# include <cppad/cppad.hpp>
# include <cppad/mixed/cppad_mixed.hpp>
# include <cppad/mixed/manage_gsl_rng.hpp>
# include <Eigen/Dense>
namespace {
using CppAD::log;
using CppAD::AD;
//
using CppAD::mixed::d_sparse_rcv;
using CppAD::mixed::a1_double;
using CppAD::mixed::d_vector;
using CppAD::mixed::a1_vector;
//
class mixed_derived : public cppad_mixed {
private:
const size_t n_fixed_;
const size_t n_random_;
const d_vector& y_;
// ----------------------------------------------------------------------
public:
// constructor
mixed_derived(
size_t n_fixed ,
size_t n_random ,
bool quasi_fixed ,
bool bool_sparsity ,
const d_vector& y ) :
cppad_mixed(
n_fixed, n_random, quasi_fixed, bool_sparsity
) ,
n_fixed_(n_fixed) ,
n_random_(n_random) ,
y_(y)
{ assert( n_fixed == 3);
assert( y_.size() == n_random_ );
}
// ----------------------------------------------------------------------
// implementation of ran_likelihood
// Note that theta[2] is not used
a1_vector ran_likelihood(
const a1_vector& theta ,
const a1_vector& u ) override
{
assert( theta.size() == n_fixed_ );
assert( u.size() == y_.size() );
a1_vector vec(1);
// initialize part of log-density that is always smooth
vec[0] = 0.0;
// sqrt_2pi = CppAD::sqrt(8.0 * CppAD::atan(1.0) );
for(size_t i = 0; i < n_random_; i++)
{ a1_double mu = u[i] + theta[0];
a1_double sigma = theta[1];
a1_double res = (y_[i] - mu) / sigma;
// p(y_i | u, theta)
vec[0] += log(sigma) + res * res / 2.0;
// following term does not depend on fixed or random effects
// vec[0] += log(sqrt_2pi);
// p(u_i | theta)
vec[0] += u[i] * u[i] / 2.0;
// following term does not depend on fixed or random effects
// vec[0] += log(sqrt_2pi);
}
return vec;
}
// implementation of fix_likelihood
a1_vector fix_likelihood(
const a1_vector& fixed_vec ) override
{
assert( fixed_vec.size() == n_fixed_ );
a1_vector vec(1);
// initialize part of log-density that is smooth
vec[0] = 0.0;
// sqrt_2pi = CppAD::sqrt( 8.0 * CppAD::atan(1.0) );
// Note that theta[2] is not included
for(size_t j = 0; j < 2; j++)
{ a1_double mu = 4.0;
a1_double sigma = 1.0;
a1_double res = (fixed_vec[j] - mu) / sigma;
// This is a Gaussian term, so entire density is smooth
vec[0] += res * res / 2.0;
// following term does not depend on fixed effects
// vec[0] += log(sqrt_2pi * sigma);
}
return vec;
}
};
}
bool sample_fixed_xam(void)
{
bool ok = true;
double inf = std::numeric_limits<double>::infinity();
double eps = 10. * std::numeric_limits<double>::epsilon();
//
// initialize gsl random number generator
size_t random_seed = CppAD::mixed::new_gsl_rng(0);
//
size_t n_data = 10;
size_t n_fixed = 3;
size_t n_random = n_data;
d_vector
fixed_lower(n_fixed), fixed_in(n_fixed), fixed_upper(n_fixed);
fixed_lower[0] = - inf; fixed_in[0] = 2.0; fixed_upper[0] = inf;
fixed_lower[1] = .01; fixed_in[1] = 0.5; fixed_upper[1] = inf;
// Set upper and lower limit for theta[2] equal so that it is bounded
// (otherwise the implicit information matrix would be singular).
fixed_lower[2] = 1.0; fixed_in[2] = 1.0; fixed_upper[2] = 1.0;
//
// explicit constriants (in addition to l1 terms)
d_vector fix_constraint_lower(0), fix_constraint_upper(0);
//
d_vector data(n_data), random_in(n_random);
for(size_t i = 0; i < n_data; i++)
{ data[i] = double(i + 1);
random_in[i] = 0.0;
}
// object that is derived from cppad_mixed
bool quasi_fixed = true;
bool bool_sparsity = true;
mixed_derived mixed_object(
n_fixed, n_random, quasi_fixed, bool_sparsity, data
);
mixed_object.initialize(fixed_in, random_in);
// optimize the fixed effects using quasi-Newton method
std::string fixed_ipopt_options =
"Integer print_level 0\n"
"String sb yes\n"
"String derivative_test adaptive\n"
"String derivative_test_print_all yes\n"
"Numeric tol 1e-8\n"
"Integer max_iter 15\n"
;
std::string random_ipopt_options =
"Integer print_level 0\n"
"String sb yes\n"
"String derivative_test second-order\n"
"Numeric tol 1e-8\n"
;
d_vector random_lower(n_random), random_upper(n_random);
for(size_t i = 0; i < n_random; i++)
{ random_lower[i] = -inf;
random_upper[i] = +inf;
}
// optimize fixed effects
d_vector fixed_scale = fixed_in;
CppAD::mixed::fixed_solution solution = mixed_object.optimize_fixed(
fixed_ipopt_options,
random_ipopt_options,
fixed_lower,
fixed_upper,
fix_constraint_lower,
fix_constraint_upper,
fixed_scale,
fixed_in,
random_lower,
random_upper,
random_in
);
//
// check that none of the constraints are active
// (Note that the Lagragian w.r.t. theta[2] will be zero because
// it does not affect the objective).
ok &= solution.fixed_lag.size() == n_fixed;
for(size_t i = 0; i < n_fixed; i++)
ok &= solution.fixed_lag[i] == 0.0;
ok &= solution.fix_con_lag.size() == 0;
ok &= solution.ran_con_lag.size() == 0;
//
// corresponding optimal random effects
d_vector random_opt = mixed_object.optimize_random(
random_ipopt_options,
solution.fixed_opt,
random_lower,
random_upper,
random_in
);
//
// compute corresponding information matrix
d_vector& fixed_opt = solution.fixed_opt;
d_sparse_rcv hes_fixed_obj_rcv =
mixed_object.hes_fixed_obj(fixed_opt, random_opt);
//
// sample from the posterior for fixed effects
size_t n_sample = 20000;
d_vector sample( n_sample * n_fixed );
std::string error_msg = mixed_object.sample_fixed(
sample,
hes_fixed_obj_rcv,
solution,
fixed_lower,
fixed_upper
);
ok &= error_msg == "";
//
typedef Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > matrix;
//
// compute sample covariance matrix
matrix sample_cov = matrix::Zero(n_fixed, n_fixed);
for(size_t i = 0; i < n_sample; i++)
{ matrix diff(n_fixed, 1);
for(size_t j = 0; j < n_fixed; j++)
diff(j, 0) = sample[ i * n_fixed + j] - solution.fixed_opt[j];
sample_cov += diff * diff.transpose();
}
sample_cov *= 1.0 / double(n_sample);
//
matrix info_mat(n_fixed-1, n_fixed-1);
// note theta[2] does not have any non-zero terms in Hessian
size_t K = ( (n_fixed-1) * n_fixed ) / 2;
ok &= K == hes_fixed_obj_rcv.nnz();
for(size_t k = 0; k < K; k++)
{ size_t i = hes_fixed_obj_rcv.row()[k];
size_t j = hes_fixed_obj_rcv.col()[k];
info_mat(i, j) = hes_fixed_obj_rcv.val()[k];
info_mat(j, i) = hes_fixed_obj_rcv.val()[k];
}
matrix cov_mat = info_mat.inverse();
//
for(size_t i = 0; i < n_fixed; i++)
{ for(size_t j = 0; j < n_fixed; j++)
{ double value = sample_cov(i, j);
if( i == 2 || j == 2 )
ok &= std::fabs(value) < eps;
else
{ double check = cov_mat(i, j);
double scale = std::sqrt( cov_mat(i, i) * cov_mat(j, j) );
ok &= std::fabs(value - check) / scale < .05;
}
}
}
//
if( ! ok )
std::cout << "\nrandom_seed = " << random_seed << "\n";
//
CppAD::mixed::free_gsl_rng();
return ok;
}
