Hidden linear regression

#include <Eigen/Dense>

/* @brief Solve the linear regression problem (fitting a line to data)
 * for data points $(t_i,y_i)$, $i = 1,\dots,n$
 * @param[in] t An $n$-dimensional vector containing one side of input data
 * @param[in] y An $n$-dimensional vector containing the other side of input
 * data
 * @param[out] x The vector of parameters $(x_1,x_2)$, intercept and slope of
 * the line fitted
 */
/* SAM_LISTING_BEGIN_0 */
Eigen::VectorXd linReg(const Eigen::VectorXd &t, const Eigen::VectorXd &y) {
  // TODO: (3-1.d) Use the method of normal equations to solve the 1D linear
  // regression problem in least-square sense.
  // Note: The tests anticipate the outputs in the order (alpha, beta), and not
  // (beta, alpha).
  
  // START
  
  //get size of t 
  int n = t.size();
  
  //declare and initialize A
  Eigen::MatrixXd A = Eigen::MatrixXd::Zero(n,2);
  for (int i=0; i<n;i++){
    //first column of A is all 1
    A(i,0) = 1;
    //second column of A is ti
    A(i,1) = t(i);
  }
  
  //declare and initialize b
  Eigen::VectorXd b = y;
  
  //declare and initialize ATA
  Eigen::MatrixXd ATA = A.transpose() * A;
  
   //declare and initialize ATb
  Eigen::VectorXd ATb = A.transpose() * b;
  
  //declare and initialize x
  Eigen::VectorXd x = ATA.lu().solve(ATb);
  
  // END

  return x;
}
/* SAM_LISTING_END_0 */

/* @brief Solve the linearized exponential problem
 * for data points $(t_i,y_i)$, $i = 1,\dots,n$
 * @param[in] t An $n$-dimensional vector containing one side of input data
 * @param[in] y An $n$-dimensional vector containing the other side of input
 * data
 * @param[out] x The vector of parameters $(x_1,x_2)$, intercept and slope of
 * the line fitted to the linearized problem
 */
/* SAM_LISTING_BEGIN_1 */
Eigen::VectorXd expFit(const Eigen::VectorXd &t, const Eigen::VectorXd &y) {
  // TODO: (3-1.e) Implement least square estimate of alpha and beta using
  // the previously implemented function linReg().
  // Note: You don't need to have implemented linReg() to solve this subproblem.
  // The tests will compile and use the master solution for linReg().
  // Note: The tests anticipate the outputs in the order (alpha, beta), and not
  // (beta, alpha).
  
  // START
  
  Eigen::VectorXd b = y.array().log().matrix();
  //compute least squares
  Eigen::VectorXd x_temp = linReg(t,b);
  
  //return x with exp of first value (alpha)
  Eigen::VectorXd x_temp_exp = x_temp.array().exp().matrix();
  
  // Dummy code. We need this otherwise the code won't compile. 
  Eigen::Vector2d x;
  //first element is alpha from exp vector to eliminate the ln
  x(0) = x_temp_exp(0);
  //second element is beta from original vector
  x(1) = x_temp(1);

  // END

  return x;
}
/* SAM_LISTING_END_1 */

// END