/// Normal-distribution percent point function, or inverse of the Normal CDF.
/// This function(prob,mu,sig) == X where prob = NormalCDF(X,mu,sig).
/// Ref http://www.itl.nist.gov/div898/handbook/ 1.3.6.6.1
/// @param prob probability or significance level of the test, >=0 and < 1
/// @param mu mean of the sample (location parameter of the distribution)
/// @param sig std dev of the sample (scale parameter of the distribution)
/// @return X the statistic at this probability
double invNormalCDF(double prob, const double& mu, const double& sig)
throw(Exception)
{
try {
if(prob < 0 || prob >= 1)
GPSTK_THROW(Exception("Invalid probability argument"));
if(sig <= 0.0) GPSTK_THROW(Exception("Non-positive sigma"));
static const double eps(1000000*std::numeric_limits<double>().epsilon());
if(prob < eps)
return 0.0;
if(1.0-prob < eps)
GPSTK_THROW(Exception("Invalid probability -- too close to 1.0"));
// find X such that prob == NormalCDF(X,mu,sig); use bracket method
// we know 0.5 = NormalCDF(muwhen prob = 0.5, X = mu
// also invNormalCDF(1-prob,mu,sig) = 2*mu - invNormalCDF(prob,mu,sig)
// so make alpha >= 0.5
bool swap(false);
double alpha(prob);
if(prob < 0.5) { swap=true; alpha=1.0-prob; }
// we know a0 = NormalCDF(X0,mu,sig) where a0=0.5,X0=mu and alpha >= 0.5
double X,X0(mu),X1,a;
// first find X1 such that a1 = NormalCDF(X1,mu,sig) and a1 > alpha
X1 = 2.0;
while((a = NormalCDF(X1,mu,sig)) <= alpha) { X1 *= 2.0; }
// bracket
int niter(0); // iteration count
while(1) {
X = (X0+X1)/2.0;
a = NormalCDF(X,mu,sig);
//LOG(INFO) << "LOOP invNormalCDF X = " << niter << " " << std::fixed
//<< std::setprecision(15) << X0 << " < " << X << " < " << X1
//<< " and a = " << a << " alpha = " << alpha << " a-alpha = "
//<< std::scientific << std::setprecision(2) << a-alpha << " eps " << eps
//<< " fabs(a-alpha)-eps " << ::fabs(alpha-a)-eps;
if(::fabs(alpha-a) < eps) break;
if(a > alpha) { X1 = X; } else { X0 = X; }
if(++niter > 100) GPSTK_THROW(Exception("Failed to converge"));
}
return (swap ? 2.0*mu-X : X);
}
catch(Exception& e) { GPSTK_RETHROW(e); }
}
TWxTestResult WxTest(const TVector<double>& baseline,
const TVector<double>& test) {
TVector<double> diffs;
for (ui32 i = 0; i < baseline.size(); i++) {
const double i1 = baseline[i];
const double i2 = test[i];
const double diff = i1 - i2;
if (diff != 0) {
diffs.push_back(diff);
}
}
if (diffs.size() < 2) {
TWxTestResult result;
result.PValue = 0.5;
result.WMinus = result.WPlus = 0;
return result;
}
Sort(diffs.begin(), diffs.end(), [&](double x, double y) {
return Abs(x) < Abs(y);
});
double w_plus = 0;
double w_minus = 0;
double n = diffs.size();
for (int i = 0; i < n; ++i) {
double sum = 0;
double weight = 0;
int j = i;
double signPlus = 0;
double signMinus = 0;
for (j = i; j < n && diffs[j] == diffs[i]; ++j) {
sum += (j + 1);
++weight;
signPlus += diffs[i] >= 0;
signMinus += diffs[i] < 0;
}
const double meanRank = sum / weight;
w_plus += signPlus * meanRank;
w_minus += signMinus * meanRank;
i = j - 1;
}
TWxTestResult result;
result.WPlus = w_plus;
result.WMinus = w_minus;
const double w = result.WPlus - result.WMinus;
if (n > 16) {
double z = w / sqrt(n * (n + 1) * (2 * n + 1) * 1.0 / 6);
result.PValue = 2 * (1.0 - NormalCDF(Abs(z)));
} else {
result.PValue = 2 * CalcLevelOfSignificanceWXMPSR(Abs(w), (int) n);
}
result.PValue = 1.0 - result.PValue;
return result;
}