**2. Materials and Methods**

The focus of our study was climate of the Caucasian region (southern Russia), whose territory in the context of the article was limited to 41.28–47.14 degrees north latitude (◦N) and 38.58–48.17 degrees east longitude (◦E).

To study the climate in different regions of southern Russia, we used data from meteorological instrumental observations (1961–2011) by 20 weather stations, of the state observational network of Roshydromet and provided by the North Caucasian Administration for Hydrometeorology and Environmental Monitoring. (Table 1 and Figure 1). The data of the time series were homogeneous, throughout the period under study the location of the stations remained constant (outside populated areas), and the so-called urban warming did not affect them. The average, maximum, and minimum seasonal and annual temperatures in the south of Russia were investigated.


**Table 1.** Geographical location of weather stations inside the Caucasian region.

In previous studies [31,32], trends in the amount of precipitation and daily maximums of precipitation were analyzed in the Caucasus region, an analysis of the temperature regime was added in this study. In the series of temperatures, averaged values, anomalies (deviations of the observed value from the norm), and trends for the four seasons and the calendar year (January–December) were considered. The climatic norm was considered to be the mean multi-year value of the considered climate variable for the base period of 1961–1990 [33]. The anomalies were calculated for each year as the difference between the current value and the norm of the corresponding climate variable (average 1961–1990). In the series of mean temperature and sum precipitation, the data were averaged within the calendar seasons of each year (the winter season included December of the previous year) and for the year as a whole. Maximum and minimum temperatures were defined as the largest and lowest values for a certain period (month). The absolute maximum (minimum) was the largest (smallest) value was observed at least once in a month. We used absolute maxima and minima for each month of the season during the period 1961–2011.

**Figure 1.** Geographical location of weather stations inside the Caucasian region.

Time series were investigated by statistical methods, as well as by means of the STATISTICA, SPSS 15.0 programs [34–36], spatial fields of distributions were constructed using the geoinformation system Golden Software Surfer 8 [37]. Linear trends characterizing the trend of the considered value over the entire observation period from 1961 to 2011, and from 1976 (the beginning of global warming) were built in Excel. The estimation of the linear trend coefficients was considered the least squares method degrees per decade, ◦C/10 years.

To accept the hypothesis regarding the presence of a statistically significant linear trend, a 95% significance level (α) was adopted and determined through a determination coefficient *R*<sup>2</sup> characterizing the share of the trend in the explained variance (*D*, %). Using the coefficient of determination *R*2, it was possible to check the significance of the trend. For this, the F-criterion was determined:

$$F = \frac{R^2}{1 - R^2} \frac{n - k - 1}{k} ,\tag{1}$$

where *k* is the number of trend equation coefficients. The constructed linear regression trend was significant with a level of significance α, if the inequality held:

$$F \ge F\_{1-a;k;n-k-1} \tag{2}$$

Quantile *F*1−*α*;*k*;*n*−*k*−<sup>1</sup> calculated in Excel by expressing FINV(*α*; *k*; *n* − *k* − 1).

The lower threshold value of the coefficient of determination, which determines the statistical significance of the trend at a 95% confidence interval, was *D* = 8% (for *n* = 51, 1961–2011).

Climate of the Caucasus region, which includes various climatic zones (Table 1 and Figure 1), was primarily determined by its position in the temperate latitudinal climate zone.

The North Caucasus mountain system prevents the movement of cold air masses from north to south, and warm from south-west and west to north-east and east. A complex local circulation is created in the mountains with the separation of the two temperate zone regions: the Atlantic-continental (plain, foothill) and the mountain (high-mountain). At the same time, atmospheric processes in the region are complicated by local factors, namely the complex orography of the North Caucasus. Due to the complexity of climate formation in such a complex orographically heterogeneous terrain, the correlation of meteorological parameters of stations located in different climatic zones was of interest.

The spatial structures of the air temperature fields and precipitation fields were analyzed from weather station data in different climatic zones, and spatial correlation relationships between them were determined depending on the scale of the distance between them.

The study assessed the persistence of climate change. As its integral characteristic, the rescaled range method (*R*/*S* analysis) and fractal properties of time series (Hurst exponent *H*), were used [38–44]. By using the rescaled range method for the first time, the British hydrologist Harold Hurst studied the rise of the Nile River, as well as the sequence of measurements of atmospheric temperature, rainfall, river flow parameters, thickness of annual wood growth rings, and other natural processes [38]. The method is based on the analysis of the range *R* of the meteorological parameter (the largest and smallest value in the segment under study), and the standard deviation *S* and its dependence on the period of the studied time *t*. The Hurst exponent can distinguish a random series from a nonrandom one, even if the random series is not Gaussian (that is, not normally distributed).

To calibrate the time series, Hurst introduced a dimensionless ratio by dividing the range *R* by the standard deviation *S* of the observations. The range of *Rn* is the difference between the maximum and minimum levels of accumulated deviation *Xn*.

$$R\_n = \max\left(X\_k - \frac{k}{n}X\_n\right) - \min\left(X\_k - \frac{k}{n}X\_n\right),\tag{3}$$

where *Xn* = *x*<sup>1</sup> + *x*<sup>2</sup> + ... + *xn*, *n* ≥ 1;

*Xn*—accumulated deviation in n steps (*t* periods); *Rn*—deviation range in *n* steps, where

*Sn* = *n* ∑ *k*=1 *xk*−*x*<sup>2</sup> *n <sup>n</sup>* —empirical standard deviation; *xn* = *Xn*/*n*—empirical average; *Rn*/*Sn*—normalized range of accumulated sums *Rk*, *k* ≤ *n*.

Based on the formula for Brownian motion, the system displacement (normalized range) Hurst proposed to calculate using the following relation:

$$R\_n / S\_n = (at)^H,\tag{4}$$

from where

$$H = \frac{\ln(R\_n/S\_n)}{\ln(at)},\tag{5}$$

where *H* is the Hurst exponent, varying from 0 to 1; *Rn*/*Sn* is the rescaled range; *t* is the studied period, and *a* is a constant. The value of the coefficient *H* characterizes the ratio of the strength of the trend (deterministic factor) to noise level (random factor). The indicator *H* is a tool for determining the systems persistence behavior and gives an answer to the question of what the next value of the investigated series would be, more or less than the current value. The processes for which *H* = 0.5 have an independent data distribution, and are characterized by the absence of a trend (classical Brownian motion). Time sequences for which *H* is greater than 0.5 are classified as persistent, preserving the existing trend. If the increments were positive for some time in the past—there was an increase—then on the average there will be an increase, this corresponds to a good predictability of the series. Thus, for a process with *H* > 0.5 there is a tendency to increase in the future, the effect of long-term memory is preserved. The case 0 < *H* < 0.5 is characterized by antipersistence and is characterized by an alternating tendency.

On a large empirical material, it was found that the Hurst index value of the series of various natural processes is grouped in the interval *H* = 0.72–0.74. [38,41]. The question of why this is so remains open. Note that our average value of the Hurst index for time series of temperatures *H* = 0.74, also fell in this interval.
