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

lasso.cpp

View page source

Lasso on Fixed Effects: Example and Test

Model

We are given a set of times \(\{ t_i \W{:} i = 0 , \ldots , N-1 \}\) and

\begin{eqnarray} q( \theta, s ) & = & \theta_0 (s / N) + \theta_1 \sin ( 2 \pi s ) + \theta_2 \cos ( 2 \pi s ) \\ z_i & = & q( \theta , t_i ) + e_i \\ \B{p} ( e_i | \theta ) & \sim & \B{N} ( 0, \sigma ) \end{eqnarray}

The idea in Lasso is that one or more of the components of \(\theta\) are zero and using the Laplace prior we can recover this fact. We use \(\B{L} ( \mu , \sigma )\) to denote the Laplace distribution with mean \(\mu\) and standard deviation \(\sigma\).

\[\B{p} ( \theta ) \sim \B{L} ( 0 , \delta )\]

The corresponding fixed likelihood g(theta) is

\[g( \theta ) = \sum_{i=0}^{N-1} \left[ \log ( \sigma \sqrt{2 \pi} ) + \left( \frac{ z_i - q( \theta , t_i ) }{2 \sigma} \right)^2 \right] + \sum_{j=0}^2 \left[ \log \left( \delta \sqrt{2} \right) + \sqrt{2} \; \left| \frac{\theta_j}{\delta} \right| \right]\]

# include <cppad/cppad.hpp>
# include <cppad/mixed/cppad_mixed.hpp>
# include <cppad/mixed/manage_gsl_rng.hpp>
# include <gsl/gsl_randist.h>

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:
      size_t                n_fixed_;
      double                sigma_;
      double                delta_;
      const d_vector&       t_;
      const d_vector&       z_;
   public:
      // constructor
      mixed_derived(
         size_t                 n_fixed        ,
         size_t                 n_random       ,
         double                 sigma          ,
         double                 delta          ,
         const d_vector&        t              ,
         const d_vector&        z              ) :
         cppad_mixed(n_fixed, n_random)  ,
         n_fixed_(n_fixed)               ,
         sigma_(sigma)                   ,
         delta_(delta)                   ,
         t_(t)                           ,
         z_(z)
      {  assert(n_fixed == 3);
         assert( t.size() == z.size() );
      }
      // implementation of fix_likelihood as p(z|theta) * p(theta)
      virtual a1_vector fix_likelihood(
         const a1_vector&         fixed_vec  )
      {  size_t N = t_.size();

         // initialize log-density
         a1_vector vec(1 + n_fixed_);
         vec[0] = a1_double(0.0);

         // compute this factors once
         a1_double   pi     = a1_double( 4.0 * CppAD::atan(1.0) );
         a1_double sqrt_2   = a1_double( CppAD::sqrt( 2.0 ) );
         // a1_double sqrt_2pi = CppAD::sqrt( 2.0 * pi );

         // Data terms p(z|theta)
         for(size_t i = 0; i < N; i++)
         {  a1_double q_i   = 0.0;
            q_i += fixed_vec[0] * t_[i] / double(N);
             q_i += fixed_vec[1] * sin( 2.0 * pi * t_[i] );
             q_i += fixed_vec[2] * cos( 2.0 * pi * t_[i] );
            a1_double res  = ( z_[i] - q_i ) / sigma_;
            vec[0] += res * res / 2.0;
            // following term does not depend on the fixed effects
            // vec[0] += log(sigma_ * sqrt_2pi);
         }

         // Prior terms p(theta)
         for(size_t j = 0; j < n_fixed_; j++)
         {  // following term does not depend on the fixed effects
            // vec[0] += log( delta_ * sqrt_2 );
            vec[1 + j] = sqrt_2 * fixed_vec[j] / delta_;
         }
         return vec;
      }
   };
}

bool lasso_xam(void)
{
   bool   ok         = true;
   double inf         = std::numeric_limits<double>::infinity();
   // size_t random_seed = CppAD::mixed::new_gsl_rng(0);
   CppAD::mixed::new_gsl_rng(0);
   gsl_rng* rng       = CppAD::mixed::get_gsl_rng();

   // fixed effects
   size_t n_fixed  = 3;
   d_vector
      fixed_lower(n_fixed), fixed_in(n_fixed), fixed_upper(n_fixed);
   for(size_t j = 0; j < n_fixed; j++)
   {  fixed_lower[j] = - inf;
      fixed_in[j]    = 0.0;
      fixed_upper[j] = inf;
   }
   //
   // no random effects
   size_t n_random = 0;
   d_vector random_in(0);
   d_vector random_lower(n_random), random_upper(n_random);
   std::string random_ipopt_options = "";
   //
   // no constraints
   d_vector fix_constraint_lower(0), fix_constraint_upper(0);
   //
   size_t n_data = 50;
   double sigma  = 0.1;
   double pi     = 4.0 * std::atan(1.0);
   d_vector z(n_data), t(n_data);
   for(size_t i = 0; i < n_data; i++)
   {  t[i] = double(i) / double(n_data - 1) - 0.5;
      //
      // simulation theta_0 = 0, theta_1 = 1, theta_2 = 0
      double q_i = 0.0 * t[i] / double(n_data);
      q_i       += 1.0 * sin(2.0 * pi *t[i]);
      q_i       += 0.0 * cos(2.0 * pi *t[i]);
      double e_i = gsl_ran_gaussian(rng, sigma);
      z[i]       = q_i + e_i;
   }

   // object that is derived from cppad_mixed
   double delta     = 0.002;
   mixed_derived mixed_object(n_fixed, n_random, sigma, delta, t, z);
   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"
   ;
   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;
   //
   // coefficients that should be zero
   ok &= fabs( fixed_out[0] ) <= 5e-8;
   ok &= fabs( fixed_out[2] ) <= 5e-8;
   //
   // non-zero coefficient has shrunk (due to prior)
   ok &= fixed_out[1] < 1.0;
   ok &= 0.5 < fixed_out[1];
   //
   if( ! ok )
      std::cout << "\nfixed_out = " << fixed_out << "\n";
   //
   CppAD::mixed::free_gsl_rng();
   return ok;
}