## 2 Materials and Methods
### Method for the Non-Stationary Hydrological Time Series Analysis and Recommendations  
#### 2.2.1 Estimation of the Change Point in Sequences of Random Variables of River Runoff  
The main task of the non-stationary time series analysis is the task of detecting the point of change (CPD). To estimate the state change point, it is necessary to consider this point as a parameter of a non-stationary stochastic model and apply one of the estimation methods, for example, the Bayesian method. Let us consider as an example one of the most common cases when the parameters of probabilistic models for homogeneous (stationary) sections are known. There is a sequence of random variables \(x_{1},x_{2},\ldots,x_{n}\), which is divided into two parts at the point \(r\) (\(1\leq r\leq n\)), and \(x_{i1}\) is distributed according to the law \(F_{1}\) (\(x/\Theta_{1}\)), \(i=1,\cdot\cdot\cdot,r\), and \(x_{i2}\) is distributed according to the law \(F_{2}\) (\(x/\Theta_{2}\)), \(i=r+1,\ldots,n\), and \(F_{1}\) (\(x/\Theta_{1}\)) \(\
eq F_{2}\) (\(x/\Theta_{2}\)). Let us consider the case, following [21], when the distribution functions \(F_{1}\) and \(F_{2}\) are known, and it is necessary to draw a conclusion about the point of change in the state of the process. Having set the a priori density \(\rho_{0}(r)\) as a uniform distribution, we determine the a posteriori distribution density of a possible change point for the available observational data \(x_{1},\ldots,x_{n}\) based on the general formula for the total probability (Bayes formula):  
\[\rho(\Theta|x)=\frac{\rho(x|\Theta)\cdot\rho(\Theta)}{\overset{\
eg}{\rho}(x)} \tag{2}\]  
where \(\rho(x|\Theta)\) is the likelihood of data \(x\) for a given value of the parameters \(\Theta_{1}\) and \(\Theta_{2}\). The method as applied to fluctuations in river flow is considered in detail in [22; 23; 24]. 2.2 Bayesian Method for Determining the Design Characteristics of Runoff in Non-Stationary Conditions  
To calculate runoff characteristics in the presence of violations of the homogeneity of time series due to climate change, the time series is divided into several homogeneous parts, each of which is characterized by its own set of parameters. The date of runoff transition from one homogeneous state to another is determined by the methods described in Section 2.2.1. The determination of the design characteristics of river runoff in non-stationary conditions is based on the construction of a distribution law in the form of a sum of several distribution laws, weighted by the duration of conditionally stationary regimes. The calculated distribution curve for the entire time series is constructed as the sum of distribution laws with weights proportional to the sample lengths. The method is known as \"composite distribution curves\". To determine the calculated values for each of the homogeneous parts of the series, the distribution curve \(P_{i}(x)\) is determined. The composite distribution curve is calculated by the formula:  
\[P(x)=1.0-(1.0-P_{1}(x))\frac{n_{1}}{N}-(1.0-P_{2}(x))\frac{n_{2}}{N}-(1.0-P_{3 }(x))\frac{n_{3}}{N}, \tag{3}\]  
where \(N\) is the length of the entire time series and \(n_{i}\) are the lengths of the \(i\)-th homogeneous parts of the series. To obtain a predictive runoff distribution, the Bayesian methodology is allowed to be used for solving statistical problems for two or three conditionally stationary periods. In contrast to the method of \"composite distribution curves\", where the final solution considers only the weight coefficients of individual homogeneous populations, the Bayesian solution additionally considers the errors in sample estimates of distribution parameters. With the Bayesian approach, for each of the homogeneous parts of the series, the parameter \(\theta\) is determined, which can be considered as the mean value, variance, and asymmetry, as well as the relationship between them. The parameter \(\theta\) is a random variable since it is calculated from a series \(y\) (sample) of limited length.