Stochastic correlation, regression probabilistic-statistical models of laminated composite structures with material-energy-saving polyurethane thermal insulation

. After testing according to the developed state methods and procedures for polyurethane foam thermal insulation of laminated composite structures with metal facings, the issues of correlation, probabilistic-statistical regression models between the signs of elasticity in tension Eshear and signs of elasticity in shear Gshear were investigated. It is shown that the test results are random stochastic values. Their variability, depending on the type of tests and the parameter under study, is in the acceptable average values. The presence of a significant number of each of the characteristics, for example, Eshear - Gshift, predetermined the use of correlation tables to establish the fact of a relationship. Modules of elasticity under tension and shear were determined. The correlation coefficient between Eshear and Gshift is 0.659. With the help of a computer program, correlation tables were built and the calculation of the probabilistic-statistical interaction of characteristics with different reference points was made. When considering one-dimensional aggregates, the laws of distribution of positive values of the characteristics of foams can be adopted of various types. The connection between phenomena can be not only linear, but also non-linear. In this case, nonlinear correlation can be realized in the form of parabolic and other equations of a certain degree. The calculation of the interaction is carried out on the basis of second-order parabolic equations. Approximation in the form of second-order equations does not greatly improve the convergence, but the possibilities for extrapolating statistical models are reduced.


Introduction
In works [1 -3], the authors, on the basis of numerous tests of samples, including full-scale (one-span, two-span), according to the methods and procedures approved by state standards, reliably showed that the test results are random stochastic values. Their variability (variation) depending on the type of tests and physical parameters (strength, elasticity, creep, thermal conductivity, etc.). It is in acceptable averages (сi d 33%, where сi is the variance with Sheppard's corrections). Almost homogeneous for each individual parameter Ri, Oi, 'i, etc. This is due to a typical continuous technological process on the same equipment.
Correlation here is understood as the dependence of the variability (variation) of a trait on the surrounding conditions according to Galton, Pearson -a ratio, a correspondence, which can be functional, correlative.

Research methods
In what follows, we will consider only complex correlations, the study of which solves a wide range of issues [4 -7]. A preliminary analysis of the simulated population is carried out, the presence of the fact of connection, its direction and form is established, the degree of tightness of the connection between the parameters is measured. The construction of a probabilistic-statistical regression model (analytical expression of the relationship) is carried out, an assessment of the adequacy of the model, its economic interpretation and practical use are given.
Let us emphasize once again the essential scientific significance of correlation and regression. The presence of a significant amount of each of the features, for example, elasticity Eshear -Gshift at N = 68, predetermined the use of correlation tables to establish the fact of the presence of a relationship ( Table 1). The factorial attribute of elasticity Eshear, i is located in columns, the attribute Gshift, i is located in rows. At the intersection of rows and columns, the frequency of repetition of the combination Eshear, Gshift is indicated, the total value of which 6nE, 6nG determines the limiting laws of the probability distribution. The values of the tensile modulus of elasticity ‫ܧ‬ ሜ = 99.27 Pa · 10 5 , the shear modulus ‫ܩ‬ ሜ = 37.56 Pa · 10 5 and their root-mean-square values ܵ ா ೞೌ = 20.19 and ܵ ீ = 8.64 have been reliably established. Determination of fluctuations (changes) of elastic characteristics from these data with сEp = 20.19 / 99.27 = 0.203; сG = 8.64 / 37.56 = 0.23, we are convinced that the correlation of features Eshear and Gshift is Gaussian normal.

Research results
Let us refer to table 1 of the distributions of the probabilities of the tensile modulus of elasticity Eshear and the shear modulus Gshift with the number of interaction pairs equal to 68.
As you can see from table 1, the probabilities are located from right to left. They are centered around a single straight line. Therefore, it is possible to hypothesize about the linear form of the interaction between the elastic characteristics. An analytical expression must be found for this interaction. And so that the sum of the squares of the deviations from the straight line reaches the smallest value. This will be the best fit of the theoretical curve to the experimental results. The equation of a straight line of regression of Eshear on Gshift (the indices for Ep and Gsdv will be omitted for convenience in the future) can be found from the test results: with the number of equidistant intervals in i = (1, 2 .... 8) and j = (1, 2.... 8). For simplicity of calculations, let's move on to conditional options using the following formulas: , -"false zeros" adopted for the intervals with the highest frequency -G6 = 39.9 Pa·10 5 and E7 = 112.0 Pa·10 5 ; C1 and C2 -reduction factors for G and E, respectively, equal to 11.9 Pa · 10 5 and 4.3 Pa · 10 5 .  , -"false zeros" adopted for the intervals with the highest frequency -G6 = 39.9 Pa·10 5 and E7 = 112.0 Pa·10 5 ; C1 and C2 -reduction factors for G and E, respectively, equal to 11.9 Pa · 10 5 and 4.3 Pa · 10 5 .
For convenience, the correlation table, taking into account the "false zeros", is combined with the values and obtained by the formula (2), they are shown on the side panels of the table: We determine the average values of ‫ܩ‬ ሜ ᇱ and ‫ܧ‬ ሜ ᇱ , taking into account the "false zeros" (where the dash above G and E means the average values): Here n is the number of test pairs equal to 68. Let us find the auxiliary values ( ' Let's determine the values of the standard deviations taking into account the "false zeros": Then the values of the standard deviations are equal: SG = SG'C1 = 2,01 4,3 = 8,64 Па10 5 ; SE=SE'C2=1,7 11,9=20,19 Па10 5 Here C1, C2 are the values of the reduction coefficients equal to 4.3 and 11.9 Pa 10 5 , respectively. Let's calculate the correlation coefficient rEG: For example, for a cell nGE is equal to 2, the product of ¦nGE· Gc = 2 · (-5) = -10 and ¦nGE·Е' = 2 · (-6) = -12. Summing the values of ¦nGE· Gc and ¦nGE·Е' over the rows and columns of the correlation table, we obtain the values of U and -: For example, for the first column: -= ¦nGE·Е' = -12 -5 = -17, for the second line: These amounts are written in the sidebars of Table 1: Summing up the values of the obtained series taking into account the signs, we have: The convergence of the sums means that all calculations were performed correctly. Then the sought correlation coefficient according to the formula (3) will be equal to: The interaction between Е and G is found from the equation: Substituting the values of the obtained parameters into the equation, we have:   Here Еshear theor and Еshift exper are the theoretical and experimental values of the tensile modulus.
The above algorithm was used to compose a computer program. According to the program, correlation tables were obtained and the calculation of the probabilistic-statistical interaction between characteristics with different reference points was made. At the same time, the values of the reduction factors varied within wide limits. This was achieved by varying the number of intervals for i and j. The calculation of the probabilistic-statistical interaction was carried out for both square and rectangular initial matrices with the number of intervals from five to twelve. In this case, as a rule, the parameters of equations (1 -4)mean values, standard deviations, correlation coefficients, free terms were somewhat different from each other. For example, for this calculation with a 5 x 5 matrix, they were equal: G = 39.19 Pa10 5 ; E = 102.3 Pa10 5 ; SG = 8.4 Pa·10 5 ; SE = 20.32 Pa·10 5 ; rE = 0.583. The grouping of test results affects the final results, but not significantly. The calculation of the characteristics of interaction with a new reference point increases the reliability of the calculations, allows for a comprehensive assessment of the phenomena under study. It is especially needed when considering non-linear correlation in a limited number of trials.
As we have established when considering one-dimensional aggregates, the distribution laws of essentially positive quantities, which are all the characteristics of foamed plastics, can be adopted in various forms. This implies one important point -the correlation between phenomena can be not only linear (in the case of normal laws or those close to them), but also nonlinear. In this case, the nonlinear correlation can be realized in the form of parabolic equations of a certain degree: у = ао + а1х 1 + а2х 2 + . . . акх n , nonparabolic correlation equations: power-law у = ах b , exponential у = аb x , logarithmic y=а + blg х, polynomials, etc. [8 -12]. As noted in [1,4,13], the type of the correlation equation is determined by the nature of the probabilistic-statistical interaction between the studied random variables. As a rule, this character is determined by the normal correlation in the plane. In the case of nonlinear correlation, the problem will be solved according to the principle proposed by P.L. Chebyshev [4,7] -a gradual increase in the order of the correlation equation. First, we calculate the probabilistic-statistical interaction using the first-order correlation equations «n». With their insufficient convergence, especially with insignificant correlation coefficients, we increase the order of the equation to (n+1). If the interaction in this case, the approximation is not enough, we pass to equations of order (n+2), etc. We will assume that if the results obtained by the theoretical calculation are "covered" by the experimental confidence limits, then the approximation at a given level of significance is D = 0,05; D = 0,01 is permissible and meets the assigned tasks The decision on the probabilistic-statistical interaction between characteristics using first-order equations was given earlier. Consider the calculation of the interaction based on second-order parabolic equations: In this case, we will use the same experimental results as for linear correlation. The parameters of function (5) will be found from the condition of minimizing the mean square of deviations [5,7,12]: It follows from this condition that all partial derivatives of the sum Q with respect to the required parameters must be equal to zero: Differentiation and transformation of formulas (6 -7) leads to a system of equations of the form [4,6,8,10]: Let us estimate the probabilistic-statistical nonlinear interaction between the elastic and shear moduli. Using the results given in correlation table 1 and formulas (8). The calculation results are shown in Table 2.  (7) Substituting the obtained values of the sums (the last row of Table 2) into equation ( Therefore, the probabilistic-statistical interaction between E and G is determined by a second-order nonlinear equation: Еi = -0,043Gi 2 + 4,25 Gi + 3,61. A comparison of the experimental and theoretical results obtained using this formula is given below: The convergence of experimental and theoretical results with a nonlinear correlation between E and G improved on average by 1 -2% at each interval in comparison with the linear correlation. The obtained theoretical results are completely "covered" by the confidence limits for the experimental results with the coefficient of СЕ variability -15 -20%. Taking into account that the approximation of theoretical results to experimental ones does not exceed 2% on average and is "covered" by confidence limits, taking into account random fluctuations of experimental results, we believe that further approximation in the form of parabolas of the third or fourth order does not make sense. According to the given algorithm in the form (8), a computer program was compiled. The program is designed in such a way that at the beginning the parameters of the linear correlation equations are determined and the linear probabilistic-statistical interaction is calculated. Using some necessary results of this calculation, we determine the parameters of nonlinear correlation equations. For comparison, the experimental conditional means are determined. When analyzing the results of marginal distributions for two characteristics and parameters of the obtained linear and nonlinear equations, a decision is made on the type of connection and on a further increase in the order of parabolic equations.
As shown by the analysis, the most expedient is linear correlation, since the interaction between characteristics is determined by the normal laws of their distribution. The second approximation in the form of second-order equations improves convergence by only one to two percent, but the extrapolation capabilities of statistical models are reduced.

Conclusions
1. Tests of elastic properties of polyurethane foam thermal insulation Ep,, Gсдв were carried out according to methods, procedures, state standards, developed by the authors with the number of interaction pairs N = 68.
2. The test results are grouped in the form of a correlation table using the Sturges interval widths.
4. An estimate of the correlation coefficient ‫ݎ‬ ாீ = 0,659 is given for a straight-line relationship between Ep and Gсдв . It is compared with the curvilinear correlation of the parabolic type with a difference of no more than two percent. A comparison table is provided. 5. A regression analytical model Ep -Gsdv is given, which makes it possible to predict the development of Ep, Gsdv by linear probabilistic-statistical interaction.