Provide several likelihoods to estimate parameters using bayesian methods.
likelihood_log_ciidn(sim_list, obs_list)
likelihood_log_ciidn_corr(sim_list, obs_list)The value of the likelihood given the observed and simulated values of the variables.
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).