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
|
### -----------------------------------------------------------------
### Train the Dirichlet mixture model from matrice
### Exported!
setMethod("dmmEM", signature(x="matrix"),
function(x, K=6, alg=c("C", "R")){
alg <- match.arg(alg)
if(alg == "C"){
fit <- lapply(1:K, dmn, count=x, verbose=TRUE)
lplc <- sapply(fit, laplace)
best <- fit[[which.min(lplc)]]
ans <- list(alpha0=fitted(best), pmix=mixturewt(best)$pi,
ll=laplace(best))
}else{
ans <- dirichletMixtureEMEstimation(t(x), K, alpha0=NULL,
pmix=NULL)
}
return(ans)
}
)
setMethod("dmmEM", signature(x="PFMatrixList"),
function(x, K=6, alg=c("C", "R")){
allMatrix <- do.call(cbind, Matrix(x))
#dirichletMixtureEMEstimation(t(allMatrix), K, alpha0, pmix)
dmmEM(t(allMatrix), K, alg=alg)
}
)
setMethod("dmmEM", signature(x="ANY"),
function(x, K=6, alg=c("C", "R")){
allMatrix <- getMatrixSet(x, opts=list(all=TRUE))
dmmEM(allMatrix, K, alg=alg)
}
)
dirichletMixtureEMEstimation <- function(inputMatrix, K,
alpha0=NULL, pmix=NULL){
## inputMatrix: the samples summarized as counts for each of A letters, N x A
## A <- 4 for DNA. This matrix should be concatenated by all matrices
## from Jaspar.
## K: the number of sought component.
## alpha0: the estimated Dirichlet parameters A x K
## pmix: mixing proportions 1 x K
K = as.integer(K)
N <- nrow(inputMatrix)
A <- ncol(inputMatrix)
rowSumsInputMatrix <- rowSums(inputMatrix)
oN <- rep(1, N)
oK <- rep(1, K)
oA <- rep(1, A)
gam <- matrix(0, nrow=N, ncol=K)
contrib_n <- matrix(0, nrow=A, ncol=K)
contrib_d <- rep(0, K)
## random initialization
if(is.null(alpha0)){
epsilon <- 0.1
alpha0 <- epsilon * (2 * matrix(runif(A * K), nrow=A, ncol=K) - 1) + 2
}else{
stopifnot(all(dim(alpha0) == c(nrow(inputMatrix), K)))
}
Alpha0 <- colSums(alpha0)
if(is.null(pmix)){
pmix <- rep(1, K) / K
}else{
stopifnot(sum(pmix) == 1)
}
## minimum alpha0 value
alpha0_min <- 1e-8
ftol <- 1e-10
iteouter_max <- 10000
iteinner_max <- 1
dll_min <- 1e-6
ite <- 0;
dll <- Inf;
ll <- numeric(iteouter_max)
while(ite < iteouter_max && (dll > dll_min || ite == 1)){
ite <- ite + 1;
## E-step
contrib <- log(pmix) + lgamma(Alpha0) - colSums(lgamma(alpha0))
for(k in 1:K){
alpha0_k <- alpha0[ , k]
gam[ ,k] <- rowSums(lgamma(inputMatrix +
matrix(rep(alpha0_k, N), nrow=N, ncol=A,
byrow=TRUE))) -
lgamma(rowSumsInputMatrix + rep(Alpha0[k], N))
}
gam <- gam + matrix(rep(contrib, N), nrow=N, ncol=K, byrow=TRUE)
maxgam <- apply(gam, 1, max)
gam <- exp(gam - matrix(rep(maxgam, K), nrow=N, ncol=K, byrow=FALSE))
ll[ite] <- sum(maxgam + log(rowSums(gam)))
dll <- ll[ite] - ll[max(ite-1,1)]
sumgam <- 1 / rowSums(gam)
gam <- gam * matrix(rep(sumgam, K), nrow=N, ncol=K, byrow=FALSE)
## M-step
pmix <- colSums(gam) / N
dalpha0 <- Inf
iteinner <- 0
while(sum(abs(dalpha0)) > ftol && iteinner < iteinner_max){
iteinner = iteinner + 1
for(k in 1:K){
alpha0_k <- alpha0[ , k]
contrib_n[ , k] <- psigamma(t(inputMatrix) +
matrix(rep(alpha0_k, N), nrow=A,
ncol=N, byrow=FALSE)) %*% gam[ , k]
contrib_d[k] <- sum(psigamma(rowSumsInputMatrix + Alpha0[k]) *
gam[ ,k]) - pmix[k] * N * psigamma(Alpha0[k])
}
contrib_n <- contrib_n - N * psigamma(alpha0) *
matrix(rep(pmix, A), nrow=A, ncol=length(pmix), byrow=TRUE)
dalpha0 <- pmax(alpha0 * (contrib_n /
matrix(rep(contrib_d, A), nrow=A,
ncol=length(contrib_d), byrow=TRUE)),
alpha0_min) - alpha0
alpha0 <- alpha0 + dalpha0
Alpha0 <- colSums(alpha0)
}
if(isTRUE(all.equal(ite %% iteouter_max / 10, 0))){
cat("Iteration: ", ite, "\n")
print(dll)
print(sum(abs(dalpha0)))
print(pmix)
print(alpha0)
}
}
return(list(alpha0=alpha0, pmix=pmix, ll=ll))
}
repmat <- function(a,n,m) {kronecker(matrix(1,n,m),a)}
## Sampling from Dirichlet distribution is implemented in gtools::rdirichlet.
#dirichletSample <- function(a, n=1){
## Sample from Dirichlet distribution.
## DIRICHLET_SAMPLE(a) returns a probability vector sampled from a
## Dirichlet distribution with parameter vector A.
## DIRICHLET_SAMPLE(a,n) returns N samples, collected into a matrix, each
## vector having the same orientation as A.
## References:
## [1] L. Devroye, "Non-Uniform Random Variate Generation",
## Springer-Verlag, 1986
## This is essentially a generalization of the method for Beta rv's.
## Theorem 4.1, p.594
#rgamma(n*length(a), rep(a, n))
#}
### -----------------------------------------------------------------
### sample the matrix from Dirichlet mixture model
### Exported!
setMethod("rPWMDmm", signature(x="matrix"),
function(x, alpha0, pmix, N=1, W=6){
PWMrandomizeBayes(x, alpha0, pmix, N, W)
}
)
setMethod("rPWMDmm", signature(x="PFMatrix"),
function(x, alpha0, pmix, N=1, W=6){
PWMrandomizeBayes(Matrix(x), alpha0, pmix, N, W)
}
)
setMethod("rPWMDmm", signature(x="PFMatrixList"),
function(x, alpha0, pmix, N=1, W=6){
allMatrix <- do.call(cbind, Matrix(x))
PWMrandomizeBayes(allMatrix, alpha0, pmix, N, W)
}
)
#setMethod("rPWMDmm", signature(x="ANY"),
# function(x, alpha0, pmix, N=1, W=6){
# allMatrix <- getMatrixSet(x, opts=list(all=TRUE))
# rPWMDmm(allMatrix, alpha0, pmix, N, W)
# }
# )
PWMrandomizeBayes <- function(PCM, alpha0, pmix, N=1, W=6){
## generates N (default 1) random PWM of width drawn from
## the posterior distribution
## of PWMs. The posterior is propertional to the
## Dirichlet prior distribution Dprior (which
## might be a mixture) times the mulitnomial likelihood with count matrix PCM.
A <- nrow(PCM)
Win <- ncol(PCM)
### parameters of prior
#alpha0 <- Dprior$alpha0
##alpha0 is the estimated Dirichlet parameters A x K
#pmix <- Dprior$pmix
## pmix mixing proportions 1 x K
K <- length(pmix)
### accumulative distribution for mixing proportions
apmix <- cumsum(pmix)
PWM = list()
for(n in 1:N){
PWM[[n]] <- matrix(NA, ncol=W, nrow=nrow(alpha0))
if(W == Win){
for(w in 1:W){
## draw from the mixture component
k = which(runif(1) < apmix)[1]
## draw from component k of dirichlet posterior
PWM[[n]][ ,w] <-
rdirichlet(n=1, alpha=(alpha0[ ,k] + PCM[ ,w]))
}
}else{
for(w in 1:W){
## draw from the mixture component
k = which(runif(1) < apmix)[1]
## draw from component k of dirichlet posterior and use random
## column in PCM
PWM[[n]][, w] <-
rdirichlet(n=1, alpha=(alpha0[ ,k] +
PCM[ ,ceiling(Win * runif(1))]))
}
}
}
## Make sure colSums is almost 1.
stopifnot(all(sapply(PWM, function(x){all(sapply(colSums(x),
all.equal, 1))})))
return(PWM)
}
|
