Correlation Analysis of Experimental Data Applied in the Study of the Extraction Process the correlation analysis of experimental data

The article examines the application of correlation analysis of experimental data in research into the process of extracting bioactive compounds and antioxidant activity in plant extracts from berries and grape pomace. The correlation analysis of the experimental data allowed the establishment of the second order statistical characteristics (autocorrelation function, intercorrelation function and correlation coefficient). Based on the correlation analysis of the experimental data, it was shown that the influencing factor and the measured parameters have zero correlation coefficients for all types of researched extracts. This indicates that they are not independent. Therefore, related mathematical models can be deduced.


Introduction
The correlation analysis uses second-order statistical characteristics (second-order probability density) and ensures the assessment of the correlation of the experimental data (whether independent or not and what is the nature of the dependence in the latter case) and the estimation of their nonlinear character (existence of a nonlinear component in experimental series) [1].
Although first-order statistical features that use first-order probability density are frequently used, they do not give a complete picture of the character of a deterministic series. Indeed, two experimental series may have the same mean and dispersion, but their character of variation may be different [2] [3].
The aim of this study is to apply the correlation analysis of experimental data

Materials
In this study, for the correlation analysis were used experimental data from the extraction of bioactive compounds from chokeberry, rosehip, rowan, hawthorn, and grape marc. Both berries and grape marc are known for their rich content in bioactive compounds [4] [5] [6] [7] [8]. In the obtained hydroalcoholic extracts (ethyl alcohol) the content of biologically active compounds (polyphenol index and total anthocyanin content) and the antioxidant activity were determined by 3 different methods ( Hydrogen peroxide scavenging activity in the acidic medium % P7 Hydrogen peroxide scavenging activity in the basic medium % P8 Note: AAE-ascorbic acid equivalents; TE-trolox equivalents; ME-malvidol glycoside equivalents; c.u.-conventional units; DPPH-2,2-diphenyl-1-picrylhydrazyl.

Methods
The autocorrelation function (R xx ) was applied to characterize the internal structure of any series x, which in the discrete domain such as experimental data is determined by knowing that the n values of any series x i are arranged at equal time intervals h and as a result x i = x (ih). Considering the time shift τ = rh, the expression of the autocorrelation function results (with m the maximum possible shift): The autocorrelation function shows the degree of correlation of the experimental data. Thus, a perfect symmetry on the discrete time axis τ of the graph of the autocorrelation function shows the existence of a perfect linear dependence between data at different time points. In addition, the slower the autocorrelation function tends to zero, the better autocorrelation of the data is obtained.
The statistical properties of two random experimental series x and y are characterized by the intercorrelation function, denoted R xy , which in the case of a finite discrete series, has the expression (with m the maximum possible shift and τ = r·Δt): Frequently, for determining the intercorrelation (interdependence) type between two quantities, that can be linear or nonlinear, direct or indirect, the correlation coefficient (Pearson's coefficient) is used, which for two series x and y is determined from the relation [3] [9]: with values In expression (3) a maximum possible intercorrelation (a perfect linear dependence) is for ρ 2 = 1. If ρ = 1. Then there is a perfect direct linear dependence, and if ρ = −1 then there is a perfect indirect linear dependence. If 0 < ρ ≤ 1 there is a direct dependence, and if −1 ≤ ρ < 0-there is an indirect dependence (when x decreases, y increases and vice versa). Finally, if ρ = 0, then the two targeted quantities are independent. Therefore, the further ρ 2 is from the unit value (without reaching the zero value), the more accentuated the nonlinearity.
Correlation analysis was performed in the MATLAB program version R2020b (MathWorks, Inc., Natick, MA, USA) [10]. The results presented for the autocorrelation function remain valid for the intercorrelation function as well. If the graph of the intercorrelation function does not suddenly tend to zero, then there is a good intercorrelation of the experimental data. Also, the symmetry of the graph of the intercorrelation function indicates the degree of linear dependence between the two quantities.

Results and Discussions
An example of the intercorrelation function is shown in Figure 2 for chokeberry and grape marc extracts. The graphs in Figure 2(a) and Figure 2 Similarly, Figure 3 shows the values of the correlation coefficients for the other 4 types of plant extracts (sea buckthorn, rosehip, rowan, and hawthorn).
And in these cases, all the correlation coefficients have subunitary and non-zero To illustrate the significance of the correlation coefficient from Figure 3, parameter P2 from rosehip extracts (Figure 3(b)) and parameter P6 from hawthorn extracts (Figure 3(d)) were chosen. The graphs in Figure 5 confirm the aspect mentioned above, that between the measured parameters there are dependencies not only between the alcohol concentration and the determined parameters. Indeed, the non-zero values of the correlation coefficients (here in hawthorn and chokeberry extracts) show that the parameters are not independent.
Analogous to the graph in Figure 4, Figure 6 provides two examples with the Journal of Applied Mathematics and Physics significance of the correlation coefficient set in Figure 5. Thus, Figure 6(a) confirms the existence of an indirect quasilinear dependence between parameters P1 and P5 in hawthorn extracts, the correlation coefficient having a negative value close to the unit value. Similarly, Figure 6(b) confirms the existence of a direct quasilinear dependence between parameters P1 and P3 in the chokeberry extracts, the correlation coefficient having a positive value close to the unit.
The fact that the measured parameters are not independent of each other allows the establishment of mathematical models not only between the concentration of ethyl alcohol (as an influencing factor) and these parameters but even between the measured parameters [11] [12] [13] [14].

Conclusions
The correlation analysis of the experimental data allowed to establish the second order statistical characteristics (autocorrelation function, intercorrelation function and correlation coefficient). The correlation of the experimental data was assessed whether they are independent or dependent and what is the nature of the dependence. The nonlinear nature of the experimental series was estimated.
Based on the correlation analysis of the experimental data it was shown that the influencing factor (ethyl alcohol concentration) and the measured parame- For rosehip and rowan extracts, the correlation coefficients between the measured parameters and the concentration of ethyl alcohol are negative, which means that on the whole of the measurements for these products there is an indirect dependence between the mentioned quantities.
The non-zero values of the correlation coefficients between the various parameters show that in all 6 types of extracts there are interdependencies between the measured parameters, so they are not independent. The non-unit values of the correlation coefficients between the various measured parameters indicate that there are nonlinear interdependencies between them.