I have set up a Metropolis-Hastings algorithm, and now I am trying to run the algorithm using parallel computing. I have set up a single-chain function
library(parallel)
library(foreach)
library(mvtnorm)
library(doParallel)
n<-100
mX <- 1:n
vY <- rnorm(n)
chains <- 4
iter <- n
p <- 2
#Loglikelihood
post <- function(y, theta) dmvnorm(t(y), rep(0,length(y)), theta[1]*exp(- abs(matrix(rep(mX,n),n) - matrix(rep(mX,each=n),n))/theta[2]),log=TRUE)
geninits <- function() list(theta = runif(p, 0, 1))
dist <- 0.01
jump <- function(x, dist) exp(log(x) + rmvnorm(1,rep(0,p),diag(rep(dist,p))))
MCsingle <- function(){ # This is part of a larger function, so no input are needed
inits <- geninits()
theta.post <- matrix(NA,nrow=p,ncol=iter)
for (i in 1:p) theta.post[i,1] <- inits$theta[i]
for (t in 2:iter){
theta_star <- c(jump(theta.post[, t-1],dist))
pstar <- post(vY, theta = theta_star) # post is the loglikelihood using dmvnorm.
pprev <- post(vY, theta = theta.post[,t-1])
r <- min(exp(pstar - pprev) , 1)
accept <- rbinom(1, 1, prob = r)
if (accept == 1){
theta.post[, t] <- theta_star
} else {
theta.post[, t] <- theta.post[, t-1]
}
}
return(theta.post)
}
, which returns an p x iter matrix, with p parameters and iter iterations.
cl<-makeCluster(4)
registerDoParallel(cl)
posterior <- foreach(c = 1:chains) %dopar% {
MCsingle() }
UPDATE: When I tried to simplify the problem the code suddenly seemed to work. Even though I purposely tried to make errors, the code ran perfectly and the results were as wanted. So for others with similar problems unfortunately I cannot give an answer.
A follow-up question:
My initial purpose was to built up an entire function, such that
MCmulti <- function(mX,vY,iter,chains){
posterior <- foreach(c = 1:chains) %dopar% {
MCsingle() }
return(posterior)
}
but the foreach-loop does not seem to read all the required functions like:
Error in FUN() : task 1 failed - "could not find function "geninits""
Can anybody answer how to implement custom functions inside a foreach loop? Am I to input it as MCmulti <- function(FUN,...) FUN() and call MCmulti(MCsingle,...) ?
I have the 3 following functions which I would like to make faster, I assume apply functions are the best way to go, but I have never used apply functions, so I have no idea what to do. Any type of hints, ideas and code snippets will be much appreciated.
n, T, dt are global parameters and par is a vector of parameters.
Function 1: is a function to create an m+1,n matrix containing poisson distributed jumps with exponentially distributed jump sizes. My troubles here is because I have 3 loops and I am not sure how to incorporate the if statement in the inner loop. Also I have no idea if it is at all possible to use apply functions on the outer layers of the loops only.
jump <- function(t=0,T=T,par){
jump <- matrix(0,T/dt+1,n) # initializing output matrix
U <- replicate(n,runif(100,t,T)) #matrix used to decide when the jumps will happen
Y <-replicate(n,rexp(100,1/par[6])) #matrix with jump sizes
for (l in 1:n){
NT <- rpois(1,par[5]*T) #number of jumps
k=0
for (j in seq(t,T,dt)){
k=k+1
if (NT>0){
temp=0
for (i in 1:NT){
u <- vector("numeric",NT)
if (U[i,l]>j){ u[i]=0
}else u[i]=1
temp=temp+Y[i,l]*u[i]
}
jump[k,l]=temp
}else jump[k,l]=0
}
}
return(jump)
}
Function 2: calculates a default intensity, based on Brownian motions and the jumps from function 1. Here my trouble is how to use apply functions when the variable used for the calculation is the values from the row above in the output matrix AND how to get the right values from the external matrices which are used in the calculations (BMz_C & J)
lambda <- function(t=0,T=T,par,fit=0){
lambda <- matrix(0,m+1,n) # matrix to hold intesity path output
lambda[1,] <- par[4] #initializing start value of the intensity path.
J <- jump(t,T,par) #matrix containing jumps
for(i in 2:(m+1)){
dlambda <- par[1]*(par[2]-max(lambda[i-1,],0))*dt+par[3]*sqrt(max(lambda[i- 1,],0))*BMz_C[i,]+(J[i,]-J[i-1,])
lambda[i,] <- lambda[i-1,]+dlambda
}
return(lambda)
}
Function 3: calculates a survival probability based on the intensity from function 2. Here a() and B() are functions that return numerical values. My problem here is that the both value i and j are used because i is not always an integer which thus can to be used to reference the external matrix. I have earlier tried to use i/dt, but sometimes it would overwrite one line and skip the next lines in the matrix, most likely due to rounding errors.
S <- function(t=0,T=T,par,plot=0, fit=0){
S <- matrix(0,(T-t)/dt+1,n)
if (fit > 0) S.fit <- matrix(0,1,length(mat)) else S.fit <- 0
l=lambda(t,T,par,fit)
j=0
for (i in seq(t,T,dt)){
j=j+1
S[j,] <- a(i,T,par)*exp(B(i,T,par)*l[j,])
}
return(S)
}
Sorry for the long post, any help for any of the functions will be much appreciated.
EDIT:
First of all thanks to digEmAll for the great reply.
I have now worked on vectorising function 2. First I tried
lambda <- function(t=0,T=T,par,fit=0){
lambda <- matrix(0,m+1,n) # matrix to hold intesity path input
J <- jump(t,T,par,fit)
lambda[1,] <- par[4]
lambda[2:(m+1),] <- sapply(2:(m+1), function(i){
lambda[i-1,]+par[1]*(par[2]-max(lambda[i-1,],0))*dt+par[3]*sqrt(max(lambda[i-1,],0))*BMz_C[i,]+(J[i,]-J[i-1,])
})
return(lambda)
}
but it would only produce the first column. So I tried a two step apply function.
lambda <- function(t=0,T=T,par,fit=0){
lambda <- matrix(0,m+1,n) # matrix to hold intesity path input
J <- jump(t,T,par,fit)
lambda[1,] <- par[4]
lambda[2:(m+1),] <- sapply(1:n, function(l){
sapply(2:(m+1), function(i){
lambda[i-1,l]+par[1]*(par[2]-max(lambda[i-1,l],0))*dt+par[3]*sqrt(max(lambda[i-1,l],0))*BMz_C[i,l]+(J[i,l]-J[i-1,l])
})
})
return(lambda)
}
This seems to work, but only on the first row, all rows after that have an identical non-zero value, as if lambda[i-1] is not used in the calculation of lambda[i], does anyone have an idea how to manage that?
I'm going to explain to you, setp-by-step, how to vectorize the first function (one possible way of vectorization, maybe not the best one for your case).
For the others 2 functions, you can simply apply the same concepts and you should be able to do it.
Here, the key concept is: start to vectorize from the innermost loop.
1) First of all, rpois can generate more than one random value at a time but you are calling it n-times asking one random value. So, let's take it out of the loop obtaining this:
jump <- function(t=0,T=T,par){
jump <- matrix(0,T/dt+1,n)
U <- replicate(n,runif(100,t,T))
Y <-replicate(n,rexp(100,1/par[6]))
NTs <- rpois(n,par[5]*T) # note the change
for (l in 1:n){
NT <- NTs[l] # note the change
k=0
for (j in seq(t,T,dt)){
k=k+1
if (NT>0){
temp=0
for (i in 1:NT){
u <- vector("numeric",NT)
if (U[i,l]>j){ u[i]=0
}else u[i]=1
temp=temp+Y[i,l]*u[i]
}
jump[k,l]=temp
}else jump[k,l]=0
}
}
return(jump)
}
2) Similarly, it is useless/inefficient to call seq(t,T,dt) n-times in the loop since it will always generate the same sequence. So, let's take it out of the loop and store into a vector, obtainig this:
jump <- function(t=0,T=T,par){
jump <- matrix(0,T/dt+1,n)
U <- replicate(n,runif(100,t,T))
Y <-replicate(n,rexp(100,1/par[6]))
NTs <- rpois(n,par[5]*T)
js <- seq(t,T,dt) # note the change
for (l in 1:n){
NT <- NTs[l]
k=0
for (j in js){ # note the change
k=k+1
if (NT>0){
temp=0
for (i in 1:NT){
u <- vector("numeric",NT)
if (U[i,l]>j){ u[i]=0
}else u[i]=1
temp=temp+Y[i,l]*u[i]
}
jump[k,l]=temp
}else jump[k,l]=0
}
}
return(jump)
}
3) Now, let's have a look at the innermost loop:
for (i in 1:NT){
u <- vector("numeric",NT)
if (U[i,l]>j){ u[i]=0
}else u[i]=1
temp=temp+Y[i,l]*u[i]
}
this is equal to :
u <- as.integer(U[1:NT,l]<=j)
temp <- sum(Y[1:NT,l]*u)
or, in one-line:
temp <- sum(Y[1:NT,l] * as.integer(U[1:NT,l] <= j))
hence, now the function can be written as :
jump <- function(t=0,T=T,par){
jump <- matrix(0,T/dt+1,n)
U <- replicate(n,runif(100,t,T))
Y <-replicate(n,rexp(100,1/par[6]))
NTs <- rpois(n,par[5]*T)
js <- seq(t,T,dt)
for (l in 1:n){
NT <- NTs[l]
k=0
for (j in js){
k=k+1
if (NT>0){
jump[k,l] <- sum(Y[1:NT,l]*as.integer(U[1:NT,l]<=j)) # note the change
}else jump[k,l]=0
}
}
return(jump)
}
4) Again, let's have a look at the current innermost loop:
for (j in js){
k=k+1
if (NT>0){
jump[k,l] <- sum(Y[1:NT,l]*as.integer(U[1:NT,l]<=j)) # note the change
}else jump[k,l]=0
}
as you can notice, NT does not depend on the iteration of this loop, so the inner if can be moved outside, as follows:
if (NT>0){
for (j in js){
k=k+1
jump[k,l] <- sum(Y[1:NT,l]*as.integer(U[1:NT,l]<=j)) # note the change
}
}else{
for (j in js){
k=k+1
jump[k,l]=0
}
}
this seems worse than before, well yes it is, but now the 2 conditions can be turned into one-liner's (note the use of sapply¹):
if (NT>0){
jump[1:length(js),l] <- sapply(js,function(j){ sum(Y[1:NT,l]*as.integer(U[1:NT,l]<=j)) })
}else{
jump[1:length(js),l] <- 0
}
obtaining the following jump function:
jump <- function(t=0,T=T,par){
jump <- matrix(0,T/dt+1,n)
U <- replicate(n,runif(100,t,T))
Y <-replicate(n,rexp(100,1/par[6]))
NTs <- rpois(n,par[5]*T)
js <- seq(t,T,dt)
for (l in 1:n){
NT <- NTs[l]
if (NT>0){
jump[1:length(js),l] <- sapply(js,function(j){ sum(Y[1:NT,l]*as.integer(U[1:NT,l]<=j)) })
}else{
jump[1:length(js),l] <- 0
}
}
return(jump)
}
5) finally we can get rid of the last loop, using again the sapply¹ function, obtaining the final jump function :
jump <- function(t=0,T=T,par){
U <- replicate(n,runif(100,t,T))
Y <-replicate(n,rexp(100,1/par[6]))
js <- seq(t,T,dt)
NTs <- rpois(n,par[5]*T)
jump <- sapply(1:n,function(l){
NT <- NTs[l]
if (NT>0){
sapply(js,function(j){ sum(Y[1:NT,l]*as.integer(U[1:NT,l]<=j)) })
}else {
rep(0,length(js))
}
})
return(jump)
}
(¹)
sapply function is pretty easy to use. For each element of the list or vector passed in the X parameter, it applies the function passed in the FUN parameter, e.g. :
vect <- 1:3
sapply(X=vect,FUN=function(el){el+10}
# [1] 11 12 13
since by default the simplify parameter is true, the result is coerced to the simplest possible object. So, for example in the previous case the result becomes a vector, while in the following example result become a matrix (since for each element we return a vector of the same size) :
vect <- 1:3
sapply(X=vect,FUN=function(el){rep(el,5)})
# [,1] [,2] [,3]
# [1,] 1 2 3
# [2,] 1 2 3
# [3,] 1 2 3
# [4,] 1 2 3
# [5,] 1 2 3
Benchmark :
The following benchmark just give you an idea of the speed gain, but the actual performances may be different depending on your input parameters.
As you can imagine, jump_old corresponds to your original function 1, while jump_new is the final vectorized version.
# let's use some random parameters
n = 10
m = 3
T = 13
par = c(0.1, 0.2, 0.3, 0.4, 0.5, 0.6)
dt <- 3
set.seed(123)
system.time(for(i in 1:5000) old <- jump_old(T=T,par=par))
# user system elapsed
# 12.39 0.00 12.41
set.seed(123)
system.time(for(i in 1:5000) new <- jump_new(T=T,par=par))
# user system elapsed
# 4.49 0.00 4.53
# check if last results of the 2 functions are the same:
isTRUE(all.equal(old,new))
# [1] TRUE
I've done searching similar problems and I have a vague idea about what should I do: to vectorize everything or use apply() family. But I'm a beginner on R programming and both of the above methods are quite confusing.
Here is my source code:
x<-rlnorm(100,0,1.6)
j=0
k=0
i=0
h=0
lambda<-rep(0,200)
sum1<-rep(0,200)
constjk=0
wj=0
wk=0
for (h in 1:200)
{
lambda[h]=2+h/12.5
N=ceiling(lambda[h]*max(x))
for (j in 0:N)
{
wj=(sum(x<=(j+1)/lambda[h])-sum(x<=j/lambda[h]))/100
for (k in 0:N)
{
constjk=dbinom(k, j + k, 0.5)
wk=(sum(x<=(k+1)/lambda[h])-sum(x<=k/lambda[h]))/100
sum1[h]=sum1[h]+(lambda[h]/2)*constjk*wk*wj
}
}
}
Let me explain a bit. I want to collect 200 sum1 values (that's the first loop), and for every sum1 value, it is the summation of (lambda[h]/2)*constjk*wk*wj, thus the other two loops. Most tedious is that N changes with h, so I have no idea how to vectorize the j-loop and the k-loop. But of course I can vectorize the h-loop with lambda<-seq() and N<-ceiling(), and that's the best I can do. Is there a way to further simplify the code?
Your code can be perfectly verctorized with 3 nested sapply calls. It might be a bit hard to read for the untrained eye, but the essence of it is that instead of adding one value at a time to sum1[h] we calculate all the terms produced by the innermost loop in one go and sum them up.
Although this vectorized solution is faster than your tripple for loop, the improvement is not dramatical. If you plan to use it many times I suggest you implement it in C or Fortran (with regular for loops), which improves the speed a lot. Beware though that it has high time complexity and will scale badly with increased values of lambda, ultimatelly reaching a point when it is not possible to compute within reasonable time regardless of the implementation.
lambda <- 2 + 1:200/12.5
sum1 <- sapply(lambda, function(l){
N <- ceiling(l*max(x))
sum(sapply(0:N, function(j){
wj <- (sum(x <= (j+1)/l) - sum(x <= j/l))/100
sum(sapply(0:N, function(k){
constjk <- dbinom(k, j + k, 0.5)
wk <- (sum(x <= (k+1)/l) - sum(x <= k/l))/100
l/2*constjk*wk*wj
}))
}))
})
Btw, you don't need to predefine variables like h, j, k, wj and wk. Especially since not when vectorizing, as assignments to them inside the functions fed to sapply will create overlayered local variables with the same name (i.e. ignoring the ones you predefied).
Let`s wrap your simulation in a function and time it:
sim1 <- function(num=20){
set.seed(42)
x<-rlnorm(100,0,1.6)
j=0
k=0
i=0
h=0
lambda<-rep(0,num)
sum1<-rep(0,num)
constjk=0
wj=0
wk=0
for (h in 1:num)
{
lambda[h]=2+h/12.5
N=ceiling(lambda[h]*max(x))
for (j in 0:N)
{
wj=(sum(x<=(j+1)/lambda[h])-sum(x<=j/lambda[h]))/100
for (k in 0:N)
{
set.seed(42)
constjk=dbinom(k, j + k, 0.5)
wk=(sum(x<=(k+1)/lambda[h])-sum(x<=k/lambda[h]))/100
sum1[h]=sum1[h]+(lambda[h]/2)*constjk*wk*wj
}
}
}
sum1
}
system.time(res1 <- sim1())
# user system elapsed
# 5.4 0.0 5.4
Now let's make it faster:
sim2 <- function(num=20){
set.seed(42) #to make it reproducible
x <- rlnorm(100,0,1.6)
h <- 1:num
sum1 <- numeric(num)
lambda <- 2+1:num/12.5
N <- ceiling(lambda*max(x))
#functions for wj and wk
wjfun <- function(x,j,lambda,h){
(sum(x<=(j+1)/lambda[h])-sum(x<=j/lambda[h]))/100
}
wkfun <- function(x,k,lambda,h){
(sum(x<=(k+1)/lambda[h])-sum(x<=k/lambda[h]))/100
}
#function to calculate values of sum1
fun1 <- function(N,h,x,lambda) {
sum1 <- 0
set.seed(42) #to make it reproducible
#calculate constants using outer
const <- outer(0:N[h],0:N[h],FUN=function(j,k) dbinom(k, j + k, 0.5))
wk <- numeric(N[h]+1)
#loop only once to calculate wk
for (k in 0:N[h]){
wk[k+1] <- (sum(x<=(k+1)/lambda[h])-sum(x<=k/lambda[h]))/100
}
for (j in 0:N[h])
{
wj <- (sum(x<=(j+1)/lambda[h])-sum(x<=j/lambda[h]))/100
for (k in 0:N[h])
{
sum1 <- sum1+(lambda[h]/2)*const[j+1,k+1]*wk[k+1]*wj
}
}
sum1
}
for (h in 1:num)
{
sum1[h] <- fun1(N,h,x,lambda)
}
sum1
}
system.time(res2 <- sim2())
#user system elapsed
#1.25 0.00 1.25
all.equal(res1,res2)
#[1] TRUE
Timings for #Backlin`s code (with 20 interations) for comparison:
user system elapsed
3.30 0.00 3.29
If this is still too slow and you cannot or don't want to use another language, there is also the possibility of parallelization. As far as I see the outer loop is embarrassingly parallel. There are some nice and easy packages for parallelization.