CPPpriorElicit            package:BayHaz            R Documentation

_F_u_n_c_t_i_o_n _t_o _S_e_t _H_y_p_e_r_p_a_r_a_m_e_t_e_r_s _o_f _C_P_P _P_r_i_o_r_s

_D_e_s_c_r_i_p_t_i_o_n:

     A function to set the hyperparameters of a CPP prior distribution,
     following the procedure described in La Rocca (2005).

_U_s_a_g_e:

     CPPpriorElicit(r0 = 1, H = 1, T00 = 1, M00 = 1, extra = 0)

_A_r_g_u_m_e_n_t_s:

      r0: prior mean hazard rate (r_0)

       H: corresponding coefficient of variation

     T00: time-horizon of interest (T_infty)

     M00: number of extremes within the time-horizon in a "typical"
          hazard rate trajectory (M_infty)

   extra: number of additional CPP jumps (compared with default)

_D_e_t_a_i_l_s:

     A CPP prior hazard rate is defined, for 0<t<T_infty, by

          rho(t)=xi_0 k_0(t)+sum_{j=1}^{F} xi_j k(t-sigma_j)

     where:

        *  sigma_j is the time of the j-th jump of a CPP process with
           gamma distributed jump-sizes

        *  xi_j is the j-th jump-size of the above process

        *  k is a zero-mean Gaussian density (kernel)

        *  F is a positive integer such that (with high probability)
           sigma_{F+1} is much larger than T_infty

        *  xi_0 is an independent random variable with the same
           distribution as xi_j

        *  k_0 is a suitable function such that the mean of rho(t) does
           not depend on t

     The elicitation procedure makes the mean of rho(t) identically
     equal to r_0 and its standard deviation approximately equal to
     Hr_0. An exponential distribution is selected for the jump-sizes.
     The kernel bandwidth choice is based on M_infty (and T_infty).

_V_a_l_u_e:

     A list with nine components: 

      r0: prior mean hazard rate (copy of the input argument)

       H: corresponding coefficient of variation (copy of the input
          argument)

     T00: time-horizon of interest (copy of the input argument)

     M00: number of extremes within the time-horizon in a "typical"
          hazard rate trajectory (copy of the input argument)

       a: shape parameter of the jump-size distribution (always equal
          to 1)

      sd: standard deviation of the Gaussian kernel (bandwidth)

       q: expected number of CPP jumps per time unit

       b: rate parameter of the jump-size distribution

       F: maximum number of jumps within the time-horizon (with high
          probability)

_N_o_t_e:

     As the default value of 'F' is computed _a priori_, additional
     jumps may be needed _a posteriori_.

_R_e_f_e_r_e_n_c_e_s:

     Luca La Rocca (2005). On Bayesian Nonparametric Estimation of
     Smooth Hazard Rates with a View to Seismic Hazard Assessment.
     _Research Report n. 38-05_, Department of Social, Cognitive and
     Quantitative Sciences, Reggio Emilia, Italy.

_S_e_e _A_l_s_o:

     'BayHaz-package', 'CPPpriorSample', 'CPPpostSample'

_E_x_a_m_p_l_e_s:

     # ten events per century with unit coefficient of variation
     # fifty year time horizon with a couple of extremes in a "typical" trajectory
     hypars<-CPPpriorElicit(r0 = 0.1, H = 1, T00 = 50, M00 = 2)

