SJpearson: Simulate joint given marginals and Pearson correlations.

In SimJoint: Simulate Joint Distribution

Description

Reorder elements in each column of a matrix such that the column-wise Pearson correlations approximate a given correlation matrix. Use xSJpearson() for the freedom of supplying the noise matrix, which can let the dependency structure of the result joint distribution be characterized by a certain copula. See the copula section in the package vignette for details.

Usage

SJpearson(
  X,
  cor,
  stochasticStepDomain = as.numeric(c(0, 1)),
  errorType = "meanSquare",
  seed = 123L,
  maxCore = 7L,
  convergenceTail = 8L,
  iterLimit = 100000L,
  verbose = TRUE
  )

Arguments

X	An N x K numeric matrix of K marginal distributions (samples). Columns are sorted.
cor	A K x K correlation matrix. The matrix should be positive semi-definite.
stochasticStepDomain	A numeric vector of size 2. Range of the stochastic step ratio for correcting the correlation matrix in each iteration. Default [0, 1]. See the package vignette for more details.
errorType	Cost function for convergence test. "meanRela": average absolute relative error between elements of the target correlation matrix and the correlation matrix approximated in each iteration. "maxRela": maximal absolute relative error. "meanSquare": mean squared error. Default.
seed	An integer or an integer vector of size 4. A single integer seeds a pcg64 generator the usual way. An integer vector of size 4 supplies all the bits for a pcg64 object. Default 123.
maxCore	An integer. Maximal threads to invoke. Default 7. Better be no greater than the total number of virtual cores on machine.
convergenceTail	An integer. If the last convergenceTail iterations resulted in equal cost function values, return. Default 8.
iterLimit	An integer. The maximal number of iterations. Default 100000.
verbose	A boolean value. TRUE prints progress.

Details

Algorithms are detailed in the package vignette.

Value

A list of size 2.

X	A numeric matrix of size N x K, the simulated joint distribution.
cor	Pearson correlation matrix of X.

Examples

# #############################################################################
# Commented code blocks either require external source, or would exceed
# execution time constraint for CRAN check.
# #############################################################################


# =============================================================================
# Benchmark against R package `SimMultiCorrData`. Use the same example
# from <https://cran.r-project.org/web/packages/SimMultiCorrData/
#       vignettes/workflow.html>.
# =============================================================================
set.seed(123)
N = 10000L # Sample size.
K = 10L    # 10 marginals.
# Sample from 3 PDFs, 2 nonparametric PMFs, 5 parametric PMFs:
marginals = cbind(
  rnorm(N), rchisq(N, 4), rbeta(N, 4, 2),
  SimJoint::LHSpmf(data.frame(val = 1:3, P = c(0.3, 0.45, 0.25)), N,
               seed = sample(1e6L, 1)),
  SimJoint::LHSpmf(data.frame(val = 1:4, P = c(0.2, 0.3, 0.4, 0.1)), N,
               seed = sample(1e6L, 1)),
  rpois(N, 1), rpois(N, 5), rpois(N, 10),
  rnbinom(N, 3, 0.2), rnbinom(N, 6, 0.8))
# The seeding for `LHSpmf()` is unhealthy, but OK for small examples.


marginals = apply(marginals, 2, function(x) sort(x))


# Create the example target correlation matrix `Rey`:
set.seed(11)
Rey <- diag(1, nrow = K)
for (i in 1:nrow(Rey)) {
  for (j in 1:ncol(Rey)) {
    if (i > j) Rey[i, j] <- runif(1, 0.2, 0.7)
    Rey[j, i] <- Rey[i, j]
  }
}


system.time({result = SimJoint::SJpearson(
  X = marginals, cor = Rey, errorType = "meanSquare", seed = 456,
  maxCore = 1, convergenceTail = 8, verbose = FALSE)})
# user  system elapsed
# 0.30    0.00    0.29
# One the same platform, single-threaded speed (Intel i7-4770 CPU
# @ 3.40GHz, 32GB RAM, Windows 10, g++ 4.9.3 -Ofast, R 3.5.2) is more
# than 50 times faster than `SimMultiCorrData::rcorrvar()`:
# user  system elapsed
# 16.05   0.34   16.42


# Check error statistics.
summary(as.numeric(round(cor(result$X) - Rey, 6)))
# Min.          1st Qu.     Median       Mean    3rd Qu.       Max.
# -0.000365   -0.000133  -0.000028  -0.000047  0.000067    0.000301


# Post simulation optimization further reduce the errors:
resultOpt = SimJoint::postSimOpt(
  X = result$X, cor = Rey, convergenceTail = 10000)
summary(as.numeric(round(cor(resultOpt$X) - Rey, 6)))
# Min.        1st Qu.    Median      Mean   3rd Qu.      Max.
# -7.10e-05 -3.10e-05 -1.15e-05 -6.48e-06  9.00e-06  7.10e-05


# Max error magnitude is less than 1% of that from
# `SimMultiCorrData::rcorrvar()`:
# Min.          1st Qu.     Median       Mean    3rd Qu.       Max.
# -0.008336   -0.001321          0  -0.000329   0.001212    0.00339
# This table is reported in Step 4, correlation methods 1 or 2.




# =============================================================================
# Use the above example and benchmark against John Ruscio & Walter
# Kaczetow (2008) iteration. The R code released with their paper was
# erroneous. A corrected version is given by Github user "nicebread":
# <https://gist.github.com/nicebread/4045717>, but his correction was
# incomprehensive and can only handle 2-dimensional instances. Please change
# Line 32 to `Target.Corr <- rho` and source the file.
# =============================================================================
# # Test Ruscio-Kaczetow's code.
# set.seed(123)
# RuscioKaczetow = GenData(Pop = as.data.frame(marginals),
#                          Rey, N = 1000) # By default, the function takes 1000
# # samples from each marginal population of size 10000.
# summary(round(as.numeric(cor(RuscioKaczetow) - Rey), 6))
# # Min.         1st Qu.    Median      Mean   3rd Qu.      Max.
# # -0.183274 -0.047461  -0.015737 -0.008008  0.027475  0.236662


result = SimJoint::SJpearson(
  X = apply(marginals, 2, function(x) sort(sample(x, 1000, replace = TRUE))),
  cor = Rey, errorType = "maxRela", maxCore = 2) # CRAN does not allow more
# than 2 threads for running examples.
summary(round(as.numeric(cor(result$X) - Rey), 6))
# Min.           1st Qu.     Median       Mean    3rd Qu.       Max.
# -0.0055640  -0.0014850 -0.0004810 -0.0007872  0.0000000  0.0025920
resultOpt = SimJoint::postSimOpt(
  X = result$X, cor = Rey, convergenceTail = 10000)
summary(as.numeric(round(cor(resultOpt$X) - Rey, 6)))
# Min.          1st Qu.     Median       Mean    3rd Qu.       Max.
# -6.240e-04 -2.930e-04 -2.550e-05 -6.532e-05  1.300e-04  5.490e-04




# =============================================================================
# Benchmark against R package `GenOrd`
# <https://cran.r-project.org/web/packages/GenOrd/index.html> using the
# example above Statistics cannot be collected because it has been running
# for more than 10 hours.
# =============================================================================
# # Library `GenOrd` should have been installed and attached.
# system.time({resultGenOrd = ordsample(
#   N, marginal = lapply(1L : K, function(x) (1 : (N - 1)) / N), Rey,
#   support = as.data.frame(marginals))})




# =============================================================================
# Benchmark against R package `EnvStats` using its manual example on Page 1156
# of <https://cran.r-project.org/web/packages/EnvStats/EnvStats.pdf>. The
# function `simulateVector()` imposes rank correlations.
# =============================================================================
# # Library `EnvStats` should have been installed and attached.
# cor.mat = matrix(c(1, 0.8, 0, 0.5, 0.8, 1, 0, 0.7, 0, 0, 1, 0.2, 0.5,
#                    0.7, 0.2, 1), 4, 4)
# pareto.rns <- simulateVector(100, "pareto", list(location = 10, shape = 2),
#                              sample.method = "LHS", seed = 56)
# mat <- simulateMvMatrix(
#   1000, distributions = c(Normal = "norm", Lognormal = "lnormAlt",
#                           Beta = "beta", Empirical = "emp"),
#   param.list = list(Normal = list(mean=10, sd=2),
#                     Lognormal = list(mean=10, cv=1),
#                     Beta = list(shape1 = 2, shape2 = 3),
#                     Empirical = list(obs = pareto.rns)),
#   cor.mat = cor.mat, seed = 47, sample.method = "LHS")
#
#
# round(cor(mat, method = "spearman"), 2)
# #          Normal Lognormal    Beta Empirical
# #Normal      1.00    0.78     -0.01      0.47
# #Lognormal   0.78    1.00     -0.01      0.67
# #Beta       -0.01   -0.01      1.00      0.19
# #Empirical   0.47    0.67      0.19      1.00
#
#
# # Imposing rank correlations is equivalent to imposing Pearson correlations
# # on ranks.
# set.seed(123)
# marginals = cbind(sort(rnorm(1000, 10, 2)),
#                   sort(rlnormAlt(1000, 10, 1)),
#                   sort(rbeta(1000, 2, 3)),
#                   sort(sample(pareto.rns, 1000, replace = TRUE)))
# marginalsRanks = cbind(1:1000, 1:1000, 1:1000, 1:1000)
# # Simulate the joint for ranks:
# tmpResult = SimJoint::SJpearson(
#   X = marginalsRanks, cor = cor.mat, errorType = "meanSquare", seed = 456,
#   maxCore = 2, convergenceTail = 8, verbose = TRUE)$X
# # Reorder `marginals` by ranks.
# result = matrix(mapply(function(x, y) y[as.integer(x)],
#                        as.data.frame(tmpResult),
#                        as.data.frame(marginals), SIMPLIFY = TRUE), ncol = 4)
# round(cor(result, method = "spearman"), 2)
# # 1.0  0.8  0.0  0.5
# # 0.8  1.0  0.0  0.7
# # 0.0  0.0  1.0  0.2
# # 0.5  0.7  0.2  1.0




# ============================================================================
# Play random numbers.
# ============================================================================
set.seed(123)
N = 2000L
K = 20L
# The following essentially creates a mixture distribution.
marginals = c(runif(10000L, -2, 2), rgamma(10000L, 2, 2), rnorm(20000L))
marginals = matrix(sample(marginals, length(marginals)), ncol = K)
# This operation made the columns comprise samples from the same
# mixture distribution.
marginals = apply(marginals, 2, function(x) sort(x))


# May take a while to generate valid correlation matrix.
while(TRUE)
{
  targetCor = matrix(runif(K * K, -0.1, 0.4), ncol = K)
  targetCor[lower.tri(targetCor)] = t(targetCor)[lower.tri(t(targetCor))]
  diag(targetCor) = 1
  if(all(eigen(targetCor)$values >= 0)) break
}


result = SimJoint::SJpearson(
  X = marginals, cor = targetCor, errorType = "meanSquare", seed = 456,
  maxCore = 2, convergenceTail = 8, verbose = TRUE)
resultOpt = SimJoint::postSimOpt(
  X = result$X, cor = targetCor, convergenceTail = 10000)


# Visualize errors and correlation matrices.
par(mfrow = c(2, 2))
hist(resultOpt$cor - targetCor, breaks = K * K, main = NULL,
     xlab = "Error")
hist(resultOpt$cor / targetCor - 1, breaks = K * K, main = NULL,
     xlab = "Relative error")
zlim = range(range(targetCor[targetCor < 1]),
             range(resultOpt$cor[resultOpt$cor < 1]))
col = colorRampPalette(c("blue", "red", "yellow"))(K * K)
tmp = targetCor[, K : 1L]
image(tmp, xaxt = "n", yaxt = "n", zlim = zlim, bty = "n",
      main = "Target cor", col = col)
tmp = resultOpt$cor[, K : 1L]
image(tmp, xaxt = "n", yaxt = "n", zlim = zlim, bty = "n",
      main = "Cor reached", col = col)
par(mfrow = c(1, 1))




# =============================================================================
# An example where the functional relationships between marginals are highly
# nonlinear and the target correlations are hard to impose. Other packages
# would fail or report theoretical infeasibility.
# =============================================================================
set.seed(123)
N = 10000L
K = 10L


# Several 2-parameter PDFs in R:
marginals = list(rbeta, rcauchy, rf, rgamma, rnorm, runif, rweibull)
Npdf = length(marginals)


if(Npdf >= K) chosenMarginals =
  marginals[sample(Npdf, K, replace = TRUE)] else chosenMarginals =
  marginals[c(1L : Npdf, sample(Npdf, K - Npdf, replace = TRUE))]


# Sample from the marginal PDFs.
marginals = as.matrix(as.data.frame(lapply(chosenMarginals, function(f)
{
  para = sort(runif(2, 0.1, 10))
  rst = f(N, para[1], para[2])
  sort(rst)
})))
dimnames(marginals) = NULL


frechetUpperCor = cor(marginals) # The correlation matrix should be
# upper-bounded by that of the perfectly rank-correlated
# joint (Frechet upper bound). The lower bound is characterized by
# d-countercomonotonicity and depends not only on marginals.
cat("Range of maximal correlations between marginals:",
    range(frechetUpperCor[frechetUpperCor < 1]))
# Two perfectly rank-correlated marginals can have a Pearson
# correlation below 0.07. This is due to high nonlinearities
# in marginal PDFs.


# Create a valid correlation matrix upper-bounded by `frechetUpperCor`.
while(TRUE)
{
  targetCor = sapply(frechetUpperCor, function(x)
    runif(1, -0.1, min(0.3, x * 0.8)))
  targetCor = matrix(targetCor, ncol = K)
  targetCor[lower.tri(targetCor)] = t(targetCor)[lower.tri(t(targetCor))]
  diag(targetCor) = 1
  if(min(eigen(targetCor)$values) >= 0) break # Stop once the correlation
  # matrix is semi-positive definite. This loop could run for
  # a long time if we do not bound the uniform by 0.3.
}


result = SimJoint::SJpearson(
  X = marginals, cor = targetCor, stochasticStepDomain = c(0, 1),
  errorType = "meanSquare", seed = 456, maxCore = 2, convergenceTail = 8)
# resultOpt = SimJoint::postSimOpt( # Could take many seconds.
#   X = result$X, cor = targetCor, convergenceTail = 10000)
#
#
# # Visualize errors and correlation matrices.
# par(mfrow = c(2, 2))
# hist(resultOpt$cor - targetCor, breaks = K * K, main = NULL,
#      xlab = "Error")
# hist(resultOpt$cor / targetCor - 1, breaks = K * K, main = NULL,
#      xlab = "Relative error")
# zlim = range(range(targetCor[targetCor < 1]),
#              range(resultOpt$cor[resultOpt$cor < 1]))
# col = colorRampPalette(c("blue", "red", "yellow"))(K * K)
# tmp = targetCor[, K : 1L]
# image(tmp, xaxt = "n", yaxt = "n", zlim = zlim, bty = "n",
#       main = "Target cor", col = col)
# tmp = resultOpt$cor[, K : 1L]
# image(tmp, xaxt = "n", yaxt = "n", zlim = zlim, bty = "n",
#       main = "Cor reached", col = col)
# par(mfrow = c(1, 1))


# Different `errorType` could make a difference.
result = SimJoint::SJpearson(
  X = marginals, cor = targetCor, stochasticStepDomain = c(0, 1),
  errorType = "maxRela", seed = 456, maxCore = 2, convergenceTail = 8)
# resultOpt = SimJoint::postSimOpt(
#   X = result$X, cor = targetCor, convergenceTail = 10000)
#
#
# # Visualize errors and correlation matrices.
# par(mfrow = c(2, 2))
# hist(resultOpt$cor - targetCor, breaks = K * K, main = NULL,
#      xlab = "Error")
# hist(resultOpt$cor / targetCor - 1, breaks = K * K, main = NULL,
#      xlab = "Relative error")
# zlim = range(range(targetCor[targetCor < 1]),
#              range(resultOpt$cor[resultOpt$cor < 1]))
# col = colorRampPalette(c("blue", "red", "yellow"))(K * K)
# tmp = targetCor[, K : 1L]
# image(tmp, xaxt = "n", yaxt = "n", zlim = zlim, bty = "n",
#       main = "Target cor", col = col)
# tmp = resultOpt$cor[, K : 1L]
# image(tmp, xaxt = "n", yaxt = "n", zlim = zlim, bty = "n",
#       main = "Cor reached", col = col)
# par(mfrow = c(1, 1))

SimJoint documentation built on Dec. 11, 2021, 9:29 a.m.