/
varmix.py
209 lines (177 loc) · 6.26 KB
/
varmix.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
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
'''
Created on 14 May 2013
@author: James McInerney
'''
#implementation of variational Gaussian mixture models
from numpy import *
from matplotlib.pyplot import *
from numpy.linalg.linalg import inv, det
from scipy.special.basic import digamma
import time
from viz import create_cov_ellipse
ROOT = 'PATH_TO_WHERE_YOU_WANT_YOUR_ANIMATIONS_TO_BE_SAVED_TO'
def gen(K,N,XDim):
#K: number of components
#N: number of data points
mu = array([random.multivariate_normal(zeros(XDim),10*eye(XDim)) for _ in range(K)])
cov = [0.1*eye(XDim) for _ in range(K)]
q = random.dirichlet(ones(K)) #component coefficients
X = zeros((N,XDim)) #observations
Z = zeros((N,K)) #latent variables
for n in range(N):
#decide which component has responsibility for this data point:
Z[n,:] = random.multinomial(1, q)
k = Z[n,:].argmax()
X[n,:] = random.multivariate_normal(mu[k,:],cov[k])
return X
def run(X,K,VERBOSE=True):
#X: observations
(N,XDim) = shape(X)
#hyperparams:
alpha0 = 0.1 #prior coefficient count (for Dir)
beta0 = (1e-20)*1. #variance of mean (smaller: broader the means)
v0 = XDim+1. #2. #degrees of freedom in inverse wishart
m0 = zeros(XDim) #prior mean
W0 = (1e0)*eye(XDim) #prior cov (bigger: smaller covariance)
#params:
#Z = ones((N,K))/float(K) #uniform initial assignment
Z = array([random.dirichlet(ones(K)) for _ in range(N)])
ion()
fig = figure(figsize=(10,10))
ax_spatial = fig.add_subplot(1,1,1) #http://stackoverflow.com/questions/3584805/in-matplotlib-what-does-111-means-in-fig-add-subplot111
circs = []
itr, max_itr = 0, 200
while itr < max_itr:
#M-like-step
NK = Z.sum(axis=0)
vk = v0 + NK + 1.
xd = calcXd(Z,X)
S = calcS(Z,X,xd,NK)
betak = beta0 + NK
m = calcM(K,XDim,beta0,m0,NK,xd,betak)
W = calcW(K,W0,xd,NK,m0,XDim,beta0,S)
#E-like-step
mu = Muopt(X,XDim,NK,betak,m,W,xd,vk,N,K) #eqn 10.64 Bishop
invc = Invcopt(W,vk,XDim,K) #eqn 10.65 Bishop
pik = Piopt(alpha0,NK) #eqn 10.66 Bishop
Z = Zopt(XDim, pik, invc, mu, N, K) #eqn 10.46 Bishop
if VERBOSE:
print 'itr %i'%itr
print 'means',m
print 'Z',Z
print 'mu',mu
print 'invc',invc
print 'exp(pik)',exp(pik)
print 'NK',NK
if itr==0:
sctX = scatter(X[:,0],X[:,1])
sctZ = scatter(m[:,0],m[:,1],color='r')
else:
#ellipses to show covariance of components
for circ in circs: circ.remove()
circs = []
for k in range(K):
circ = create_cov_ellipse(S[k], m[k,:],color='r',alpha=0.3) #calculate params of ellipses (adapted from http://stackoverflow.com/questions/12301071/multidimensional-confidence-intervals)
circs.append(circ)
#add to axes:
ax_spatial.add_artist(circ)
#make sure components with NK=0 are not visible:
if NK[k]<=alpha0: m[k,:] = m[NK.argmax(),:] #put over point that obviously does have assignments
sctZ.set_offsets(m)
draw()
#time.sleep(0.1)
savefig(ROOT + 'animation/%04d.png'%itr)
itr += 1
if VERBOSE:
#keep display:
time.sleep(360)
return m,invc,pik,Z
def calcXd(Z,X):
#weighted means (by component responsibilites)
(N,XDim) = shape(X)
(N1,K) = shape(Z)
NK = Z.sum(axis=0)
assert N==N1
xd = zeros((K,XDim))
for n in range(N):
for k in range(K):
xd[k,:] += Z[n,k]*X[n,:]
#safe divide:
for k in range(K):
if NK[k]>0: xd[k,:] = xd[k,:]/NK[k]
return xd
def calcS(Z,X,xd,NK):
(N,K)=shape(Z)
(N1,XDim)=shape(X)
assert N==N1
S = [zeros((XDim,XDim)) for _ in range(K)]
for n in range(N):
for k in range(K):
B0 = reshape(X[n,:]-xd[k,:], (XDim,1))
L = dot(B0,B0.T)
assert shape(L)==shape(S[k]),shape(L)
S[k] += Z[n,k]*L
#safe divide:
for k in range(K):
if NK[k]>0: S[k] = S[k]/NK[k]
return S
def calcW(K,W0,xd,NK,m0,XDim,beta0,S):
Winv = [None for _ in range(K)]
for k in range(K):
Winv[k] = inv(W0) + NK[k]*S[k]
Q0 = reshape(xd[k,:] - m0, (XDim,1))
q = dot(Q0,Q0.T)
Winv[k] += (beta0*NK[k] / (beta0 + NK[k]) ) * q
assert shape(q)==(XDim,XDim)
W = []
for k in range(K):
try:
W.append(inv(Winv[k]))
except linalg.linalg.LinAlgError:
print 'Winv[%i]'%k, Winv[k]
raise linalg.linalg.LinAlgError()
return W
def calcM(K,XDim,beta0,m0,NK,xd,betak):
m = zeros((K,XDim))
for k in range(K): m[k,:] = (beta0*m0 + NK[k]*xd[k,:]) / betak[k]
return m
def Muopt(X,XDim,NK,betak,m,W,xd,vk,N,K):
Mu = zeros((N,K))
for n in range(N):
for k in range(K):
A = XDim / betak[k] #shape: (k,)
B0 = reshape((X[n,:] - m[k,:]),(XDim,1))
B1 = dot(W[k], B0)
l = dot(B0.T, B1)
assert shape(l)==(1,1),shape(l)
Mu[n,k] = A + vk[k]*l #shape: (n,k)
return Mu
def Piopt(alpha0,NK):
alphak = alpha0 + NK
pik = digamma(alphak) - digamma(alphak.sum())
return pik
def Invcopt(W,vk,XDim,K):
invc = [None for _ in range(K)]
for k in range(K):
dW = det(W[k])
print 'dW',dW
if dW>1e-30: ld = log(dW)
else: ld = 0.0
invc[k] = sum([digamma((vk[k]+1-i) / 2.) for i in range(XDim)]) + XDim*log(2) + ld
return invc
def Zopt(XDim, exp_ln_pi, exp_ln_gam, exp_ln_mu, N, K):
Z = zeros((N,K)) #ln Z
for k in range(K):
Z[:,k] = exp_ln_pi[k] + 0.5*exp_ln_gam[k] - 0.5*XDim*log(2*pi) - 0.5*exp_ln_mu[:,k]
#normalise ln Z:
Z -= reshape(Z.max(axis=1),(N,1))
Z1 = exp(Z) / reshape(exp(Z).sum(axis=1), (N,1))
return Z1
if __name__ == "__main__":
#generate synthetic data:
X = gen(30,200,2)
#run VB on the data:
K1 = 20 # num components in inference
mu,invc,pik,Z = run(X,K1)
print 'mu',mu
print 'NK',Z.sum(axis=0)