// Original code from which Stan's code is derived:
// Copyright (c) 2013, Joachim Vandekerckhove.
// All rights reserved.
//
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// modification, are permitted
// provided that the following conditions are met:
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#ifndef STAN_MATH_PRIM_MAT_PROB_WIENER_LPDF_HPP
#define STAN_MATH_PRIM_MAT_PROB_WIENER_LPDF_HPP

#include <stan/math/prim/scal/fun/constants.hpp>
#include <stan/math/prim/scal/fun/square.hpp>
#include <stan/math/prim/scal/fun/value_of.hpp>
#include <stan/math/prim/scal/err/check_consistent_sizes.hpp>
#include <stan/math/prim/scal/err/check_bounded.hpp>
#include <stan/math/prim/scal/err/check_finite.hpp>
#include <stan/math/prim/scal/err/check_not_nan.hpp>
#include <stan/math/prim/scal/err/check_positive.hpp>
#include <stan/math/prim/scal/meta/return_type.hpp>
#include <stan/math/prim/scal/meta/include_summand.hpp>
#include <boost/math/distributions.hpp>
#include <algorithm>
#include <cmath>
#include <string>

namespace stan {
  namespace math {

    /**
     * The log of the first passage time density function for a (Wiener)
     *  drift diffusion model for the given \f$y\f$,
     * boundary separation \f$\alpha\f$, nondecision time \f$\tau\f$,
     * relative bias \f$\beta\f$, and drift rate \f$\delta\f$.
     * \f$\alpha\f$ and \f$\tau\f$ must be greater than 0, and
     * \f$\beta\f$ must be between 0 and 1. \f$y\f$ should contain
     * reaction times in seconds (strictly positive) with
     * upper-boundary responses.
     *
     * @param y A scalar variate.
     * @param alpha The boundary separation.
     * @param tau The nondecision time.
     * @param beta The relative bias.
     * @param delta The drift rate.
     * @return The log of the Wiener first passage time density of
     *  the specified arguments.
     */
    template <bool propto,
              typename T_y, typename T_alpha, typename T_tau,
              typename T_beta, typename T_delta>
    typename return_type<T_y, T_alpha, T_tau, T_beta, T_delta>::type
    wiener_lpdf(const T_y& y, const T_alpha& alpha, const T_tau& tau,
               const T_beta& beta, const T_delta& delta) {
      static const char* function("wiener_lpdf(%1%)");

      using std::log;
      using std::exp;
      using std::pow;

      static const double WIENER_ERR = 0.000001;
      static const double PI_TIMES_WIENER_ERR = pi() * WIENER_ERR;
      static const double LOG_PI_LOG_WIENER_ERR =
        LOG_PI + log(WIENER_ERR);
      static const double
        TWO_TIMES_SQRT_2_TIMES_SQRT_PI_TIMES_WIENER_ERR =
        2.0 * SQRT_2_TIMES_SQRT_PI * WIENER_ERR;
      static const double LOG_TWO_OVER_TWO_PLUS_LOG_SQRT_PI =
        LOG_TWO / 2 + LOG_SQRT_PI;
      static const double SQUARE_PI_OVER_TWO = square(pi()) * 0.5;
      static const double TWO_TIMES_LOG_SQRT_PI = 2.0 * LOG_SQRT_PI;

      if (!(stan::length(y)
            && stan::length(alpha)
            && stan::length(beta)
            && stan::length(tau)
            && stan::length(delta)))
        return 0.0;

      typedef typename return_type<T_y, T_alpha, T_tau,
                                   T_beta, T_delta>::type T_return_type;
      T_return_type lp(0.0);

      check_not_nan(function, "Random variable", y);
      check_not_nan(function, "Boundary separation", alpha);
      check_not_nan(function, "A-priori bias", beta);
      check_not_nan(function, "Nondecision time", tau);
      check_not_nan(function, "Drift rate", delta);
      check_finite(function, "Boundary separation", alpha);
      check_finite(function, "A-priori bias", beta);
      check_finite(function, "Nondecision time", tau);
      check_finite(function, "Drift rate", delta);
      check_positive(function, "Random variable", y);
      check_positive(function, "Boundary separation", alpha);
      check_positive(function, "Nondecision time", tau);
      check_bounded(function, "A-priori bias", beta , 0, 1);
      check_consistent_sizes(function, "Random variable", y,
                             "Boundary separation", alpha,
                             "A-priori bias", beta,
                             "Nondecision time", tau, "Drift rate", delta);

      size_t N = std::max(max_size(y, alpha, beta), max_size(tau, delta));
      if (!N) return 0.0;

      VectorView<const T_y> y_vec(y);
      VectorView<const T_alpha> alpha_vec(alpha);
      VectorView<const T_beta> beta_vec(beta);
      VectorView<const T_tau> tau_vec(tau);
      VectorView<const T_delta> delta_vec(delta);

      size_t N_y_tau = max_size(y, tau);
      for (size_t i = 0; i < N_y_tau; ++i) {
        if (y_vec[i] <= tau_vec[i]) {
          std::stringstream msg;
          msg << ", but must be greater than nondecision time = " << tau_vec[i];
          std::string msg_str(msg.str());
          domain_error(function, "Random variable", y_vec[i], " = ",
                       msg_str.c_str());
        }
      }

      if (!include_summand<propto, T_y, T_alpha, T_tau, T_beta, T_delta>::value)
        return 0;

      for (size_t i = 0; i < N; i++) {
        typename scalar_type<T_beta>::type one_minus_beta = 1.0 - beta_vec[i];
        typename scalar_type<T_alpha>::type alpha2 = square(alpha_vec[i]);
        T_return_type x = (y_vec[i] - tau_vec[i]) / alpha2;
        T_return_type kl, ks, tmp = 0;
        T_return_type k, K;
        T_return_type sqrt_x = sqrt(x);
        T_return_type log_x = log(x);
        T_return_type one_over_pi_times_sqrt_x = 1.0 / pi() * sqrt_x;

        // calculate number of terms needed for large t:
        // if error threshold is set low enough
        if (PI_TIMES_WIENER_ERR * x < 1) {
          // compute bound
          kl = sqrt(-2.0 * SQRT_PI * (LOG_PI_LOG_WIENER_ERR + log_x)) / sqrt_x;
          // ensure boundary conditions met
          kl = (kl > one_over_pi_times_sqrt_x) ? kl : one_over_pi_times_sqrt_x;
        } else {
          kl = one_over_pi_times_sqrt_x;  // set to boundary condition
        }
        // calculate number of terms needed for small t:
        // if error threshold is set low enough
        T_return_type tmp_expr0
          = TWO_TIMES_SQRT_2_TIMES_SQRT_PI_TIMES_WIENER_ERR * sqrt_x;
        if (tmp_expr0 < 1) {
          // compute bound
          ks = 2.0 +  sqrt_x * sqrt(-2 * log(tmp_expr0));
          // ensure boundary conditions are met
          T_return_type sqrt_x_plus_one = sqrt_x + 1.0;
          ks = (ks > sqrt_x_plus_one) ? ks : sqrt_x_plus_one;
        } else {  // if error threshold was set too high
          ks = 2.0;  // minimal kappa for that case
        }
        if (ks < kl) {  // small t
          K = ceil(ks);  // round to smallest integer meeting error
          T_return_type tmp_expr1 = (K - 1.0) / 2.0;
          T_return_type tmp_expr2 = ceil(tmp_expr1);
          for (k = -floor(tmp_expr1); k <= tmp_expr2; k++)
            tmp += (one_minus_beta + 2.0 * k) *
              exp(-(square(one_minus_beta + 2.0 * k)) * 0.5 / x);
          tmp = log(tmp) - LOG_TWO_OVER_TWO_PLUS_LOG_SQRT_PI - 1.5 * log_x;
        } else {  // if large t is better...
          K = ceil(kl);  // round to smallest integer meeting error
          for (k = 1; k <= K; ++k)
            tmp += k * exp(-(square(k)) * (SQUARE_PI_OVER_TWO * x))
              * sin(k * pi() * one_minus_beta);
          tmp = log(tmp) + TWO_TIMES_LOG_SQRT_PI;
        }

        // convert to f(t|v,a,w) and return result
        lp += delta_vec[i] * alpha_vec[i] * one_minus_beta
          - square(delta_vec[i]) * x * alpha2 / 2.0
          - log(alpha2) + tmp;
      }
      return lp;
    }

    template <typename T_y, typename T_alpha, typename T_tau,
              typename T_beta, typename T_delta>
    inline
    typename return_type<T_y, T_alpha, T_tau, T_beta, T_delta>::type
    wiener_lpdf(const T_y& y, const T_alpha& alpha, const T_tau& tau,
               const T_beta& beta, const T_delta& delta) {
      return wiener_lpdf<false>(y, alpha, tau, beta, delta);
    }

  }
}
#endif
