As r �� 0, the correlation exponent, d, is defined asC(r)��rd.(4)It is apparent that the correlation exponent, d, is given by the slope coefficient of ln C(r) versus ln r. According to (ln r, ln C(r)), d can be obtained by the least squares method (LSM) using a log-log grid (as shown in Figure Regorafenib Sigma 2).Figure 2A plot of ln C(r) versus ln (r).To detect the chaotic behavior of the system, the correlation exponent has to be plotted as a function of the embedding dimension (as shown in Figure 3). Figure 3The correlation exponent (d) versus embedding dimension (m).If the system is purely random (e.g., white noise), the correlation exponent increases as the embedding dimension increases, without reaching the saturation value.

If there are deterministic dynamics in the system, the correlation exponent reaches the saturation value, which means that it remains approximately constant as the embedding dimension increases. The saturated correlation exponent is called the correlation dimension (CD) of the attractor. The CD belongs to the invariants of the motion on the attractor. It is generally assumed that the CD equals the number of degrees of freedom of the system, and higher embedding dimensions are therefore redundant. For example, to describe the position of the point on the plane (two-dimensional system), the third dimension is not necessary because it is redundant. In addition, the CD value is often fractal and represented as a nonintegral dimension, which is typical for chaotic dynamical systems that are very sensitive to initial conditions.

The CD value provides the information regarding the dimension of the phase-space required for embedding the attractor. It is important for determining the number of dimensions necessary to embed the attractor and the number of variables present in the evolution of the process.We used the previous correlation dimension method to analyze the chaotic and fractal characteristics for the temperature dynamics in this study.3.2. Correlation Analysis and Stepwise RegressionCorrelation and regression analyses are the two commonly useful methods in various disciplines of geography [24], which were Cilengitide used to check the correlations between the CD value with geographical location and elevation in this study.The correlation analysis is one of the most useful classical statistics, which is a statistical measurement of the correlationship between two variables. Possible correlations range from +1 to �C1. A zero correlation indicates that there is no relationship between the variables. A negative correlation indicates that as one variable goes up, the other goes down. A positive correlation indicates that both variables move in the same direction together.