prim/fun/gp_exponential_cov.hpp

#ifndef STAN_MATH_PRIM_FUN_GP_EXPONENTIAL_COV_HPP
#define STAN_MATH_PRIM_FUN_GP_EXPONENTIAL_COV_HPP

#include <stan/math/prim/meta.hpp>
#include <stan/math/prim/err.hpp>
#include <stan/math/prim/fun/distance.hpp>
#include <stan/math/prim/fun/divide_columns.hpp>
#include <stan/math/prim/fun/Eigen.hpp>
#include <stan/math/prim/fun/exp.hpp>
#include <stan/math/prim/fun/size.hpp>
#include <stan/math/prim/fun/square.hpp>
#include <cmath>
#include <vector>

namespace stan {
namespace math {

/**
 * Returns a Matern exponential covariance Matrix.
 *
 * \f[ k(x, x') = \sigma^2 exp(-\frac{d(x, x')^2}{2l^2}) \f]
 *
 * where d(x, x') is the Euclidean distance.
 *
 * @tparam T_x type for each scalar
 * @tparam T_s type of parameter sigma
 * @tparam T_l type of parameter length scale
 *
 * @param x std::vector of scalars that can be used in stan::math::distance
 * @param sigma standard deviation or magnitude
 * @param length_scale length scale
 * @throw std::domain error if sigma <= 0, l <= 0, or x is nan or inf
 *
 */
template <typename T_x, typename T_s, typename T_l>
inline typename Eigen::Matrix<return_type_t<T_x, T_s, T_l>, Eigen::Dynamic,
                              Eigen::Dynamic>
gp_exponential_cov(const std::vector<T_x> &x, const T_s &sigma,
                   const T_l &length_scale) {
  using std::exp;
  using std::pow;

  size_t x_size = stan::math::size(x);
  Eigen::Matrix<return_type_t<T_x, T_s, T_l>, Eigen::Dynamic, Eigen::Dynamic>
      cov(x_size, x_size);
  if (x_size == 0) {
    return cov;
  }

  const char *function = "gp_exponential_cov";
  size_t x_obs_size = stan::math::size(x[0]);
  for (size_t i = 0; i < x_size; ++i) {
    check_size_match(function, "x row", x_obs_size, "x's other row",
                     stan::math::size(x[i]));
  }

  for (size_t i = 0; i < x_size; ++i) {
    check_not_nan(function, "x", x[i]);
  }

  check_positive_finite(function, "magnitude", sigma);
  check_positive_finite(function, "length scale", length_scale);

  T_s sigma_sq = square(sigma);
  T_l neg_inv_l = -1.0 / length_scale;

  size_t block_size = 10;
  for (size_t jb = 0; jb < x_size; jb += block_size) {
    for (size_t ib = jb; ib < x_size; ib += block_size) {
      size_t j_end = std::min(x_size, jb + block_size);
      for (size_t j = jb; j < j_end; ++j) {
        cov(j, j) = sigma_sq;
        size_t i_end = std::min(x_size, ib + block_size);
        for (size_t i = std::max(ib, j + 1); i < i_end; ++i) {
          cov.coeffRef(j, i) = cov.coeffRef(i, j)
              = sigma_sq * exp(neg_inv_l * distance(x[i], x[j]));
        }
      }
    }
  }
  return cov;
}

/**
 * Returns a Matern exponential covariance matrix.
 *
 * \f[ k(x, x') = \sigma^2 exp(-\sum_{k=1}^K\frac{d(x, x')}{l_k}) \f]
 *
 * where d(x, x') is the Euclidean distance.
 *
 * @tparam T_x type for each scalar
 * @tparam T_s type for each parameter sigma
 * @tparam T_l type for each length scale parameter
 *
 * @param x std::vector of Eigen::vectors of scalars
 * @param sigma standard deviation that can be used in stan::math::square
 * @param length_scale std::vector of length scales
 * @throw std::domain error if sigma <= 0, l <= 0, or x is nan or inf
 */
template <typename T_x, typename T_s, typename T_l>
inline typename Eigen::Matrix<return_type_t<T_x, T_s, T_l>, Eigen::Dynamic,
                              Eigen::Dynamic>
gp_exponential_cov(const std::vector<Eigen::Matrix<T_x, -1, 1>> &x,
                   const T_s &sigma, const std::vector<T_l> &length_scale) {
  using std::exp;
  using std::pow;

  size_t x_size = stan::math::size(x);
  Eigen::Matrix<return_type_t<T_x, T_s, T_l>, Eigen::Dynamic, Eigen::Dynamic>
      cov(x_size, x_size);
  if (x_size == 0) {
    return cov;
  }

  const char *function = "gp_exponential_cov";
  for (size_t n = 0; n < x_size; ++n) {
    check_not_nan(function, "x", x[n]);
  }

  check_positive_finite(function, "magnitude", sigma);
  check_positive_finite(function, "length scale", length_scale);

  size_t l_size = length_scale.size();
  check_size_match(function, "x dimension", x[0].size(),
                   "number of length scales", l_size);

  std::vector<Eigen::Matrix<return_type_t<T_x, T_l>, -1, 1>> x_new
      = divide_columns(x, length_scale);

  T_s sigma_sq = square(sigma);
  size_t block_size = 10;
  for (size_t jb = 0; jb < x_size; jb += block_size) {
    for (size_t ib = jb; ib < x_size; ib += block_size) {
      size_t j_end = std::min(x_size, jb + block_size);
      for (size_t j = jb; j < j_end; ++j) {
        cov(j, j) = sigma_sq;
        size_t i_end = std::min(x_size, ib + block_size);
        for (size_t i = std::max(ib, j + 1); i < i_end; ++i) {
          return_type_t<T_x, T_l> dist = distance(x_new[i], x_new[j]);
          cov.coeffRef(j, i) = cov.coeffRef(i, j) = sigma_sq * exp(-dist);
        }
      }
    }
  }
  return cov;
}

/**
 * Returns a Matern exponential cross covariance matrix
 *
 * \f[ k(x, x') = \sigma^2 exp(-\frac{d(x, x')}{l}) \f]
 *
 * where d(x, x') is the Euclidean distance
 * This function is for the cross covariance matrix needed to
 * compute the posterior predictive distribution
 *
 * @tparam T_x1 first type of scalars contained in vector x1
 * @tparam T_x2 second type of scalars contained in vector x2
 * @tparam T_s type of parameter sigma, marginal standard deviation
 * @tparam T_l type of parameter length scale
 *
 * @param x1 std::vector of scalars that can be used in squared_distance
 * @param x2 std::vector of scalars that can be used in squared_distance
 * @param length_scale length scale
 * @param sigma standard deviation that can be used in stan::math::square
 * @throw std::domain error if sigma <= 0, l <= 0, or x1, x2 are nan or inf
 */
template <typename T_x1, typename T_x2, typename T_s, typename T_l>
inline typename Eigen::Matrix<return_type_t<T_x1, T_x2, T_s, T_l>,
                              Eigen::Dynamic, Eigen::Dynamic>
gp_exponential_cov(const std::vector<T_x1> &x1, const std::vector<T_x2> &x2,
                   const T_s &sigma, const T_l &length_scale) {
  using std::exp;
  using std::pow;

  size_t x1_size = stan::math::size(x1);
  size_t x2_size = stan::math::size(x2);
  Eigen::Matrix<return_type_t<T_x1, T_x2, T_s, T_l>, Eigen::Dynamic,
                Eigen::Dynamic>
      cov(x1_size, x2_size);
  if (x1_size == 0 || x2_size == 0) {
    return cov;
  }

  const char *function = "gp_exponential_cov";
  size_t x1_obs_size = stan::math::size(x1[0]);
  for (size_t i = 0; i < x1_size; ++i) {
    check_size_match(function, "x1's row", x1_obs_size, "x1's other row",
                     stan::math::size(x1[i]));
  }
  for (size_t i = 0; i < x2_size; ++i) {
    check_size_match(function, "x1's row", x1_obs_size, "x2's other row",
                     stan::math::size(x2[i]));
  }

  for (size_t n = 0; n < x1_size; ++n) {
    check_not_nan(function, "x1", x1[n]);
  }
  for (size_t n = 0; n < x2_size; ++n) {
    check_not_nan(function, "x2", x2[n]);
  }

  check_positive_finite(function, "magnitude", sigma);
  check_positive_finite(function, "length scale", length_scale);

  T_s sigma_sq = square(sigma);
  T_l neg_inv_l = -1.0 / length_scale;
  size_t block_size = 10;

  for (size_t ib = 0; ib < x1_size; ib += block_size) {
    for (size_t jb = 0; jb < x2_size; jb += block_size) {
      size_t j_end = std::min(x2_size, jb + block_size);
      for (size_t j = jb; j < j_end; ++j) {
        size_t i_end = std::min(x1_size, ib + block_size);
        for (size_t i = ib; i < i_end; ++i) {
          cov(i, j) = sigma_sq * exp(neg_inv_l * distance(x1[i], x2[j]));
        }
      }
    }
  }
  return cov;
}

/**
 * Returns a Matern exponential cross covariance matrix
 *
 * \f[ k(x, x') = \sigma^2 exp(-\sum_{k=1}^K\frac{d(x, x')}{l_k}) \f]
 *
 * where \f$d(x, x')\f$ is the Euclidean distance
 *
 * This function is for the cross covariance matrix needed
 * to compute the posterior predictive density.
 *
 * @tparam T_x1 first type of std::vector of scalars
 * @tparam T_x2 second type of std::vector of scalars
 * @tparam T_s type of parameter sigma, marginal standard deviation
 * @tparam T_l type of parameter length scale
 *
 * @param x1 std::vector of Eigen vectors of scalars
 * @param x2 std::vector of Eigen vectors of scalars
 * @param length_scale parameter length scale
 * @param sigma standard deviation that can be used in stan::math::square
 * @throw std::domain error if sigma <= 0, l <= 0, or x1, x2 are nan or inf
 */
template <typename T_x1, typename T_x2, typename T_s, typename T_l>
inline typename Eigen::Matrix<return_type_t<T_x1, T_x2, T_s, T_l>,
                              Eigen::Dynamic, Eigen::Dynamic>
gp_exponential_cov(const std::vector<Eigen::Matrix<T_x1, -1, 1>> &x1,
                   const std::vector<Eigen::Matrix<T_x2, -1, 1>> &x2,
                   const T_s &sigma, const std::vector<T_l> &length_scale) {
  using std::exp;
  using std::pow;

  size_t x1_size = stan::math::size(x1);
  size_t x2_size = stan::math::size(x2);
  size_t l_size = stan::math::size(length_scale);
  Eigen::Matrix<return_type_t<T_x1, T_x2, T_s, T_l>, Eigen::Dynamic,
                Eigen::Dynamic>
      cov(x1_size, x2_size);
  if (x1_size == 0 || x2_size == 0) {
    return cov;
  }

  const char *function = "gp_exponential_cov";
  for (size_t n = 0; n < x1_size; ++n) {
    check_not_nan(function, "x1", x1[n]);
  }
  for (size_t n = 0; n < x2_size; ++n) {
    check_not_nan(function, "x2", x2[n]);
  }

  check_positive_finite(function, "magnitude", sigma);
  check_positive_finite(function, "length scale", length_scale);

  for (size_t i = 0; i < x1_size; ++i) {
    check_size_match(function, "x1's row", stan::math::size(x1[i]),
                     "number of length scales", l_size);
  }
  for (size_t i = 0; i < x2_size; ++i) {
    check_size_match(function, "x2's row", stan::math::size(x2[i]),
                     "number of length scales", l_size);
  }

  T_s sigma_sq = square(sigma);
  T_l temp;

  std::vector<Eigen::Matrix<return_type_t<T_x1, T_l>, -1, 1>> x1_new
      = divide_columns(x1, length_scale);
  std::vector<Eigen::Matrix<return_type_t<T_x2, T_l>, -1, 1>> x2_new
      = divide_columns(x2, length_scale);

  size_t block_size = 10;

  for (size_t ib = 0; ib < x1_size; ib += block_size) {
    for (size_t jb = 0; jb < x2_size; jb += block_size) {
      size_t j_end = std::min(x2_size, jb + block_size);
      for (size_t j = jb; j < j_end; ++j) {
        size_t i_end = std::min(x1_size, ib + block_size);
        for (size_t i = ib; i < i_end; ++i) {
          cov(i, j) = sigma_sq * exp(-distance(x1_new[i], x2_new[j]));
        }
      }
    }
  }
  return cov;
}
}  // namespace math
}  // namespace stan
#endif