\(\newcommand{\B}[1]{ {\bf #1} }\) \(\newcommand{\R}[1]{ {\rm #1} }\) \(\newcommand{\W}[1]{ \; #1 \; }\)

data_mismatch.cpp

View page source

Random Effects Variance May Cause Data Mismatch

Model

\[\B{p}( z | \theta ) \sim \B{N} ( \theta , \sigma_z^2 )\]

\[\B{p}( y | \theta ) \sim \B{N} [ \exp( u ) \theta , \sigma_y^2 ]\]

\[\B{p}( u | \theta ) \sim \B{N} ( 0 , \sigma_u^2 )\]

The fixed likelihood g(theta) is

\[g( \theta ) = \frac{1}{2} \left[ \log ( 2 \pi \sigma_z^2 ) + ( z - \theta )^2 \right]\]

The random likelihood f(theta, u) is

\[f(\theta , u ) = \frac{1}{2} \left[ \log ( 2 \pi \sigma_u^2 ) + u^2 / \sigma_u^2 + \log ( 2 \pi \sigma_y^2 ) + [ y - \exp(u) \theta ]^2 / \sigma_y^2 \right]\]

Mismatch

In the case where \(y = z\), one might expect the solution \(\theta = z\) and \(u = 0\) because all the data residuals are zero, \(\B{p}(u | \theta )\) is maximal, and there is no prior for \(\theta\). This example demonstrates that \(\theta = z\) and \(u = 0\) may not be optimal for the this case. To be specific it shows that the derivative of L(theta) may be non-zero.

Theory

See the Laplace Approximation for Mixed Effects Models section for the theory behind the calculations below:

Derivatives

\begin{eqnarray} g_\theta ( \theta ) & = & ( \theta - z ) \sigma_z^{-2} \\ f_\theta ( \theta , u ) & = & [ \exp(u) \theta - y ] \exp(u) \sigma_y^{-2} \\ f_u ( \theta , u ) & = & u / \sigma_u^2 + [ \exp(u) \theta - y ] \exp(u) \theta \sigma_y^{-2} \\ f_{u,u} ( \theta , u ) & = & \sigma_u^{-2} + [ 2 \exp(u) \theta - y ] \exp(u) \theta \sigma_y^{-2} \\ f_{u,\theta} ( \theta , u ) & = & [ 2 \exp(u) \theta - y ] \exp(u) \sigma_y^{-2} \\ \hat{u}_\theta ( \theta ) & = & - f_{u,\theta} [ \theta , \hat{u} ( \theta ) ] / f_{u,u} [ \theta , \hat{u} ( \theta ) ] \\ f_{u,u,u} ( \theta , u ) & = & [ 4 \exp(u) \theta - y ] \exp(u) \theta \sigma_y^{-2} \\ f_{u,u,\theta} ( \theta , u ) & = & [ 4 \exp(u) \theta - y ] \exp(u) \sigma_y^{-2} \end{eqnarray}

Objective

\begin{eqnarray} h( \theta , u ) & = & \frac{1}{2} \log f_{u,u} ( \theta , u ) + f( \theta , u ) - \log( 2 \pi ) \\ h_\theta ( \theta , u ) & = & \frac{1}{2} f_{u,u,\theta} ( \theta , u ) / f_{u,u} ( \theta , u ) + f_\theta ( \theta , u ) \\ h_u ( \theta , u ) & = & \frac{1}{2} f_{u,u,u} ( \theta , u ) / f_{u,u} ( \theta , u ) + f_u ( \theta , u ) \\ L( \theta ) & = & h [ \theta , \hat{u} ( \theta ) ] + g ( \theta ) \\ L_\theta ( \theta ) & = & h_\theta [ \theta , \hat{u} ( \theta ) ] + h_u [ \theta , \hat{u} ( \theta ) ] \hat{u}_\theta ( \theta ) + g_\theta ( \theta ) \end{eqnarray}

# include <cppad/cppad.hpp>
# include <cppad/mixed/cppad_mixed.hpp>

namespace {
    using CppAD::log;
    using CppAD::exp;
    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 double y_, z_;
        const double sigma_u_, sigma_y_, sigma_z_;
    public:
        // constructor
        mixed_derived(
            size_t                 n_fixed       ,
            size_t                 n_random      ,
            double                 y             ,
            double                 z             ,
            double                 sigma_u       ,
            double                 sigma_y       ,
            double                 sigma_z       ) :
            cppad_mixed(n_fixed, n_random) ,
            y_(y)                          ,
            z_(z)                          ,
            sigma_u_(sigma_u)              ,
            sigma_y_(sigma_y)              ,
            sigma_z_(sigma_z )
        {   assert( n_fixed == 1 );
            assert( n_random == 1 );
        }
        // implementation of fix_likelihood as p(z|theta) * p(theta)
        a1_vector fix_likelihood(
            const a1_vector&         fixed_vec  ) override
        {
            a1_double theta = fixed_vec[0];

            // initialize log-density
            a1_vector vec(1);
            vec[0] = 0.0;

            // compute this factor once
            // sqrt_2pi = CppAD::sqrt( 8.0 * CppAD::atan(1.0) );

            // Data term p(z|theta)
            a1_double res  = (z_ - theta) / sigma_z_;
            vec[0]    += res * res / 2.0;
            // the following term does not depend on fixed effects
            // vec[0]    += log(sqrt_2pi * sigma_z_ );

            // prior term p(theta)

            return vec;
        }
        // ------------------------------------------------------------------
        // implementation of ran_likelihood as p(y|theta, u) * p(u|theta)
        a1_vector ran_likelihood(
            const a1_vector&         fixed_vec  ,
            const a1_vector&         random_vec ) override
        {
            a1_double theta = fixed_vec[0];
            a1_double u     = random_vec[0];

            // initialize log-density
            a1_vector vec(1);
            vec[0] = 0.0;

            // compute this factors once
            // sqrt_2pi = CppAD::sqrt( 8.0 * CppAD::atan(1.0) );

            // Data term p(y|theta,u)
            a1_double res  = (y_ - exp(u) * theta) / sigma_y_;
            vec[0]    += res * res / 2.0;
            // the following term does not depend on fixed or random effects
            // vec[0]    += log(sqrt_2pi * sigma_y_ );

            // prior term p(u|theta)
            res        = u / sigma_u_;
            vec[0]    += res * res / 2.0;
            // the following term does not depend on fixed or random effects
            // vec[0]    += log(sqrt_2pi * sigma_u_ );

            return vec;
        }
        // ==================================================================
        // Routines used to check that objective derivative is zero at solution
        // g(thata)    = (theta - z)^2           / (2.0 * sigma_z * sigma_z)
        // f(theta, u) = ( exp(u) * theta - y)^2 / (2.0 * sigma_y * sigma_y )
        //             + u^2  / (2.0 * sigma_u * sigma_u);
        double g_theta(double theta)
        {   return (theta - z_) / (sigma_z_ * sigma_z_); }
        //
        double f_theta(double theta, double u)
        {   double num_theta    = 2.0 * (exp(u) * theta - y_) * exp(u);
            double ratio_theta  = num_theta / (2.0 * sigma_y_ * sigma_y_);
            return ratio_theta;
        }
        double f_u(double theta, double u)
        {   double num_u    = 2.0 * (exp(u) * theta - y_) * exp(u) * theta;
            double ratio_u  = num_u / (2.0 * sigma_y_ * sigma_y_);
            return ratio_u + u / (sigma_u_ * sigma_u_);
        }
        double f_uu(double theta, double u)
        {   double term     = exp(u) * theta;
            double num_uu   = 2.0 * (term * term + (term - y_) * term);
            double ratio_uu = num_uu / (2.0 * sigma_y_ * sigma_y_);
            return ratio_uu + 1.0 / (sigma_u_ * sigma_u_);
        }
        double f_utheta(double theta, double u)
        {   double num_utheta    = 2.0 * (exp(u) * theta - y_) * exp(u);
            num_utheta          += 2.0 * exp(u) * exp(u) * theta;
            double ratio_utheta  = num_utheta / (2.0 * sigma_y_ * sigma_y_);
            return ratio_utheta;
        }
        double uhat_theta(double theta, double uhat)
        {   double ret = - f_utheta(theta, uhat) / f_uu(theta, uhat);
            return ret;
        }
        double f_uuu(double theta, double u)
        {   double term = exp(u) * theta;
            // term_u = term
            // num_uu = 2.0 * term * (2.0 * term  - y)
            //        = 4.0 * term^2 - 2.0 * term * y
            double num_uuu  = 8.0 * term * term  - 2.0 * term * y_;
            double ratio_uuu = num_uuu / (2.0 * sigma_y_ * sigma_y_);
            return ratio_uuu;
        }
        double f_uutheta(double theta, double u)
        {   double term = exp(u) * theta;
            // term_theta = exp(u)
            // num_uu = 2.0 * term * (2.0 * term  - y)
            //        = 4.0 * term^2 - 2.0 * term * y
            double num_uutheta = 8.0 * term * exp(u) - 2.0 * y_ * exp(u);
            double ratio_uutheta = num_uutheta / (2.0 * sigma_y_ * sigma_y_);
            return ratio_uutheta;
        }
        double h_theta(double theta, double u)
        {   double ret = 0.5 * f_uutheta(theta, u) / f_uu(theta, u);
            ret       += f_theta(theta, u);
            return ret;
        }
        double h_u(double theta, double u)
        {   double ret = 0.5 * f_uuu(theta, u) / f_uu(theta, u);
            ret       += f_u(theta, u);
            return ret;
        }
        double L_theta(double theta , double uhat)
        {   double ret = h_theta(theta, uhat);
            ret       += h_u(theta, uhat) * uhat_theta(theta, uhat);
            ret       += g_theta(theta);
            return ret;
        }
    };

}

bool data_mismatch_xam(void)
{
    bool   ok = true;
    double inf = std::numeric_limits<double>::infinity();

    size_t n_fixed  = 1;
    size_t n_random = 1;
    double z        = 0.05;
    double y        = z;
    double sigma_u  = 0.1;
    double sigma_y  = 0.1 * y;
    double sigma_z  = 0.1 * z;
    d_vector fixed_in(n_fixed), random_in(n_random);
    d_vector fixed_lower(n_fixed), fixed_upper(n_fixed);
    fixed_lower[0] = -inf;
    fixed_in[0]    = z;
    fixed_upper[0] = + inf;
    random_in[0]   = 0.0;
    //
    // no constraints
    d_vector fix_constraint_lower(0), fix_constraint_upper(0);
    //
    //
    // object that is derived from cppad_mixed
    mixed_derived mixed_object(
        n_fixed, n_random,
        y, z, sigma_u, sigma_y, sigma_z

    );
    mixed_object.initialize(fixed_in, random_in);
    //
    // compute the derivative of the objective at the starting point
    double theta_in   = fixed_in[0];
    double u_in       = random_in[0];
    double L_theta_in = mixed_object.L_theta(theta_in, u_in);
    //
    // derivative corresponding to theta = z = y is greater than 1
    ok &= fabs( L_theta_in ) > 1.0;
    //
    // 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"
        "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;
    }
    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
    );
    d_vector fixed_out = solution.fixed_opt;
    //
    d_vector random_out = mixed_object.optimize_random(
        random_ipopt_options, fixed_out, random_lower, random_upper, random_in
    );
    //
    // compute the derivative of the objective at the final point
    double theta_out   = fixed_out[0];
    double u_out       = random_out[0];
    double L_theta_out = mixed_object.L_theta(theta_out, u_out);
    //
    // derivative corresponding to solution is less that 1e-7
    ok &= fabs( L_theta_out ) <= 1e-7;
    //
    // Now demonstrate that the solution is still close to the expected values
    ok &= fabs( theta_out / z - 1.0 ) <= 1e-2;
    ok &= fabs( u_out ) <= 1e-2;
    //
    return ok;
}