| Title: | Simulate Joint Distribution |
|---|---|
| Description: | Simulate multivariate correlated data given nonparametric marginals and their joint structure characterized by a Pearson or Spearman correlation matrix. The simulator engages the problem from a purely computational perspective. It assumes no statistical models such as copulas or parametric distributions, and can approximate the target correlations regardless of theoretical feasibility. The algorithm integrates and advances the Iman-Conover (1982) approach <doi:10.1080/03610918208812265> and the Ruscio-Kaczetow iteration (2008) <doi:10.1080/00273170802285693>. Package functions are carefully implemented in C++ for squeezing computing speed, suitable for large input in a manycore environment. Precision of the approximation and computing speed both substantially outperform various CRAN packages to date. Benchmarks are detailed in function examples. A simple heuristic algorithm is additionally designed to optimize the joint distribution in the post-simulation stage. The heuristic demonstrated good potential of achieving the same level of precision of approximation without the enhanced Iman-Conover-Ruscio-Kaczetow. The package contains a copy of Permuted Congruential Generator. |
| Authors: | Charlie Wusuo Liu |
| Maintainer: | Charlie Wusuo Liu <[email protected]> |
| License: | GPL-3 |
| Version: | 0.3.12 |
| Built: | 2024-11-10 06:37:35 UTC |
| Source: | CRAN |
Create a matrix where columns are (Pearson) uncorrelated.
decor(seedMat)decor(seedMat)
seedMat |
A matrix where the number of rows is no less than the number of columns. The function will change |
Algorithm: for i = 2 to ncol(seedMat), the function replaces the first i - 1 elements of the ith column with values such that the new ith column becomes uncorrelated with the first i - 1 columns.
None.
set.seed(123) X = matrix(rnorm(1000), ncol = 10) corMat = cor(X) summary(corMat[corMat < 1]) # Off-diagonal. # Min. 1st Qu. Median Mean 3rd Qu. Max. # -0.19271 -0.05648 -0.02272 -0.01303 0.01821 0.24521 SimJoint::decor(X) corMat2 = cor(X) summary(corMat2[corMat2 < 1]) # Min. 1st Qu. Median Mean 3rd Qu. Max. # -2.341e-17 -3.627e-18 3.766e-18 4.018e-18 1.234e-17 3.444e-17set.seed(123) X = matrix(rnorm(1000), ncol = 10) corMat = cor(X) summary(corMat[corMat < 1]) # Off-diagonal. # Min. 1st Qu. Median Mean 3rd Qu. Max. # -0.19271 -0.05648 -0.02272 -0.01303 0.01821 0.24521 SimJoint::decor(X) corMat2 = cor(X) summary(corMat2[corMat2 < 1]) # Min. 1st Qu. Median Mean 3rd Qu. Max. # -2.341e-17 -3.627e-18 3.766e-18 4.018e-18 1.234e-17 3.444e-17
Export all the bits needed for seeding Permuted Congruential Generator.
exportRandomState(seed)exportRandomState(seed)
seed |
An integer or an integer vector of size 4 (128 bits). See <http://www.pcg-random.org/>. |
The returned integer vector Value supplies all the bits necessary for determining the state of a pcg64 generator. Value can seed for all functions that need a RNG in this package. It will change after the function call, ready for seeding the pcg64 generator in the next function call.
An integer vector of size 4.
# Make a random PMF. set.seed(456) val = seq(0, 15, len = 100) pmf = data.frame( val = val, P = dgamma(val, shape = 2, scale = 2) + runif(100) * 0.1) pmf$P = pmf$P / sum(pmf$P) completeRandomState = SimJoint::exportRandomState(456) # `completeRandomState` comprises all the bits of a pcg64 # engine seeded by 456. It is similar to R's `.Random.seed`. pmfSample1 = SimJoint::LHSpmf(pmf, 1000, completeRandomState) pmfSample2 = SimJoint::LHSpmf(pmf, 1000, completeRandomState) pmfSample3 = SimJoint::LHSpmf(pmf, 1000, completeRandomState) # `completeRandomState` is changed in each run of `LHSpmf()`. targetCor = rbind( c(1, 0.3, 0.5), c(0.3, 1, 0.3), c(0.5, 0.3, 1)) result = SimJoint::SJpearson( X = cbind(sort(pmfSample1), sort(pmfSample2), sort(pmfSample3)), cor = targetCor, seed = completeRandomState, errorType = "maxRela") cor(result$X)# Make a random PMF. set.seed(456) val = seq(0, 15, len = 100) pmf = data.frame( val = val, P = dgamma(val, shape = 2, scale = 2) + runif(100) * 0.1) pmf$P = pmf$P / sum(pmf$P) completeRandomState = SimJoint::exportRandomState(456) # `completeRandomState` comprises all the bits of a pcg64 # engine seeded by 456. It is similar to R's `.Random.seed`. pmfSample1 = SimJoint::LHSpmf(pmf, 1000, completeRandomState) pmfSample2 = SimJoint::LHSpmf(pmf, 1000, completeRandomState) pmfSample3 = SimJoint::LHSpmf(pmf, 1000, completeRandomState) # `completeRandomState` is changed in each run of `LHSpmf()`. targetCor = rbind( c(1, 0.3, 0.5), c(0.3, 1, 0.3), c(0.5, 0.3, 1)) result = SimJoint::SJpearson( X = cbind(sort(pmfSample1), sort(pmfSample2), sort(pmfSample3)), cor = targetCor, seed = completeRandomState, errorType = "maxRela") cor(result$X)
Sample from a probability mass function (PMF) via Latin hypercube sampling.
LHSpmf(pmf, sampleSize, seed)LHSpmf(pmf, sampleSize, seed)
pmf |
A 2-column data frame as a PMF. The 1st column is sorted and contains value points. The 2nd column contains probabilities. Probabilities should sum up to 1. |
sampleSize |
Sample size. |
seed |
An integer vector of size 1 or 4. Both seed a |
Random samples from pmf as a numeric vector of size sampleSize.
# Make a random PMF. val = seq(0, 15, len = 100) pmf = data.frame(val = val, P = dgamma(val, shape = 2, scale = 2)) pmf$P = pmf$P / sum(pmf$P) pmfSample = SimJoint::LHSpmf(pmf, 1000, 123) hist(pmfSample, breaks = 200)# Make a random PMF. val = seq(0, 15, len = 100) pmf = data.frame(val = val, P = dgamma(val, shape = 2, scale = 2)) pmf$P = pmf$P / sum(pmf$P) pmfSample = SimJoint::LHSpmf(pmf, 1000, 123) hist(pmfSample, breaks = 200)
Impose the target correlation matrix via a heuristic algorithm.
postSimOpt( X, cor, Xcor = matrix(), acceptProb = 1, seed = 123L, convergenceTail = 10000L )postSimOpt( X, cor, Xcor = matrix(), acceptProb = 1, seed = 123L, convergenceTail = 10000L )
X |
An |
cor |
A |
Xcor |
The |
acceptProb |
A numeric vector of probabilities that sum up to 1. In each iteration, the entry having the largest error in the current correlation matrix will be selected with probability |
seed |
An integer or an integer vector of size 4. A single integer seeds a |
convergenceTail |
An integer. If the last |
Algorithms are detailed in the package vignette. Examples of usage also appeared in functions like SJpearson().
A list of size 2.
X |
A numeric matrix of size |
cor |
Pearson correlation matrix of |
# ============================================================================= # Use one of the examples for `SJpearson()` # ============================================================================= 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 highly nonlinear functional # relationships between 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)$values) >= 0) break # Stop once the correlation # matrix is semi-positive definite. This loop could run for # a long time if we do not bound the uniform by 0.3. } result = SimJoint::SJpearson( X = marginals, cor = targetCor, stochasticStepDomain = c(0, 1), errorType = "meanSquare", seed = 456, maxCore = 1, convergenceTail = 8) # # Code blocks are commented due to execution time constraint by CRAN check. # system.time({postOptResult = SimJoint::postSimOpt( # X = result$X, cor = targetCor, convergenceTail = 10000)}) # # user system elapsed # # 6.66 0.00 6.66 # # system.time({directOptResult = SimJoint::postSimOpt( # X = marginals, cor = targetCor, convergenceTail = 10000)}) # # user system elapsed # # 8.48 0.00 8.48 # # sum((result$cor - targetCor) ^ 2) # # [1] 0.02209447 # sum((resultOpt$cor - targetCor) ^ 2) # # [1] 0.0008321346 # sum((directOptResult$cor - targetCor) ^ 2) # # [1] 0.02400257# ============================================================================= # Use one of the examples for `SJpearson()` # ============================================================================= set.seed(123) N = 10000L K = 10L # Several 2-parameter PDFs in R: marginals = list(rbeta, rcauchy, rf, rgamma, rnorm, runif, rweibull) Npdf = length(marginals) if(Npdf >= K) chosenMarginals = marginals[sample(Npdf, K, replace = TRUE)] else chosenMarginals = marginals[c(1L : Npdf, sample(Npdf, K - Npdf, replace = TRUE))] # Sample from the marginal PDFs. marginals = as.matrix(as.data.frame(lapply(chosenMarginals, function(f) { para = sort(runif(2, 0.1, 10)) rst = f(N, para[1], para[2]) sort(rst) }))) dimnames(marginals) = NULL frechetUpperCor = cor(marginals) # The correlation matrix should be # upper-bounded by that of the perfectly rank-correlated # joint (Frechet upper bound). The lower bound is characterized by # d-countercomonotonicity and depends not only on marginals. cat("Range of maximal correlations between marginals:", range(frechetUpperCor[frechetUpperCor < 1])) # Two perfectly rank-correlated marginals can have a Pearson # correlation below 0.07. This is due to highly nonlinear functional # relationships between marginal PDFs. # Create a valid correlation matrix upper-bounded by `frechetUpperCor`. while(TRUE) { targetCor = sapply(frechetUpperCor, function(x) runif(1, -0.1, min(0.3, x * 0.8))) targetCor = matrix(targetCor, ncol = K) targetCor[lower.tri(targetCor)] = t(targetCor)[lower.tri(t(targetCor))] diag(targetCor) = 1 if(min(eigen(targetCor)$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 = 1, convergenceTail = 8) # # Code blocks are commented due to execution time constraint by CRAN check. # system.time({postOptResult = SimJoint::postSimOpt( # X = result$X, cor = targetCor, convergenceTail = 10000)}) # # user system elapsed # # 6.66 0.00 6.66 # # system.time({directOptResult = SimJoint::postSimOpt( # X = marginals, cor = targetCor, convergenceTail = 10000)}) # # user system elapsed # # 8.48 0.00 8.48 # # sum((result$cor - targetCor) ^ 2) # # [1] 0.02209447 # sum((resultOpt$cor - targetCor) ^ 2) # # [1] 0.0008321346 # sum((directOptResult$cor - targetCor) ^ 2) # # [1] 0.02400257
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.
SJpearson( X, cor, stochasticStepDomain = as.numeric(c(0, 1)), errorType = "meanSquare", seed = 123L, maxCore = 7L, convergenceTail = 8L, iterLimit = 100000L, verbose = TRUE )SJpearson( X, cor, stochasticStepDomain = as.numeric(c(0, 1)), errorType = "meanSquare", seed = 123L, maxCore = 7L, convergenceTail = 8L, iterLimit = 100000L, verbose = TRUE )
X |
An |
cor |
A |
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.
|
seed |
An integer or an integer vector of size 4. A single integer seeds a |
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 |
iterLimit |
An integer. The maximal number of iterations. Default 100000. |
verbose |
A boolean value. |
Algorithms are detailed in the package vignette.
A list of size 2.
X |
A numeric matrix of size |
cor |
Pearson correlation matrix of |
# ############################################################################# # 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))# ############################################################################# # 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))
Sample from marginal probability mass functions via Latin hypercube sampling and then simulate the joint distribution with Pearson correlations. Use xSJpearsonPMF() 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.
SJpearsonPMF( PMFs, sampleSize, cor, stochasticStepDomain = as.numeric(c(0, 1)), errorType = "meanSquare", seed = 123L, maxCore = 7L, convergenceTail = 8L, iterLimit = 100000L, verbose = TRUE )SJpearsonPMF( PMFs, sampleSize, cor, stochasticStepDomain = as.numeric(c(0, 1)), errorType = "meanSquare", seed = 123L, maxCore = 7L, convergenceTail = 8L, iterLimit = 100000L, verbose = TRUE )
PMFs |
A list of data frames. Each data frame has 2 columns, a value vector and a probability vector. Probabilities should sum up to 1. Let the size of |
sampleSize |
An integer. The sample size |
cor |
A |
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.
|
seed |
An integer or an integer vector of size 4. A single integer seeds a |
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 |
iterLimit |
An integer. The maximal number of iterations. Default 100000. |
verbose |
A boolean value. |
Algorithms are detailed in the package vignette.
A list of size 2.
X |
A numeric matrix of size |
cor |
Pearson correlation matrix of |
# ============================================================================= # 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. # 3 PDFs, 2 nonparametric PMFs, 5 parametric PMFs: PMFs = c( apply(cbind(rnorm(N), rchisq(N, 4), rbeta(N, 4, 2)), 2, function(x) data.frame(val = sort(x), P = 1.0 / N)), list(data.frame(val = 1:3 + 0.0, P = c(0.3, 0.45, 0.25))), list(data.frame(val = 1:4 + 0.0, P = c(0.2, 0.3, 0.4, 0.1))), apply(cbind(rpois(N, 1), rpois(N, 5), rpois(N, 10), rnbinom(N, 3, 0.2), rnbinom(N, 6, 0.8)), 2, function(x) data.frame(val = as.numeric(sort(x)), P = 1.0 / N)) ) # Create the target correlation matrix `Rey`: set.seed(11) Rey <- diag(1, nrow = 10) 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::SJpearsonPMF( PMFs = PMFs, sampleSize = N, cor = Rey, errorType = "meanSquare", seed = 456, maxCore = 2, convergenceTail = 8, verbose = TRUE)}) # Check relative errors. summary(as.numeric(abs(result$cor / Rey - 1))) # ============================================================================= # Play with random nonparametric PMFs. # ============================================================================= set.seed(123) N = 2000L K = 20L # Create totally random nonparametric PMFs: PMFs = lapply(1L : K, function(x) { p = runif(2, 1, 10) result = data.frame( val = sort(rnorm(200)), P = runif(200)) result$P = result$P / sum(result$P) result }) # Create a valid correlation matrix upper-bounded by `frechetUpperCor`. while(TRUE) { targetCor = matrix(runif(K * K, -0.1, 0.3), ncol = K) targetCor[lower.tri(targetCor)] = t(targetCor)[lower.tri(t(targetCor))] diag(targetCor) = 1 if(min(eigen(targetCor)$values) >= 0) break # Break once the correlation # matrix is semi-positive definite. This loop could be running for quite # a long time if we do not bound `runif()`. } result = SimJoint::SJpearsonPMF( PMFs = PMFs, sampleSize = N, cor = targetCor, stochasticStepDomain = c(0, 1), errorType = "meanSquare", seed = 456, maxCore = 2, convergenceTail = 8) # Visualize errors and correlation matrices. par(mfrow = c(2, 2)) hist(result$cor - targetCor, breaks = K * K, main = NULL, xlab = "Error", cex.lab = 1.5, cex.axis = 1.25) hist(result$cor / targetCor - 1, breaks = K * K, main = NULL, xlab = "Relative error", ylab = "", cex.lab = 1.5, cex.axis = 1.25) zlim = range(range(targetCor[targetCor < 1]), range(result$cor[result$cor < 1])) col = colorRampPalette(c("blue", "red", "yellow"))(K * K) tmp = targetCor[, K : 1L] image(tmp, xaxt = "n", yaxt = "n", zlim = zlim, bty = "n", main = "Target cor", col = col) tmp = result$cor[, K : 1L] image(tmp, xaxt = "n", yaxt = "n", zlim = zlim, bty = "n", main = "Cor reached", col = col) par(mfrow = c(1, 1))# ============================================================================= # 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. # 3 PDFs, 2 nonparametric PMFs, 5 parametric PMFs: PMFs = c( apply(cbind(rnorm(N), rchisq(N, 4), rbeta(N, 4, 2)), 2, function(x) data.frame(val = sort(x), P = 1.0 / N)), list(data.frame(val = 1:3 + 0.0, P = c(0.3, 0.45, 0.25))), list(data.frame(val = 1:4 + 0.0, P = c(0.2, 0.3, 0.4, 0.1))), apply(cbind(rpois(N, 1), rpois(N, 5), rpois(N, 10), rnbinom(N, 3, 0.2), rnbinom(N, 6, 0.8)), 2, function(x) data.frame(val = as.numeric(sort(x)), P = 1.0 / N)) ) # Create the target correlation matrix `Rey`: set.seed(11) Rey <- diag(1, nrow = 10) 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::SJpearsonPMF( PMFs = PMFs, sampleSize = N, cor = Rey, errorType = "meanSquare", seed = 456, maxCore = 2, convergenceTail = 8, verbose = TRUE)}) # Check relative errors. summary(as.numeric(abs(result$cor / Rey - 1))) # ============================================================================= # Play with random nonparametric PMFs. # ============================================================================= set.seed(123) N = 2000L K = 20L # Create totally random nonparametric PMFs: PMFs = lapply(1L : K, function(x) { p = runif(2, 1, 10) result = data.frame( val = sort(rnorm(200)), P = runif(200)) result$P = result$P / sum(result$P) result }) # Create a valid correlation matrix upper-bounded by `frechetUpperCor`. while(TRUE) { targetCor = matrix(runif(K * K, -0.1, 0.3), ncol = K) targetCor[lower.tri(targetCor)] = t(targetCor)[lower.tri(t(targetCor))] diag(targetCor) = 1 if(min(eigen(targetCor)$values) >= 0) break # Break once the correlation # matrix is semi-positive definite. This loop could be running for quite # a long time if we do not bound `runif()`. } result = SimJoint::SJpearsonPMF( PMFs = PMFs, sampleSize = N, cor = targetCor, stochasticStepDomain = c(0, 1), errorType = "meanSquare", seed = 456, maxCore = 2, convergenceTail = 8) # Visualize errors and correlation matrices. par(mfrow = c(2, 2)) hist(result$cor - targetCor, breaks = K * K, main = NULL, xlab = "Error", cex.lab = 1.5, cex.axis = 1.25) hist(result$cor / targetCor - 1, breaks = K * K, main = NULL, xlab = "Relative error", ylab = "", cex.lab = 1.5, cex.axis = 1.25) zlim = range(range(targetCor[targetCor < 1]), range(result$cor[result$cor < 1])) col = colorRampPalette(c("blue", "red", "yellow"))(K * K) tmp = targetCor[, K : 1L] image(tmp, xaxt = "n", yaxt = "n", zlim = zlim, bty = "n", main = "Target cor", col = col) tmp = result$cor[, K : 1L] image(tmp, xaxt = "n", yaxt = "n", zlim = zlim, bty = "n", main = "Cor reached", col = col) par(mfrow = c(1, 1))
Reorder elements in each column of a matrix such that the column-wise Spearman correlations approximate a given correlation matrix.
SJspearman( X, cor, stochasticStepDomain = as.numeric(c(0, 1)), errorType = "meanSquare", seed = 123L, maxCore = 7L, convergenceTail = 8L, iterLimit = 100000L, verbose = TRUE )SJspearman( X, cor, stochasticStepDomain = as.numeric(c(0, 1)), errorType = "meanSquare", seed = 123L, maxCore = 7L, convergenceTail = 8L, iterLimit = 100000L, verbose = TRUE )
X |
An |
cor |
A |
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.
|
seed |
An integer or an integer vector of size 4. A single integer seeds a |
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 |
iterLimit |
An integer. The maximal number of iterations. Default 100000. |
verbose |
A boolean value. |
Algorithms are detailed in the package vignette.
A list of size 2.
X |
A numeric matrix of size |
cor |
Spearman correlation matrix of |
# ============================================================================= # 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), LHSpmf(data.frame(val = 1:3, P = c(0.3, 0.45, 0.25)), N, seed = sample(1e6L, 1)), 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 target correlation matrix `Rey` treated as Spearman # correlations. set.seed(11) Rey <- diag(1, nrow = 10) 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] } } result = SimJoint::SJspearman( X = marginals, cor = Rey, errorType = "meanSquare", seed = 456, maxCore = 1, convergenceTail = 8, verbose = TRUE) # Check relative errors. summary(as.numeric(abs(cor(result$X, method = "spearman") / Rey - 1))) # Another way to impose rank correlation is to supply rank matrix # to SJpearson(): system.time({reorderedRanks = SimJoint::SJpearson( X = apply(marginals, 2, function(x) rank(x)), cor = Rey, errorType = "meanSquare", seed = 456, maxCore = 1, convergenceTail = 8, verbose = TRUE)}) # Reordering according to ranks: result = apply(rbind(reorderedRanks$X, marginals), 2, function(x) { x[(N + 1L) : (2L * N)][as.integer(x[1L : N])] }) # Check the relative errors. summary(as.numeric(abs(cor(result, method = "spearman") / Rey - 1)))# ============================================================================= # 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), LHSpmf(data.frame(val = 1:3, P = c(0.3, 0.45, 0.25)), N, seed = sample(1e6L, 1)), 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 target correlation matrix `Rey` treated as Spearman # correlations. set.seed(11) Rey <- diag(1, nrow = 10) 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] } } result = SimJoint::SJspearman( X = marginals, cor = Rey, errorType = "meanSquare", seed = 456, maxCore = 1, convergenceTail = 8, verbose = TRUE) # Check relative errors. summary(as.numeric(abs(cor(result$X, method = "spearman") / Rey - 1))) # Another way to impose rank correlation is to supply rank matrix # to SJpearson(): system.time({reorderedRanks = SimJoint::SJpearson( X = apply(marginals, 2, function(x) rank(x)), cor = Rey, errorType = "meanSquare", seed = 456, maxCore = 1, convergenceTail = 8, verbose = TRUE)}) # Reordering according to ranks: result = apply(rbind(reorderedRanks$X, marginals), 2, function(x) { x[(N + 1L) : (2L * N)][as.integer(x[1L : N])] }) # Check the relative errors. summary(as.numeric(abs(cor(result, method = "spearman") / Rey - 1)))
Sample from marginal probability mass functions via Latin hypercube sampling and then simulate the joint distribution with Spearman correlations.
SJspearmanPMF( PMFs, sampleSize, cor, stochasticStepDomain = as.numeric(c(0, 1)), errorType = "meanSquare", seed = 123L, maxCore = 7L, convergenceTail = 8L, iterLimit = 100000L, verbose = TRUE )SJspearmanPMF( PMFs, sampleSize, cor, stochasticStepDomain = as.numeric(c(0, 1)), errorType = "meanSquare", seed = 123L, maxCore = 7L, convergenceTail = 8L, iterLimit = 100000L, verbose = TRUE )
PMFs |
A list of data frames. Each data frame has 2 columns, a value vector and a probability vector. Probabilities should sum up to 1. Let the size of |
sampleSize |
An integer. The sample size |
cor |
A |
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.
|
seed |
An integer or an integer vector of size 4. A single integer seeds a |
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 |
iterLimit |
An integer. The maximal number of iterations. Default 100000. |
verbose |
A boolean value. |
Algorithms are detailed in the package vignette.
A list of size 2.
X |
A numeric matrix of size |
cor |
Spearman correlation matrix of |
# ============================================================================= # Play with completely random nonparametric PMFs. # ============================================================================= set.seed(123) N = 2000L K = 20L # Create totally random nonparametric PMFs: PMFs = lapply(1L : K, function(x) { p = runif(2, 1, 10) result = data.frame( val = sort(rnorm(200)), P = runif(200)) result$P = result$P / sum(result$P) result }) # Create a valid correlation matrix upper-bounded by `frechetUpperCor`. while(TRUE) { targetCor = matrix(runif(K * K, -0.1, 0.3), ncol = K) targetCor[lower.tri(targetCor)] = t(targetCor)[lower.tri(t(targetCor))] diag(targetCor) = 1 if(min(eigen(targetCor)$values) >= 0) break # Break once the correlation # matrix is semi-positive definite. This loop could be running for quite # a long time if we do not bound `runif()`. } result = SimJoint::SJspearmanPMF( PMFs = PMFs, sampleSize = N, cor = targetCor, stochasticStepDomain = c(0, 1), errorType = "meanSquare", seed = 456, maxCore = 1, convergenceTail = 8) # Visualize errors and correlation matrices. par(mfrow = c(2, 2)) hist(result$cor - targetCor, breaks = K * K, main = NULL, xlab = "Error", cex.lab = 1.5, cex.axis = 1.25) hist(result$cor / targetCor - 1, breaks = K * K, main = NULL, xlab = "Relative error", ylab = "", cex.lab = 1.5, cex.axis = 1.25) zlim = range(range(targetCor[targetCor < 1]), range(result$cor[result$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 = result$cor[, K : 1L] image(tmp, xaxt = "n", yaxt = "n", zlim = zlim, bty = "n", main = "Cor reached", col = col) par(mfrow = c(1, 1))# ============================================================================= # Play with completely random nonparametric PMFs. # ============================================================================= set.seed(123) N = 2000L K = 20L # Create totally random nonparametric PMFs: PMFs = lapply(1L : K, function(x) { p = runif(2, 1, 10) result = data.frame( val = sort(rnorm(200)), P = runif(200)) result$P = result$P / sum(result$P) result }) # Create a valid correlation matrix upper-bounded by `frechetUpperCor`. while(TRUE) { targetCor = matrix(runif(K * K, -0.1, 0.3), ncol = K) targetCor[lower.tri(targetCor)] = t(targetCor)[lower.tri(t(targetCor))] diag(targetCor) = 1 if(min(eigen(targetCor)$values) >= 0) break # Break once the correlation # matrix is semi-positive definite. This loop could be running for quite # a long time if we do not bound `runif()`. } result = SimJoint::SJspearmanPMF( PMFs = PMFs, sampleSize = N, cor = targetCor, stochasticStepDomain = c(0, 1), errorType = "meanSquare", seed = 456, maxCore = 1, convergenceTail = 8) # Visualize errors and correlation matrices. par(mfrow = c(2, 2)) hist(result$cor - targetCor, breaks = K * K, main = NULL, xlab = "Error", cex.lab = 1.5, cex.axis = 1.25) hist(result$cor / targetCor - 1, breaks = K * K, main = NULL, xlab = "Relative error", ylab = "", cex.lab = 1.5, cex.axis = 1.25) zlim = range(range(targetCor[targetCor < 1]), range(result$cor[result$cor < 1])) col = colorRampPalette(c("blue", "red", "yellow"))(K * K) tmp = targetCor[, K : 1L] image(tmp, xaxt = "n", yaxt = "n", zlim = zlim, bty = "n", main = "Target cor", col = col) tmp = result$cor[, K : 1L] image(tmp, xaxt = "n", yaxt = "n", zlim = zlim, bty = "n", main = "Cor reached", col = col) par(mfrow = c(1, 1))
Users specify the uncorrelated random source instead of using a permuted X to left-multiply the correlation matrix decomposition. See the package vignette for more details.
xSJpearson( X, cor, noise, stochasticStepDomain = as.numeric(c(0, 1)), errorType = "meanSquare", seed = 123L, maxCore = 7L, convergenceTail = 8L, iterLimit = 100000L, verbose = TRUE )xSJpearson( X, cor, noise, stochasticStepDomain = as.numeric(c(0, 1)), errorType = "meanSquare", seed = 123L, maxCore = 7L, convergenceTail = 8L, iterLimit = 100000L, verbose = TRUE )
X |
An |
cor |
A |
noise |
An |
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.
|
seed |
An integer or an integer vector of size 4. A single integer seeds a |
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 |
iterLimit |
An integer. The maximal number of iterations. Default 100000. |
verbose |
A boolean value. |
Algorithms are detailed in the package vignette.
A list of size 2.
X |
A numeric matrix of size |
cor |
Pearson correlation matrix of |
# ============================================================================= # 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 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::xSJpearson( X = marginals, cor = Rey, noise = matrix(runif(N * K), ncol = K), errorType = "meanSquare", seed = 456, maxCore = 1, convergenceTail = 8, verbose = TRUE)}) summary(as.numeric(round(cor(result$X) - Rey, 6)))# ============================================================================= # Use the same example from <https://cran.r-project.org/web/packages/ # SimMultiCorrData/vignettes/workflow.html>. # ============================================================================= set.seed(123) N = 10000L # Sample size. K = 10L # 10 marginals. # Sample from 3 PDFs, 2 nonparametric PMFs, 5 parametric PMFs: marginals = cbind( rnorm(N), rchisq(N, 4), rbeta(N, 4, 2), SimJoint::LHSpmf(data.frame(val = 1:3, P = c(0.3, 0.45, 0.25)), N, seed = sample(1e6L, 1)), SimJoint::LHSpmf(data.frame(val = 1:4, P = c(0.2, 0.3, 0.4, 0.1)), N, seed = sample(1e6L, 1)), rpois(N, 1), rpois(N, 5), rpois(N, 10), rnbinom(N, 3, 0.2), rnbinom(N, 6, 0.8)) # The seeding for `LHSpmf()` is unhealthy, but OK for small examples. marginals = apply(marginals, 2, function(x) sort(x)) # Create the target correlation matrix `Rey`: set.seed(11) Rey <- diag(1, nrow = K) for (i in 1:nrow(Rey)) { for (j in 1:ncol(Rey)) { if (i > j) Rey[i, j] <- runif(1, 0.2, 0.7) Rey[j, i] <- Rey[i, j] } } system.time({result = SimJoint::xSJpearson( X = marginals, cor = Rey, noise = matrix(runif(N * K), ncol = K), errorType = "meanSquare", seed = 456, maxCore = 1, convergenceTail = 8, verbose = TRUE)}) summary(as.numeric(round(cor(result$X) - Rey, 6)))
Sample from marginal probability mass functions via Latin hypercube sampling and then simulate the joint distribution with Pearson correlations. Users specify the uncorrelated random source instead of using permuted marginal samples to left-multiply the correlation matrix decomposition.
xSJpearsonPMF( PMFs, sampleSize, cor, noise, stochasticStepDomain = as.numeric(c(0, 1)), errorType = "meanSquare", seed = 123L, maxCore = 7L, convergenceTail = 8L, iterLimit = 100000L, verbose = TRUE )xSJpearsonPMF( PMFs, sampleSize, cor, noise, stochasticStepDomain = as.numeric(c(0, 1)), errorType = "meanSquare", seed = 123L, maxCore = 7L, convergenceTail = 8L, iterLimit = 100000L, verbose = TRUE )
PMFs |
A list of data frames. Each data frame has 2 columns, a value vector and a probability vector. Probabilities should sum up to 1. Let the size of |
sampleSize |
An integer. The sample size |
cor |
A |
noise |
An |
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.
|
seed |
An integer or an integer vector of size 4. A single integer seeds a |
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 |
iterLimit |
An integer. The maximal number of iterations. Default 100000. |
verbose |
A boolean value. |
Algorithms are detailed in the package vignette.
A list of size 2.
X |
A numeric matrix of size |
cor |
Pearson correlation matrix of |
# ============================================================================= # 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. # 3 PDFs, 2 nonparametric PMFs, 5 parametric PMFs: PMFs = c( apply(cbind(rnorm(N), rchisq(N, 4), rbeta(N, 4, 2)), 2, function(x) data.frame(val = sort(x), P = 1.0 / N)), list(data.frame(val = 1:3 + 0.0, P = c(0.3, 0.45, 0.25))), list(data.frame(val = 1:4 + 0.0, P = c(0.2, 0.3, 0.4, 0.1))), apply(cbind(rpois(N, 1), rpois(N, 5), rpois(N, 10), rnbinom(N, 3, 0.2), rnbinom(N, 6, 0.8)), 2, function(x) data.frame(val = as.numeric(sort(x)), P = 1.0 / N)) ) # Create the target correlation matrix `Rey`: set.seed(11) Rey <- diag(1, nrow = 10) 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::xSJpearsonPMF( PMFs = PMFs, sampleSize = N, noise = matrix(runif(N * K), ncol = K), cor = Rey, errorType = "meanSquare", seed = 456, maxCore = 1, convergenceTail = 8, verbose = TRUE)}) # Check relative errors. summary(as.numeric(abs(result$cor / Rey - 1)))# ============================================================================= # Use the same example from <https://cran.r-project.org/web/packages/ # SimMultiCorrData/vignettes/workflow.html>. # ============================================================================= set.seed(123) N = 10000L # Sample size. K = 10L # 10 marginals. # 3 PDFs, 2 nonparametric PMFs, 5 parametric PMFs: PMFs = c( apply(cbind(rnorm(N), rchisq(N, 4), rbeta(N, 4, 2)), 2, function(x) data.frame(val = sort(x), P = 1.0 / N)), list(data.frame(val = 1:3 + 0.0, P = c(0.3, 0.45, 0.25))), list(data.frame(val = 1:4 + 0.0, P = c(0.2, 0.3, 0.4, 0.1))), apply(cbind(rpois(N, 1), rpois(N, 5), rpois(N, 10), rnbinom(N, 3, 0.2), rnbinom(N, 6, 0.8)), 2, function(x) data.frame(val = as.numeric(sort(x)), P = 1.0 / N)) ) # Create the target correlation matrix `Rey`: set.seed(11) Rey <- diag(1, nrow = 10) for (i in 1:nrow(Rey)) { for (j in 1:ncol(Rey)) { if (i > j) Rey[i, j] <- runif(1, 0.2, 0.7) Rey[j, i] <- Rey[i, j] } } system.time({result = SimJoint::xSJpearsonPMF( PMFs = PMFs, sampleSize = N, noise = matrix(runif(N * K), ncol = K), cor = Rey, errorType = "meanSquare", seed = 456, maxCore = 1, convergenceTail = 8, verbose = TRUE)}) # Check relative errors. summary(as.numeric(abs(result$cor / Rey - 1)))