double AnalyticIntegral(
SpkModel &model ,
const std::valarray<int> &N ,
const std::valarray<double> &y ,
const std::valarray<double> &alpha ,
const std::valarray<double> &L ,
const std::valarray<double> &U ,
size_t individual )
{
double Pi = 4. * atan(1.);
// set the fixed effects
model.setPopPar(alpha);
// set the individual
model.selectIndividual(individual);
// index in y where measurements for this individual starts
size_t start = 0;
size_t i;
for(i = 0; i < individual; i++)
{ assert( N[i] >= 0 );
start += N[i];
}
// number of measurements for this individual
assert( N[individual] >= 0 );
size_t Ni = N[individual];
// data for this individual
std::valarray<double> yi = y[ std::slice( start, Ni, 1 ) ];
// set the random effects = 0
std::valarray<double> b(1);
b[0] = 0.;
model.setIndPar(b);
std::valarray<double> Fi(Ni);
model.dataMean(Fi);
// model for the variance of random effects
std::valarray<double> D(1);
model.indParVariance(D);
// model for variance of the measurements given random effects
std::valarray<double> Ri( Ni * Ni );
model.dataVariance(Ri);
// determine bHat
double residual = 0;
size_t j;
for(j = 0; j < Ni; j++)
{ assert( Ri[j * Ni + j] == Ri[0] );
residual += (yi[j] - Fi[j]);
}
std::valarray<double> bHat(1);
bHat[0] = residual * D[0] / (Ni * D[0] + Ri[0]);
// now evaluate the Map Bayesian objective at its optimal value
model.setIndPar(bHat);
model.dataMean(Fi);
double sum = bHat[0] * bHat[0] / D[0] + log( 2. * Pi * D[0] );
for(j = 0; j < Ni; j++)
{ sum += (yi[j] - Fi[j]) * (yi[j] - Fi[j]) / Ri[0];
sum += log( 2. * Pi * Ri[0] );
}
double GHat = .5 * sum;
// square root of Hessian of the Map Bayesian objective
double rootHessian = sqrt( 1. / D[0] + Ni / Ri[0]);
// determine the values of beta that correspond to the limits
double Gamma_L = (L[0] - bHat[0]) * rootHessian;
double Gamma_U = (U[0] - bHat[0]) * rootHessian;
// compute the upper tails corresponding to the standard normal
double Q_L = gsl_sf_erf_Q(Gamma_L);
double Q_U = gsl_sf_erf_Q(Gamma_U);
// analytic value of the integral
double factor = exp(-GHat) * sqrt(2. * Pi) / rootHessian;
return factor * (Q_L - Q_U);
}
double gaussCdf(double x)
{
// return 0.5*(1+gsl_sf_erf(x)*M_SQRT1_2);
return 1 - gsl_sf_erf_Q(x);
};
static VALUE Error_erf_Q(VALUE self, VALUE x) {
return rb_float_new(gsl_sf_erf_Q(NUM2DBL(x)));
}
