swNoData

Gibbs sampling was originally designed by Geman and Geman (1984) for drawing updates from the Gibbs distribution, hence the name. However, single-site Gibbs sampling exhibits poor mixing due to the posterior correlation between the pixel labels. Thus it is very slow to converge when the correlation (controlled by the inverse temperature β) is high.

The algorithm of Swendsen and Wang (1987) addresses this problem by forming clusters of neighbouring pixels, then updating all of the labels within a cluster to the same value. When simulating from the prior, such as a Potts model without an external field, this algorithm is very efficient.

The SW function in the PottsUtils package is implemented in a combination of R and C. The swNoData function in bayesImageS is implemented using RcppArmadillo, which gives it a speed advantage. It is worth noting that the intention of bayesImageS is not to replace PottsUtils. Rather, an efficient Swendsen-Wang algorithm is used as a building block for implementations of ABC (Grelaud et al. 2009), path sampling (Gelman and Meng 1998), and the exchange algorithm (Murray, Ghahramani, and MacKay 2006). These other algorithms will be covered in future posts.

There are two things that we want to keep track of in this simulation study: the speed of the algorithm and the distribution of the summary statistic. We will be using system.time(..) to measure both CPU and elapsed (wall clock) time taken for the same number of iterations, for a range of inverse temperatures:

beta <- seq(0,2,by=0.1)
tmMx.PU <- tmMx.bIS <- matrix(nrow=length(beta),ncol=2)
rownames(tmMx.PU) <- rownames(tmMx.bIS) <- beta
colnames(tmMx.PU) <- colnames(tmMx.bIS) <- c("user","elapsed")

We will discard the first 100 iterations as burn-in and keep the remaining 500.

iter <- 600
burn <- 100
samp.PU <- samp.bIS <- matrix(nrow=length(beta),ncol=iter-burn)

The distribution of pixel labels can be summarised by the sufficient statistic of the Potts model:

S(z) = ∑i ∼ ℓ ∈ 𝒩δ(zi, zℓ)

where i ∼ ℓ ∈ 𝒩 are all of the pairs of neighbours in the lattice (ie. the cliques) and δ(u, v) is 1 if u = v and 0 otherwise (the Kronecker delta function). swNoData returns this automatically, but with SW we will need to use the function sufficientStat to calculate the sufficient statistic for the labels.

library(bayesImageS)

mask <- matrix(1,50,50)
neigh <- getNeighbors(mask, c(2,2,0,0))
block <- getBlocks(mask, 2)
edges <- getEdges(mask, c(2,2,0,0))

n <- sum(mask)
k <- 2
bcrit <- log(1 + sqrt(k))
maxSS <- nrow(edges)

for (i in 1:length(beta)) {
  if (requireNamespace("PottsUtils", quietly = TRUE)) {
    tm <- system.time(result <- PottsUtils::SW(iter,n,k,edges,beta=beta[i]))
    tmMx.PU[i,"user"] <- tm["user.self"]
    tmMx.PU[i,"elapsed"] <- tm["elapsed"]
    res <- sufficientStat(result, neigh, block, k)
    samp.PU[i,] <- res$sum[(burn+1):iter]
    print(paste("PottsUtils::SW",beta[i],tm["elapsed"],median(samp.PU[i,])))
   } else {
      print("PottsUtils::SW unavailable on this platform.")
   }

  
  # bayesImageS
  tm <- system.time(result <- swNoData(beta[i],k,neigh,block,iter))
  tmMx.bIS[i,"user"] <- tm["user.self"]
  tmMx.bIS[i,"elapsed"] <- tm["elapsed"]
  samp.bIS[i,] <- result$sum[(burn+1):iter]
  print(paste("bayesImageS::swNoData",beta[i],tm["elapsed"],median(samp.bIS[i,])))
}
## [1] "PottsUtils::SW 0 0.0329999999999995 2451"
## [1] "bayesImageS::swNoData 0 0.103 2452"
## [1] "PottsUtils::SW 0.1 0.353 2575"
## [1] "bayesImageS::swNoData 0.1 0.147000000000002 2574"
## [1] "PottsUtils::SW 0.2 0.310000000000002 2699"
## [1] "bayesImageS::swNoData 0.2 0.155000000000001 2698"
## [1] "PottsUtils::SW 0.3 0.290000000000003 2831"
## [1] "bayesImageS::swNoData 0.3 0.161000000000001 2831"
## [1] "PottsUtils::SW 0.4 0.265999999999998 2968"
## [1] "bayesImageS::swNoData 0.4 0.166999999999998 2974"
## [1] "PottsUtils::SW 0.5 0.244000000000003 3129"
## [1] "bayesImageS::swNoData 0.5 0.169 3133"
## [1] "PottsUtils::SW 0.6 0.221999999999998 3308.5"
## [1] "bayesImageS::swNoData 0.6 0.169 3309"
## [1] "PottsUtils::SW 0.7 0.198 3513"
## [1] "bayesImageS::swNoData 0.7 0.165000000000003 3524"
## [1] "PottsUtils::SW 0.8 0.169 3779"
## [1] "bayesImageS::swNoData 0.8 0.158999999999999 3772.5"
## [1] "PottsUtils::SW 0.9 0.146000000000001 4184"
## [1] "bayesImageS::swNoData 0.9 0.145000000000003 4163"
## [1] "PottsUtils::SW 1 0.123000000000001 4520.5"
## [1] "bayesImageS::swNoData 1 0.134 4526"
## [1] "PottsUtils::SW 1.1 0.110999999999997 4680"
## [1] "bayesImageS::swNoData 1.1 0.127000000000002 4677.5"
## [1] "PottsUtils::SW 1.2 0.103000000000002 4763"
## [1] "bayesImageS::swNoData 1.2 0.123000000000001 4760"
## [1] "PottsUtils::SW 1.3 0.100000000000001 4810"
## [1] "bayesImageS::swNoData 1.3 0.120000000000001 4815"
## [1] "PottsUtils::SW 1.4 0.0920000000000023 4843"
## [1] "bayesImageS::swNoData 1.4 0.116999999999997 4845"
## [1] "PottsUtils::SW 1.5 0.0879999999999974 4862"
## [1] "bayesImageS::swNoData 1.5 0.114000000000001 4863.5"
## [1] "PottsUtils::SW 1.6 0.0860000000000021 4877"
## [1] "bayesImageS::swNoData 1.6 0.112000000000002 4877"
## [1] "PottsUtils::SW 1.7 0.0839999999999996 4884"
## [1] "bayesImageS::swNoData 1.7 0.111000000000001 4883.5"
## [1] "PottsUtils::SW 1.8 0.0820000000000007 4890"
## [1] "bayesImageS::swNoData 1.8 0.109000000000002 4889"
## [1] "PottsUtils::SW 1.9 0.0799999999999983 4893"
## [1] "bayesImageS::swNoData 1.9 0.107000000000003 4893"
## [1] "PottsUtils::SW 2 0.0790000000000006 4896"
## [1] "bayesImageS::swNoData 2 0.105999999999998 4896"

Here is the comparison of elapsed times between the two algorithms (in seconds):

summary(tmMx.PU)
##       user           elapsed      
##  Min.   :0.0330   Min.   :0.0330  
##  1st Qu.:0.0860   1st Qu.:0.0860  
##  Median :0.1120   Median :0.1110  
##  Mean   :0.1533   Mean   :0.1552  
##  3rd Qu.:0.2210   3rd Qu.:0.2220  
##  Max.   :0.3130   Max.   :0.3530
summary(tmMx.bIS)
##       user           elapsed      
##  Min.   :0.1030   Min.   :0.1030  
##  1st Qu.:0.1130   1st Qu.:0.1120  
##  Median :0.1270   Median :0.1270  
##  Mean   :0.1342   Mean   :0.1343  
##  3rd Qu.:0.1590   3rd Qu.:0.1590  
##  Max.   :0.1690   Max.   :0.1690
boxplot(tmMx.PU[,"elapsed"],tmMx.bIS[,"elapsed"],ylab="seconds elapsed",names=c("SW","swNoData"))

On average, swNoData using RcppArmadillo (Eddelbuettel and Sanderson 2014) is seven times faster than SW.

s_z <- c(samp.PU,samp.bIS)
s_x <- rep(beta,times=iter-burn)
s_a <- rep(1:2,each=length(beta)*(iter-burn))
s.frame <- data.frame(s_z,c(s_x,s_x),s_a)
names(s.frame) <- c("stat","beta","alg")
s.frame$alg <- factor(s_a,labels=c("SW","swNoData"))
  if (requireNamespace("lattice", quietly = TRUE)) {
    lattice::xyplot(stat ~ beta | alg, data=s.frame)
  }

plot(c(s_x,s_x),s_z,pch=s_a,xlab=expression(beta),ylab=expression(S(z)))
abline(v=bcrit,col="red")

The overlap between the two distributions is almost complete, although it is a bit tricky to verify that statistically. The relationship between beta and S(z) is nonlinear and heteroskedastic.

rowMeans(samp.bIS) - rowMeans(samp.PU)
##  [1]  -0.040  -0.340  -1.404  -0.392   1.554   1.546   2.064  10.660  -3.618
## [10] -18.178   7.728  -2.438  -1.772   4.414   1.834   2.222   0.544  -0.576
## [19]  -1.232   0.202  -0.376
apply(samp.PU, 1, sd)
##  [1] 36.698374 35.921500 35.748666 36.849151 37.962016 38.662288 42.819998
##  [8] 48.798967 54.238319 69.861555 44.101127 31.817644 24.592360 19.228992
## [15] 15.688572 12.071654 10.052622  8.320042  5.945174  5.315147  4.304819
apply(samp.bIS, 1, sd)
##  [1] 35.416699 33.803422 37.107419 36.668399 36.605667 42.091585 45.678606
##  [8] 48.604568 52.184749 68.799173 45.259994 32.174729 24.749038 19.682361
## [15] 15.629154 12.256048  9.503787  8.781097  6.338919  5.241626  4.554782
s.frame$beta <- factor(c(s_x,s_x))
  if (requireNamespace("PottsUtils", quietly = TRUE)) {
    s.fit <- aov(stat ~ alg + beta, data=s.frame)
    summary(s.fit)
    TukeyHSD(s.fit,which="alg")
  }
##   Tukey multiple comparisons of means
##     95% family-wise confidence level
## 
## Fit: aov(formula = stat ~ alg + beta, data = s.frame)
## 
## $alg
##                 diff        lwr      upr    p adj
## swNoData-SW 0.114381 -0.8183198 1.047082 0.810044

References

Eddelbuettel, Dirk, and Conrad Sanderson. 2014. “RcppArmadillo: Accelerating R with High-Performance C++ Linear Algebra.” Comput. Stat. Data Anal. 71: 1054–63. https://doi.org/10.1016/j.csda.2013.02.005.
Gelman, Andrew, and Xiao-Li Meng. 1998. “Simulating Normalizing Constants: From Importance Sampling to Bridge Sampling to Path Sampling.” Statist. Sci. 13 (2): 163–85. https://doi.org/10.1214/ss/1028905934.
Geman, Stuart, and Donald Geman. 1984. “Stochastic Relaxation, Gibbs Distributions and the Bayesian Restoration of Images.” IEEE Trans. PAMI 6: 721–41.
Grelaud, Aude, Christian P. Robert, Jean-Michel Marin, François Rodolphe, and Jean-François Taly. 2009. “ABC Likelihood-Free Methods for Model Choice in Gibbs Random Fields.” Bayesian Analysis 4 (2): 317–36. https://doi.org/10.1214/09-BA412.
Murray, Iain, Zoubin Ghahramani, and David J. C. MacKay. 2006. “MCMC for Doubly-Intractable Distributions.” In Proc. 22nd Conf. UAI, 359–66. Arlington, VA: AUAI Press.
Swendsen, Robert H., and Jian-Sheng Wang. 1987. “Nonuniversal Critical Dynamics in Monte Carlo Simulations.” Phys. Rev. Lett. 58: 86–88. https://doi.org/10.1103/PhysRevLett.58.86.