GaussianDistribution.cpp

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#include "GaussianDistribution.h"
#include "Message.h"
#include <string>
#include <iostream>

using namespace std;
namespace Isis {
  GaussianDistribution::GaussianDistribution(const double mean, const double standardDeviation) {
    p_mean = mean;
    p_stdev = standardDeviation;
  }

  double GaussianDistribution::Probability(const double value) {
    // P(x) = 1/(sqrt(2*pi)*sigma)*e^(-1/2*((x-mu)/sigma)^2)
    return std::exp(-0.5 * std::pow((value - p_mean) / p_stdev, 2)) / (std::sqrt(2 * PI) * p_stdev);
  }

  double GaussianDistribution::CumulativeDistribution(const double value) {
    // returns for the two extremes
    if(value == DBL_MIN) {
      return 0.0;
    }
    else if(value == DBL_MAX) {
      return 1.0;
    }

    // normalize the value and calculate the area under the
    //  pdf's curve.
    double x = (value - p_mean) / p_stdev;
    double sum = 0.0,
           pre = -1.0;
    double fact = 1.0;  // fact = n!
    // use taylor series to compute the area to machine precesion
    //  i.e. if an iteration has no impact on the sum, none of the following
    //  ones will since they are monotonically decreasing
    for(int n = 0; pre != sum; n++) {
      pre = sum;
      // the nth term is x^(2n+1)/( (2n+1)*n!*(-2)^n )
      sum += std::pow(x, 2 * n + 1) / (fact * (2 * n + 1) * std::pow(-2.0, n));
      fact *= n + 1;
    }

    // return the percentage (100% based)
    return 50.0 + 100.0 / std::sqrt(2.0 * PI) * sum;
  }

  double GaussianDistribution::InverseCumulativeDistribution(const double percent) {
    // the cutoff values used in the ICDF algorithm
    static double lowCutoff = 2.425;
    static double highCutoff = 97.575;

    if((percent < 0.0) || (percent > 100.0)) {
      string m = "Argument percent outside of the range 0 to 100 in";
      m += " [GaussianDistribution::InverseCumulativeDistribution]";
      throw IException(IException::Programmer, m, _FILEINFO_);
    }

    // for information on the following algorithm, go to the website
    //  specified above

    double x;

    if(percent == 0.0) {
      return DBL_MIN;
    }
    else if(percent == 100.0) {
      return DBL_MAX;
    }

    if(percent < lowCutoff) {
      double q = std::sqrt(-2.0 * std::log(percent / 100.0));
      x = C(q) / D(q);
    }
    else if(percent < highCutoff) {
      double q = percent / 100.0 - 0.5,
             r = q * q;
      x = A(r) * q / B(r);
    }
    else {
      double q = std::sqrt(-2.0 * std::log(1.0 - percent / 100.0));
      x = -1.0 * C(q) / D(q);
    }

    // refine the calculated value using one iteration of Halley's method
    //   for full machine precision
    double e = CumulativeDistribution(p_stdev * x + p_mean) - percent; // the error amount
    double u = e * std::sqrt(2.0 * PI) * std::exp(-0.5 * x * x);
    x = x - u / (1 + 0.5 * x * u);

    return p_stdev * x + p_mean;
  }

  // The following methods are used to compute the ICDF
  //  see the InverseCumulativeDistribution method for
  //  more information

  double GaussianDistribution::A(const double x) {
    static const double a[6] = { -39.69683028665376,
                                 220.9460984245205,
                                 -275.9285104469687,
                                 138.3577518672690,
                                 -30.66479806614716,
                                 2.506628277459239
                               };

    return((((a[0] * x + a[1]) * x + a[2]) * x + a[3]) * x + a[4]) * x + a[5];
  }

  double GaussianDistribution::B(const double x) {
    static const double b[6] = { -54.47609879822406,
                                 161.5858368580409,
                                 -155.6989798598866,
                                 66.80131188771972,
                                 -13.28068155288572,
                                 1.0
                               };

    return((((b[0] * x + b[1]) * x + b[2]) * x + b[3]) * x + b[4]) * x + b[5];
  }

  double GaussianDistribution::C(const double x) {
    static const double c[6] = { -0.007784894002430293,
                                 -0.3223964580411365,
                                 -2.400758277161838,
                                 -2.549732539343734,
                                 4.374664141464968,
                                 2.938163982698783
                               };

    return((((c[0] * x + c[1]) * x + c[2]) * x + c[3]) * x + c[4]) * x + c[5];
  }

  double GaussianDistribution::D(const double x) {
    static const double d[5] = { 0.007784695709041462,
                                 0.3224671290700398,
                                 2.445134137142996,
                                 3.754408661907416,
                                 1.0
                               };

    return(((d[0] * x + d[1]) * x + d[2]) * x + d[3]) * x + d[4];
  }
}