Description Usage Arguments Details Value Note Author(s) References Examples
Generates a sequence of random variables using ARMS. For multivariate densities, ARMS is used along randomly selected straight lines through the current point.
1 |
y.start |
initial point |
myldens |
univariate or multivariate log target density |
indFunc |
indicator function of the convex support of the target density |
n.sample |
desired sample size |
... |
parameters passed to |
Strictly speaking, the support of the target density must be a bounded convex set.
When this is not the case, the following tricks usually work.
If the support is not bounded, restrict it to a bounded set having probability
practically one.
A workaround, if the support is not convex, is to consider the convex set
generated by the support
and define myldens
to return log(.Machine$double.xmin)
outside
the true support (see the last example.)
The next point is generated along a randomly selected line through the current point using arms.
Make sure the value returned by myldens
is never smaller than
log(.Machine$double.xmin)
, to avoid divisions by zero.
An n.sample
by length(y.start)
matrix, whose rows are the
sampled points.
The function is based on original C code by W. Gilks for the univariate case, http://www.mrc-bsu.cam.ac.uk/pub/methodology/adaptive_rejection/.
Giovanni Petris GPetris@uark.edu
Gilks, W.R., Best, N.G. and Tan, K.K.C. (1995) Adaptive rejection Metropolis sampling within Gibbs sampling (Corr: 97V46 p541-542 with Neal, R.M.), Applied Statistics 44:455–472.
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 | #### ==> Warning: running the examples may take a few minutes! <== ####
## Not run:
set.seed(4521222)
### Univariate densities
## Unif(-r,r)
y <- arms(runif(1,-1,1), function(x,r) 1, function(x,r) (x>-r)*(x<r), 5000, r=2)
summary(y); hist(y,prob=TRUE,main="Unif(-r,r); r=2")
## Normal(mean,1)
norldens <- function(x,mean) -(x-mean)^2/2
y <- arms(runif(1,3,17), norldens, function(x,mean) ((x-mean)>-7)*((x-mean)<7),
5000, mean=10)
summary(y); hist(y,prob=TRUE,main="Gaussian(m,1); m=10")
curve(dnorm(x,mean=10),3,17,add=TRUE)
## Exponential(1)
y <- arms(5, function(x) -x, function(x) (x>0)*(x<70), 5000)
summary(y); hist(y,prob=TRUE,main="Exponential(1)")
curve(exp(-x),0,8,add=TRUE)
## Gamma(4.5,1)
y <- arms(runif(1,1e-4,20), function(x) 3.5*log(x)-x,
function(x) (x>1e-4)*(x<20), 5000)
summary(y); hist(y,prob=TRUE,main="Gamma(4.5,1)")
curve(dgamma(x,shape=4.5,scale=1),1e-4,20,add=TRUE)
## Gamma(0.5,1) (this one is not log-concave)
y <- arms(runif(1,1e-8,10), function(x) -0.5*log(x)-x,
function(x) (x>1e-8)*(x<10), 5000)
summary(y); hist(y,prob=TRUE,main="Gamma(0.5,1)")
curve(dgamma(x,shape=0.5,scale=1),1e-8,10,add=TRUE)
## Beta(.2,.2) (this one neither)
y <- arms(runif(1), function(x) (0.2-1)*log(x)+(0.2-1)*log(1-x),
function(x) (x>1e-5)*(x<1-1e-5), 5000)
summary(y); hist(y,prob=TRUE,main="Beta(0.2,0.2)")
curve(dbeta(x,0.2,0.2),1e-5,1-1e-5,add=TRUE)
## Triangular
y <- arms(runif(1,-1,1), function(x) log(1-abs(x)), function(x) abs(x)<1, 5000)
summary(y); hist(y,prob=TRUE,ylim=c(0,1),main="Triangular")
curve(1-abs(x),-1,1,add=TRUE)
## Multimodal examples (Mixture of normals)
lmixnorm <- function(x,weights,means,sds) {
log(crossprod(weights, exp(-0.5*((x-means)/sds)^2 - log(sds))))
}
y <- arms(0, lmixnorm, function(x,...) (x>(-100))*(x<100), 5000, weights=c(1,3,2),
means=c(-10,0,10), sds=c(1.5,3,1.5))
summary(y); hist(y,prob=TRUE,main="Mixture of Normals")
curve(colSums(c(1,3,2)/6*dnorm(matrix(x,3,length(x),byrow=TRUE),c(-10,0,10),c(1.5,3,1.5))),
par("usr")[1], par("usr")[2], add=TRUE)
### Bivariate densities
## Bivariate standard normal
y <- arms(c(0,2), function(x) -crossprod(x)/2,
function(x) (min(x)>-5)*(max(x)<5), 500)
plot(y, main="Bivariate standard normal", asp=1)
## Uniform in the unit square
y <- arms(c(0.2,.6), function(x) 1,
function(x) (min(x)>0)*(max(x)<1), 500)
plot(y, main="Uniform in the unit square", asp=1)
polygon(c(0,1,1,0),c(0,0,1,1))
## Uniform in the circle of radius r
y <- arms(c(0.2,0), function(x,...) 1,
function(x,r2) sum(x^2)<r2, 500, r2=2^2)
plot(y, main="Uniform in the circle of radius r; r=2", asp=1)
curve(-sqrt(4-x^2), -2, 2, add=TRUE)
curve(sqrt(4-x^2), -2, 2, add=TRUE)
## Uniform on the simplex
simp <- function(x) if ( any(x<0) || (sum(x)>1) ) 0 else 1
y <- arms(c(0.2,0.2), function(x) 1, simp, 500)
plot(y, xlim=c(0,1), ylim=c(0,1), main="Uniform in the simplex", asp=1)
polygon(c(0,1,0), c(0,0,1))
## A bimodal distribution (mixture of normals)
bimodal <- function(x) { log(prod(dnorm(x,mean=3))+prod(dnorm(x,mean=-3))) }
y <- arms(c(-2,2), bimodal, function(x) all(x>(-10))*all(x<(10)), 500)
plot(y, main="Mixture of bivariate Normals", asp=1)
## A bivariate distribution with non-convex support
support <- function(x) {
return(as.numeric( -1 < x[2] && x[2] < 1 &&
-2 < x[1] &&
( x[1] < 1 || crossprod(x-c(1,0)) < 1 ) ) )
}
Min.log <- log(.Machine$double.xmin) + 10
logf <- function(x) {
if ( x[1] < 0 ) return(log(1/4))
else
if (crossprod(x-c(1,0)) < 1 ) return(log(1/pi))
return(Min.log)
}
x <- as.matrix(expand.grid(seq(-2.2,2.2,length=40),seq(-1.1,1.1,length=40)))
y <- sapply(1:nrow(x), function(i) support(x[i,]))
plot(x,type='n',asp=1)
points(x[y==1,],pch=1,cex=1,col='green')
z <- arms(c(0,0), logf, support, 1000)
points(z,pch=20,cex=0.5,col='blue')
polygon(c(-2,0,0,-2),c(-1,-1,1,1))
curve(-sqrt(1-(x-1)^2),0,2,add=TRUE)
curve(sqrt(1-(x-1)^2),0,2,add=TRUE)
sum( z[,1] < 0 ) # sampled points in the square
sum( apply(t(z)-c(1,0),2,crossprod) < 1 ) # sampled points in the circle
## End(Not run)
|
Min. 1st Qu. Median Mean 3rd Qu. Max.
-1.998639 -1.006130 0.002898 0.006704 1.024718 1.999930
Min. 1st Qu. Median Mean 3rd Qu. Max.
6.208 9.292 9.975 9.978 10.679 14.128
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.000622 0.286766 0.685792 0.997736 1.380835 8.698386
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.3733 2.9347 4.1502 4.5097 5.6906 15.2753
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.000394 0.075437 0.265410 0.535382 0.711538 6.345985
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.000378 0.156981 0.900456 0.639593 0.999988 0.999988
Min. 1st Qu. Median Mean 3rd Qu. Max.
-0.991684 -0.283172 0.004016 0.005851 0.296995 0.989903
Min. 1st Qu. Median Mean 3rd Qu. Max.
-14.613 -2.662 1.592 1.890 9.110 14.969
[1] 501
[1] 499
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.