*6.1. RME Estimation of Model Parameters and Measurement Noises*

The temporal evolution of the lake area *S*[*t*] is described by the following dynamic regression equation with two influencing factors, the average annual temperature *T*[*t*] and the annual precipitation *R*[*t*]:

$$\begin{aligned} \mathcal{S}[t] &= \quad a\_0 + \sum\_{k=1}^p a\_k \mathcal{S}[t-k] + a\_{(p+1)} T[t] + a\_{(p+2)} (\mathcal{R}[t], \\ \mathcal{S}[t] &= \quad \mathcal{S}[t] + \mathcal{S}[t]. \end{aligned} \tag{34}$$

The model parameters and measurement noises are assumed to be random and of the interval type:

$$\begin{aligned} a\_k \in \mathcal{A}\_k &= [a\_k^-, a\_k^+], \quad k = 0, \\ \mathbf{a} &= \{a\_{0'}, \dots, a\_{p'}, a\_{p+1'}, a\_{p+2}\} \in \mathcal{A} = \bigcup\_{k=0}^{p+2} \mathcal{A}\_k. \end{aligned}$$

The probabilistic properties of the parameters are characterized by a PDF *P*(**a**).

The variable *v*ˆ[*t*] is the observed output of the model, and the values of the random measurement noise *ξ*[*t*] at different time instants *t* may belong to different ranges:

$$\mathfrak{J}[t] \in \Xi\_t = [\mathfrak{J}^-[t], \mathfrak{J}^+[t]],\tag{35}$$

with a PDF *Qt*(*ξ*[*t*]), (*t* = 0, ... , *N*), where *N* denotes the length of the observation interval. The order *p* = 4 and the parameter ranges for the dynamic randomized regression model (34) (see Table 1 below) were calculated based on real data using the empirical correlation functions and the least-square estimates of the residual variances.

**Table 1.** Parameter ranges for the model.


For the training collection L, the model can be written in the vector–matrix form

$$\begin{array}{rcl}\hat{\mathbf{S}} &=& \hat{\mathbf{S}}\mathbf{a} + a\_5 \mathbf{T} + a\_6 \mathbf{R}, \\ \hat{\mathbf{v}} &=& \hat{\mathbf{S}} + \tilde{\mathbf{y}}, \end{array} \tag{36}$$

with the matrix

$$
\hat{S} = \begin{pmatrix}
1 & \mathring{S}[3] & \cdots & \mathring{S}[0] \\
1 & \mathring{S}[4] & \cdots & \mathring{S}[1] \\
\cdots & \cdots & \cdots & \cdots & \cdots \\
1 & \mathring{S}[23] & \cdots & \mathring{S}[20]
\end{pmatrix} \tag{37}
$$

and the vectors **S**ˆ = [*S*ˆ[4], ... , *S*ˆ[24]], **T** = [*T*[4], ... , *T*[24]], **R** = [*R*[4], ... , *R*[24]], and **v**ˆ = [*v*[4],..., *v*[24]]; *ξ* = [*ξ*[4],..., *ξ*[24]].

The RME estimation procedure yielded the following entropy-optimal PDFs of the model parameters (36) and measurement noises:

$$\begin{split}P^\*(\mathsf{a},\lambda) &= \prod\_{k=0}^6 \frac{\exp(-q\_k a\_k)}{\mathcal{P}\_k(\lambda)}, \; \mathcal{P}\_k(\lambda) = \int\_{\mathcal{A}\_{\parallel}} \exp(-q\_k a\_k) da\_{k\prime} \\ q\_0 &= \sum\_{t=4}^{24} \lambda\_{n\prime} \quad q\_k = \sum\_{t=p}^{24} \lambda\_{n\prime} S[t-k], \quad k = 1, \ldots, 4, \\ q\_5 &= \sum\_{t=4}^{24} \lambda\_t T[t], \quad q\_6 = \sum\_{t=p}^{24} \lambda\_t R[t], \\ Q^\*(\vec{\xi}, \lambda) &= \frac{\exp(-\bar{\lambda}\,\vec{\xi})}{\mathcal{Q}}, \quad \mathcal{Q} = \int\_{\Xi} \exp(-\bar{\lambda}\,\vec{\xi}) d\vec{\xi}, \quad \lambda = \frac{q\_0}{20}. \end{split} \tag{38}$$

Note that *S*[*t* − *k*], *T*[*t*], and *R*[*t*] are the data from the collection L. The two-dimensional sections of the function *P*∗(**a**) and the function *Q*∗(*ξ*) are shown in Figure 1.

(**a**) Two-dimensional section of function *P*∗(**a**) (**b**) Function *Q*∗(*ξ*). **Figure 1.** Two-dimensional section of the function *P*∗ and the function *Q*∗.
