\(\newcommand{\B}[1]{ {\bf #1} }\) \(\newcommand{\R}[1]{ {\rm #1} }\) \(\newcommand{\W}[1]{ \; #1 \; }\)
optimize_fixed.cpp¶
View page sourceOptimize Fixed Effects: Example and Test¶
Model¶
\[\B{p}( y_i | \theta , u ) \sim \B{N} ( u_i + \theta_0 , \theta_1^2 )\]
\[\B{p}( u_i | \theta ) \sim \B{N} ( 0 , 1 )\]
\[\B{p}( \theta ) \sim \B{N} ( 4 , 1 )\]
It follows that the Laplace approximation is exact and
\[\B{p}( y_i | \theta ) \sim \B{N} \left( \theta_0 , 1 + \theta_1^2 \right)\]
The constraints on the fixed effect are
\[- \infty \leq \theta_0 \leq + \infty
\R{\; and \;}
0.1 \leq \theta_1 \leq 100\]
Objective¶
The corresponding objective for the fixed effects is equivalent to:
\[F( \theta ) = \frac{1}{2} \left[
( \theta_0 - 4 )^2 + ( \theta_1 - 4 )^2 +
N \log \left( 1 + \theta_1^2 \right) +
( 1 + \theta_1^2)^{-1} \sum_{i=0}^{N-1} ( y_i - \theta_0 )^2
\right]\]
First Order Partials¶
The first order partial derivatives of the objective are:
\[F_0 ( \theta )
=
( \theta_0 - 4 )
-
( 1 + \theta_1^2)^{-1} \sum_{i=0}^{N-1} ( y_i - \theta_0 )\]
\[F_1 ( \theta )
=
( \theta_1 - 4 )
+
N \left( 1 + \theta_1^2 \right)^{-1} \theta_1
-
( 1 + \theta_1^2)^{-2} \theta_1 \sum_{i=0}^{N-1} ( y_i - \theta_0 )^2\]
Optimizer Trace¶
This example uses the optimizer trace information; see trace_vec .
Optimizer Warm Start¶
This example uses the optimizer warm start information; see warm_start .
Source Code¶
# include <cppad/cppad.hpp>
# include <cppad/mixed/cppad_mixed.hpp>
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 == 2);
assert( y_.size() == n_random_ );
}
// ----------------------------------------------------------------------
// implementation of ran_likelihood
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) );
for(size_t j = 0; j < n_fixed_; 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;
}
};
// derivative of objective
d_vector objective_fixed(
const d_vector& data ,
const d_vector& theta )
{ d_vector dF(2);
//
// compute partials of F
double sum = 0.0;
double sumsq = 0.0;
for(size_t i = 0; i < data.size(); i++)
{ sum += theta[0] - data[i];
sumsq += (theta[0] - data[i]) * (theta[0] - data[i]);
}
double den = 1.0 + theta[1] * theta[1];
dF[0] = (theta[0] - 4.0) + sum / den;
dF[1] = theta[1] - 4.0;
dF[1] += double(data.size()) * theta[1] / den;
dF[1] -= sumsq * theta[1] / (den * den);
//
return dF;
}
}
bool optimize_fixed_xam(void)
{
bool ok = true;
double inf = std::numeric_limits<double>::infinity();
double tol = 1e-8;
size_t n_data = 10;
size_t n_fixed = 2;
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;
//
// explicit constraints (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"
"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 with tolerance 1e-3
std::string temp_string = fixed_ipopt_options + "Numeric tol 1e-3\n";
d_vector fixed_scale = fixed_in;
CppAD::mixed::fixed_solution solution = mixed_object.optimize_fixed(
temp_string,
random_ipopt_options,
fixed_lower,
fixed_upper,
fix_constraint_lower,
fix_constraint_upper,
fixed_scale,
fixed_in,
random_lower,
random_upper,
random_in
);
d_vector fixed_out = solution.fixed_opt;
// ------------------------------------------------------------------
// continue optimization, from previous, with new tolerance of 1e-8
temp_string = fixed_ipopt_options + "Numeric tol 1e-8\n";
fixed_in = fixed_out;
solution = mixed_object.optimize_fixed(
temp_string,
random_ipopt_options,
fixed_lower,
fixed_upper,
fix_constraint_lower,
fix_constraint_upper,
fixed_scale,
fixed_in,
random_lower,
random_upper,
random_in,
solution.warm_start
);
fixed_out = solution.fixed_opt;
// ------------------------------------------------------------------
// deriative of objective at fixed_in and fixed_out
d_vector dF_scale = objective_fixed(data, fixed_scale);
d_vector dF_out = objective_fixed(data, fixed_out);
// scaling for objective
double scale = std::max(
std::fabs( dF_scale[0] ), std::fabs( dF_scale[1] )
);
scale = 1.0 / scale;
// Note that no constraints are active, (not even the l1 terms)
// so the partials should be zero.
ok &= fabs( scale * dF_out[0] ) <= 5. * tol;
ok &= fabs( scale * dF_out[1] ) <= 5. * tol;
return ok;
}