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testRandDists.cc

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// -*- C++ -*-
// $Id: testRandDists.cc,v 1.11 2011/07/11 15:55:45 garren Exp $
// ----------------------------------------------------------------------

// ----------------------------------------------------------------------
//
// testRandDists -- tests of the correctness of random distributions 
//
// Usage:
//   testRandDists < testRandDists.dat > testRandDists.log
//
// Currently tested:
//      RandGauss
//      RandGeneral
//
// M. Fischler      5/17/99     Reconfigured to be suitable for use with
//                              an automated validation script - will return 
//                              0 if validation is OK, or a mask indicating
//                              where problems were found.
// M. Fischler      5/18/99     Added test for RandGeneral.
// Evgenyi T.       5/20/99     Vetted for compilation on various CLHEP/CERN
//                              platforms.
// M. Fischler      5/26/99     Extended distribution test to intervals of .5 
//                              sigma and to moments up to the sixth.
// M. Fischler     10/29/99     Added validation for RandPoisson.
// M. Fischler     11/09/99     Made gammln static to avoid (harmless) 
//                              confusion with the gammln in RandPoisson.
// M. Fischler      2/04/99     Added validation for the Q and T versions of
//                              Poisson and Gauss
// M. Fischler     11/04/04     Add kludge to gaussianTest to deal with
//                              different behaviour under optimization on
//                              some compilers (gcc 2.95.2)
//                              This behaviour was only seen with stepwise 
//                              RandGeneral and appears to be solely a 
//                              function of the test program.
// 
// ----------------------------------------------------------------------

#include "CLHEP/Units/GlobalPhysicalConstants.h"  // used to provoke shadowing warnings
#include "CLHEP/Random/Randomize.h"
#include "CLHEP/Random/RandGaussQ.h"
#include "CLHEP/Random/RandGaussT.h"
#include "CLHEP/Random/RandPoissonQ.h"
#include "CLHEP/Random/RandPoissonT.h"
#include "CLHEP/Random/RandSkewNormal.h"
#include "CLHEP/Random/defs.h"
#include <iostream>
#include <iomanip>
#include <cmath>                // double abs()
#include <stdlib.h>             // int abs()
#include <cstdlib>              // for exit()

using std::cin;
using std::cout;
using std::cerr;
using std::endl;
using std::setprecision;
using namespace CLHEP;
//#ifndef _WIN32
//using std::exp;
//#endif

// Tolerance of deviation from expected results
static const double REJECT = 4.0;

// Mask bits to form a word indicating which if any dists were "bad"
static const int GaussBAD    = 1 << 0;
static const int GeneralBAD  = 1 << 1;
static const int PoissonBAD  = 1 << 2;
static const int GaussQBAD   = 1 << 3;
static const int GaussTBAD   = 1 << 4;
static const int PoissonQBAD = 1 << 5;
static const int PoissonTBAD = 1 << 6;
static const int SkewNormalBAD = 1 << 7;


// **********************
//
// SECTION I - General tools for the various tests
//
// **********************

static
double gammln(double x) {
        // Note:  This uses the gammln algorith in Numerical Recipes.
        // In the "old" RandPoisson there is a slightly different algorithm, 
        // which mathematically is identical to this one.  The advantage of
        // the modified method is one fewer division by x (in exchange for
        // doing one subtraction of 1 from x).  The advantage of the method 
        // here comes when .00001 < x < .65:  In this range, the alternate
        // method produces results which have errors 10-100 times those
        // of this method (though still less than 1.0E-10).  If we package
        // either method, we should use the reflection formula (6.1.4) so
        // that the user can never get inaccurate results, even for x very 
        // small.  The test for x < 1 is as costly as a divide, but so be it.

  double y, tmp, ser;
  static double c[6] = {
         76.18009172947146,
        -86.50532032941677,
         24.01409824083091,
         -1.231739572450155,
          0.001208650973866179,
         -0.000005395239384953 };
  y = x;
  tmp = x + 5.5;
  tmp -= (x+.5)*std::log(tmp);
  ser = 1.000000000190015;
  for (int i = 0; i < 6; i++) {
    ser += c[i]/(++y);
  }
  double ans = (-tmp + std::log (std::sqrt(CLHEP::twopi)*ser/x));
  return ans;
}

static
double gser(double a, double x) {
  const int ITMAX = 100;
  const double EPS = 1.0E-8;
  double ap = a;
  double sum = 1/a;
  double del = sum;
  for (int n=0; n < ITMAX; n++) {
    ap++;
    del *= x/ap;
    sum += del;
    if (std::fabs(del) < std::fabs(sum)*EPS) {
      return sum*std::exp(-x+a*std::log(x)-gammln(a));
    }
  }
  cout << "Problem - inaccurate gser " << a << ", " << x << "\n";
  return sum*std::exp(-x+a*std::log(x)-gammln(a));
}

static
double gcf(double a, double x) {
  const int ITMAX = 100;
  const double EPS = 1.0E-8;
  const double VERYSMALL = 1.0E-100;
  double b = x+1-a;
  double c = 1/VERYSMALL;
  double d = 1/b;
  double h = d;
  for (int i = 1; i <= ITMAX; i++) {
    double an = -i*(i-a);
    b += 2;
    d = an*d + b;
    if (std::fabs(d) < VERYSMALL) d = VERYSMALL;
    c = b + an/c;
    if (std::fabs(c) < VERYSMALL) c = VERYSMALL;
    d = 1/d;
    double del = d*c;
    h *= del;
    if (std::fabs(del-1.0) < EPS) {
      return std::exp(-x+a*std::log(x)-gammln(a))*h;
    }
  }
  cout << "Problem - inaccurate gcf " << a << ", " << x << "\n";
  return std::exp(-x+a*std::log(x)-gammln(a))*h;
}

static
double gammp (double a, double x) {
  if (x < a+1) {
    return gser(a,x);
  } else {
    return 1-gcf(a,x);
  }
}


// **********************
//
// SECTION II - Validation of specific distributions
//
// **********************

// ------------
// gaussianTest
// ------------

bool gaussianTest ( HepRandom & dist, double mu, 
                    double sigma, int nNumbers ) {

  bool good = true;
  double worstSigma = 0;

// We will accumulate mean and moments up to the sixth,
// The second moment should be sigma**2, the fourth 3 sigma**4,
// the sixth 15 sigma**6.  The expected variance in these is 
// (for the m-th moment with m even) (2m-1)!! (m-1)!!**2 / n
// (for the m-th moment with m odd)  (2m-1)!!  m!!**2    / n
// We also do a histogram with bins every half sigma.

  double sumx = 0;
  double sumx2 = 0;
  double sumx3 = 0;
  double sumx4 = 0;
  double sumx5 = 0;
  double sumx6 = 0;
  int counts[11];
  int ncounts[11];
  int ciu;
  for (ciu = 0; ciu < 11; ciu++) {
    counts[ciu] = 0;
    ncounts[ciu] = 0;
  }

  int oldprecision = cout.precision();
  cout.precision(5);
  // hack so that gcc 4.3 puts x and u into memory instead of a register
  volatile double x;
  volatile double u;
  int ipr = nNumbers / 10 + 1;
  for (int ifire = 0; ifire < nNumbers; ifire++) {
    x = dist();         // We avoid fire() because that is not virtual 
                        // in HepRandom.
    if( x < mu - 12.0*sigma ) {
        cout << "x  = " << x << "\n";
    }
    if ( (ifire % ipr) == 0 ) {
      cout << ifire << endl;
    }
    sumx  += x;
    sumx2 += x*x;
    sumx3 += x*x*x;
    sumx4 += x*x*x*x;
    sumx5 += x*x*x*x*x;
    sumx6 += x*x*x*x*x*x;
    u = (x - mu) / sigma;
    if ( u >= 0 ) {
      ciu = (int)(2*u);
      if (ciu>10) ciu = 10;
      counts[ciu]++;
    } else {
      ciu = (int)(2*(-u));
      if (ciu>10) ciu = 10;
      ncounts[ciu]++;
    }
  }

  double mean = sumx / nNumbers;
  double u2 = sumx2/nNumbers - mean*mean;
  double u3 = sumx3/nNumbers - 3*sumx2*mean/nNumbers + 2*mean*mean*mean;
  double u4 = sumx4/nNumbers - 4*sumx3*mean/nNumbers 
                + 6*sumx2*mean*mean/nNumbers - 3*mean*mean*mean*mean;
  double u5 = sumx5/nNumbers - 5*sumx4*mean/nNumbers 
                + 10*sumx3*mean*mean/nNumbers 
                - 10*sumx2*mean*mean*mean/nNumbers
                + 4*mean*mean*mean*mean*mean;
  double u6 = sumx6/nNumbers - 6*sumx5*mean/nNumbers 
                + 15*sumx4*mean*mean/nNumbers 
                - 20*sumx3*mean*mean*mean/nNumbers
                + 15*sumx2*mean*mean*mean*mean/nNumbers
                - 5*mean*mean*mean*mean*mean*mean;

  cout << "Mean (should be close to " << mu << "): " << mean << endl;
  cout << "Second moment (should be close to " << sigma*sigma << 
                        "): " << u2 << endl;
  cout << "Third  moment (should be close to zero): " << u3 << endl;
  cout << "Fourth moment (should be close to " << 3*sigma*sigma*sigma*sigma << 
                        "): " << u4 << endl;
  cout << "Fifth moment (should be close to zero): " << u5 << endl;
  cout << "Sixth moment (should be close to " 
        << 15*sigma*sigma*sigma*sigma*sigma*sigma 
        << "): " << u6 << endl;

  // For large N, the variance squared in the scaled 2nd, 3rd 4th 5th and
  // 6th moments are roughly 2/N, 6/N, 96/N, 720/N and 10170/N respectively.  
  // Based on this, we can judge how many sigma a result represents:
  
  double del1 = std::sqrt ( (double) nNumbers ) * std::abs(mean - mu) / sigma;
  double del2 = std::sqrt ( nNumbers/2.0 ) * std::abs(u2 - sigma*sigma) / (sigma*sigma);
  double del3 = std::sqrt ( nNumbers/6.0 ) * std::abs(u3) / (sigma*sigma*sigma);
  double sigma4 = sigma*sigma*sigma*sigma;
  double del4 = std::sqrt ( nNumbers/96.0 ) * std::abs(u4 - 3 * sigma4) / sigma4;
  double del5 = std::sqrt ( nNumbers/720.0 ) * std::abs(u5) / (sigma*sigma4);
  double del6 = std::sqrt ( nNumbers/10170.0 ) * std::abs(u6 - 15*sigma4*sigma*sigma) 
                                / (sigma4*sigma*sigma);

  cout << "        These represent " << 
        del1 << ", " << del2 << ", " << del3 << ", \n" 
        <<"                        " << del4 << ", " << del5 << ", " << del6
        <<"\n                         standard deviations from expectations\n";
  if ( del1 > worstSigma ) worstSigma = del1;
  if ( del2 > worstSigma ) worstSigma = del2;
  if ( del3 > worstSigma ) worstSigma = del3;
  if ( del4 > worstSigma ) worstSigma = del4;
  if ( del5 > worstSigma ) worstSigma = del5;
  if ( del6 > worstSigma ) worstSigma = del6;

  if ( del1 > REJECT || del2 > REJECT || del3 > REJECT 
    || del4 > REJECT || del5 > REJECT || del6 > REJECT ) {
      cout << "REJECT hypothesis that this distribution is correct!!\n";
      good = false;
  }

  // The variance of the bin counts is given by a Poisson estimate (std::sqrt(npq)).

  double table[11] = {  // Table of integrated density in each range:
        .191462, // 0.0 - 0.5 sigma
        .149882, // 0.5 - 1.0 sigma
        .091848, // 1.0 - 1.5 sigma
        .044057, // 1.5 - 2.0 sigma
        .016540, // 2.0 - 2.5 sigma
        .004860, // 2.5 - 3.0 sigma
        .001117, // 3.0 - 3.5 sigma
        .000201, // 3.5 - 4.0 sigma
        2.83E-5, // 4.0 - 4.5 sigma
        3.11E-6, // 4.5 - 5.0 sigma
        3.87E-7  // 5.0 sigma and up
        };

  for (int m1 = 0; m1 < 11; m1++) {
    double expect = table[m1]*nNumbers;
    double sig = std::sqrt ( table[m1] * (1.0-table[m1]) * nNumbers );
    cout.precision(oldprecision);
    cout << "Between " << m1/2.0 << " sigma and " 
        << m1/2.0+.5 << " sigma (should be about " << expect << "):\n " 
        << "         "
        << ncounts[m1] << " negative and " << counts[m1] << " positive " << "\n";
    cout.precision(5);
    double negSigs = std::abs ( ncounts[m1] - expect ) / sig;
    double posSigs = std::abs (  counts[m1] - expect ) / sig;
    cout << "        These represent " << 
        negSigs << " and " << posSigs << " sigma from expectations\n";
    if ( negSigs > REJECT || posSigs > REJECT ) {
      cout << "REJECT hypothesis that this distribution is correct!!\n";
      good = false;
    }
    if ( negSigs > worstSigma ) worstSigma = negSigs;
    if ( posSigs > worstSigma ) worstSigma = posSigs;
  }

  cout << "\n The worst deviation encountered (out of about 25) was "
        << worstSigma << " sigma \n\n";

  cout.precision(oldprecision);

  return good;

} // gaussianTest()



// ------------
// skewNormalTest
// ------------

bool skewNormalTest ( HepRandom & dist, double k, int nNumbers ) {

  bool good = true;
  double worstSigma = 0;

// We will accumulate mean and moments up to the sixth,
// The second moment should be sigma**2, the fourth 3 sigma**4.
// The expected variance in these is 
// (for the m-th moment with m even) (2m-1)!! (m-1)!!**2 / n
// (for the m-th moment with m odd)  (2m-1)!!  m!!**2    / n

  double sumx = 0;
  double sumx2 = 0;
  double sumx3 = 0;
  double sumx4 = 0;
  double sumx5 = 0;
  double sumx6 = 0;

  int oldprecision = cout.precision();
  cout.precision(5);
  // hack so that gcc 4.3 puts x into memory instead of a register
  volatile double x;
  // calculate mean and sigma
  double delta = k / std::sqrt( 1 + k*k );
  double mu = delta/std::sqrt(CLHEP::halfpi);
  double mom2 = 1.;
  double mom3 = 3*delta*(1-(delta*delta)/3.)/std::sqrt(CLHEP::halfpi);
  double mom4 = 3.;
  double mom5 = 15*delta*(1-2.*(delta*delta)/3.+(delta*delta*delta*delta)/5.)/std::sqrt(CLHEP::halfpi);
  double mom6 = 15.;

  int ipr = nNumbers / 10 + 1;
  for (int ifire = 0; ifire < nNumbers; ifire++) {
    x = dist();         // We avoid fire() because that is not virtual 
                        // in HepRandom.
    if( x < mu - 12.0 ) {
        cout << "x  = " << x << "\n";
    }
    if ( (ifire % ipr) == 0 ) {
      cout << ifire << endl;
    }
    sumx  += x;
    sumx2 += x*x;
    sumx3 += x*x*x;
    sumx4 += x*x*x*x;
    sumx5 += x*x*x*x*x;
    sumx6 += x*x*x*x*x*x;
  }

  double mean = sumx / nNumbers;
  double u2 = sumx2/nNumbers;
  double u3 = sumx3/nNumbers;
  double u4 = sumx4/nNumbers;
  double u5 = sumx5/nNumbers;
  double u6 = sumx6/nNumbers;

  cout << "Mean (should be close to " << mu << "): " << mean << endl;
  cout << "Second moment (should be close to " << mom2 << "): " << u2 << endl;
  cout << "Third  moment (should be close to " << mom3 << "): " << u3 << endl;
  cout << "Fourth moment (should be close to " << mom4 << "): " << u4 << endl;
  cout << "Fifth moment (should be close to " << mom5 << "): " << u5 << endl;
  cout << "Sixth moment (should be close to " << mom6 << "): " << u6 << endl;

  double del1 = std::sqrt ( (double) nNumbers ) * std::abs(mean - mu);
  double del2 = std::sqrt ( nNumbers/2.0 ) * std::abs(u2 - mom2);
  double del3 = std::sqrt ( nNumbers/(15.-mom3*mom3) ) * std::abs(u3 - mom3 );
  double del4 = std::sqrt ( nNumbers/96.0 ) * std::abs(u4 - mom4);
  double del5 = std::sqrt ( nNumbers/(945.-mom5*mom5) ) * std::abs(u5 - mom5 );
  double del6 = std::sqrt ( nNumbers/10170.0 ) * std::abs(u6 - mom6);

  cout << "        These represent " << 
        del1 << ", " << del2 << ", " << del3 << ", \n" 
        <<"                        " << del4 << ", " << del5 << ", " << del6
        <<"\n                         standard deviations from expectations\n";
  if ( del1 > worstSigma ) worstSigma = del1;
  if ( del2 > worstSigma ) worstSigma = del2;
  if ( del3 > worstSigma ) worstSigma = del3;
  if ( del4 > worstSigma ) worstSigma = del4;
  if ( del5 > worstSigma ) worstSigma = del5;
  if ( del6 > worstSigma ) worstSigma = del6;
 
  if ( del1 > REJECT || del2 > REJECT || del3 > REJECT ||
       del4 > REJECT || del5 > REJECT || del6 > REJECT  ) {
      cout << "REJECT hypothesis that this distribution is correct!!\n";
      good = false;
  }

  cout << "\n The worst deviation encountered (out of about 25) was "
        << worstSigma << " sigma \n\n";

  cout.precision(oldprecision);

  return good;

} // skewNormalTest()


// ------------
// poissonTest
// ------------

class poisson {
  double mu_;
  public:
  poisson(double mu) : mu_(mu) {}
  double operator()(int r) { 
    double logAnswer = -mu_ + r*std::log(mu_) - gammln(r+1);
    return std::exp(logAnswer);
  }
};

double* createRefDist ( poisson pdist, int N,
                                int MINBIN, int MAXBINS, int clumping, 
                                int& firstBin, int& lastBin ) {

  // Create the reference distribution -- making sure there are more than
  // 20 points at each value.  The entire tail will be rolled up into one 
  // value (at each end).  We shall end up with some range of bins starting
  // at 0 or more, and ending at MAXBINS-1 or less.

  double * refdist = new double [MAXBINS];

  int c  = 0; // c is the number of the clump, that is, the member number 
              // of the refdist array.
  int ic = 0; // ic is the number within the clump; mod clumping
  int r  = 0; // r is the value of the variate.

  // Determine the first bin:  at least 20 entries must be at the level
  // of that bin (so that we won't immediately dip belpw 20) but the number
  // to enter is cumulative up to that bin.

  double start = 0;
  double binc;
  while ( c < MAXBINS ) {
    for ( ic=0, binc=0; ic < clumping; ic++, r++ ) {
      binc += pdist(r) * N;
    }
    start += binc;
    if (binc >= MINBIN) break;
    c++;
    if ( c > MAXBINS/3 ) {
      cout << "The number of samples supplied " << N << 
        " is too small to set up a chi^2 to test this distribution.\n";
      exit(-1);
    }
  }
  firstBin = c;
  refdist[firstBin] = start;
  c++;

  // Fill all the other bins until one has less than 20 items.
  double next = 0;
  while ( c < MAXBINS ) {
    for ( ic=0, binc=0; ic < clumping; ic++, r++ ) {
      binc += pdist(r) * N;
    }
    next = binc;
    if (next < MINBIN) break;
    refdist[c] = next;
    c++;
  }

  // Shove all the remaining items into last bin.
  lastBin = c-1;
  next += refdist[lastBin];
  while ( c < MAXBINS ) {
    for ( ic=0, binc=0; ic < clumping; ic++, r++ ) {
      binc += pdist(r) * N;
    }
    next += binc;
    c++;
  } 
  refdist[lastBin] = next;

  return refdist;

} // createRefDist()


bool poissonTest ( RandPoisson & dist, double mu, int N ) {

// Three tests will be done:
//
// A chi-squared test will be used to test the hypothesis that the 
// generated distribution of N numbers matches the proper Poisson distribution.
//
// The same test will be applied to the distribution of numbers "clumping"
// together std::sqrt(mu) bins.  This will detect small deviations over several 
// touching bins, when mu is not small.
//
// The mean and second moment are checked against their theoretical values.

  bool good = true;

  int clumping = int(std::sqrt(mu));
  if (clumping <= 1) clumping = 2;
  const int MINBIN  = 20;
  const int MAXBINS = 1000;
  int firstBin;
  int lastBin;
  int firstBin2;
  int lastBin2;

  poisson pdist(mu);

  double* refdist  = createRefDist( pdist, N,
                        MINBIN, MAXBINS, 1, firstBin, lastBin);
  double* refdist2 = createRefDist( pdist, N,
                        MINBIN, MAXBINS, clumping, firstBin2, lastBin2);

  // Now roll the random dists, treating the tails in the same way as we go.
  
  double sum = 0;
  double moment = 0;

  double* samples  = new double [MAXBINS];
  double* samples2 = new double [MAXBINS];
  int r;
  for (r = 0; r < MAXBINS; r++) {
    samples[r] = 0;
    samples2[r] = 0;
  }

  int r1;
  int r2;
  for (int i = 0; i < N; i++) {
    r = dist.fire();
    sum += r;
    moment += (r - mu)*(r - mu);
    r1 = r;
    if (r1 < firstBin) r1 = firstBin;
    if (r1 > lastBin)  r1 = lastBin;
    samples[r1] += 1;
    r2 = r/clumping;
    if (r2 < firstBin2) r2 = firstBin2;
    if (r2 > lastBin2)  r2 = lastBin2;
    samples2[r2] += 1;
  }

//  #ifdef DIAGNOSTIC
  int k;
  for (k = firstBin; k <= lastBin; k++) {
    cout << k << "  " << samples[k] << "  " << refdist[k] << "     " <<
        (samples[k]-refdist[k])*(samples[k]-refdist[k])/refdist[k] << "\n";
  }
  cout << "----\n";
  for (k = firstBin2; k <= lastBin2; k++) {
    cout << k << "  " << samples2[k] << "  " << refdist2[k] << "\n";
  }
// #endif // DIAGNOSTIC

  // Now find chi^2 for samples[] to apply the first test

  double chi2 = 0;
  for ( r = firstBin; r <= lastBin; r++ ) {
    double delta = (samples[r] - refdist[r]);
    chi2 += delta*delta/refdist[r];
  }
  int degFreedom = (lastBin - firstBin + 1) - 1;
  
  // and finally, p.  Since we only care about it for small values, 
  // and never care about it past the 10% level, we can use the approximations
  // CL(chi^2,n) = 1/std::sqrt(CLHEP::twopi) * ErrIntC ( y ) with 
  // y = std::sqrt(2*chi2) - std::sqrt(2*n-1) 
  // errIntC (y) = std::exp((-y^2)/2)/(y*std::sqrt(CLHEP::twopi))

  double pval;
  pval = 1.0 - gammp ( .5*degFreedom , .5*chi2 );

  cout << "Chi^2 is " << chi2 << " on " << degFreedom << " degrees of freedom."
       << "  p = " << pval << "\n";

  delete[] refdist;
  delete[] samples;

  // Repeat the chi^2 and p for the clumped sample, to apply the second test

  chi2 = 0;
  for ( r = firstBin2; r <= lastBin2; r++ ) {
    double delta = (samples2[r] - refdist2[r]);
    chi2 += delta*delta/refdist2[r];
  }
  degFreedom = (lastBin2 - firstBin2 + 1) - 1;
  double pval2;
  pval2 = 1.0 - gammp ( .5*degFreedom , .5*chi2 );

  cout << "Clumps: Chi^2 is " << chi2 << " on " << degFreedom << 
        " degrees of freedom." << "  p = " << pval2 << "\n";

  delete[] refdist2;
  delete[] samples2;

  // Check out the mean and sigma to apply the third test

  double mean = sum / N;
  double sigma = std::sqrt( moment / (N-1) );

  double deviationMean  = std::fabs(mean - mu)/(std::sqrt(mu/N));
  double expectedSigma2Variance = (2*N*mu*mu/(N-1) + mu) / N;
  double deviationSigma = std::fabs(sigma*sigma-mu)/std::sqrt(expectedSigma2Variance);

  cout << "Mean  (should be " << mu << ") is " << mean << "\n";
  cout << "Sigma (should be " << std::sqrt(mu) << ") is " << sigma << "\n";

  cout << "These are " << deviationMean << " and " << deviationSigma <<
        " standard deviations from expected values\n\n";
  
  // If either p-value for the chi-squared tests is less that .0001, or
  // either the mean or sigma are more than 3.5 standard deviations off,
  // then reject the validation.  This would happen by chance one time 
  // in 2000.  Since we will be validating for several values of mu, the
  // net chance of false rejection remains acceptable.

  if ( (pval < .0001) || (pval2 < .0001) ||
       (deviationMean > 3.5) || (deviationSigma > 3.5) ) {
    good = false;
    cout << "REJECT this distributon!!!\n";
  }

  return good;

} // poissonTest()


// **********************
//
// SECTION III - Tests of each distribution class
//
// **********************

// ---------
// RandGauss
// ---------

int testRandGauss() {

  cout << "\n--------------------------------------------\n";
  cout << "Test of RandGauss distribution \n\n";

  long seed;
  cout << "Please enter an integer seed:        ";
  cin >> seed;  cout << seed << "\n";
  if (seed == 0) {
    cout << "Moving on to next test...\n";
    return 0; 
  }

  int nNumbers;
  cout << "How many numbers should we generate: ";
  cin >> nNumbers; cout << nNumbers << "\n";

  double mu;
  double sigma;
  cout << "Enter mu: ";
  cin >> mu; cout << mu << "\n";

  cout << "Enter sigma: ";
  cin >> sigma; cout << sigma << "\n";

  cout << "\nInstantiating distribution utilizing TripleRand engine...\n";
  TripleRand eng (seed);
  RandGauss dist (eng, mu, sigma);
 
  cout << "\n Sample  fire(): \n";
  double x;
  
  x = dist.fire();
  cout << x;

  cout << "\n Testing operator() ... \n";

  bool good = gaussianTest ( dist, mu, sigma, nNumbers );

  if (good) { 
    return 0;
  } else {
    return GaussBAD;
  }

} // testRandGauss()



// ---------
// SkewNormal
// ---------

int testSkewNormal() {

  cout << "\n--------------------------------------------\n";
  cout << "Test of SkewNormal distribution \n\n";

  long seed;
  cout << "Please enter an integer seed:        ";
  cin >> seed;  cout << seed << "\n";
  if (seed == 0) {
    cout << "Moving on to next test...\n";
    return 0; 
  }

  int nNumbers;
  cout << "How many numbers should we generate: ";
  cin >> nNumbers; cout << nNumbers << "\n";

  double k;
  cout << "Enter k: ";
  cin >> k; cout << k << "\n";

  cout << "\nInstantiating distribution utilizing TripleRand engine...\n";
  TripleRand eng (seed);
  RandSkewNormal dist (eng, k);
 
  cout << "\n Sample  fire(): \n";
  double x;
  
  x = dist.fire();
  cout << x;

  cout << "\n Testing operator() ... \n";

  bool good = skewNormalTest ( dist, k, nNumbers );

  if (good) { 
    return 0;
  } else {
    return SkewNormalBAD;
  }

} // testSkewNormal()



// ---------
// RandGaussT
// ---------

int testRandGaussT() {

  cout << "\n--------------------------------------------\n";
  cout << "Test of RandGaussT distribution \n\n";

  long seed;
  cout << "Please enter an integer seed:        ";
  cin >> seed;  cout << seed << "\n";
  if (seed == 0) {
    cout << "Moving on to next test...\n";
    return 0; 
  }

  int nNumbers;
  cout << "How many numbers should we generate: ";
  cin >> nNumbers; cout << nNumbers << "\n";

  double mu;
  double sigma;
  cout << "Enter mu: ";
  cin >> mu; cout << mu << "\n";

  cout << "Enter sigma: ";
  cin >> sigma; cout << sigma << "\n";

  cout << "\nInstantiating distribution utilizing TripleRand engine...\n";
  TripleRand eng (seed);
  RandGaussT dist (eng, mu, sigma);
 
  cout << "\n Sample  fire(): \n";
  double x;
  
  x = dist.fire();
  cout << x;

  cout << "\n Testing operator() ... \n";

  bool good = gaussianTest ( dist, mu, sigma, nNumbers );

  if (good) { 
    return 0;
  } else {
    return GaussTBAD;
  }

} // testRandGaussT()



// ---------
// RandGaussQ
// ---------

int testRandGaussQ() {

  cout << "\n--------------------------------------------\n";
  cout << "Test of RandGaussQ distribution \n\n";

  long seed;
  cout << "Please enter an integer seed:        ";
  cin >> seed;  cout << seed << "\n";
  if (seed == 0) {
    cout << "Moving on to next test...\n";
    return 0; 
  }

  int nNumbers;
  cout << "How many numbers should we generate: ";
  cin >> nNumbers; cout << nNumbers << "\n";

  if (nNumbers >= 20000000) {
    cout << "With that many samples RandGaussQ need not pass validation...\n";
  }

  double mu;
  double sigma;
  cout << "Enter mu: ";
  cin >> mu; cout << mu << "\n";

  cout << "Enter sigma: ";
  cin >> sigma; cout << sigma << "\n";

  cout << "\nInstantiating distribution utilizing DualRand engine...\n";
  DualRand eng (seed);
  RandGaussQ dist (eng, mu, sigma);
 
  cout << "\n Sample  fire(): \n";
  double x;
  
  x = dist.fire();
  cout << x;

  cout << "\n Testing operator() ... \n";

  bool good = gaussianTest ( dist, mu, sigma, nNumbers );

  if (good) { 
    return 0;
  } else {
    return GaussQBAD;
  }

} // testRandGaussQ()


// ---------
// RandPoisson
// ---------

int testRandPoisson() {

  cout << "\n--------------------------------------------\n";
  cout << "Test of RandPoisson distribution \n\n";

  long seed;
  cout << "Please enter an integer seed:        ";
  cin >> seed; cout << seed << "\n";
  if (seed == 0) {
    cout << "Moving on to next test...\n";
    return 0; 
  }

  cout << "\nInstantiating distribution utilizing TripleRand engine...\n";
  TripleRand eng (seed);

  int nNumbers;
  cout << "How many numbers should we generate for each mu: ";
  cin >> nNumbers; cout << nNumbers << "\n";

  bool good = true;

  while (true) {
   double mu;
   cout << "Enter a value for mu: ";
   cin >> mu; cout << mu << "\n";
   if (mu == 0) break;

   RandPoisson dist (eng, mu);
 
   cout << "\n Sample  fire(): \n";
   double x;
  
   x = dist.fire();
   cout << x;

   cout << "\n Testing operator() ... \n";

   bool this_good = poissonTest ( dist, mu, nNumbers );
   if (!this_good) {
    cout << "\n Poisson distribution for mu = " << mu << " is incorrect!!!\n";
   }
   good &= this_good;
  } // end of the while(true)

  if (good) { 
    return 0;
  } else {
    return PoissonBAD;
  }

} // testRandPoisson()


// ---------
// RandPoissonQ
// ---------

int testRandPoissonQ() {

  cout << "\n--------------------------------------------\n";
  cout << "Test of RandPoissonQ distribution \n\n";

  long seed;
  cout << "Please enter an integer seed:        ";
  cin >> seed; cout << seed << "\n";
  if (seed == 0) {
    cout << "Moving on to next test...\n";
    return 0; 
  }

  cout << "\nInstantiating distribution utilizing TripleRand engine...\n";
  TripleRand eng (seed);

  int nNumbers;
  cout << "How many numbers should we generate for each mu: ";
  cin >> nNumbers; cout << nNumbers << "\n";

  bool good = true;

  while (true) {
   double mu;
   cout << "Enter a value for mu: ";
   cin >> mu; cout << mu << "\n";
   if (mu == 0) break;

   RandPoissonQ dist (eng, mu);
 
   cout << "\n Sample  fire(): \n";
   double x;
  
   x = dist.fire();
   cout << x;

   cout << "\n Testing operator() ... \n";

   bool this_good = poissonTest ( dist, mu, nNumbers );
   if (!this_good) {
    cout << "\n Poisson distribution for mu = " << mu << " is incorrect!!!\n";
   }
   good &= this_good;
  } // end of the while(true)

  if (good) { 
    return 0;
  } else {
    return PoissonQBAD;
  }

} // testRandPoissonQ()


// ---------
// RandPoissonT
// ---------

int testRandPoissonT() {

  cout << "\n--------------------------------------------\n";
  cout << "Test of RandPoissonT distribution \n\n";

  long seed;
  cout << "Please enter an integer seed:        ";
  cin >> seed; cout << seed << "\n";
  if (seed == 0) {
    cout << "Moving on to next test...\n";
    return 0; 
  }

  cout << "\nInstantiating distribution utilizing TripleRand engine...\n";
  TripleRand eng (seed);

  int nNumbers;
  cout << "How many numbers should we generate for each mu: ";
  cin >> nNumbers; cout << nNumbers << "\n";

  bool good = true;

  while (true) {
   double mu;
   cout << "Enter a value for mu: ";
   cin >> mu; cout << mu << "\n";
   if (mu == 0) break;

   RandPoissonT dist (eng, mu);
 
   cout << "\n Sample  fire(): \n";
   double x;
  
   x = dist.fire();
   cout << x;

   cout << "\n Testing operator() ... \n";

   bool this_good = poissonTest ( dist, mu, nNumbers );
   if (!this_good) {
    cout << "\n Poisson distribution for mu = " << mu << " is incorrect!!!\n";
   }
   good &= this_good;
  } // end of the while(true)

  if (good) { 
    return 0;
  } else {
    return PoissonTBAD;
  }

} // testRandPoissonT()


// -----------
// RandGeneral
// -----------

int testRandGeneral() {

  cout << "\n--------------------------------------------\n";
  cout << "Test of RandGeneral distribution (using a Gaussian shape)\n\n";

  bool good;
  
  long seed;
  cout << "Please enter an integer seed:        ";
  cin >> seed; cout << seed << "\n";
  if (seed == 0) {
    cout << "Moving on to next test...\n";
    return 0;
  }

  int nNumbers;
  cout << "How many numbers should we generate: ";
  cin >> nNumbers; cout << nNumbers << "\n";

  double mu;
  double sigma;
  mu = .5;      // Since randGeneral always ranges from 0 to 1
  sigma = .06;  

  cout << "Enter sigma: ";
  cin >> sigma; cout << sigma << "\n";
                // We suggest sigma be .06.  This leaves room for 8 sigma 
                // in the distribution.  If it is much smaller, the number 
                // of bins necessary to expect a good match will increase.
                // If sigma is much larger, the cutoff before 5 sigma can
                // cause the Gaussian hypothesis to be rejected. At .14, for 
                // example, the 4th moment is 7 sigma away from expectation.

  int nBins;
  cout << "Enter nBins for stepwise pdf test: ";
  cin >> nBins; cout << nBins << "\n";
                // We suggest at least 10000 bins; fewer would risk 
                // false rejection because the step-function curve 
                // does not match an actual Gaussian.  At 10000 bins,
                // a million-hit test does not have the resolving power
                // to tell the boxy pdf from the true Gaussian.  At 5000
                // bins, it does.

  double xBins = nBins;
  double* aProbFunc = new double [nBins];
  double x;
  for ( int iBin = 0; iBin < nBins; iBin++ )  {
    x = iBin / (xBins-1);
    aProbFunc [iBin] = std::exp ( - (x-mu)*(x-mu) / (2*sigma*sigma) );
  }
  // Note that this pdf is not normalized; RandGeneral does that

  cout << "\nInstantiating distribution utilizing Ranlux64 engine...\n";
  Ranlux64Engine eng (seed, 3);

 { // Open block for testing type 1 - step function pdf

  RandGeneral dist (eng, aProbFunc, nBins, 1);
  delete[] aProbFunc;

  double* garbage = new double[nBins]; 
                                // We wish to verify that deleting the pdf
                                // after instantiating the engine is fine.
  for ( int gBin = 0; gBin < nBins; gBin++ )  {
    garbage [gBin] = 1;
  }
  
  cout << "\n Sample fire(): \n";
    
  x = dist.fire();
  cout << x;

  cout << "\n Testing operator() ... \n";

  good = gaussianTest ( dist, mu, sigma, nNumbers );

  delete[] garbage;

 } // Close block for testing type 1 - step function pdf
   // dist goes out of scope but eng is supposed to stick around; 
   // by closing this block we shall verify that!  

  cout << "Enter nBins for linearized pdf test: ";
  cin >> nBins; cout << nBins << "\n";
                // We suggest at least 1000 bins; fewer would risk 
                // false rejection because the non-smooth curve 
                // does not match an actual Gaussian.  At 1000 bins,
                // a million-hit test does not resolve the non-smoothness;
                // at 300 bins it does.

  xBins = nBins;
  aProbFunc = new double [nBins];
  for ( int jBin = 0; jBin < nBins; jBin++ )  {
    x = jBin / (xBins-1);
    aProbFunc [jBin] = std::exp ( - (x-mu)*(x-mu) / (2*sigma*sigma) );
  }
  // Note that this pdf is not normalized; RandGeneral does that

  RandGeneral dist (eng, aProbFunc, nBins, 0);

  cout << "\n Sample operator(): \n";
    
  x = dist();
  cout << x;

  cout << "\n Testing operator() ... \n";

  bool good2 = gaussianTest ( dist, mu, sigma, nNumbers );
  good = good && good2;

  if (good) { 
    return 0;
  } else {
    return GeneralBAD;
  }

} // testRandGeneral()




// **********************
//
// SECTION IV - Main
//
// **********************

int main() {

  int mask = 0;
  
  mask |= testRandGauss();
  mask |= testRandGaussQ();
  mask |= testRandGaussT();

  mask |= testRandGeneral();

  mask |= testRandPoisson();
  mask |= testRandPoissonQ();
  mask |= testRandPoissonT();

  mask |= testSkewNormal();  // k = 0 (gaussian)
  mask |= testSkewNormal();  // k = -2
  mask |= testSkewNormal();  // k = 1
  mask |= testSkewNormal();  // k = 5

  return mask > 0 ? -mask : mask;
}

---