void diff2PDF_rho_u_tCopula_new(double* u, double* v, int* n, double* param, int* copula, double* out)
{
double x1, x2;
int j=0, k=1;
double t1, t2, t3, t4, M, c=0, out1=0, out2=0;
double rho = param[0];
double nu = param[1];
t1=nu+2.0;
t4=1.0-rho*rho;
for(j=0;j<*n;j++)
{
LL(copula, &k, &u[j], &v[j], &rho, &nu, &c);
c=exp(c);
x1=qt(u[j],nu,1,0);
x2=qt(v[j],nu,1,0);
M = ( nu*t4 + x1*x1 + x2*x2 - 2.0*rho*x1*x2 );
t2=nu*rho+x1*x2;
t3=x1-rho*x2;
diffPDF_rho_tCopula(&u[j], &v[j], &k, param, copula, &out1);
diffPDF_u_tCopula_new(&u[j], &v[j], &k, param, copula, &out2);
out[j]=c*(t1/M/dt(x1,nu,0) * (x2 - 2.0*t2/M*t3)) + out1/c*out2;
}
}
void diff2PDF_u_tCopula_new(double* u, double* v, int* n, double* param, int* copula, double* out)
{
double x1, x2;
int j=0, k=1;
double t1, t2, t4, t5, t6, t7, t8, t9, t11, t12, t13, M, c=0, diffPDF=0, diff_dt2=0;
double t14, t15, t16;
double rho = param[0];
double nu = param[1];
t1=nu+2.0;
t14=rho*rho;
t4=1.0-t14;
for(j=0;j<*n;j++)
{
LL(copula, &k, &u[j], &v[j], &rho, &nu, &c);
c=exp(c);
x1=qt(u[j],nu,1,0);
x2=qt(v[j],nu,1,0);
t15=x1*x1;
t16=x2*x2;
M = ( nu*t4 + t15 + t16 - 2.0*rho*x1*x2 );
t2=dt(x1,nu,0);
diffPDF_u_tCopula_new(&u[j], &v[j], &k, param, copula, &diffPDF);
diff_dt_u(&x1, &nu, &diff_dt2);
t7=x1-rho*x2;
t5=-diffPDF/t2 + diff_dt2/t2/t2*c;
t6=t1*t7/M + diff_dt2;
t11=c/t2;
t8=1.0/t2;
t9=t1/M - 2.0*t1*t7*t7/M/M;
t13=1.0+t15/nu;
t12=t8*( -(nu+1.0)/(nu+t15) + 2.0*t15* (nu+1.0)/nu/nu / t13/t13);
out[j]=t5*t6 - t11*(t8*t9 + t12);
}
}
void VineLogLikRvineDeriv(int* T, int* d, int* family, int* kk, int* ii, int* maxmat, int* matrix, int* condirect, int* conindirect, double* par, double* par2, double* data,
double* out, double* ll, double* vv, double* vv2, int* calcupdate, double* tilde_vdirect, double* tilde_vindirect, double* tilde_value, int* tcop, int* margin)
{
int i, j, k, t, m, **fam, **calc;
double sumloglik=0.0, **theta, **nu, ***tildevdirect, ***tildevindirect, *zr1, *zr2, *tildezr1, *tildezr2, *cop;
double *handle1;
double param[2];
zr1=(double*) Calloc(*T,double);
zr2=(double*) Calloc(*T,double);
handle1=(double*) Calloc(*T,double);
tildezr1=(double*) Calloc(*T,double);
tildezr2=(double*) Calloc(*T,double);
cop=(double*) Calloc(*T,double);
double ***tildevalue;
tildevalue=create_3darray(*d,*d,*T);
//Allocate memory
tildevdirect = create_3darray(*d,*d,*T);
tildevindirect = create_3darray(*d,*d,*T);
theta=create_matrix(*d,*d);
nu=create_matrix(*d,*d);
fam=create_intmatrix(*d,*d);
calc=create_intmatrix(*d,*d);
//Initialize
k=0;
for(i=0;i<(*d);i++)
{
for(j=0;j<(*d);j++)
{
theta[i][j]=par[(i+1)+(*d)*j-1] ;
nu[i][j]=par2[(i+1)+(*d)*j-1] ;
fam[i][j]=family[(i+1)+(*d)*j-1] ;
calc[i][j]=calcupdate[(i+1)+(*d)*j-1] ;
}
}
for(i=0;i<(*d);i++)
{
for(j=0;j<(*d);j++)
{
for(t=0;t<*T;t++ )
{
tildevdirect[i][j][t]=0;
tildevindirect[i][j][t]=0;
tildevalue[i][j][t]=0;
}
}
}
m=maxmat[*kk+(*d)*(*ii-1)-1];
for(t=0;t<*T;t++ )
{
zr1[t]=vv[*kk+(*d)*(*ii-1)+(*d)*(*d)*t-1];
cop[t]=exp(ll[*kk+(*d)*(*ii-1)+(*d)*(*d)*t-1]);
}
if(m == matrix[*kk+*d*(*ii-1)-1])
{
for(t=0;t<*T;t++ )
{
zr2[t]=vv[*kk+(*d)*(*d-m)+(*d)*(*d)*t-1];
}
}
else
{
for(t=0;t<*T;t++)
{
zr2[t]=vv2[*kk+(*d)*(*d-m)+(*d)*(*d)*t-1];
}
}
param[0]=theta[*kk-1][*ii-1];
param[1]=nu[*kk-1][*ii-1];
if(*tcop==1) //For the t-copula (first parameter)
{
diffhfunc_rho_tCopula(zr1,zr2,T,param,&fam[*kk-1][*ii-1],tildevdirect[*kk-2][*ii-1]);
diffhfunc_rho_tCopula(zr2,zr1,T,param,&fam[*kk-1][*ii-1],tildevindirect[*kk-2][*ii-1]);
diffPDF_rho_tCopula(zr1,zr2,T,param,&fam[*kk-1][*ii-1],tildevalue[*kk-1][*ii-1]);
for(t=0;t<*T;t++ )
{
tildevalue[*kk-1][*ii-1][t]=tildevalue[*kk-1][*ii-1][t]/cop[t];
}
}
else if(*tcop==2) // for the t-copula (second parameter)
{
diffhfunc_nu_tCopula_new(zr1,zr2,T,param,&fam[*kk-1][*ii-1],tildevdirect[*kk-2][*ii-1]);
diffhfunc_nu_tCopula_new(zr2,zr1,T,param,&fam[*kk-1][*ii-1],tildevindirect[*kk-2][*ii-1]);
diffPDF_nu_tCopula_new(zr1,zr2,T,param,&fam[*kk-1][*ii-1],tildevalue[*kk-1][*ii-1]);
for(t=0;t<*T;t++ )
{
tildevalue[*kk-1][*ii-1][t]=tildevalue[*kk-1][*ii-1][t]/cop[t];
}
}
else
{
if( *margin == 0 ) //Das ist unser bisheriger Fall mit stetigen Variablen (ohne t-copula)
{
diffhfunc_mod(zr1,zr2,T,&theta[*kk-1][*ii-1],&fam[*kk-1][*ii-1],tildevdirect[*kk-2][*ii-1]);
diffhfunc_mod2(zr2,zr1,T,&theta[*kk-1][*ii-1],&fam[*kk-1][*ii-1],tildevindirect[*kk-2][*ii-1]);
diffPDF_mod(zr1,zr2,T,&theta[*kk-1][*ii-1],&fam[*kk-1][*ii-1],tildevalue[*kk-1][*ii-1]);
for(t=0;t<*T;t++ )
{
tildevalue[*kk-1][*ii-1][t]=tildevalue[*kk-1][*ii-1][t]/cop[t];
}
}
else if( *margin== 1) // Ableitung nach dem ersten Argument = margin1
{
diffhfunc_v_mod2(zr2,zr1,T,&theta[*kk-1][*ii-1],&fam[*kk-1][*ii-1],tildevindirect[*kk-2][*ii-1]);
diffPDF_u_mod(zr1,zr2,T,&theta[*kk-1][*ii-1],&fam[*kk-1][*ii-1],tildevalue[*kk-1][*ii-1]); // hier k?nnte difflPDF stehen
for(t=0;t<*T;t++ )
{
tildevdirect[*kk-2][*ii-1][t]=cop[t];
tildevalue[*kk-1][*ii-1][t]=tildevalue[*kk-1][*ii-1][t]/cop[t];
}
}
else // Ableitung nach dem zweiten Argument = margin2
{
diffhfunc_v_mod(zr1,zr2,T,&theta[*kk-1][*ii-1],&fam[*kk-1][*ii-1],tildevdirect[*kk-2][*ii-1]);
diffPDF_v_mod(zr1,zr2,T,&theta[*kk-1][*ii-1],&fam[*kk-1][*ii-1],tildevalue[*kk-1][*ii-1]); // hier k?nnte difflPDF stehen
for(t=0;t<*T;t++ )
{
tildevindirect[*kk-2][*ii-1][t]=cop[t];
tildevalue[*kk-1][*ii-1][t]=tildevalue[*kk-1][*ii-1][t]/cop[t];
}
}
}
// add up for the final derivative
for(t=0;t<*T;t++ )
{
sumloglik+=tildevalue[*kk-1][*ii-1][t];
}
for(i=*ii-1; i>-1; i--)
{
for(k=*kk-2;k>i;k--)
{
if(calc[k][i]==1)
{
m=maxmat[(k+1)+*d*i-1];
for(t=0;t<*T;t++ )
{
zr1[t]=vv[(k+1)+(*d)*i+(*d)*(*d)*t-1];
tildezr1[t]=tildevdirect[k][i][t];
cop[t]=exp(ll[(k+1)+(*d)*i+(*d)*(*d)*t-1]);
}
if(m == matrix[(k+1)+*d*i-1])
{
for(t=0;t<*T;t++ )
{
zr2[t]=vv[(k+1)+(*d)*(*d-m)+(*d)*(*d)*t-1];
tildezr2[t]=tildevdirect[k][(*d-m)][t];
}
}
else
{
for(t=0;t<*T;t++ )
{
zr2[t]=vv2[(k+1)+(*d)*(*d-m)+(*d)*(*d)*t-1];
tildezr2[t]=tildevindirect[k][(*d-m)][t];
}
}
for(t=0;t<*T;t++ )
{
tildevdirect[k-1][i][t]=0;
tildevindirect[k-1][i][t]=0;
tildevalue[k][i][t]=0;
}
if(calc[k+1][i]==1)
{
param[0]=theta[k][i];
param[1]=nu[k][i];
if(fam[k][i]==2) //For the t-copula
{
diffPDF_u_tCopula_new(zr1,zr2,T,param,&fam[k][i],handle1);
}
else
{
diffPDF_u_mod(zr1,zr2,T,&theta[k][i],&fam[k][i],handle1);
}
for(t=0;t<*T;t++ )
{
tildevalue[k][i][t]+=handle1[t]/cop[t]*tildezr1[t];
}
if(condirect[k+(*d)*i-1]==1)
{
for(t=0;t<*T;t++ )
{
tildevdirect[k-1][i][t]+=cop[t]*tildezr1[t];
}
}
if(conindirect[k+(*d)*i-1]==1)
{
param[0]=theta[k][i];
param[1]=nu[k][i];
if(fam[k][i]==2) //For the t-copula
{
diffhfunc_v_tCopula_new(zr2,zr1,T,param,&fam[k][i],handle1);
}
else
{
diffhfunc_v_mod2(zr2,zr1,T,&theta[k][i],&fam[k][i],handle1);
}
for(t=0;t<*T;t++ )
{
tildevindirect[k-1][i][t]+=handle1[t]*tildezr1[t];
}
}
}
if(calc[k+1][(*d-m)]==1)
{
param[0]=theta[k][i];
param[1]=nu[k][i];
if(fam[k][i]==2) //For the t-copula
{
diffPDF_u_tCopula_new(zr2,zr1,T,param,&fam[k][i],handle1);
}
else
{
diffPDF_v_mod(zr1,zr2,T,&theta[k][i],&fam[k][i],handle1);
}
for(t=0;t<*T;t++ )
{
tildevalue[k][i][t]+=handle1[t]/cop[t]*tildezr2[t];
}
if(condirect[k+(*d)*i-1]==1)
{
param[0]=theta[k][i];
param[1]=nu[k][i];
if(fam[k][i]==2) //For the t-copula
{
diffhfunc_v_tCopula_new(zr1,zr2,T,param,&fam[k][i],handle1);
}
else
{
diffhfunc_v_mod(zr1,zr2,T,&theta[k][i],&fam[k][i],handle1);
}
for(t=0;t<*T;t++ )
{
tildevdirect[k-1][i][t]+=handle1[t]*tildezr2[t];
}
}
if(conindirect[k+(*d)*i-1]==1)
{
for(t=0;t<*T;t++ )
{
tildevindirect[k-1][i][t]+=cop[t]*tildezr2[t];
}
}
}
}
for(t=0;t<*T;t++ )
{
sumloglik += tildevalue[k][i][t];
}
}
}//for loops closed
*out = sumloglik;
for(i=0;i<(*d);i++)
{
for(j=0;j<(*d);j++)
{
for(t=0;t<*T;t++ )
{
tilde_vdirect[(i+1)+(*d)*j+(*d)*(*d)*t-1]=tildevdirect[i][j][t];
tilde_vindirect[(i+1)+(*d)*j+(*d)*(*d)*t-1]=tildevindirect[i][j][t];
tilde_value[(i+1)+(*d)*j+(*d)*(*d)*t-1]=tildevalue[i][j][t];
}
}
}
//Free memory:
free_matrix(theta,*d); free_matrix(nu,*d); free_intmatrix(fam,*d);
free_intmatrix(calc, *d);
free_3darray(tildevindirect,*d,*d);
free_3darray(tildevdirect,*d,*d);
free_3darray(tildevalue,*d,*d);
Free(zr1); Free(zr2); Free(tildezr1); Free(tildezr2); Free(handle1); Free(cop);
}
void diffPDF_u(double* u, double* v, int* n, double* param, int* copula, double* out)
{
int j, k=1;
double t1, t2, t3, t4, t5, t6, t7, t8, t9, t10, t11, t12, t13, t14, t15, t16, t18, t19, t20, t21, t22, t23, t24, t25, t27, t28, t29;
double t30, t33, t36;
double theta = param[0];
//double delta = param[1];
for(j=0;j<*n;j++)
{
if(*copula==0)
{
out[j]=0;
}
else if(*copula==1)
{
t1 = qnorm(u[j],0.0,1.0,1,0);
t2 = qnorm(v[j],0.0,1.0,1,0);
t3 = theta*theta;
t4 = 1.0-t3;
t5 = sqrt(t4);
t6 = pow(t1,2.0);
t7 = pow(t2,2.0);
t8 = t3*(t6+t7)-(2.0*theta*t1*t2);
t9 = exp(-t8/t4/2.0);
t10 = t9/t5;
t11 = sqrt(2.0*pi);
t12 = exp(-t6/2.0);
t13 = t11/t12;
out[j] = -t10*(theta*t13/t4)*(theta*t1-t2);
}
else if(*copula==2)
{
diffPDF_u_tCopula_new(&u[j], &v[j], &k, param, copula, &out[j]); // special function for t-copula
}
else if(*copula==3)
{
t1 = 1.0+theta;
t3 = pow(u[j]*v[j],-1.0*t1);
t4 = t1*t3;
t5 = 1/u[j];
t7 = pow(u[j],-1.0*theta);
t8 = pow(v[j],-1.0*theta);
t9 = t7+t8-1.0;
t11 = -2.0-1/theta;
t12 = pow(t9,1.0*t11);
out[j] = -t4*t1*t5*t12-t4*t12*t11*t7*theta*t5/t9;
}
else if(*copula==4)
{
t3 = log(u[j]);
t4 = pow(-t3,1.0*theta);
t5 = log(v[j]);
t6 = pow(-t5,1.0*theta);
t7 = t4+t6;
t8 = 1/theta;
t9 = pow(t7,1.0*t8);
t11 = u[j]*u[j];
t12 = 1/t11;
t13 = 1/t3;
t15 = 1/t7;
t18 = exp(-t9);
t19 = 1/v[j];
t21 = -1.0+t8;
t22 = pow(t7,2.0*t21);
t24 = theta-1.0;
t25 = pow(t3*t5,1.0*t24);
t27 = pow(t7,-1.0*t8);
t28 = t24*t27;
t29 = 1.0+t28;
t30 = t22*t25*t29;
t33 = t18*t12;
t36 = t19*t22;
out[j] = -t9*t4*t12*t13*t15*t18*t19*t30-t33*t19*t30+2.0*t33*t36*t21*t4*theta
*t13*t15*t25*t29+t33*t36*t25*t24*t13*t29-t33*t36*t25*t28*t4*t13*t15;
}
else if(*copula==5)
{
t1 = theta*theta;
t2 = exp(theta);
t3 = t2-1.0;
t5 = theta*v[j];
t6 = theta*u[j];
t8 = exp(t5+t6+theta);
t10 = exp(t5+t6);
t12 = exp(t5+theta);
t14 = exp(t6+theta);
t15 = t10-t12-t14+t2;
t16 = t15*t15;
out[j] = t1*t3*t8/t16-2.0*theta*t3*t8/t16/t15*(theta*t10-theta*t14);
}
else if(*copula==6)
{
t1 = 1.0-u[j];
t2 = pow(t1,1.0*theta);
t3 = 1.0-v[j];
t4 = pow(t3,1.0*theta);
t5 = t2*t4;
t6 = t2+t4-t5;
t8 = 1/theta-2.0;
t9 = pow(t6,1.0*t8);
t11 = t2*theta;
t12 = 1/t1;
t16 = -t11*t12+t11*t12*t4;
t19 = theta-1.0;
t20 = pow(t1,1.0*t19);
t22 = pow(t3,1.0*t19);
t23 = theta-1.0+t2+t4-t5;
t27 = t9*t20;
out[j] = t9*t8*t16/t6*t20*t22*t23-t27*t19*t12*t22*t23+t27*t22*t16;
}
}
}