"""
# Get data and compute number of classes
X, Y = blobs(600, n_blobs=2, blob_var=0.15, surplus=0)
num_classes = len(np.unique(Y))
"""
Choose parameters
"""
k = 0
var = 1.0 # exponential_euclidean's sigma^2
laplacian_normalization = 'unn'
chosen_eig_indices = [0, 1] # indices of the ordered eigenvalues to pick
# build laplacian
if k == 0:
dists = sd.cdist(X, X, metric="euclidean")
min_tree = min_span_tree(dists)
distance_threshold = dists[min_tree].max()
eps = np.exp(-distance_threshold**2.0 / (2 * var))
W = build_similarity_graph(X, var=var, k=k, eps=eps)
else:
W = build_similarity_graph(X, var=var, k=k)
L = build_laplacian(W, laplacian_normalization)
# run spectral clustering
Y_rec = spectral_clustering(L, chosen_eig_indices, num_classes=num_classes)
# Plot results
plot_clustering_result(X, Y, L, Y_rec, KMeans(num_classes).fit_predict(X))
def parameter_sensitivity(eig_max=15):
"""
TO BE COMPLETED.
A function to test spectral clustering sensitivity to parameter choice.
Used in question 2.9
"""
# the number of samples to generate
num_samples = 500
"""
Choose parameters
"""
var = 1.0 # exponential_euclidean's sigma^2
laplacian_normalization = 'unn'
#chosen_eig_indices = [0, 1, 2]
"""
Choose candidate parameters
"""
parameter_candidate = [
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
] # the number of neighbours for the graph or the epsilon threshold
parameter_performance = []
for k in parameter_candidate:
# Generate data
X, Y = two_moons(num_samples, 1, 0.02)
num_classes = len(np.unique(Y))
if k == 0: # compute epsilon
dists = sd.cdist(
X, X, 'euclidean'
) # dists[i, j] = euclidean distance between x_i and x_j
min_tree = min_span_tree(dists)
l = []
n1, m1 = min_tree.shape
for i in range(n1):
for j in range(m1):
if min_tree[i][j] == True:
l.append(dists[i][j])
distance_threshold = sorted(l)[-1]
eps = np.exp(-(distance_threshold)**2.0 / (2 * var))
W = build_similarity_graph(X, var=var, eps=eps, k=k)
else:
W = build_similarity_graph(X, k=k)
L = build_laplacian(W, laplacian_normalization)
eigenvalues, U = np.linalg.eig(L)
indexes = np.argsort(eigenvalues)
eigenvalues = eigenvalues[indexes]
U = U[:, indexes]
chosen_eig_indices = choose_eigenvalues(eigenvalues, eig_max=eig_max)
Y_rec = spectral_clustering(L, chosen_eig_indices, num_classes)
parameter_performance += [skm.adjusted_rand_score(Y, Y_rec)]
plt.figure()
plt.plot(parameter_candidate, parameter_performance)
plt.title('parameter sensitivity')
plt.show()
#parameter_sensitivity()
def find_the_bend(eig_max=15, blob_var=0.03):
"""
TO BE COMPLETED
Used in question 2.3
:return:
"""
eig_max -= 1 # to count starting from 0
# the number of samples to generate
num_samples = 600
# Generate blobs and compute number of clusters
X, Y = blobs(num_samples, 4, blob_var)
num_classes = len(np.unique(Y))
"""
Choose parameters
"""
k = 0
var = 1.0 # exponential_euclidean's sigma^2
laplacian_normalization = 'sym' # either 'unn'normalized, 'sym'metric normalization or 'rw' random-walk normalization
if k == 0: # compute epsilon
dists = sd.cdist(
X, X, 'euclidean'
) # dists[i, j] = euclidean distance between x_i and x_j
min_tree = min_span_tree(dists)
l = []
n1, m1 = min_tree.shape
for i in range(n1):
for j in range(m1):
if min_tree[i][j] == True:
l.append(dists[i][j])
#distance_threshold = sorted(l)[-1]
distance_threshold = sorted(l)[-num_classes]
eps = np.exp(-(distance_threshold)**2.0 / (2 * var))
# build laplacian
W = build_similarity_graph(X, var=var, eps=eps, k=k)
L = build_laplacian(W, laplacian_normalization)
"""
compute first 15 eigenvalues and call choose_eigenvalues() to choose which ones to use.
"""
eigenvalues, U = np.linalg.eig(L)
indexes = np.argsort(eigenvalues)
eigenvalues = eigenvalues[indexes]
U = U[:, indexes]
chosen_eig_indices = choose_eigenvalues(
eigenvalues,
eig_max=eig_max) # indices of the ordered eigenvalues to pick
plt.plot(eigenvalues, [i for i in range(len(eigenvalues))], 'r+')
"""
compute spectral clustering solution using a non-adaptive method first, and an adaptive one after (see handout)
Y_rec = (n x 1) cluster assignments [0,1,..., c-1]
"""
# run spectral clustering
Y_rec = spectral_clustering(L, chosen_eig_indices, num_classes=num_classes)
Y_rec_adaptive = spectral_clustering_adaptive(L,
num_classes=num_classes,
eig_max=eig_max)
plot_the_bend(X, Y, L, Y_rec_adaptive, eigenvalues)
def how_to_choose_epsilon(gen_pam, k):
"""
TO BE COMPLETED.
Consider the distance matrix with entries dist(x_i, x_j) (the euclidean distance between x_i and x_j)
representing a fully connected graph.
One way to choose the parameter epsilon to build a graph is to choose the maximum value of dist(x_i, x_j) where
(i,j) is an edge that is present in the minimal spanning tree of the fully connected graph. Then, the threshold
epsilon can be chosen as exp(-dist(x_i, x_j)**2.0/(2*sigma^2)).
"""
# the number of samples to generate
num_samples = 100
# the option necessary for worst_case_blob, try different values
#gen_pam = 10 # to understand the meaning of the parameter, read worst_case_blob in generate_data.py
# get blob data
# X, Y = worst_case_blob(num_samples, gen_pam)
X, Y = two_moons(num_samples)
"""
use the distance function and the min_span_tree function to build the minimal spanning tree min_tree
- var: the exponential_euclidean's sigma2 parameter
- dists: (n x n) matrix with euclidean distance between all possible couples of points
- min_tree: (n x n) indicator matrix for the edges in the minimal spanning tree
"""
var = 1.0
dists = sd.cdist(
X, X,
'euclidean') # dists[i, j] = euclidean distance between x_i and x_j
min_tree = min_span_tree(dists)
l = []
n1, m1 = min_tree.shape
for i in range(n1):
for j in range(m1):
if min_tree[i][j] == True:
l.append([(i, j), dists[i][j]])
l = sorted(l, key=lambda x: x[1], reverse=True)
#print(min_tree)
"""
set threshold epsilon to the max weight in min_tree
"""
distance_threshold = l[0][1]
eps = np.exp(-distance_threshold**2.0 / (2 * var))
"""
use the build_similarity_graph function to build the graph W
W: (n x n) dimensional matrix representing
the adjacency matrix of the graph
use plot_graph_matrix to plot the graph
"""
W = build_similarity_graph(X, var=var, eps=eps, k=k)
plot_graph_matrix(X, Y, W)
return eps, X, Y, W
#if __name__ == '__main__':
# for gp in [0,1,10,100]:
# print(gp)
# how_to_choose_epsilon(gp,0)
# for k in [0,1,2,5,10]:
# how_to_choose_epsilon(0,k)
distance_threshold = np.max(dists[min_tree])
eps = np.exp(-distance_threshold**2 / (2 * var))
"""
use the build_similarity_graph function to build the graph W
W: (n x n) dimensional matrix representing
the adjacency matrix of the graph
use plot_graph_matrix to plot the graph
"""
W = build_similarity_graph(X, var=var, eps=eps, k=0)
plot_graph_matrix(X, Y, W)
if __name__ == '__main__':
n = 300
blobs_data, blobs_clusters = blobs(n)
moons_data, moons_clusters = two_moons(n)
point_circle_data, point_circle_clusters = point_and_circle(n)
worst_blobs_data, worst_blobs_clusters = worst_case_blob(n, 1.0)
var = 1
X, Y = moons_data, moons_clusters
n_samples = X.shape[0]
dists = pairwise_distances(X).reshape((n_samples, n_samples))
min_tree = min_span_tree(dists)
eps = np.exp(-np.max(dists[min_tree])**2 / (2 * var))
W_eps = build_similarity_graph(X, var=var, eps=0.6)
W_knn = build_similarity_graph(X, k=15)
plot_graph_matrix(X, Y, W_eps)
plot_graph_matrix(X, Y, W_knn)
