Provide several least squares criteria to estimate parameters by minimizing the difference between observed and simulated values of model output variables.
crit_ols(sim_list, obs_list)
crit_wls(sim_list, obs_list, weight)
crit_log_cwss(sim_list, obs_list)
crit_log_cwss_corr(sim_list, obs_list)List of simulated variables
List of observed variables
Weights to use in the criterion to optimize. A function that takes in input a vector of observed values and the name of the corresponding variable and that must return either a single value for the weights for the given variable or a vector of values of length the length of the vector of observed values given in input.
The value of the criterion given the observed and simulated values of the variables.
The following criteria are proposed ( see html version for a better rendering of equations):
crit_ols: ordinary least squares
The sum of squared residues for each variable:
$$ \sum_{i,j,k} [Y_{ijk}-f_{jk}(X_i;\theta)]^2 $$
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.
Using this criterion, 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 and equal for the different variables \(j\).
crit_wls: weighted least squares
The weighted sum of squared residues for each variable:
$$ \sum_{i,j,k} [\frac{Y_{ijk}-f_{jk}(X_i;\theta)}{w_{i,j,k}}]^2 $$
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 \(w_{i,j,k}\) a weight.
Using this criterion, one assume that all errors (model and observations errors for all variables, dates and situations) are independent, and that the error variances are equal to \(w_{i,j,k}^2\) \(j\).
crit_log_cwss: log transformation of concentrated version of weighted sum of squares
The concentrated version of weighted sum of squares is:
$$ \prod_{j} {(\frac{1}{n_j} \sum_{i,k} [Y_{ijk}-f_{jk}(X_i;\theta)]^2 )} ^{n_j/2} $$
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 \(j\). crit_log_cwss computes the log of this equation.
Using this criterion, 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.
More details about this criterion are given in Wallach et al. (2011), equation 5.
crit_log_cwss_corr: log transformation of concentrated version of weighted sum of squares with hypothesis of high correlation between errors for different measurements over time
The original criterion is:
$$ \prod_{j} {(\frac{1}{N_j} \sum_{i} [ \frac{1}{n_{ij}} \sum_{k} (Y_{ijk}-f_{jk}(X_i;\theta))^2 ] )} ^{N_j/2} $$
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 \(i\). . crit_log_cwss_corr computes the log of this equation.
Using this criterion, 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.
More details about this criterion are given in Wallach et al.
(2011), equation 8.
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).