Author: Riccardo Porreca, Roland Schmid
Date: 2022-03-14

# Flexible and consistent simulation of a matrix of Monte Carlo variates

#### Riccardo Porreca, Roland Schmid

#### 2022-03-14

Consider the Monte Carlo simulation of a matrix of i.i.d. normal random variables. We will show how rTRNG can be used to perform a consistent (fair-playing) simulation of a subset of the variables and simulations.

## Consistent simulation in R

We rely on the TRNG engines exposed to R as reference classes by rTRNG.

    library(rTRNG)

The `mcMatR` function below performs the full sequential Monte Carlo simulation of `nrow` normal i.i.d. samples of `ncol` variables using the `yarn2` generator.

    mcMatR <- function(nrow, ncol) {
      r <- yarn2$new(12358)
      M <- matrix(rnorm_trng(nrow * ncol, engine = r),
                  nrow = nrow, ncol = ncol, byrow = TRUE)
      M
    }

A second function `mcSubMatR` relies on `jump` and `split` operations to perform only a chunk [`startRow`, `endRow`] of simulations for a subset `subCols` of the variables.

    mcSubMatR <- function(nrow, ncol,
                          startRow, endRow, subCols) {
      r <- yarn2$new(12358)
      r$jump((startRow - 1)*ncol)
      nSubCols <- endRow - startRow + 1
      S <- matrix(0.0, nrow, ncol)
      S[startRow:endRow, subCols] <-
        vapply(subCols,
               function(j) {
                 rj = r$copy()
                 rj$split(ncol, j)
                 rnorm_trng(nSubCols, engine = rj)
               },
               FUN.VALUE = numeric(nSubCols))
      S
    }

The parallel nature of the `yarn2` generator ensures the sub-simulation obtained via `mcSubMatR` is consistent with the full sequential simulation.

    rows <- 9
    cols <- 5
    M <- mcMatR(rows, cols)
    startRow <- 4
    endRow <- 6
    subCols <- c(2, 4:5)
    S <- mcSubMatR(rows, cols,
                   startRow, endRow, subCols)
    identical(M[startRow:endRow, subCols],
              S[startRow:endRow, subCols])
    ## [1] TRUE

| M.1 | M.2 | M.3 | M.4 | M.5 | S.1 | S.2 | S.3 | S.4 | S.5
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
1 | 0.20256 | -0.41401 | -0.76749 | -0.33344 | 0.10718 | 0 | 0.00000 | 0 | 0.00000 | 0.00000
2 | -2.76439 | -1.15524 | -0.39394 | -1.16604 | 1.61759 | 0 | 0.00000 | 0 | 0.00000 | 0.00000
3 | -0.42199 | -1.13148 | -0.30448 | 0.12741 | -0.16111 | 0 | 0.00000 | 0 | 0.00000 | 0.00000
4 | -0.94448 | -1.86384 | -1.03244 | -0.41155 | 1.31009 | 0 | -1.86384 | 0 | -0.41155 | 1.31009
5 | -0.09614 | -0.16366 | -0.31964 | 0.87053 | 0.77996 | 0 | -0.16366 | 0 | 0.87053 | 0.77996
6 | 1.42049 | -0.73062 | -1.19459 | -1.02146 | 0.07202 | 0 | -0.73062 | 0 | -1.02146 | 0.07202
7 | -0.61202 | 0.02906 | -0.29100 | 0.10095 | -0.74647 | 0 | 0.00000 | 0 | 0.00000 | 0.00000
8 | 1.10246 | -0.50507 | 0.01286 | 0.63140 | -1.28893 | 0 | 0.00000 | 0 | 0.00000 | 0.00000
9 | -0.08732 | -0.32545 | 0.29099 | 0.62003 | -0.94617 | 0 | 0.00000 | 0 | 0.00000 | 0.00000

## Consistent simulation with Rcpp

We now use Rcpp to define functions `mcMatRcpp` and `mcSubMatRcpp` for the full sequential simulation and the sub-simulation, respectively. The `Rcpp::depends` attribute makes sure the TRNG library and headers shipped with rTRNG are available to the C++ code. Moreover, `Rcpp::plugins(cpp11)` enforces the C++11 standard required by TRNG >= 4.22.

    // [[Rcpp::depends(rTRNG)]]
    // TRNG >= 4.22 requires C++11
    // [[Rcpp::plugins(cpp11)]]
    #include <Rcpp.h>
    #include <trng/normal_dist.hpp>
    #include <trng/yarn2.hpp>
    using namespace Rcpp;

    // [[Rcpp::export]]
    NumericMatrix mcMatRcpp(const int nrow, const int ncol) {
      NumericMatrix M(nrow, ncol);
      trng::yarn2 r(12358);
      trng::normal_dist<> normal(0.0, 1.0);
      for (int i = 0; i < nrow; i++) {
        for (int j = 0; j < ncol; j++) {
          M(i, j) = normal(r);
        }
      }
      return M;
    }

    // [[Rcpp::export]]
    NumericMatrix mcSubMatRcpp(const int nrow, const int ncol,
                               const int startRow,
                               const int endRow,
                               const IntegerVector subCols) {
      NumericMatrix M(nrow, ncol);
      trng::yarn2 r(12358), rj;
      trng::normal_dist<> normal(0.0, 1.0);
      r.jump((startRow - 1) * ncol);
      for (IntegerVector::const_iterator jSub = subCols.begin();
           jSub < subCols.end(); jSub++) {
        int j = *jSub - 1;
        rj = r;
        rj.split(ncol, j);
        for (int i = startRow - 1; i < endRow; i++) {
          M(i, j) = normal(rj);
        }
      }
      return M;
    }

As seen above for the R case, consistency of the simulation obtained via `mcSubMatRcpp` with the full sequential simulation is guaranteed.

    rows <- 9
    cols <- 5
    startRow <- 4
    endRow <- 6
    subCols <- c(2, 4:5)
    M <- mcMatRcpp(rows, cols)
    S <- mcSubMatRcpp(rows, cols, startRow, endRow, subCols)
    identical(M[startRow:endRow, subCols],
              S[startRow:endRow, subCols])
    ## [1] TRUE

| M.1 | M.2 | M.3 | M.4 | M.5 | S.1 | S.2 | S.3 | S.4 | S.5
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
1 | 0.20256 | -0.41401 | -0.76749 | -0.33344 | 0.10718 | 0 | 0.00000 | 0 | 0.00000 | 0.00000
2 | -2.76439 | -1.15524 | -0.39394 | -1.16604 | 1.61759 | 0 | 0.00000 | 0 | 0.00000 | 0.00000
3 | -0.42199 | -1.13148 | -0.30448 | 0.12741 | -0.16111 | 0 | 0.00000 | 0 | 0.00000 | 0.00000
4 | -0.94448 | -1.86384 | -1.03244 | -0.41155 | 1.31009 | 0 | -1.86384 | 0 | -0.41155 | 1.31009
5 | -0.09614 | -0.16366 | -0.31964 | 0.87053 | 0.77996 | 0 | -0.16366 | 0 | 0.87053 | 0.77996
6 | 1.42049 | -0.73062 | -1.19459 | -1.02146 | 0.07202 | 0 | -0.73062 | 0 | -1.02146 | 0.07202
7 | -0.61202 | 0.02906 | -0.29100 | 0.10095 | -0.74647 | 0 | 0.00000 | 0 | 0.00000 | 0.00000
8 | 1.10246 | -0.50507 | 0.01286 | 0.63140 | -1.28893 | 0 | 0.00000 | 0 | 0.00000 | 0.00000
9 | -0.08732 | -0.32545 | 0.29099 | 0.62003 | -0.94617 | 0 | 0.00000 | 0 | 0.00000 | 0.00000

## Consistent parallel simulation with RcppParallel

The same technique used for generating a sub-set of the simulations can be exploited for performing a parallel simulation in C++. We can embed the body of `mcSubMatRcpp` above into an `RcppParallel::Worker` for performing chunks of Monte Carlo simulations in parallel, for any subset `subCols` of the variables.

    struct MCMatWorker : public Worker
    {
      RMatrix<double> M;
      const RVector<int> subCols;

      // constructor
      MCMatWorker(NumericMatrix M,
                  const IntegerVector subCols)
        : M(M), subCols(subCols) {}

      // operator processing an exclusive range of indices
      void operator()(std::size_t begin, std::size_t end) {
        trng::yarn2 r(12358), rj;
        trng::normal_dist<> normal(0.0, 1.0);
        r.jump((int)begin * M.ncol());
        for (IntegerVector::const_iterator jSub = subCols.begin();
             jSub < subCols.end(); jSub++) {
          int j = *jSub - 1;
          rj = r;
          rj.split(M.ncol(), j);
          for (int i = (int)begin; i < (int)end; i++) {
            M(i, j) = normal(rj);
          }
        }
      }
    };
    // [[Rcpp::export]]
    NumericMatrix mcMatRcppParallel(const int nrow, const int ncol,
                                    const IntegerVector subCols) {
      NumericMatrix M(nrow, ncol);
      MCMatWorker w(M, subCols);
      parallelFor(0, M.nrow(), w);
      return M;
    }

The parallel nature of the `yarn2` generator ensures the parallel simulation is playing fair, i.e. is consistent with the sequential simulation.

    M <- mcMatRcpp(rows, cols)
    Mp <- mcMatRcppParallel(rows, cols, seq_len(ncol(M)))
    identical(M, Mp)
    ## [1] TRUE

Similarly, we can achieve a consistent parallel simulation of a subset of the variables only.

    Sp <- mcMatRcppParallel(rows, cols, subCols)
    identical(M[, subCols],
              Sp[, subCols])
    ## [1] TRUE

| M.1 | M.2 | M.3 | M.4 | M.5 | Sp.1 | Sp.2 | Sp.3 | Sp.4 | Sp.5
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
1 | 0.20256 | -0.41401 | -0.76749 | -0.33344 | 0.10718 | 0 | -0.41401 | 0 | -0.33344 | 0.10718
2 | -2.76439 | -1.15524 | -0.39394 | -1.16604 | 1.61759 | 0 | -1.15524 | 0 | -1.16604 | 1.61759
3 | -0.42199 | -1.13148 | -0.30448 | 0.12741 | -0.16111 | 0 | -1.13148 | 0 | 0.12741 | -0.16111
4 | -0.94448 | -1.86384 | -1.03244 | -0.41155 | 1.31009 | 0 | -1.86384 | 0 | -0.41155 | 1.31009
5 | -0.09614 | -0.16366 | -0.31964 | 0.87053 | 0.77996 | 0 | -0.16366 | 0 | 0.87053 | 0.77996
6 | 1.42049 | -0.73062 | -1.19459 | -1.02146 | 0.07202 | 0 | -0.73062 | 0 | -1.02146 | 0.07202
7 | -0.61202 | 0.02906 | -0.29100 | 0.10095 | -0.74647 | 0 | 0.02906 | 0 | 0.10095 | -0.74647
8 | 1.10246 | -0.50507 | 0.01286 | 0.63140 | -1.28893 | 0 | -0.50507 | 0 | 0.63140 | -1.28893
9 | -0.08732 | -0.32545 | 0.29099 | 0.62003 | -0.94617 | 0 | -0.32545 | 0 | 0.62003 | -0.94617