-
Notifications
You must be signed in to change notification settings - Fork 2
/
Copy pathvbgmm.py
189 lines (159 loc) · 6.48 KB
/
vbgmm.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
# -*- coding: utf-8 -*-
import numpy as np
import scipy.special as sp
import matplotlib.pyplot as plt
import matplotlib as mpl
class VariationalGaussianMixture():
"""Variational Bayesian estimation of a Gaussian mixture."""
def __init__(self, x, K):
self.x = x
self.N, self.D = x.shape
self.K = K
self.alpha0 = self.D + 1
self.beta0 = self.D + 1
self.nu0 = self.D + 1
self.u0 = self._get_u0()
self.V0 = self._get_V0()
self._init_params()
def _logdet(self, mat):
(sign, logdet) = np.linalg.slogdet(mat)
return logdet if sign == 1 else 0
def _get_u0(self):
return np.mean(self.x, axis=0)
def _get_V0(self):
diff = self.x - self.u0
return np.linalg.inv(self.nu0 * diff.T.dot(diff) / self.N)
def _init_u(self):
cov = np.linalg.inv(self.nu0*self.V0) / self.beta0
return np.random.multivariate_normal(self.u0, cov)
def _init_params(self):
self.alpha = np.tile(self.alpha0, self.K)
self.beta = np.tile(self.beta0, self.K)
self.nu = np.tile(self.nu0, self.K)
self.u = np.array([self._init_u() for _ in range(self.K)])
self.V = np.tile(self.V0, (self.K, 1, 1))
self.rho = np.zeros((self.N, self.K))
self.r = np.zeros((self.N, self.K))
self.eta = np.zeros(self.K)
def _update_rho(self, n, k):
sum_digamma = np.sum([sp.digamma(0.5*(self.nu[k]+1-d))
for d in range(1, self.D+1)])
diff = self.x[n, :] - self.u[k, :]
self.rho[n, k] = np.exp(sp.digamma(self.alpha[k]) +
0.5*(sum_digamma +
self._logdet(self.V[k, :]) -
self.D/self.beta[k] - self.nu[k] *
diff.dot(self.V[k, :]).dot(diff.T)
)
)
def _update_r(self, n, k):
tmpsum = np.sum(self.rho[n, :])
self.r[n, k] = self.rho[n, k] / tmpsum if tmpsum != 0 else 0
def _update_eta(self, k):
self.eta[k] = np.sum(self.r[:, k])
def _update_alpha(self, k):
self.alpha[k] = self.alpha0 + self.eta[k]
def _update_beta(self, k):
self.beta[k] = self.beta0 + self.eta[k]
def _update_nu(self, k):
self.nu[k] = self.nu0 + self.eta[k]
def _update_u(self, k):
self.u[k, :] = self.beta0*self.u0 + self.r[:, k].T.dot(self.x)
self.u[k, :] /= self.beta[k]
def _update_V(self, k):
cov_sample = np.zeros_like(self.V0)
diff_x = self.x - self.u[k]
for n in range(self.N):
cov_sample += self.r[n, k]*np.outer(diff_x[n, :], diff_x[n, :])
diff = self.u[k] - self.u0
self.V[k, :] = np.linalg.inv(
np.linalg.inv(self.V0) +
self.beta0 * np.outer(diff, diff) +
cov_sample
)
def update_params(self):
for k in range(self.K):
for n in range(self.N):
self._update_rho(n, k)
self._update_r(n, k)
self._update_eta(k)
self._update_alpha(k)
self._update_beta(k)
self._update_nu(k)
self._update_u(k)
self._update_V(k)
def fit(self, eps=1e-2, max_epochs=50, print_diff=False):
epochs = 0
while(True):
eta_prev = self.eta.copy()
self.update_params()
epochs += 1
diff = np.linalg.norm(self.eta - eta_prev)
if print_diff:
print(diff)
if (epochs > 5 and
(diff < eps or
np.abs(diff_prev - diff) < eps or
epochs >= max_epochs)):
break
diff_prev = diff
print(f'# iterations: {epochs}')
self.map_estimate()
def map_estimate(self):
self.w = np.zeros(self.K)
self.means = self.u
self.covs = np.zeros_like(self.V)
for k in range(self.K):
self.w[k] = self.alpha[k]/np.ones(self.K).dot(self.alpha)
self.covs[k, :] = np.linalg.inv(self.nu[k] * self.V[k, :])
print(f'w: {self.w}\nmeans: {self.means}\ncovs: {self.covs}')
def kl_divergence(self):
sum_kl = sp.gammaln(self.K*self.alpha0 + self.N) - \
sp.gammaln(self.K*self.alpha0)
tmp = self.r*np.log(self.r)
tmp[np.isnan(tmp)] = 0
sum_kl += np.sum(tmp)
for k in range(self.K):
sum_kl += (sp.gammaln(self.alpha0)-sp.gammaln(self.alpha[k]) +
sp.multigammaln(self.nu0*.5, self.D) -
sp.multigammaln(self.nu[k]*.5, self.D) +
0.5*(self._logdet(self.V0) * self.nu0 -
self._logdet(self.V[k, :])*self.nu[k] +
(np.log(self.beta[k])-np.log(self.beta0))*self.D))
return sum_kl
def _make_ellipses(self, k, ax):
cov = self.covs[k, :]
w, v = np.linalg.eigh(cov)
# w: 2 eigenvalues (length), v: 2 eigenvectors (rotation)
angle = np.degrees(np.arctan2(v[0, 1], v[0, 0]))
height, width = 2 * np.sqrt(w) # diameter = 2 * radius
ell = mpl.patches.Ellipse(self.means[k, :], height, width,
angle,
fc='lime', lw=0, alpha=.4)
edge = mpl.patches.Ellipse(self.means[k, :], height, width,
angle,
fc='none', ec='forestgreen', lw=3)
ax.add_artist(ell)
ax.add_artist(edge)
def plot_with_ellipses(self, figsize=(7, 7), dpi=200):
fig, ax = plt.subplots(1, 1, figsize=figsize, dpi=dpi)
ax.scatter(self.x[:, 0], self.x[:, 1],
c='deepskyblue', edgecolor='black', alpha=.5)
for k in range(self.K):
self._make_ellipses(k, ax)
ax.scatter(self.means[:, 0], self.means[:, 1],
c='orange', edgecolor='k',
marker='*', s=150)
plt.show()
def cluster_number_selection_by_kl(x, k_range, plot=False):
kls = np.empty(len(k_range))
for k in k_range:
print(f'======= {k} =========')
vgm = VariationalGaussianMixture(x, k)
vgm.fit(print_diff=False)
kls[k-1] = vgm.kl_divergence()
if plot:
vgm.plot_with_ellipses(figsize=(8, 5), dpi=100)
argmin_k = np.argmin(kls) + 1
print(f'argmin_k: {argmin_k}, min_kl: {kls[argmin_k]}')
return kls, argmin_k