On the paper is as follows: Section 2 describes the methodology, in the method to get the time series towards the creating with the fractal exponents made use of; Section three shows ourMathematics 2021, 9,3 ofresults, which includes the clustering in the stations; finally, we interpret and go over our final results in Section 4, and show some concluding remarks in Section five.K penclimateclassificationBsh(Hotsemi-arid) BWh(Hotdesert) Cwa(Humidsubtropical) Cwb(Subtropicalhighland)Figure 1. CONABIO’s clustering of R Bravo-San Juan Basin, Mexico, by utilizing the K pen climate classification [35].2. Methodology In our case study, 17 of 34 weather stations of your RBSJ Basin have been analyzed. They’re situated inside the northeast region of Mexico, covering an approximate region of 29,420 km2 . The chosen stations had been these possessing uninterrupted records inside the period of years 1964016, this in order to have homogeneity inside the high-quality with the data for the time series of all the stations. The region of study, like the 17 georeferenced stations are shown in Figure 1, tagged by their station number. For every weather station we’ve facts of your month-accumulated rainfall, which leads to 636 data points for each and every series. We applied month-to-month data since it is definitely the accessible public facts reported by CONAGUA. two.1. Hurst Exponent (Hrs) The version in the Hurst exponent we take into consideration is the rescaled range, in which the information of each station is represented as vectors ( Xk)k1:N , exactly where N could be the variety of months Fadrozole Protocol regarded for the evaluation. Every single vector ( Xk) is divided into a set of d-Ionomycin Protocol subseries of shorter time with the same quantity of data, such that all the n-th subseries, n = 1, two, . . . , d, possess a length of m. Then, in line with [36,37], the following actions are carried out for each subseries: 1. 2. Calculate the mean Mn plus the standard deviation Sn in the subseries. Establish the variation Vin of every single term with respect to the imply: Vin = Xin – Mn , for i = 1, . . . , m three. Obtain the accumulated sum of variation till the ith-term: Yin = four. (1)j =Vjni, for i = 1, . . . , m(2)Calculate the array of each subseries: Rn = max Yjn – min Yjnj=1:m j=1:m(three)Mathematics 2021, 9,four of5.Normalize the calculated ranges (that is why it is actually called rescaled range): R n / Sn (4)6.After this is accomplished for each and every subseries of length m, they’re averaged: R n / Sn = 1 dn =dRn Sn(five)7.Finally, the relation from the statistic Rn / Sn is provided by the following power law: R/Smcm H(six)where H refers to Hrs and its a dimensionless number, and c 0 can be a continuous. In this way, Hrs might be estimated by rewriting (7) in logarithmic terms: log10 R/Sm= log10 (c) H log10 (m),(7)in order that Hrs is definitely the slope with the straight line. In this function, we employed a total of d = 53 partitions, every of them having a length of m = 12 months. two.2. Higuchi Fractal Dimension (HFD) Higuchi fractal dimension (HFD) approximates the fractal dimension in a related way to Hrs, in the sense of dividing the total time series in windows, nonetheless, their procedure are some diverse. To get a time series X, with N information points [38], N-1 N -m 2 k kN -m kLm (k) =i =| X N (m ik) – X N (m (i – 1)k)|(eight)m expresses the fractal length on the subseries Xk , where k 1, . . . , k max indicates the k-th time, m 1, . . . , k will be the interval time of that subseries, and k max may be the total quantity of subseries utilised. In this perform, we used k max = 8, that is a considerable quantity about an intermediate value involving the K pen classification by semesters [35] as well as the.