Formation of Regression Model for Analysis of Complex Systems Using Methodology of Genetic Algorithms
Anatolii V. Mokshin, Vladimir V. Mokshin, Diana A. Mirziyarova
FFormation of Regression Model for Analysis of Complex SystemsUsing Methodology of Genetic Algorithms
Anatolii V. Mokshin , Vladimir V. Mokshin , and Diana A. Mirziyarova Institute of Physics, Kazan Federal University,420008 Kazan, Russia Institute for Computer Technologies and Information Protection,Kazan National Research Technical University named after A.N. Tupolev-KAI,420111 Kazan, Russia Almetyevsk branch of Kazan National Research Technical University named after A.N. Tupolev-KAI,423461 Kazan, Russia
E-mail: [email protected] study presents the approach to analyzing the evolution of an arbitrary complex systemwhose behavior is characterized by a set of different time-dependent factors. The key requirementfor these factors is only that they must contain an information about the system; it does notmatter at all what the nature (physical, biological, social, economic, etc.) of a complex system is.Within the framework of the presented theoretical approach, the problem of searching for non-linearregression models that express the relationship between these factors for a complex system understudy is solved. It will be shown that this problem can be solved using the methodology of genetic(evolutionary) algorithms. The resulting regression models make it possible to predict the mostprobable evolution of the considered system, as well as to determine the significance of some factorsand, thereby, to formulate some recommendations to drive by this system. It will be shown thatthe presented theoretical approach can be used to analyze the data (information) characterizing theeducational process in the discipline ”Physics” in the secondary school, and to develop the strategiesfor improving academic performance in this discipline.
I. DESCRIPTION OF THE APPROACHA. Factors and non-linear regression model.
Let evolution of a complex system be characterizedby a finite set of time-dependent factors. By externalfactors , we mean those factors that are affected by thesystem. Then, the internal factors are those factors thatcontain information generated by the system itself. Us-ing external factors, it is possible to drive the consideredsystem by setting the values of these factors. Internalfactors characterize the response of the system. There-fore, their values can only be adjusted through externalfactors. To clarify this, we give the following examplefrom physics. Let some polymer material undergo ex-ternal mechanical stress, namely, shear deformation. Inthis case, the force causing the shear deformation canbe considered as an external factor. The values of thisforce can be adjusted. In turn, as a result of the sheardeformation, some structural changes in this system mayoccur [8]. These structural changes will be characterizedby the internal factors: local density, local temperature,and local tangential stresses. In the general case, this sit-uation corresponds to a nonequilibrium physical process.Therefore, the relationship between these factors cannotbe described in terms of any currently known physicallaws; these relationships can be nontrivial [9, 10].In accordance with the standard methodology of re-gression analysis [4], it is necessary to introduce the con-cepts of input and output factors. By an output factor we mean a factor by which the evolution of the systemis monitored. It is necessary to note that the number of the output factors may be more than one, but it mustbe finite. All other factors x ( t ), x ( t ), x ( t ), . . . , x M ( t )will be defined as input factors . Here, M is the numberof the input factors. The choice of an output factor isdetermined by the purpose of the study. Therefore, theattribution of the factors to input or output factors maybe a matter of convention.Let there be a relationship between the output factor y ( t ) and the input factors x ( t ), x ( t ), x ( t ), . . . , x M ( t ),which is defined as the generalized Kolmogorov-Gaborpolynomial [11]: y ( t ) = a + M (cid:88) i =1 a i x i ( t ) + M (cid:88) i =1 M (cid:88) l>i a il x i ( t ) x l ( t ) + M (cid:88) i =1 M (cid:88) l>i M (cid:88) k>l a ilk x i ( t ) x l ( t ) x k ( t ) + . . . , (1)where a , a i , a ( i,l ) , . . . are weight coefficients whose val-ues are initially unknown. These coefficients determinethe contribution of the corresponding input factors x i ( t ), i = 1 , , . . . , M or the product of these input factors inEq. (1) to obtain a correct value of the output factor y ( t ).In other words, the weight coefficients can be consideredas a quantitative measure of the significance of the inputfactor or the corresponding product of the input factors.Obviously, the greater the number of the input factors,the more precisely it is possible to reproduce the targetvalues of the output factor y ( t ). However, as can be seenfrom expression (1), the greater the number of the inputfactors, the more complex this expression. In general, inorder to be able to formulate expression (1), it is neces-sary that value of M be finite. a r X i v : . [ phy s i c s . d a t a - a n ] N ov Since expression (1) contains all the possible combi-nations of the products of the input factors, then theproblem of finding expression (1) in its explicit form re-duces to finding values of the weight coefficients a , a i , a il , . . . . B. Search for the type of regression model as anoptimization problem.
Expression (1) can be rewritten in a form which willbe common for the products of the input factors x ( t ), x ( t ), x ( t ), . . . , x M ( t ): y ( t ) = a · x δ (1)1 ( t )1 · x δ (1)2 ( t )2 · x δ (1)3 ( t )3 · . . . · x δ (1) M ( t ) M ++ a · x δ (2)1 ( t )1 · x δ (2)2 ( t )2 · x δ (2)3 ( t )3 · . . . · x δ (2) M ( t ) M ++ a · x δ (3)1 ( t )1 · x δ (3)2 ( t )2 · x δ (3)3 ( t )3 · . . . · x δ (3) M ( t ) M + · · · == M (cid:88) i =1 a i · x δ ( i )1 ( t )1 · x δ ( i )2 ( t )2 · x δ ( i )3 ( t )3 · . . . · x δ ( i ) M ( t ) M == M (cid:88) i =1 a i M (cid:89) j =1 x δ ( i ) j ( t ) j , (2)where i is the order number of the product of the inputfactors, and j is the index of the input factor in the cor-responding product, and the quantity δ ( i ) j characterizesthe exponent and can take the values 0 or 1, i.e. δ ( i ) j = 0or δ ( i ) j = 1. We introduce the next notation:¯ X i ( t ) = M (cid:89) j =1 x δ ( i ) j ( t ) j == x δ ( i )1 ( t )1 · x δ ( i )2 ( t )2 · x δ ( i )3 ( t )3 · . . . · x δ ( i ) M ( t ) M , (3)where the factors in this product are written in ascend-ing order of their indices. Then, expression (1) takes theform y ( t ) = M (cid:88) i =1 a i ¯ X i ( t ) . (4)From a mathematical point of view, expression (4) is thescalar product of two vectors¯¯ a = { a , a , a , . . . , a M } (5)and ¯¯ X ( t ) = { ¯ X ( t ) , ¯ X ( t ) , ¯ X ( t ) , . . . , ¯ X M ( t ) } , (6)and can be written as y ( t ) = (¯¯ a, ¯¯ X ( t )) . (7) Then, the search for expression (1) [or equivalent ex-pression (2)], which would correctly reproduce values ofthe output factor y ( t ), is reduced to the search for theexponents δ ( i ) j in the products ¯ X i ( t ). We note that this isa typical optimization problem [12–15]. In the case whenthe number of the input factors is not large, such prob-lems are solved by the so called brute force method [16].However, when there are a sufficiently large number ofthe input factors, solution to this problem can be ob-tained using the machine learning methods, for example,using the genetic algorithms [5, 17–22]. C. Genetic Algorithms. Basic definitions andmethodology.
Genetic algorithms is aimed to solve the optimizationproblem and are based on the ideas typical for a naturalevolutionary process. This allows one to avoid to performa consequent consideration of all the possible forms of theequation for the output factor y ( t ) with various combina-tions of the input factors to find the optimal solution [5].The common GA methodology includes the use of basicnotions such as individual , genome and population as wellas the two specific operations [5]. As applied to solvingthe problem of finding the optimal regression model for y ( t ), we introduce the following definitions:(i) We define a model polynomial of the form (2) forthe output factor y ( t ) as an individual . In general case,the concrete form of an individual y ( t ) can be specifiedarbitrarily or in accordance with some rule. It is reason-able to take a model polynomial for y ( t ) as an individual,because the this quantity is of the main interest in thegiven problem and this quantity should be found directlyby means of the genetic algorithms [23].(ii) According to the GA methodology, uniqueness ofan individual should be determined by a genome . Asseen from expression (2) and Eq. (3), all input factorsare contained in each product ¯ X i ( t ), and these productsare included in the model polynomial for the output fac-tor y ( t ). Account for the concrete input factor (say, thefactor x j ) in a product ¯ X i ( t ) is due to a value in expo-nent δ at this factor; recall that this value can be 0 or1. So, for example, when we need to take into accountthe input factor x j in the product ¯ X i ( t ), we should takethe exponent δ ( i ) j = 1. If we need to exclude the inputfactor x j from consideration, then we take the exponent δ at the corresponding factor as δ ( i ) j = 0. Thus, each se-quence of products of input factors will be characterizedby a unique set of ones and zeros. Then, it is reasonableto define this unique set of zeros and units as a genome .To clarify this, we provide the following example. Letthere be four input factors x , x , x and x . Then, asome product, say, ¯ X takes the following form:¯ X = x x x x ≡ x x ≡ x x , (8)where the set of exponent values { , , , } is a binarynotation for the digit 5, i.e. for the index i = 5. Inother words, a natural correspondence appears betweenthe product index (in the given example with ¯ X , this is i = 5) and the input factors, which are included in thisproduct. Recall that the input factors in expression (8)should be written in order of increasing index.(iii) By population we mean a set of individuals: y (1) ( t ) = (cid:88) i a (1) i ¯ X i ( t ) ,y (2) ( t ) = (cid:88) i a (2) i ¯ X i ( t ) ,y (3) ( t ) = (cid:88) i a (3) i ¯ X i ( t ) , (9) · · · ,y ( s ) ( t ) = (cid:88) i a ( s ) i ¯ X i ( t ) , · · · , where s is the individual index in the population. Eachmodel polynomial y ( s ) ( t ) of set (9) can be considered assome possible solution for the output factor y ( t ).In addition, in accordance with the GA methodol-ogy, two basic evolutionary operations must be defined:crossover and mutation operations [6, 24–26].(iv) We define the crossover operation as the opera-tion of obtaining a new pair of the ”child” individuals byexchanging the right fragments of the vectors ¯¯ X ( j ) and¯¯ X ( k ) , which belong to two ”parent” individuals y ( j ) and y ( k ) of the same population [see Eqs. (6) and (7)]. Here, j and k are the indices of the individuals in the popu-lation. Namely, to generate a new pair of individuals,it is necessary to specify the so-called separation point ,which will be a natural number n , and n ∈ [1 , M + 2].Here, the largest value of n is 2 M + 2, because there are2 M various forms of the products ¯ X of the input factors[see Eq. (2)] and two contributions ¯ X ∗ = 0 and ¯ X = 1in the same population. The point corresponding to thisnatural number n will divide the vectors ¯¯ X ( j ) and ¯¯ X ( k ) into left and right parts. By swapping the right-handsides of these vectors, we get a new pair of the vectorsand, thereby, we get a pair of the ”child” individuals.This operation is illustrated by the following scheme (seeFig. 1)(v) Mutation is an operation due to which a new in-dividual with a changed gene appears [27]. Recall that,in accordance with our methodology, the gene is repre-sented as a set of ones and zeros [see, for example, (8)],and this set characterizes the presence or absence of afactor x j in a product ¯ X i . Therefore, to obtain a newindividual by means of the mutation , we need to realizeinversion of a random digit in this set: namely, zero in-verts to one, or vice versa. So, for example, in the case ofthe product ¯ X , that was given above, a possible resultof the mutation can be¯ X = x x x x ⇒ ¯ X = x x x x ≡ x x x ≡ x x x . (10) Here, the product ¯ X appears directly as a result of themutation with inverse procedure of the exponent for theinput factor x . D. Implementation of GA Methodology.
The main purpose of the above methodology is search-ing the most optimal polynomial y ( t ), which is expressedas (1) and is able to reproduce the experimental (target)values of the output factor. Let we have a dataset forthe time period [0 , T ]. To implement the GA, it is nec-essary to define two ranges that correspond to learningsampling and test sampling [6, 24]. A rigorous approachrequires solving the problem of finding the optimal sizeof these samplings. It should be noted that if the sam-pling sizes turn out to be small, then the available data inthese samplings will not be enough for the proper statis-tical analysis, and, therefore, the resulted model for y ( t )will not be of a sufficient accuracy. On the other hand,the larger the sizes of these samplings, the more com-puting resources are required. Thus, the solution to thisproblem implies finding an optimal size of the samplings.In the simplest implementation, one can set the learningand test samplings of a same size. Moreover, to havea possibility to do a forecast with the data, the overalltime range [0 , T ] can be divided into three ranges of thesame size: [0 , T / T / , T /
3] and [2
T / , T ], whichwill be associated with learning sampling, test samplingand forecast sampling , respectively.The first stage of the methodology implies the forma-tion of a trial population of individuals y ( s ) ( t ), where s = 1 , , . . . . Data corresponding to the learning sam-pling (for example, the time range [0 , T / y ( s ) ( t ). This isa learning regime . ”Experimental” data for the outputfactor y ( t ) from the test sampling (for example, fromthe time range [2 T / , T ]) should be used to test how theresulting models y ( s ) ( t ) are able to reproduce the “exper-imental” data. To estimate quantitatively the accuracyof a resulting model, a matching criterion ∆ – e.g., theFisher criterion, the standard deviation, the coefficientof variation or the coefficient of determination – shouldbe used. The matching criterion ∆ is computed for eachpolynomial y ( s ) ( t ) of a population, and, thus, the set of∆ ( s ) is formed [24].The second stage of the GA implies numerical calcu-lation of the set of vectors ¯¯ a [see expression (7)], as wellas finding the explicit form of the polynomials y ( s ) ( t ) ofa population and finding the values of the correspondingcriteria ∆ ( s ) . Based on the found values of the criterion∆, all the individuals of the population are ranked in or-der of increasing values of ∆. Namely, the individuals y ( s ) , whose the matching criterion ∆ takes the highestvalues, are at the end of the set. Then, the lower halfof the ranked population is discarded. Thus, the rankedset of values of ∆ ( s ) makes a sense of the so-called fit-ness function , which is used in the GA to determine an Figure 1. The crossover operation acceptable solution domain [5].At the third stage of the algorithm, a population isreplenished by means of the crossover operation for theindividuals and the mutation applied for the genes, aswas described above. We note that the crossover oper-ation applied for a random pair of individuals from theremaining half of the population is carried out until theoriginal size of the population will be restored. The mu-tation for different pairs of individuals is performed witha certain given probability.The second and third stages are cyclically repeateduntil the first polynomial y ( s ) of the population rankedby the values of the matching criterion will correspondswith the required accuracy to “experimental” data forthe output factor y ( t ). II. ANALYSIS OF DATA CHARACTERIZINGTHE EDUCATIONAL PROCESS IN THEDISCIPLINE OF ”PHYSICS”
Applying the machine learning methods in an educa-tional process represents a completely new research field,which can be aimed at improving the efficiency and qual-ity of education [28]. The interest in using the machinelearning algorithms for solving various educational prob-lems is growing, as evidenced by the regular appearanceof new studies offering various original approaches [29–33]. In particular, in the review paper [29] it is proposedto use these methods to predict student performance. Atthe same time, it is proposed in Ref. [29] to use thesemethods to make various forecasts on the basis of aninformation relating to the social, age and psychologicalfeatures of students. It is noteworthy that the ideas givenin Ref. [29] are consistent with the ideas proposed beforein Refs. [30, 31]. It is important to note that apply-ing the machine learning methods in a completely differ-ent context for the educational process was discussed inRefs. [32, 33]. Namely, as demonstrated in Refs. [32, 33],it is possible to use the machine learning methods onthe basis of the so-called flexible tree-based algorithmsto provide an accurate and maximally objective assess-ment of the schoolchildren and students performance onthe basis of the test results. The corresponding results exp
Figure 2. Time-dependent factors for the H-class: input factor x indicating the number of students in a lesson, and outputfactor y which indicate the number of schoolchildren whoreceived the assessments “good” and “excellent” in a lesson.Note that the number of schoolchildren in the full class is24. Solid (blue) lines represent experimental data; red starsconnected by solid (red) line is a model result with Eq. (11).The learning, test and forecast samplings are separated byvertical dashed lines. were obtained from a comparative analysis of schoolchil-dren/student performance from nine different countries.In this work, we are implementing the GA methodol-ogy presented in Section I to analyze the data for theeducational process related with the physics discipline inthe concrete secondary school in Kazan (Russia). Theanalyzed temporal period is covered the data for one anda half academic years. This period includes 97 lessons forthe humanitarian designated class (H-class) and the 146lessons for the class designated to natural sciences (NS-class). The data contained the following information,namely, (i) the information about the students assess-ments for each lesson, (ii) the information about whetherthe test work, independent work, laboratory work was ina lesson, whether a demonstration experiment was givenduring a lesson, and also (iii) the information about thetype of hometask for the schoolchildren.The main aim of this study is to evaluate the effi-ciency of teaching physics in these concrete lessons atthis school. As a result of this, the factor y characteriz-ing the number of schoolchildren received the assessments”good” and ”excellent” during a lesson was chosen as anoutput factor. All other factors were assigned to the in-put factors (for detail, see Tab. I). As seen from thistable, all the input factors x i , i = 1 , , . . . ,
11, canbe grouped as the follows: (a) the group characterizingthe organization of a lesson, (b) the group characteriz-ing the assessments of academic performance and (c) thegroup that characterizes the type of homework. Such adivision of the input factors into the groups may makeit possible to understand how the certain groups of fac-tors influence the academic performance. In addition, itis worth noting that we artificially introduced two addi-tional noise factors, x and x , which represented thearrays of random natural numbers. This was done in or-der to verify the accuracy of the algorithm used. So, ifthe algorithm works correctly, then the algorithm shouldrecognize that these noise factors are not associated withthe physics teaching data. Note that the factors x , x , x , x , x , x , x and x can only take two wordings:”yes” or ”no”. In our numerical analysis, we apply thenext correspondence between these wordings and the nu-meric denotations: 0 – ”no” and 1 – ”yes”.Let us consider the results of analysis for the H-class.According to the algorithm presented in Section I D, onthe basis of the data corresponding to the H-class thefollowing regression model was obtained: y ( t ) = a (1) + a (2) x ( t ) x ( t ) + a (3) x ( t ) x ( t ) ++ a (4) x ( t ) x ( t ) + a (5) x ( t ) x ( t ) ++ a (6) x ( t ) + a (7) x ( t ) x ( t ) + a (8) x ( t ) ++ a (9) x ( t ) + a (10) x ( t ) x ( t ) ++ a (11) x ( t ) x ( t ) + a (12) x ( t ) x ( t ) ++ a (13) x ( t ) x ( t ) + a (14) x ( t ) ++ a (15) x ( t ) x ( t ) + a (16) x ( t ) x ( t ) ++ a (17) x ( t ) + a (18) x ( t ) x ( t ) + (11)+ a (19) x ( t ) x ( t ) + a (20) x ( t ) x ( t ) ++ a (21) x ( t ) x ( t ) + a (22) x ( t ) x ( t ) ++ a (23) x ( t ) x ( t ) + a (24) x ( t ) x ( t ) ++ a (25) x ( t ) x ( t ) + a (26) x ( t ) x ( t ) ++ a (27) x ( t ) + a (28) x ( t ) x ( t ) ++ a (29) x ( t ) x ( t ) + a (30) x ( t ) ++ a (31) x ( t ) x ( t ) + a (32) x ( t ) x ( t ) ++ a (33) x ( t ) + a (34) x ( t ) x ( t ) ++ a (35) x ( t ) x ( t ) + a (36) x ( t ) x ( t ) ++ a (37) x ( t ) x ( t ) + a (38) x ( t ) x ( t ) ++ a (39) x ( t ) x ( t ) + a (40) x ( t ) x ( t ) ++ a (41) x ( t ) x ( t ) + a (42) x ( t ) x ( t ) . This regression model was obtained as follows. Sincethe total time domain was T = 97 corresponding to 97lessons, then we have defined the learning sampling ofthe size [0 ,
60] lessons and the test sampling of the size[60 ,
92] lessons. Further, the sampling of the size [92 , y ( t ) for the test sampling aswell as for the forecast sampling.As a result of constructing the regression model for y ( t ) [see Eq. (11)], the occurrence frequency P ( x i ) ofeach the input factor x i , i = 1 , , . . . ,
13, was obtained.In fact, the occurrence frequency, normalized to unity,indicates the probability of the appearance of a certaininput factor during the procedure for finding the modelpolynomial y ( t ). The found frequencies for the inputfactors are presented in Fig. 3(a), and the number neareach symbol (circle) indicates the index of the input fac-tor corresponding the occurrence frequency. Obviously,the higher the occurrence frequency of the input factor x i , the more significant this factor for the behaviour ofthe output factor y ( t ). Fig. 3(b) shows the same oc-currence frequencies in order of decreasing their values.As seen, the frequencies are grouped into three separateregions. The region with the highest frequency values lo-cated in the upper left corner in Fig. 3(b) contains the in-put factors x and x that have the greatest impact on thebehaviour of the output factor y ( t ). Recall that thesefactors characterize whether the laboratory and indepen-dent works were given in a lesson. The second group ofthe input factors, which can be designated as less sig-nificant factors, is located in the middle of the graphshown in Fig. 3(b). This group includes the input fac-tors that provide the following information: whether thetest work was given in a lesson ( x ), whether new knowl-edge was given by teacher in a lesson ( x ), the numberof schoolchildren who received the assessment ”satisfac-tory” in a lesson ( x ), and, finally, whether the demon-stration experiment was given in a lesson ( x ). Otherfactors located in the lower right corner in Fig. 3(b) ap-pears to be the least significant . It is important to notethat the similar results were obtained from the analysisfor the NS-class.Based on the performed analysis, we come to the fol-lowing interesting conclusions:(i) It turned out that the factors characterizing the organization of the lesson , namely, the factors associ-ated with test work, independent work and laboratorywork, most strongly affect the academic performance ofschoolchildren in physics in this secondary school.(ii) It was obtained the nontrivial result that the aca-demic performance is not strongly dependent on howmany schoolchildren attend in a lesson. Thus, there isa contradiction with the well known opinion that theless students are present at the lesson, the higher theiracademic performance should be. Nevertheless, it is im-portant to take into account that the non-trivial result Figure 3. (a) Occurrence frequency P ( x i ) of the considered input factor x i depending on the index of this factor. Note thatthese frequencies were obtained from the procedure of the regression model formation [see Eq. (11)]. Various groups of theinput factors are separated by vertical dashed lines. (b) Occurrence frequencies presented in order of decreasing their values. obtained by us in this study follows from the data for theconcrete discipline in the concrete school, and, therefore,this conclusion is relevant only to this data.(iii) As turned out, the GA algorithm allows one to findthe mathematical model for the academic performance,on the basis of which a realistic forecast for the nearestfuture can be done.(iv) Unexpectedly, the group of factors related withthe homework has a little effect on the student academicperformance.(v) Finally, the noise factors x ( t ) and x ( t ) haveno impact on the academic performance. This conclu-sion follows directly from the obtained distribution P ( x i )presented in Fig. 3(b): these noise factors have the low-est values of the occurrence frequency and are at theright bottom part of the distribution. Consequently, theGA algorithm recognizes these factors as not related tothe real input factors for the considered complex system.This is evidence that the algorithm is resistant to artifi-cially introduced ”disturbances”. III. CONCLUSION
In this study, we present the approach for analyzing theevolution of a complex system. The approach is basedon the GA-algorithms and allows one to solve the spe-cific problems associated with identifying the significantfactors by means of which it is possible to drive by a com- plex system, with optimizing the dynamics of a complexsystem, and with predicting the evolution of a system.As an example, we have considered the data associatedwith teaching the discipline “Physics” in the concrete sec-ondary school in Kazan (Russia). On the basis of the re-gression model generated for the schoolchildren academicperformance, the some results were obtained. In partic-ular, as shown, it is possible to identify the main factorsaffecting the educational process, namely, teaching thediscipline “Physics” in a concrete school. In addition, asit turned out, it is possible to make a probabilistic fore-cast of student performance for some immediate periodof time, provided that the general educational scheme forthe discipline remains unchanged. The results generatedby means of the GA-algorithm for the education data canbe applied to yield the recommendations to improve theefficiency of a specific educational process. It should benoted that the original approach presented in this studycan be applied to any other school discipline. Finally, thisapproach can be generalized to solve the problems asso-ciated with improving the efficiency of the educationalinstitutions as a whole.
ACKNOWLEDGEMENTS
This work is supported by the Russian Foundation forBasic Research (the project No. 18-02-00407). [1] M. Gell-Mann, What is complexity? Complexity. , 16(1995).[2] A. Bunde, M.I. Bogachev, Nonlinear memory and risk es-timation in financial records, in Econophysics approachesto large-scale business data and financial crisis (edited by M. Takayasu et al.), New-York, Springer. (2010).[3] V.V. Sychev, Complex thermodynamic systems, New-York, Springer, 2009.[4] D.A. Freedman, Statistical Models: Theory and Practice,Cambridge, Cambridge University Press, 2009. [5] L.A. Gladkov, V.V. Kureichik, V.M Kureichik, GeneticAlgorithms, Moscow, Fizmatlit, 2006.[6] V.V. Mokshin, I.R. Saifudinov, L.M. Sharnin, M.V.Trusfus, P.I. Tutubalin, Parallel genetic algorithm offeature selection for complex system analysis, J. Phys.:Conf. Series. , 012089 (2018).[7] P.I. Tutubalin, V.V Mokshin, Structural and FunctionalModel of the On-board Expert Control System for aProspective Unmanned Aerial Vehicle, Lecture Notes inElectrical Engineering. , 262 (2020).[8] A.V. Mokshin, B.N. Galimzyanov, J.-L. Barrat, Ex-tension of Classical Nucleation Theory for UniformlySheared Systems, Phys. Rev. E. 87 (2013) 062307.[9] L.F. Cugliandolo, Dynamics of glassy systems,arXiv:cond-mat/0210312. (2002).[10] Z. Jiang, T. Zhao, S. Wang, F. Ren, A novel risk assess-ment and analysis method for correlation in a complexsystem based on multi-dimensional theory, Applied Sci-ences (Switzerland). , 3007 (2020).[11] A.N. Kolmogorov, W. L. Doyle and I. Selin, Interpolationand extrapolation of stationary random sequences, SantaMonica, CA: RAND Corporation, 1962.[12] D.E. Kvasov, Y.D. Sergeyev, Deterministic approachesfor solving practical black-box global optimization prob-lems, Advances in Engineering Software. , 58 (2015).[13] H. Tang, X. Guo, L. Xie, S. Xue, Experimental vali-dation of optimal parameter and uncertainty estimationfor structural systems using a shued complex evolutionmetropolis algorithm, Applied Sciences (Switzerland). ,4959 (2019).[14] H. Hu, Y. Li, Y. Bai, J. Zhang, M. Liu, The ImprovedAntlion Optimizer and Artificial Neural Network for Chi-nese Influenza Prediction, Complexity. , 1480392(2019).[15] XF. Li, ZM. Lu, Optimizing the controllability of arbi-trary networks with genetic algorithm, Physica A: Statis-tical Mechanics and its Applications. , 422 (2016).[16] G. Lindfield, J. Penny, Numerical Methods, Cambridge,Academic Press, 2012.[17] C. Guo, Z. Yang, X. Wu, T. Tan, K. Zhao, Application ofan adaptive multi-population parallel genetic algorithmwith constraints in electromagnetic tomography with in-complete projections, Applied Sciences (Switzerland). ,2611 (2019).[18] N.S. Pyko, S.A. Pyko, O.A. Markelov, O.V. Mamontov,M.I. Bogachev, Quantification of the feedback regulationby digital signal analysis methods: Application to bloodpressure control efficacy, Applied Sciences (Switzerland) , 209 (2020).[19] I. Grigorenko, M. E. Garcia, Ground-state wave func-tions of two-particle systems determined using quantumgenetic algorithms, Physica A: Statistical Mechanics and its Applications. , 439 (2001).[20] C. O. Stoico, D. G. Renzi, F. Vericat, A genetic algorithmfor the 1D electron gas, Physica A: Statistical Mechanicsand its Applications. , 159 (2008).[21] J. Wu, X. Shao, J. Li, G. Huang, Scale-free properties ofinformation flux networks in genetic algorithms, PhysicaA: Statistical Mechanics and its Applications. , 1692(2012).[22] Z. Li, J. Liu, A multi-agent genetic algorithm for commu-nity detection in complex networks, Physica A: Statisti-cal Mechanics and its Applications. , 336 (2016).[23] J.R. Koza, Genetic programming: A paradigm for ge-netically breeding populations of computer programsto solve problems, Stanford, Computer Science Depart-ment, 1990.[24] A.V. Mokshin, V.V. Mokshin, L.M. Sharnin, Adaptivegenetic algorithms used to analyze behavior of complexsystem, Commun. Nonlinear Sci. Numer. Simulat. ,174 (2019).[25] M. Srinivas, L.M. Patnaik, Adaptive probabilities ofcrossover and mutation in genetic algorithms, IEEETrans. Syst. Man Cybern. , 656 (1994).[26] S. L. Podvalny, M. I. Chizhov, P. Y. Gusev, K. Y. Gu-sev, The Crossover Operator of a Genetic Algorithm asApplied to the Task of a Production Planning, ProcediaComputer Science. , 603 (2019).[27] Y. Lei, S. Zhang, X. Li, C.M. Zhou, MATLAB GeneticAlgorithm Toolbox and Applications, Xian, XiDian Uni-versity Press, 2014.[28] M. Chassignol, A. Khoroshavin, A. Klimova, A. Bily-atdinova, Artificial Intelligence trends in education: anarrative overview, Procedia Computer Science. , 16(2018).[29] S. B. Kotsiantis, Use of machine learning techniques foreducational proposes: a decision support system for fore-casting students grades, Artificial Intelligence Review. , 331 (2011).[30] P. Cortez, A. Silva, Using data mining to predict sec-ondary school student performance, EUROSIS, A. Britoand J. Teixeira (Eds.). (2008).[31] A. A. Saa, Educational Data Mining & Students Per-formance Prediction, International Journal of AdvancedComputer Science and Applications. , 212 (2016).[32] C. Masci, G. Johnes, T. Agasisti, Student and schoolperformance across countries: A machine learning ap-proach, European Journal of Operational Research. ,1072 (2018).[33] S.K. Althaf Hussain Basha, Y.R. Ramesh Kumar, A.Govardhan, M. Zaheer Ahmed, Predicting Student Aca-demic Performance Using Temporal Association Mining,International Journal of Information Science and Educa-tion. , 21 (2012). Table I. Input and output factors characterizing a current lessonFactornotation Factor Description Group of factors x Factor indicating whether new knowledge was given by teacher in a lesson x Number of schoolchildren in a lesson x Factor indicating whether test work was in a lesson Organization x Factor indicating whether independent work was in a lesson of a lesson x Factor indicating whether demonstration experiment was given x Factor indicating whether laboratory work was in a lesson x Number of schoolchildren who received the assessment ”satisfactory” Assessments x Number of schoolchildren who received the assessment ”unsatisfactory” of academic performance x Have tasks been set for homework? x Have repetition of theoretical knowledge been given as homework? Homework x Have tasks associated with the search of an additional informationbeen given as homework? x Noise factor Noise factors x Noise factor y1