11.1.1. Distances and likelihoods used to compare observations and model predictions

There are many ways to quantify the agreement between the reference observations and the model predictions, given a parameter vector \(\theta\). Depending on the framework adopted (deterministic or Bayesian), different tools are required. In a deterministic setting, a distance is used to measure how far the prediction is from the observation. In contrast, within the Bayesian framework, the analogous concept is the likelihood, a function that evaluates the probability of observing the data given a parameter vector.

Since the number of variables \(n_Y\) used to perform the calibration can be greater than one, it might be useful to introduce the coefficients \(\lbrace \omega_j \rbrace_{j \in [1,n_Y]}\) to weight the contribution of each variable relative to the others. The following lists the distance functions available in Uranie:

• L1 distance function (sometimes called Manhattan distance): \( \displaystyle d(\mathbf{y}, \mathbf{f_\theta}(\mathbf{x})) = \sum_{j=1}^{n_{Y}} \omega_{j} \times \bigg ( \sum_{i=1}^{n} | \mathbf{y}^j_i - \mathbf{f_\theta}(\mathbf{x})^j_i | \bigg ) \, ;\)

• Least squares distance function: \( \displaystyle d(\mathbf{y},\mathbf{f_\theta}(\mathbf{x})) = \sum_{j=1}^{n_{Y}} \sqrt{ \omega_{j} \sum_{i=1}^{n} (\mathbf{y}^j_i - \mathbf{f_\theta} (\mathbf{x})^j_i )^2 } \, ;\)

• Relative least squares distance function: \( \displaystyle d(\mathbf{y},\mathbf{f_\theta} (\mathbf{x})) = \sum_{j=1}^{n_{Y}} \sqrt{ \omega_{j} \sum_{i=1}^{n} \bigg (\frac{ \mathbf{y}^j_i - \mathbf{f_\theta}(\mathbf{x})^j_i}{\mathbf{y}^j_i} \bigg )^2 } \, ;\)

• Weighted least squares distance function: \( \displaystyle d(\mathbf{y},\mathbf{f_\theta} (\mathbf{x})) = \sum_{j=1}^{n_{Y}} \sqrt{ \omega_{j} \sum_{i=1}^{n} \psi_i^j\times ( \mathbf{y}^j_i - \mathbf{f_\theta}(\mathbf{x})^j_i )^2 } \, ,\) where the coefficients \(\lbrace \psi_i^j \rbrace_{i \in [1,n]}\) are associated with the \(j\)-th variable and are used to weight each observation with respect to the others;

• Mahalanobis distance function: \(\displaystyle d(\mathbf{y},\mathbf{f_\theta} (\mathbf{x})) = \sum_{j=1}^{n_{Y}} \sqrt{ \omega_{j} ( \mathbf{y}^j - \mathbf{f_\theta} (\mathbf{x})^j )^T \Sigma^{-1} (\mathbf{y}^j - \mathbf{f_\theta}(\mathbf{x})^j ) } \) where \(\Sigma\) is the covariance matrix of the observations.

Regarding the likelihood functions already implemented, only the Gaussian log-likelihood for independent parameters is available, as it is the most commonly used. Its expression follows:

• Gaussian log-likelihood for independent parameters: \( \displaystyle \textit{log-}\mathcal{L} \left(\theta | \mathbf{x},\mathbf{y}\right) = -\frac{1}{2}\sum_{j=1}^{n_{Y}}\sum_{i=1}^{n}\left( \log\left( 2 \pi \left(\sigma_i^j\right)^2 \right)+\left(\frac{\mathbf{y}^j_i - \mathbf{f_\theta}(\mathbf{x})^j_i }{\sigma_i^j}\right)^2\right)\) where the coefficients \(\lbrace \sigma_i^j \rbrace_{i \in [1,n]}\) are the standard deviations of each observation associated to the \(j\)-th variable.

If needed, it is still possible to define a custom likelihood (or distance).

For more details on the implementation of the distances and likelihoods, or on how to implement your own, see Defining data, distance and likelihood functions.