R/calculate_hessian.R

In MEIGOR: MEIGO - MEtaheuristics for bIoinformatics Global Optimization

#' @include calculate_posterior.R
#' 
#' @title Calculate the hessian of the posterior landscape.
#' 
#' @description Note that this is very expensive -- O(n^2) where n is the number of
#'    parameters being estimated. The calculation is performed using
#'    second order central finite differences.
#' 
#' 
#' 
#' @param options: list with entries as explained below. 
#'    Options set -- defines the problem and sets some
#'    parameters to control the MCMC algorithm.
#' @param model: List of model parameters - to estimate. 
#'    The parameter objects must each have a
#'    'value' attribute containing the parameter's numerical value.
#' @param estimate_params: list. 
#'    List of parameters to estimate, all of which must also be listed
#'    in 'options$model$parameters'.
#' @param initial_values: list of float, optional. 
#'    Starting values for parameters to estimate. If omitted, will use
#'    the nominal values from 'options$model$parameters'
#' @param step_fn: callable f(output), optional. 
#'    User callback, called on every MCMC iteration.
#' @param likelihood_fn: callable f(output, position). 
#'    User likelihood function.
#' @param prior_fn: callable f(output, position), optional. 
#'    User prior function. If omitted, a flat prior will be used.
#' @param nsteps: int. 
#'    Number of MCMC iterations to perform.
#' @param use_hessian: logical, optional. 
#'    Wheter to use the Hessian to guide the walk. Defaults to FALSE.
#' @param rtol: float or list of float, optional. 
#'    Relative tolerance for ode solver.
#' @param atol: float or list of float, optional. 
#'    Absolute tolerance for ode solver.    
#' @param norm_step_size: float, optional. 
#'    MCMC step size. Defaults to a reasonable value.
#' @param hessian_period: int, optional. 
#'    Number of MCMC steps between Hessian recalculations. Defaults
#'    to a reasonable but fairly large value, as Hessian calculation is expensive.
#' @param hessian_scale: float, optional. 
#'    Scaling factor used in generating Hessian-guided steps. Defaults to a
#'    reasonable value.
#' @param sigma_adj_interval: int, optional. 
#'    How often to adjust 'output$sig_value' while annealing to meet
#'    'accept_rate_target'. Defaults to a reasonable value.
#' @param anneal_length: int, optional. 
#'    Length of initial "burn-in" annealing period. Defaults to 10% of
#'    'nsteps', or if 'use_hessian' is TRUE, to 'hessian_period' (i.e. 
#'    anneal until first hessian is calculated)
#' @param T_init: float, optional. 
#'    Initial temperature for annealing. Defaults to a resonable value.
#' @param accept_rate_target: float, optional. 
#'    Desired acceptance rate during annealing. Defaults to a reasonable value.
#'    See also 'sigma_adj_interval' above.
#' @param sigma_max: float, optional. 
#'    Maximum value for 'output$sig_value'. Defaults to a resonable value.
#' @param sigma_min: float, optional. 
#'    Minimum value for 'output$sig_value'. Defaults to a resonable value. 
#' @param sigma_step: float, optional. 
#'    Increment for 'output$sig_value' adjustments. Defaults to a resonable 
#'    value. To eliminate adaptive step size, set sigma_step to 1.
#' @param thermo_temp: float in the range [0,1], optional. 
#'    Temperature for thermodynamic integration support. Used to scale
#'    likelihood when calculating the posterior value. Defaults to 1,
#'    i.e. no effect.
#' 
#' 
#' @param output: List of output values with entries as explained below. 
#' @param num_estimate: int. 
#'    Number of parameters to estimate.
#' @param estimate_idx: list of int.  
#'    Indices of parameters to estimate in the model's full parameter list.
#' @param initial_values: list of float.  
#'    Starting values for parameters to estimate, taken from the parameters'
#'    nominal values in the model or explicitly specified in 'options'.
#' @param initial_position: list of float. 
#'    Starting position of the MCMC walk in parameter space (log10 of 'initial_values').
#' @param position: list of float. 
#'    Current position of MCMC walk in parameter space, i.e. the most
#'    recently accepted move.
#' @param test_position: list of float. 
#'    Proposed MCMC mmove.
#' @param acceptance: int. 
#'    Number of accepted moves.
#' @param T: float.  
#'    Current value of the simulated annealing temperature.
#' @param T_decay: float.  
#'    Constant for exponential decay of 'T', automatically calculated such 
#'    that T will decay from 'options$T_init' down to 1 over the first
#'    'options$anneal_length' setps.
#' @param sig_value: float.  
#'    Current value of sigma, the scaling factor for the proposal distribution.
#'    The MCMC algorithm dynamically tunes this to maintain the aaceptance
#'    rate specified in 'options$accept_rate_target'.
#' @param iter: int.  
#'    Current MCMC step number.
#' @param start_iter: int.  
#'    Starting MCMC step number.
#' @param ode_options: list.  
#'    Options for the ODE integrator, currently just 'rtol' for relative
#'    tolerance and 'atol' for absolute tolerance.
#' @param initial_prior: float. 
#'    Starting prior value, i.e. the value at 'initial_position'.
#' @param initial_likelihood: float.  
#'    Starting likelihood value, i.e. the value at 'initial_position'.
#' @param initial_posterior: float.  
#'    Starting posterior value, i.e. the value at 'initial_position'.
#' @param accept_prior: float.  
#'    Current prior value i.e. the value at 'position'.
#' @param accept_likelihood: float.  
#'    Current likelihood value i.e. the value at 'position'.
#' @param accept_posterior: float.  
#'    Current posterior value i.e. the value at 'position'.
#' @param test_prior: float.  
#'    Prior value at 'test_position'.
#' @param test_likelihood: float.  
#'    Likelihood value at 'test_position'.
#' @param test_posterior: float.  
#'    Posterior value at 'test_position'.
#' @param hessian: array of float. 
#'    Current hessian of the posterior landscape. Size is 
#'    'num_estimate' x 'num_estimate'.
#' @param positions: array of float.  
#'    Trace of all proposed moves. Size is 'num_estimate' x 'nsteps'.
#' @param priors: array of float. 
#'    Trace of all priors corresponding to 'positions'. Length is 'nsteps'.
#' @param likelihoods: array of float. 
#'    Trace of all likelihoods corresponding to 'positions'. Length is 'nsteps'.
#' @param posteriors: array of float. 
#'    Trace of all posteriors corresponding to 'positions'. Length is 'nsteps'.
#' @param alphas: array of float. Trace of 'alpha' parameter and calculated values. Length is 'nsteps'.
#' @param sigmas: array of float. Trace of 'sigma' parameter and calculated values. Length is 'nsteps'.
#' @param delta_posteriors: array of float. Trace of 'delta_posterior' parameter and calculated values. Length is 'nsteps'.
#' @param ts: array of float. Trace of 'T' parameter and calculated values. Length is 'nsteps'.
#' @param accepts: logical array. 
#'    Trace of wheter each proposed move was accepted or not.
#'    Length is 'nsteps'.
#' @param rejects: logical array. Trace of wheter each proposed move was rejected or not. Length is 'nsteps'.
#' @param hessians: array of float. 
#'    Trace of all hessians. Size is 'num_estimate' x 'num_estimate' x 'num_hessians'
#'    where 'num_hessians' is the actual number of hessians to be calculated.
#' 
#' 
#' 
#' @param position: list of float, optional. 
#'    log10 of the values of the parameters being estimated.
#'    (see the 'cur_params' method for details)
#'    
#'    
#' @return Returns the hessian of the posterior landscape.
#'    

calculate_hessian = function(output,options,position=NULL){
  if (is.null(position)){
    position = output$position
  }
  d = 0.1
  hessian = matrix(0,nrow = output$num_estimate, ncol = output$num_estimate)
  # iterate over diagonal
  for(i in 1:output$num_estimate){
    position_f = position  
    position_b = position  
    position_f[i] = position_f[i] + d
    position_b[i] = position_b[i] - d
    f = calculate_posterior(output,options,position_f)[1]
    c = calculate_posterior(output,options,position)[1]
    b = calculate_posterior(output,options,position_b)[1]
    hessian[i,i] = (f - 2*c + b)/d^2
  }
  # iterate over elements above diagonal
  for(i in 1:(output$num_estimate-1)){
    for(j in (i+1):(output$num_estimate)){
      position_ff = position
      position_fb = position
      position_bf = position
      position_bb = position
      position_ff[i] = position_ff[i]+d
      position_ff[j] = position_ff[j]+d
      position_fb[i] = position_fb[i]+d
      position_fb[j] = position_fb[j]-d
      position_bf[i] = position_bf[i]-d
      position_bf[j] = position_bf[j]+d
      position_bb[i] = position_bb[i]-d
      position_bb[j] = position_bb[j]-d
      ff = calculate_posterior(output,options,position_ff)[1]
      fb = calculate_posterior(output,options,position_fb)[1]
      bf = calculate_posterior(output,options,position_bf)[1]
      bb = calculate_posterior(output,options,position_bb)[1]
      hessian[i,j] = (ff - fb - bf + bb)/(4*d^2)
      # hessian is symmetric, so mirror the values across the diagonal
      hessian[j,i] = hessian[i,j]
    }
  }
  if (any(is.nan(hessian)) || any(is.infinite(hessian))){
    stop("Numerical instability encountered in Hessian calculation")
  }
  return(hessian)
}