Likelihoods: Likelihoods

In SticsRPacks/CroptimizR: A Package to Estimate Parameters of Crop Models

Likelihoods

Description

Provide several likelihoods to estimate parameters using bayesian methods.

Usage

likelihood_log_ciidn(sim_list, obs_list)

likelihood_log_ciidn_corr(sim_list, obs_list)

Arguments

sim_list	List of simulated variables
obs_list	List of observed variables

Details

The following log-likelihoods are proposed ( see html version for a better rendering of equations):

• likelihood_log_ciidn: log transformation of concentrated version of iid normal likelihood
  The concentrated version of iid normal likelihood is:

  \prod_{j} ({\sum_{i,k} [Y_{ijk}-f_{jk}(X_i;\theta)]^2 )}^{-(n_j/2+2)}

  where Y_{ijk} is the observed value for the k^{th} time point of the j^{th} variable in the i^{th} situation, f_{jk}(X_i;\theta) the corresponding model prediction, and n_j the number of measurements of variable j.
  likelihood_log_ciidn computes the log of this equation.
  Here, one assume that all errors (model and observations errors for all variables, dates and situations) are independent, and that the error variance is constant over time but may be different between variables j. These error variances are automatically estimated.

• likelihood_log_ciidn_corr: log transformation of concentrated version of iid normal likelihood but with hypothesis of high correlation between errors for different measurements over time
  The concentrated version of iid normal likelihood is:

  \prod_{j} {( \sum_{i} [ \frac{1}{n_{ij}} \sum_{k} (Y_{ijk}-f_{jk}(X_i;\theta))^2 ] )} ^{-(N_j/2+2)}

  where Y_{ijk} is the observed value for the k^{th} time point of the j^{th} variable in the i^{th} situation, f_{jk}(X_i;\theta) the corresponding model prediction, N_j the number of situations including at least one observation of variable j, and n_{ij} the number of observation of variable j on situation i.
  likelihood_log_ciidn_corr computes the log of this equation.
  Here, one still assume that errors in different situations or for different variables in the same situation are independent. However, errors for different observations over time of the same variable in the same situation are assumed to be highly correlated. In this way, each situation contributes a single term to the overall sum of squared errors regardless of the number of observations which may be useful in case one have situations with very heterogeneous number of dates of observations.

sim_list and obs_list must have the same structure (i.e. same number of variables, dates, situations, ... use internal function intersect_sim_obs before calling the criterion functions).

Value

The value of the likelihood given the observed and simulated values of the variables.

SticsRPacks/CroptimizR documentation built on Dec. 16, 2024, 11:54 a.m.