mined package

Description

Generate minimum energy design (MinED) samples from an unnormalized probability density function. The asymptotic distribution of MinED samples converges to the target distribution and therefore, MinED can be viewed as a deterministic sample from the target distribution. The details of MinED and the algorithm used for generating it can be found in Joseph, Dasgupta, Tuo, and Wu (2015) and Joseph, Wang, Gu, Lv, and Tuo (2019). This research is supported by a U.S. National Science Foundation grant DMS-1712642.

Details

Important functions in this package are: mined generates Minimum Energy Design samples from an unnormalized density function, SelectMinED selects Minimum Energy Design samples from candidate points, and Lattice generates good rank-1 lattice rules.

Author(s)

Dianpeng Wang and V. Roshan Joseph

Maintainer: Dianpeng Wang <wdp@bit.edu.cn>

References

Joseph, V. R., Dasgupta, T., Tuo, R., and Wu, C. F. J. (2015). "Sequential Exploration of Complex Surfaces Using Minimum Energy Designs". Technometrics, 57, 64-74.

Joseph, V. R., Wang, D., Gu, L., Lv, S., and Tuo, R. (2019). "Deterministic Sampling of Expensive Posteriors Using Minimum Energy Designs", Technometrics, 61, 297-308, arXiv:1712.08929, DOI:10.1080/00401706.2018.1552203.

Good lattice points

Description

Generate good rank-1 lattice points with prime number of points by using the fast component-by-component construction algorithm of Nuyens and Cools (2006). Refer Nuyens (2007) for more details.

Usage

Lattice(n, p)

Arguments

Value

An n-by-p matrix containing the good lattice points.

Author(s)

Dianpeng Wang <wdp@bit.edu.cn> and V. Roshan Joseph <roshan@gatech.edu>

References

Nuyens, D. and Cools, R. (2006). "Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces.", Mathematics of Computation, 75, 903-920.

Nuyens, D. (2007). "Fast Construction of Good Lattice Rules.", Ph.D Thesis, Katholieke Universiteit Leuven, Leuven, Belgium.

Examples

library(mined)
res <- Lattice(101, 2)
plot(res[, 1], res[, 2], col='red',xlab='First dimension', ylab='Second dimension', pch=15)

Select Minimum Energy Design samples from a candidate set

Description

Select MinED samples from candidates by optimizing the generalized MinED criterion in Joseph et al. (2019).

Usage

SelectMinED(candidates, candlf, n, gamma=1, s=2)

Arguments

Details

This function select MinED samples from a given set of candidate samples. The function is used internally in the mined function repeatedly for K times, where K is the number of annealing steps in the algorithm. Refer to Joseph et al., (2018) for more details.

Value

The value returned from the function is a list containing the following components:

Author(s)

Dianpeng Wang <wdp@bit.edu.cn> and V. Roshan Joseph <roshan@gatech.edu>

References

Joseph, V. R., Wang, D., Gu, L., Lv, S., and Tuo, R. (2019). "Deterministic Sampling of Expensive Posteriors Using Minimum Energy Designs", Technometrics, 61, 297-308, arXiv:1712.08929, DOI:10.1080/00401706.2018.1552203.

See Also

mined

Examples

cand <- matrix(runif(10000, min = -4, max = 4), ncol = 1)
candlf <- log(dnorm(cand))
res <- mined::SelectMinED(cand, as.vector(candlf), 150, 1.0, 2.0)
print(res)
par(mfrow=c(1,2))
hist(cand)
hist(res$points)

Minimum Energy Design

Description

Generate MinED samples from an unnormalized density function.

Usage

mined(initial, logf, K_iter = 0)

Arguments

Details

This is the main function of the package, which is used for generating the MinED samples. The MinED sample can be viewed as a deterministic sample from the probability density specified in the mined function. Since only the unnormalized density is needed to generate the MinED samples, this method could be used in Bayesian computation to approximate the posterior. The method uses few evaluations of the unnormalized posterior compared to random sampling-based methods and therefore, it will be useful when the evaluations are expensive or time consuming.

There are many parameters that control the performance of the algorithm, which are fixed at some reasonable values as specified in Joseph et al. (2019). The only thing user need to choose is the region for scaling the variables in [0,1]^p. Ideally it should be the highercube containing the highest posterior density region with good coverage. However, the algorithm is robust to this choice to some extend as it can shrink or expand from the intial region. Therefore, it can be chosen based on user's guessed range of each variable.

Value

The value returned from the function is a list containing the following components:

Author(s)

Dianpeng Wang <wdp@bit.edu.cn> and V. Roshan Joseph <roshan@gatech.edu>

References

Joseph, V. R., Wang, D., Gu, L., Lv, S., and Tuo, R. (2019). "Deterministic Sampling of Expensive Posteriors Using Minimum Energy Designs", Technometrics, 61, 297-308, arXiv:1712.08929, DOI:10.1080/00401706.2018.1552203.

Examples

require(mined)
p <- 2
n <- 109 # largest prime number less than 100+5p
initial <- Lattice(n, p)

# suppose x1 is in [-40,40] and x2 in [-25,10]
logf <- function(para)
{
  l1 <- -40
  u1 <- 40
  l2 <- -25
  u2 <- 10
  x1 <- l1 + (u1 - l1) * para[1]
  x2 <- l2 + (u2 - l2) * para[2]
  val <- -.5 * (x1 ^2 / 100 + (x2+ .03 * x1^2 -3)^2)
  return(val)
}

res <- mined::mined(initial, logf, K_iter = 8)
dim(res$points)
dim(res$cand)

x1 <- seq(0, 1, length.out = 200)
x2 <- seq(0, 1, length.out = 200)
y <- matrix(0.0, 200, 200)
for(i in 1:200)
{
  for(j in 1:200)
  {
    y[i, j] = logf(c(x1[i], x2[j]))
  }
}
image(x1, x2, exp(y), col = cm.colors(5), xlab = expression(x[1]), ylab = expression(x[2]))
points(res$cand[, 1], res$cand[, 2], pch = 11, col = rgb(red = 0, green = 0, blue = 1, 
       alpha = 0.35), cex = .25)
points(res$points[, 1], res$points[, 2], pch = 17, col = 'black', cex = .75)
legend("bottom", c('Candidates points', 'MinED samples'), pch = c(11, 17), 
        col = c(rgb(red = 0, green = 0, blue = 1, alpha = 0.35), 'black'), 
        inset = .02, bg = 'transparent', bty = 'n')