static void target_func(double* p, double *f, double* g, int n, void* data) {
int i;
lbfgsb_param *param = (lbfgsb_param*)data;
double weight = 0, error = 0, consts = 0.0, consts2 = 0.0;
double sum = 0.0;
for(i = 0; i < param->moneyness_size; i++) {
double x = param->log_moneyness[i];
double sig = param->quotes[i];
double var_vec = sig*sig;
double var_cal = svi(p, x);
error += exp(x*3.0) * pow(var_cal - var_vec, 2);
consts += user_min(var_cal, 0.0) * -0.1;
}
// calculate constraints 1
consts = exp(user_max(p[1] * (1.0 + fabs(p[4])) * param->T - 4.0, 0.0) ) - 1.0;
consts2= exp(consts) - 1.0;
*f = error + consts + consts2;
}
/*************************************************************************
Internal linear regression subroutine
*************************************************************************/
static void lrinternal(const ap::real_2d_array& xy,
const ap::real_1d_array& s,
int npoints,
int nvars,
int& info,
linearmodel& lm,
lrreport& ar)
{
ap::real_2d_array a;
ap::real_2d_array u;
ap::real_2d_array vt;
ap::real_2d_array vm;
ap::real_2d_array xym;
ap::real_1d_array b;
ap::real_1d_array sv;
ap::real_1d_array t;
ap::real_1d_array svi;
ap::real_1d_array work;
int i;
int j;
int k;
int ncv;
int na;
int nacv;
double r;
double p;
double epstol;
lrreport ar2;
int offs;
linearmodel tlm;
epstol = 1000;
//
// Check for errors in data
//
if( npoints<nvars||nvars<1 )
{
info = -1;
return;
}
for(i = 0; i <= npoints-1; i++)
{
if( ap::fp_less_eq(s(i),0) )
{
info = -2;
return;
}
}
info = 1;
//
// Create design matrix
//
a.setbounds(0, npoints-1, 0, nvars-1);
b.setbounds(0, npoints-1);
for(i = 0; i <= npoints-1; i++)
{
r = 1/s(i);
ap::vmove(&a(i, 0), &xy(i, 0), ap::vlen(0,nvars-1), r);
b(i) = xy(i,nvars)/s(i);
}
//
// Allocate W:
// W[0] array size
// W[1] version number, 0
// W[2] NVars (minus 1, to be compatible with external representation)
// W[3] coefficients offset
//
lm.w.setbounds(0, 4+nvars-1);
offs = 4;
lm.w(0) = 4+nvars;
lm.w(1) = lrvnum;
lm.w(2) = nvars-1;
lm.w(3) = offs;
//
// Solve problem using SVD:
//
// 0. check for degeneracy (different types)
// 1. A = U*diag(sv)*V'
// 2. T = b'*U
// 3. w = SUM((T[i]/sv[i])*V[..,i])
// 4. cov(wi,wj) = SUM(Vji*Vjk/sv[i]^2,K=1..M)
//
// see $15.4 of "Numerical Recipes in C" for more information
//
t.setbounds(0, nvars-1);
svi.setbounds(0, nvars-1);
ar.c.setbounds(0, nvars-1, 0, nvars-1);
vm.setbounds(0, nvars-1, 0, nvars-1);
if( !rmatrixsvd(a, npoints, nvars, 1, 1, 2, sv, u, vt) )
{
info = -4;
return;
}
if( ap::fp_less_eq(sv(0),0) )
{
//
// Degenerate case: zero design matrix.
//
for(i = offs; i <= offs+nvars-1; i++)
{
lm.w(i) = 0;
}
ar.rmserror = lrrmserror(lm, xy, npoints);
ar.avgerror = lravgerror(lm, xy, npoints);
ar.avgrelerror = lravgrelerror(lm, xy, npoints);
ar.cvrmserror = ar.rmserror;
ar.cvavgerror = ar.avgerror;
ar.cvavgrelerror = ar.avgrelerror;
ar.ncvdefects = 0;
ar.cvdefects.setbounds(0, nvars-1);
ar.c.setbounds(0, nvars-1, 0, nvars-1);
for(i = 0; i <= nvars-1; i++)
{
for(j = 0; j <= nvars-1; j++)
{
ar.c(i,j) = 0;
}
}
return;
}
if( ap::fp_less_eq(sv(nvars-1),epstol*ap::machineepsilon*sv(0)) )
{
//
// Degenerate case, non-zero design matrix.
//
// We can leave it and solve task in SVD least squares fashion.
// Solution and covariance matrix will be obtained correctly,
// but CV error estimates - will not. It is better to reduce
// it to non-degenerate task and to obtain correct CV estimates.
//
for(k = nvars; k >= 1; k--)
{
if( ap::fp_greater(sv(k-1),epstol*ap::machineepsilon*sv(0)) )
{
//
// Reduce
//
xym.setbounds(0, npoints-1, 0, k);
for(i = 0; i <= npoints-1; i++)
{
for(j = 0; j <= k-1; j++)
{
r = ap::vdotproduct(&xy(i, 0), &vt(j, 0), ap::vlen(0,nvars-1));
xym(i,j) = r;
}
xym(i,k) = xy(i,nvars);
}
//
// Solve
//
lrinternal(xym, s, npoints, k, info, tlm, ar2);
if( info!=1 )
{
return;
}
//
// Convert back to un-reduced format
//
for(j = 0; j <= nvars-1; j++)
{
lm.w(offs+j) = 0;
}
for(j = 0; j <= k-1; j++)
{
r = tlm.w(offs+j);
ap::vadd(&lm.w(offs), &vt(j, 0), ap::vlen(offs,offs+nvars-1), r);
}
ar.rmserror = ar2.rmserror;
ar.avgerror = ar2.avgerror;
ar.avgrelerror = ar2.avgrelerror;
ar.cvrmserror = ar2.cvrmserror;
ar.cvavgerror = ar2.cvavgerror;
ar.cvavgrelerror = ar2.cvavgrelerror;
ar.ncvdefects = ar2.ncvdefects;
ar.cvdefects.setbounds(0, nvars-1);
for(j = 0; j <= ar.ncvdefects-1; j++)
{
ar.cvdefects(j) = ar2.cvdefects(j);
}
ar.c.setbounds(0, nvars-1, 0, nvars-1);
work.setbounds(1, nvars);
matrixmatrixmultiply(ar2.c, 0, k-1, 0, k-1, false, vt, 0, k-1, 0, nvars-1, false, 1.0, vm, 0, k-1, 0, nvars-1, 0.0, work);
matrixmatrixmultiply(vt, 0, k-1, 0, nvars-1, true, vm, 0, k-1, 0, nvars-1, false, 1.0, ar.c, 0, nvars-1, 0, nvars-1, 0.0, work);
return;
}
}
info = -255;
return;
}
for(i = 0; i <= nvars-1; i++)
{
if( ap::fp_greater(sv(i),epstol*ap::machineepsilon*sv(0)) )
{
svi(i) = 1/sv(i);
}
else
{
svi(i) = 0;
}
}
for(i = 0; i <= nvars-1; i++)
{
t(i) = 0;
}
for(i = 0; i <= npoints-1; i++)
{
r = b(i);
ap::vadd(&t(0), &u(i, 0), ap::vlen(0,nvars-1), r);
}
for(i = 0; i <= nvars-1; i++)
{
lm.w(offs+i) = 0;
}
for(i = 0; i <= nvars-1; i++)
{
r = t(i)*svi(i);
ap::vadd(&lm.w(offs), &vt(i, 0), ap::vlen(offs,offs+nvars-1), r);
}
for(j = 0; j <= nvars-1; j++)
{
r = svi(j);
ap::vmove(vm.getcolumn(j, 0, nvars-1), vt.getrow(j, 0, nvars-1), r);
}
for(i = 0; i <= nvars-1; i++)
{
for(j = i; j <= nvars-1; j++)
{
r = ap::vdotproduct(&vm(i, 0), &vm(j, 0), ap::vlen(0,nvars-1));
ar.c(i,j) = r;
ar.c(j,i) = r;
}
}
//
// Leave-1-out cross-validation error.
//
// NOTATIONS:
// A design matrix
// A*x = b original linear least squares task
// U*S*V' SVD of A
// ai i-th row of the A
// bi i-th element of the b
// xf solution of the original LLS task
//
// Cross-validation error of i-th element from a sample is
// calculated using following formula:
//
// ERRi = ai*xf - (ai*xf-bi*(ui*ui'))/(1-ui*ui') (1)
//
// This formula can be derived from normal equations of the
// original task
//
// (A'*A)x = A'*b (2)
//
// by applying modification (zeroing out i-th row of A) to (2):
//
// (A-ai)'*(A-ai) = (A-ai)'*b
//
// and using Sherman-Morrison formula for updating matrix inverse
//
// NOTE 1: b is not zeroed out since it is much simpler and
// does not influence final result.
//
// NOTE 2: some design matrices A have such ui that 1-ui*ui'=0.
// Formula (1) can't be applied for such cases and they are skipped
// from CV calculation (which distorts resulting CV estimate).
// But from the properties of U we can conclude that there can
// be no more than NVars such vectors. Usually
// NVars << NPoints, so in a normal case it only slightly
// influences result.
//
ncv = 0;
na = 0;
nacv = 0;
ar.rmserror = 0;
ar.avgerror = 0;
ar.avgrelerror = 0;
ar.cvrmserror = 0;
ar.cvavgerror = 0;
ar.cvavgrelerror = 0;
ar.ncvdefects = 0;
ar.cvdefects.setbounds(0, nvars-1);
for(i = 0; i <= npoints-1; i++)
{
//
// Error on a training set
//
r = ap::vdotproduct(&xy(i, 0), &lm.w(offs), ap::vlen(0,nvars-1));
ar.rmserror = ar.rmserror+ap::sqr(r-xy(i,nvars));
ar.avgerror = ar.avgerror+fabs(r-xy(i,nvars));
if( ap::fp_neq(xy(i,nvars),0) )
{
ar.avgrelerror = ar.avgrelerror+fabs((r-xy(i,nvars))/xy(i,nvars));
na = na+1;
}
//
// Error using fast leave-one-out cross-validation
//
p = ap::vdotproduct(&u(i, 0), &u(i, 0), ap::vlen(0,nvars-1));
if( ap::fp_greater(p,1-epstol*ap::machineepsilon) )
{
ar.cvdefects(ar.ncvdefects) = i;
ar.ncvdefects = ar.ncvdefects+1;
continue;
}
r = s(i)*(r/s(i)-b(i)*p)/(1-p);
ar.cvrmserror = ar.cvrmserror+ap::sqr(r-xy(i,nvars));
ar.cvavgerror = ar.cvavgerror+fabs(r-xy(i,nvars));
if( ap::fp_neq(xy(i,nvars),0) )
{
ar.cvavgrelerror = ar.cvavgrelerror+fabs((r-xy(i,nvars))/xy(i,nvars));
nacv = nacv+1;
}
ncv = ncv+1;
}
if( ncv==0 )
{
//
// Something strange: ALL ui are degenerate.
// Unexpected...
//
info = -255;
return;
}
ar.rmserror = sqrt(ar.rmserror/npoints);
ar.avgerror = ar.avgerror/npoints;
if( na!=0 )
{
ar.avgrelerror = ar.avgrelerror/na;
}
ar.cvrmserror = sqrt(ar.cvrmserror/ncv);
ar.cvavgerror = ar.cvavgerror/ncv;
if( nacv!=0 )
{
ar.cvavgrelerror = ar.cvavgrelerror/nacv;
}
}