void MasterElementV<METRAITS>::JxW(
const Mapping<typename ME2MPTraits<METRAITS>::value> *mapping,
const mdata_type mdata[],
const intgRule *intg, mdata_type result[]) const
{
std::vector<mdata_type> Jx(intg->npoints());
mapping->Jx(intg->npoints(), mdata, intg->locations(), &Jx[0]);
for (UInt i = 0; i < intg->npoints(); i++) result[i] = intg->weights()[i]*Jx[i];
}
static void residual(const MODEL::Properties& p,
JM flux_jacob[],
RV& res)
{
const Real gamma_minus_3 = p.gamma - 3.;
const Real uu = p.u * p.u;
const Real uv = p.u * p.v;
const Real uw = p.u * p.w;
const Real vv = p.v * p.v;
const Real vw = p.v * p.w;
const Real ww = p.w * p.w;
JM& Jx = flux_jacob[XX];
// Jx(0,0) = 0.0;
Jx(0,1) = 1.0;
// Jx(0,2) = 0.0;
// Jx(0,3) = 0.0;
// Jx(0,4) = 0.0;
Jx(1,0) = p.half_gm1_v2 - uu;
Jx(1,1) = -gamma_minus_3*p.u;
Jx(1,2) = -p.gamma_minus_1*p.v;
Jx(1,3) = -p.gamma_minus_1*p.w;
Jx(1,4) = p.gamma_minus_1;
Jx(2,0) = -uv;
Jx(2,1) = p.v;
Jx(2,2) = p.u;
// Jx(2,3) = 0.0;
// Jx(2,4) = 0.0;
Jx(3,0) = -uw;
Jx(3,1) = p.w;
// Jx(3,2) = 0.0;
Jx(3,3) = p.u;
// Jx(3,4) = 0.0;
Jx(4,0) = p.u*(p.half_gm1_v2 - p.H);
Jx(4,1) = -p.gamma_minus_1*uu + p.H;
Jx(4,2) = -p.gamma_minus_1*uv;
Jx(4,3) = -p.gamma_minus_1*uw;
Jx(4,4) = p.gamma*p.u;
JM& Jy = flux_jacob[YY];
// Jy(0,0) = 0.0;
// Jy(0,1) = 0.0;
Jy(0,2) = 1.0;
// Jy(0,3) = 0.0;
// Jy(0,4) = 0.0;
Jy(1,0) = -uv;
Jy(1,1) = p.v;
Jy(1,2) = p.u;
// Jy(1,3) = 0.0;
// Jy(1,4) = 0.0;
Jy(2,0) = p.half_gm1_v2 - vv;
Jy(2,1) = -p.gamma_minus_1*p.u;
Jy(2,2) = -gamma_minus_3*p.v;
Jy(2,3) = -p.gamma_minus_1*p.w;
Jy(2,4) = p.gamma_minus_1;
Jy(3,0) = -vw;
// Jy(3,1) = 0.0;
Jy(3,2) = p.w;
Jy(3,3) = p.v;
// Jy(3,4) = 0.0;
Jy(4,0) = p.v*(p.half_gm1_v2 - p.H);
Jy(4,1) = -p.gamma_minus_1*uv;
Jy(4,2) = -p.gamma_minus_1*vv + p.H;
Jy(4,3) = -p.gamma_minus_1*vw;
Jy(4,4) = p.gamma*p.v;
JM& Jz = flux_jacob[ZZ];
// Jz(0,0) = 0.0;
// Jz(0,1) = 0.0;
// Jz(0,2) = 0.0;
Jz(0,3) = 1.0;
// Jz(0,4) = 0.0;
Jz(1,0) = -uw;
Jz(1,1) = p.w;
// Jz(1,2) = 0.0;
Jz(1,3) = p.u;
// Jz(1,4) = 0.0;
Jz(2,0) = -vw;
// Jz(2,1) = 0.0;
Jz(2,2) = p.w;
Jz(2,3) = p.v;
// Jz(2,4) = 0.0;
Jz(3,0) = p.half_gm1_v2 - ww;
Jz(3,1) = -p.gamma_minus_1*p.u;
Jz(3,2) = -p.gamma_minus_1*p.v;
Jz(3,3) = -gamma_minus_3*p.w;
Jz(3,4) = p.gamma_minus_1;
Jz(4,0) = p.w*(p.half_gm1_v2 - p.H);
Jz(4,1) = -p.gamma_minus_1*uw;
Jz(4,2) = -p.gamma_minus_1*vw;
Jz(4,3) = -p.gamma_minus_1*ww + p.H;
Jz(4,4) = p.gamma*p.w;
res = Jx * p.grad_vars.col(XX) + Jy * p.grad_vars.col(YY) + Jz * p.grad_vars.col(ZZ);
}
// Simple Gauss-Newton method for ARMA(1,1) MLE
// y: observation
// WolfCoe: Wolfe condition coefficienti
// LineSize: line search step size
// FuncTol: function value tolerance
// OptiTol: first order tolerance
// StepTol: step tolerance
// MaxIter: total number of iterations
void ARMA11::fit_mle(vec y, double WolfCoe, double LineSize, double FuncTol, double OptiTol, double StepTol, int MaxIter)
{
// Set initial parameters as method of moments estimates
ARMA11 temp(mu, phi, psi, sigma);
vec x = temp.residual(y);
// Initialize other parameters
ARMA11 temp_next = temp;
vec x_next = x;
double alpha;
mat J(3,y.n_elem);
vec Jx(3);
vec p(3);
double L;
double L_next;
double L_next_approx;
// Print iteration info
cout.precision(5);
cout << setw(5) << "Iter" << setw(8) << "Alpha" << setw(10) << "Grad" << setw(10) << "FuncDiff" << setw(10) << "StepSize" << setw(10) << "LLH" << setw(10) << endl;
for (int k=0; k<MaxIter; k++) {
// Compute Jacobian matrix
J = temp.Jacobian(x,y);
Jx = J*x;
if (k>1 && (norm(Jx,2)<OptiTol || abs(L_next-L)/abs(L)<FuncTol || norm(alpha*p,2)<StepTol)) {
break;
}
// Gauss-Newton direction
// Approxmiate Hessian with J*J' for least square problems
p = -solve(J*J.t(),Jx);
alpha = 1.0;
// Update next parameters
temp_next.mu = temp.mu+alpha*p[0];
temp_next.phi = temp.phi+alpha*p[1];
temp_next.psi = temp.psi+alpha*p[2];
x_next = temp_next.residual(y);
// Minus log-likelihood function
L = L_next;
L_next = 0.5*dot(x_next,x_next);
// Taylor expansion of L_next
L_next_approx = 0.5*dot(x,x)+WolfCoe*alpha*dot(p,Jx);
// Line search step size alpha
// Wolfe condition: L_next<=L_next_approx
while (L_next>L_next_approx) {
alpha = LineSize*alpha;
temp_next.mu = temp.mu+alpha*p[0];
temp_next.phi = temp.phi+alpha*p[1];
temp_next.psi = temp.psi+alpha*p[2];
x_next = temp_next.residual(y);
L_next = 0.5*dot(x_next,x_next);
L_next_approx = 0.5*dot(x,x)+WolfCoe*alpha*dot(p,Jx);
}
temp = temp_next;
x = x_next;
if (k>0) {
cout << setw(5) << fixed << k << setw(8) << fixed << alpha << setw(10) << fixed << norm(Jx,2) << setw(10) << fixed << (L_next-L)/abs(L) << setw(10) << fixed << norm(alpha*p,2) << setw(10) << fixed << L_next << endl;
}
}
mu = temp.mu;
phi = temp.phi;
psi = temp.psi;
sigma = sqrt(dot(x,x)/y.n_elem);
}