*6.2. Testing*

Testing was performed using the data from the collection T , which included the lake area *S*[*t*], the average annual temperature *T*[*t*], and the annual precipitation *R*[*t*], *t* = 25, ... , 35. An ensemble of trajectories of the model's observed output *v*[*t*] was generated using Monte Carlo simulations and sampling of the entropy-optimal PDFs

*P*∗(**a**), *Q*∗*ξ* on the testing interval. In addition, the trajectory of the empirical means *v*¯[*t*] and the dimensions of the empirical standard deviation area were calculated.

The quality of RME estimation was characterized by the absolute and relative errors:

$$AbsErr = \sqrt{\sum\_{t=26}^{35} \left(\mathcal{S}[t] - \mathcal{\sigma}[t]\right)^2} = 0.3446,\tag{39}$$

$$RelErr = \frac{\sqrt{\sum\_{t=26}^{35} \left(S[t] - \vartheta[t]\right)^2}}{\sqrt{\sum\_{t=26}^{35} S^2[t]} + \sqrt{\sum\_{t=26}^{35} \vartheta^2[t]}} = 0.0089. \tag{40}$$

The generated ensemble of the trajectories is shown in Figure 2.

**Figure 2.** Ensemble of the trajectories (gray domain), the standard deviation area (dark gray domain), the empirical mean trajectory, and the lake area data.

## **7. Discussion**

Given an available data collection, the RME procedure allows estimation of the PDFs of a model's random parameters under measurement noises corresponding to the maximum uncertainty (maximum entropy). In addition, this procedure needs no assumptions about the structure of the estimated PDFs or the statistical properties of the data and measurement noises.

An entropy-optimal model can be simulated by sampling the PDFs to generate an empirical ensemble of a model's output trajectories and to calculate its empirical characteristics (the mean and median trajectories, the standard deviation area, interquartile sets, and others).

The RME procedure was illustrated with an example of the estimation of the parameters of a linear regression model for the evolution of the thermokarst lake area in Western Siberia. In this example, the procedure demonstrated a good estimation accuracy.

However, these positive features of the procedure were achieved with computational costs. Despite their analytical structure, the RME estimates of the PDFs depend on Lagrange multipliers, which are determined by solving the balance equations with the so-called integral components (the mathematical expectations of random parameters and measurement noises). Calculating the values of multidimensional integrals may require appropriate computing resources.
