forked from lingxuez/URSM
/
e_step_gibbs.py
378 lines (314 loc) · 14.8 KB
/
e_step_gibbs.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
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
## A class to perform Gibbs sampling
##
## Copyright Lingxue Zhu (lzhu@cmu.edu).
## All Rights Reserved.
import logging
import numpy as np
from numpy import linalg
import scipy as scp
from scipy import special
import pypolyagamma as ppg
import os
#######################
## Gibbs sampler for bulk data
#######################
class LogitNormalGibbs_BK(object):
## constructor: initialize with parameters
def __init__(self,
A, ## profile matrix
alpha, ## mixture propotion prior
BKexpr, ## M-by-N, bulk expression
iMarkers=None ## indices of marker genes
):
## data: unchanged throughout sampling
self.BKexpr, self.iMarkers = BKexpr, iMarkers
(self.M, self.N, self.K) = (BKexpr.shape[0], A.shape[0], A.shape[1])
# logging.debug("M=%d, N=%d, K=%d", M, N, K)
## read depths
self.BKrd = BKexpr.sum(axis=1)
## parameters: can only be changed by self.update_parameters()
self.A = np.array(A, dtype=float, copy=True)
self.alpha = np.array(alpha, dtype=float, copy=True)
def init_gibbs(self):
"""initialize latent variable values"""
## initialize Z in a way such that it captures the marker information
## without markers, Z's are zero so the first W is drawn according to alpha
self.Z = np.zeros([self.M, self.N, self.K])
## add marker information if any
if self.iMarkers is not None:
## set Z's
logging.debug("\t\tZ is initialized with Marker info.")
for index in range(self.iMarkers.shape[0]):
(i, k) = self.iMarkers[index, :]
self.Z[:, i, k] = self.BKexpr[:, i] + self.alpha[k]
## use Z's to initialize W's
self.W = self.Z.sum(axis=1).transpose()
self.W /= self.W.sum(axis=0)[np.newaxis, :]
else:
self.W = np.full([self.K, self.M], 1.0/self.K, dtype=float)
def init_suffStats(self):
"""initialize sufficient statistics"""
## E[mean_j Z_jik] and E[mean_j log W_kj]
self.suff_stats = {} ## sufficient statistics
self.suff_stats["exp_Zik"] = np.zeros([self.N, self.K], dtype=float)
self.suff_stats["exp_Zjk"] = np.zeros([self.M, self.K], dtype=float)
self.suff_stats["exp_logW"] = np.zeros([self.K, self.M], dtype=float)
self.suff_stats["exp_W"] = np.zeros([self.K, self.M], dtype=float)
def update_suffStats(self, sample):
"""update sufficient stats using current values of latent variables"""
## E[sum_j Z_jik]
self.suff_stats["exp_Zik"] += (self.Z).sum(axis=0) / float(sample)
## E[sum_i Z_jik]
self.suff_stats["exp_Zjk"] += (self.Z).sum(axis=1) / float(sample)
## E[mean_j log W_kj]
self.suff_stats["exp_logW"] += np.log(self.W) / float(sample)
## E[W]
self.suff_stats["exp_W"] += self.W / float(sample)
def update_parameters(self, A, alpha):
"""update parameters"""
self.A = np.array(A, dtype=float, copy=True)
self.alpha = np.array(alpha, dtype=float, copy=True)
#########################
## draw Gibbs samples
#########################
def gibbs(self, burnin=100, sample=100, thin=1, mean_approx=True):
"""Gibbs sampling cycle"""
## initialize sufficient statistics
self.init_suffStats()
if mean_approx:
## use one-step update for W based on expectation
## this is equivalent to NMF algorithm
logging.debug("\tE-step (one-step mean-update) for bulk samples started.")
## do not re-initialize W; use the value from last iteration.
## only update W once.
self.get_nmf_W()
self.draw_Z_mean()
self.update_suffStats(sample=1)
else:
## use the proper gibbs sampling
logging.debug("\tE-step for bulk samples started.")
## burn in
for giter in xrange(burnin):
self.gibbs_cycle()
## sampling
for giter in xrange(sample*thin):
self.gibbs_cycle()
if giter % thin == 0:
## update sufficient statistics
self.update_suffStats(sample)
def gibbs_cycle(self):
"""
perform one cycle of Gibbs sampling.
use one-step update for W based on expectation
this is equivalent to NMF algorithm
"""
## draw W first because Z may be carefully initialized with marker info
self.draw_W()
self.draw_Z()
def draw_Z(self):
"""Z: M x N x K, counts"""
for j in xrange(self.M):
for i in range(self.N):
pval = self.W[:, j]*self.A[i, :]
self.Z[j, i, :] = np.random.multinomial(n=self.BKexpr[j, i],
pvals = pval/self.AW[i, j])
def draw_W(self):
"""W: K x M, proportions"""
post_alpha = self.Z.sum(axis=1)
for j in xrange(self.M):
self.W[:, j] = np.random.dirichlet(self.alpha + post_alpha[j, :])
## update AW: N x M
self.AW = np.dot(self.A, self.W)
def draw_Z_mean(self):
"""Z: M x N x K, expected"""
self.AW = np.dot(self.A, self.W) ## N-by-M
for j in xrange(self.M):
for i in range(self.N):
pval = self.W[:, j]*self.A[i, :]
self.Z[j, i, :] = pval * self.BKexpr[j, i] /self.AW[i, j]
def get_nmf_W(self):
self.AW = np.dot(self.A, self.W) ## N-by-M
for k in range(self.K):
multiplier = self.BKexpr * self.A[:, k].transpose() / self.AW.transpose()
self.W[k, :] = self.W[k, :] * multiplier.sum(axis=1) + self.alpha[k] - 1
## re-normalize such that each column sums to 1
self.W /= self.W.sum(axis=0)[np.newaxis, :]
#######################
## Gibbs sampler for single cell
#######################
class LogitNormalGibbs_SC(object):
## constructor: initialize with parameters
def __init__(self,
A, ## profile matrix
pkappa, ## [mean, var] for kappa
ptau, ## [mean, var] for tau
SCexpr, ## L-by-N, single cell expression
G, ## L-by-1, single cell types
itype ## cell ids in each type
):
## data: never changed
(self.SCexpr, self.G, self.L) = (SCexpr, G, SCexpr.shape[0])
(self.N, self.K) = A.shape
self.SCrd = SCexpr.sum(axis=1) ## read depths
self.itype = itype
## parameters: can only be changed by self.update_parameters()
self.A = np.array(A, dtype=float, copy=True)
self.pkappa = np.array(pkappa, dtype=float, copy=True)
self.ptau = np.array(ptau, dtype=float, copy=True)
## zero-expressed entries
self.izero = np.where(self.SCexpr==0)
## for sampling from Polya-Gamma
# self.ppgs = ppg.PyPolyaGamma(seed=0)
num_threads = ppg.get_omp_num_threads()
seeds = np.random.randint(2**16, size=num_threads)
self.ppgs = self.initialize_polya_gamma_samplers()
def initialize_polya_gamma_samplers(self):
if "OMP_NUM_THREADS" in os.environ:
self.num_threads = int(os.environ["OMP_NUM_THREADS"])
else:
self.num_threads = ppg.get_omp_num_threads()
assert self.num_threads > 0
# Choose random seeds
seeds = np.random.randint(2**16, size=self.num_threads)
return [ppg.PyPolyaGamma(seed) for seed in seeds]
def init_gibbs(self):
"""initialize latent variable values"""
self.kappa = np.full([1,self.L], self.pkappa[0], dtype=float)
self.tau = np.full([1,self.L], self.ptau[0], dtype=float)
self.S = np.reshape(np.random.binomial(1, 0.5, size=self.L*self.N),
[self.L, self.N])
## note: use broadcasting
self.psi = np.transpose(self.kappa + self.tau * self.A[:, self.G])
self.w = np.ones([self.L, self.N], dtype=float)
## when expression > 0, it's known for sure that S=1
ipos = np.where(self.SCexpr>0)
self.S[ipos] = 1
## keep track of A[:, G]*S to reduce computation time
self.sum_AS = (np.transpose(self.A[:, self.G]) * self.S).sum(axis=1)
def init_suffStats(self):
"""initialize sufficient statistics to be zeros"""
self.suff_stats = {}
## posterior expectations
self.suff_stats["exp_S"] = np.zeros([self.L, self.N], dtype=float) ## E[S]
self.suff_stats["exp_kappa"] = np.zeros([1, self.L], dtype=float) ## E[kappa]
self.suff_stats["exp_tau"] = np.zeros([1, self.L], dtype=float) ## E[tau]
self.suff_stats["exp_kappasq"] = np.zeros([1, self.L], dtype=float) ## E[kappa^2]
self.suff_stats["exp_tausq"] = np.zeros([1, self.L], dtype=float) ## E[tau^2]
## part of coefficient for A: E[tau_l*(S_il-0.5) - kappa_l*tau_l*w_il]
self.suff_stats["coeffA"] = np.zeros([self.N, self.K], dtype=float)
## coefficient for A^2: E[- tau_l^2 * w_il]
self.suff_stats["coeffAsq"] = np.zeros([self.N, self.K], dtype=float)
## elbo that doesn't involve A
self.suff_stats["exp_elbo_const"] = 0
def update_suffStats(self, sample):
"""Update sufficient stats using current values of latent variables"""
self.suff_stats["exp_S"] += self.S / float(sample)
self.suff_stats["exp_kappa"] += self.kappa / sample
self.suff_stats["exp_tau"] += self.tau / sample
self.suff_stats["exp_kappasq"] += np.square(self.kappa) / sample
self.suff_stats["exp_tausq"] += np.square(self.tau) / sample
## sum_il E[- kappa_l^2 * w_il/2 + (S_il-0.5)*kappa_l ]
self.suff_stats["exp_elbo_const"] += (-self.w * \
np.transpose(np.square(self.kappa))).sum() / (2.0*sample)
self.suff_stats["exp_elbo_const"] += ((self.S - 0.5) * np.transpose(self.kappa)).sum()/ \
(sample)
## E[tau_l*(S_il-0.5) - kappa_l*tau_l*w_il]
coeffA = (self.S - 0.5) * np.transpose(self.tau) - \
self.w * np.transpose(self.tau * self.kappa)
## E[- tau_l^2 * w_il]/2
coeffAsq = (-self.w * np.transpose(np.square(self.tau))) / 2.0
## sum over l, mean over gibbs samples
for k in xrange(self.K):
if len(self.itype[k]) > 0:
self.suff_stats["coeffA"][:, k] += coeffA[self.itype[k],:].sum(axis=0) / \
float(sample)
self.suff_stats["coeffAsq"][:, k] += coeffAsq[self.itype[k],:].sum(axis=0) / \
float(sample)
def update_parameters(self, A, pkappa, ptau):
"""update parameters"""
self.A = np.array(A, dtype=float, copy=True)
self.pkappa = np.array(pkappa, dtype=float, copy=True)
self.ptau = np.array(ptau, dtype=float, copy=True)
## update psi and sum_AS due to updated parameter A
self.update_psi()
self.sum_AS = (np.transpose(self.A[:, self.G]) * self.S).sum(axis=1)
#########################
## draw Gibbs samples
#########################
def gibbs(self, burnin=100, sample=100, thin=1):
"""Gibbs sampling cycle"""
logging.debug("\tE-step for single cells started.")
## initialize sufficient statistics
self.init_suffStats()
## re-start gibbs chain
self.init_gibbs()
## burnin
for giter in xrange(burnin):
self.gibbs_cycle()
## sampling
for giter in xrange(sample*thin):
self.gibbs_cycle()
if giter % thin == 0:
## update sufficient statistics
self.update_suffStats(sample)
def gibbs_cycle(self):
"""One cycle through latent variables in Gibbs sampling"""
self.draw_w()
self.draw_S()
self.draw_kappa_tau()
self.update_psi()
def update_psi(self):
"""psi: L-by-N; logistic(psi) is the dropout probability"""
self.psi = np.transpose(self.kappa + self.tau * self.A[:, self.G])
def draw_w(self):
"""w: L-by-N; augmented latent variable"""
ns = np.ones(self.N, dtype=np.float)
## draw polya gamma parallelly
for l in xrange(self.L):
ppg.pgdrawvpar(self.ppgs, ns, self.psi[l, :], self.w[l, :])
def draw_S(self):
"""S: L-by-N; binary variables"""
## only update the entries where self.SCexpr==0
for index in xrange(len(self.izero[0])):
(l, i) = (self.izero[0][index], self.izero[1][index])
A_curr = self.A[i, self.G[l]]
sum_other = self.sum_AS[l] - A_curr * self.S[l, i]
if sum_other == 0:
b = scp.special.expit(self.psi[l][i])
else:
b = scp.special.expit(self.psi[l][i] -
self.SCrd[l] * np.log(1 + A_curr/sum_other))
## note: scp.stats.bernoulli.rv is slow!!!
self.S[l][i] = np.random.binomial(1, b)
## update sum_AS
self.sum_AS[l] = sum_other + A_curr * self.S[l, i]
def draw_kappa_tau(self):
"""kappa, tau: scalars that defines psi"""
for l in xrange(self.L):
A_curr = self.A[:, self.G[l]]
# ## precision matrix
# offdiag = sum(self.w[l, :] * A_curr)
# diag = [sum(self.w[l, :]) + self.pkappa[1],
# sum(self.w[l, :] * np.square(A_curr)) + self.ptau[1]]
# PP = np.array([[diag[0], offdiag], [offdiag, diag[1]]])
# ## PP * mP = bP: solve for the mean vector mP
# bP = np.array([sum(self.S[l,:]) - self.N/2.0 +\
# self.pkappa[0]*self.pkappa[1],
# self.sum_AS[l] - 0.5 + \
# self.ptau[0]*self.ptau[1]])
# mP = np.linalg.solve(PP, bP)
# ## draw (kappa, tau) ~ Gaussian(mP, PP^-1)
# newdraw = np.random.multivariate_normal(mP, np.linalg.inv(PP))
## compute the covariance matrix
offdiag = sum(self.w[l, :] * A_curr)
diag = [sum(self.w[l, :]) + 1.0 / self.pkappa[1],
sum(self.w[l, :] * np.square(A_curr)) + 1.0 / self.ptau[1]]
det = diag[0] * diag[1] - offdiag * offdiag
sigma_mat = np.array([[diag[1], -offdiag], [-offdiag, diag[0]]]) / det
bP = np.array([sum(self.S[l,:]) - self.N/2.0 +\
self.pkappa[0] / self.pkappa[1],
self.sum_AS[l] - 0.5 + \
self.ptau[0] / self.ptau[1]])
mP = np.dot(sigma_mat, bP)
newdraw = np.random.multivariate_normal(mP, sigma_mat)
(self.kappa[0, l], self.tau[0, l]) = newdraw