Research on predictive control of students’ performance in PE classes based on the mathematical model of multiple linear regression equation

An important reason why GA can be widely used is its global convergence. Due to the diversity of the GA group, it searches in all directions as much as possible. This is a great improvement over the previous gradient method that only searches in one direction. Moreover, GA does not need to have continuity and differentiability restrictions on optimisation problems. In the end, the dynamic optimisation of college sports performance evaluation can be realised. On this basis, prediction is made based on the multivariate nonlinear model of college sports performance, ensuring high prediction accuracy and strong objectivity. The prediction model process based on GA optimised college sports performance evaluation proposed in this research is shown in Figure 1:

Fig. 1

GA uses goodness-of-fit R2 to evaluate the performance and prediction accuracy of college sports performance evaluation and converts this objective function into a fitness function. The algorithm starts by randomly generating a group. Each group of chromosomes in the group represents the student's college sports performance level. Each group of chromosomes is evaluated according to the fitness function, and the corresponding fitness value is obtained. The greater the fitness of the chromosome, the more the representative college sports performance evaluation has been optimised and the prediction effect is better. According to the fitness value, the probability of each chromosome being selected in the selection operation can be calculated. According to the selection probability, a random traversal sampling method is used to select a group of chromosomes to form a new population. According to the crossover probability, the chromosome is selected for GA crossover operation, and finally, according to the mutation probability, the mutation operation is performed on some of the gene positions on the chromosome. This operation makes the college sports performance grade set represented by the chromosome diversity in the entire search process and has a great played an optimisation role, thereby ensuring that the optimal solution can be found. The end condition of the algorithm is to set a maximum number of iterations, epochal, to ensure that the solution obtained by GA after the end condition is reached is the optimal solution [3].

Coding is the prerequisite for GA to solve the problem. This study uses integer coding for college sports performance grades. Before chromosome coding, first of all, the range [C_min, C_max] of the C value of all students’ college sports performance grades should be determined. In general, the optimal number of clusters will not exceed Cmax≤N {C_{\max}} \le \sqrt N . Therefore, the value range of C can be set to [2, N \sqrt N ].

Each chromosome represents a set of students’ college sports performance grades. The length of the chromo-some is the number of students’ homes. The genes in the chromosomes represent the college sports performance grades, and the same genes indicate that the college sports performance grades are of the same category. Take an integer k in the value range of C, which means that the students in the set contain k college sports performance levels. The chromosome can be expressed as: [Z₁, Z₂, Z₃,...,Z_N], 0 ≤ Z_i ≤ k − 1.

For example, in this study, N = 62 students are selected as the research object, so the best college sports performance level is 2 ≤ C ≤ 8. If k = 6, then the chromosome code is: (5, 3, 5, 4, 3, 3, 0, 3, 2,...,1, 3, 4), the length is 62.

According to the code of the chromosome, this code is converted into a dummy variable, to avoid the ‘dummy variable trap’, Use k − 1 dummy variables D₁, D₂,...D(k − 1) to represent k categories (as shown in Table 1), perform multiple nonlinear regression analysis according to model (3), and convert the regression model goodness of fit R2 into the objective function shown in (4): (4) obj=1−R2=SEESST=∑(yi−y^i)2∑(yi−y^i)2 obj = 1 - {R^2} = {{SEE} \over {SST}} = {{\sum {{({y_i} - {{\hat y}_i})}^2}} \over {\sum {{({y_i} - {{\hat y}_i})}^2}}}

y_i is the observed value, y_i is the fitted value, and y is the mean value.

The fitness function is usually used to convert the objective function value to a relative fitness value. To prevent premature convergence, the fitness value can be calculated according to the order of the objective function value in the population. Sort according to the individual objective function value obj from small to large. According to the sequence number of the sort, each level of the individual is given a fitness value. Non-dominated solutions with the same sort are assigned the same fitness value. Equation (5) calculates: (5) FinV(i)=2−MAX+2(MAX−1)xi−1Nid−1 FinV(i) = 2 - MAX + 2(MAX - 1){{{x_i} - 1} \over {{N_{id}} - 1}}

In the formula: MAX represents the selection pressure difference, generally between [1, 2]; x_i is the position of individual i in the ordered population; N_id is the population number; FinV (i) represents the fitness value of the individual at position i. In this study, the pressure difference is chosen to be E. Since the higher the R2 value, the more accurate the prediction, so the fitness function gives a higher fitness value to the chromosomes with good final prediction results; conversely, the chromosomes with poor prediction accuracy are given a lower fitness value. The essence of using GA to optimise forecasts is to optimise the goodness of fit R2.

The selection operator is a GA that determines how to select a certain number of good individuals from the parent population based on the set generation gap (GGAP) to inherit into the next generation population. In order to improve global convergence and computational efficiency, the selection method uses random traversal sampling (SUS). SUS is a single-state sampling algorithm with zero deviation and minimum individual expansion. It replaces the single selection pointer used in the roulette method. SUS uses S pointers of equal distance, where S refers to the number of selections required. The population is randomly arranged, S pointers [ptr, ptr + 1, ptr + 2,..., ptr + s − 1] determine S individuals, and pointer ptr + i(i = 0, 1,...,S − 1) is determined by a random number generated in [1/S, i + 1/S]. Assuming that S = 6 individuals are selected from 10 individuals and the random position of the first pointer is 0.04 (Figure 2), then the distance between the pointers is 1/6 = 0.17, so it can be based on the position and cumulative probability of the pointer per the interval can determine the selected individuals are: 1, 2, 3, 4, 7, 8.

Fig. 2

Using uniform mutation, its operation refers to replacing the original gene value at each locus in the individual coding string with a random number that is uniformly distributed within a certain range with a certain small probability, that is, depending on the parent individual the mutation probability Pm is operated to prevent premature convergence from producing a locally optimal solution instead of the overall optimal solution [4].

The specific operation processes of uniform mutation are: 1. Specify each locus in the individual code string as a mutation point in turn; 2. For each mutation point, take a random number from the value range of the corresponding gene with the mutation probability Pm Replace the original value.

Single-point crossover means that only one crossover point is randomly set in the individual code string, and then part of the chromosomes of two paired individuals are exchanged at this point. Here, a crossover position is randomly set for individuals in the group, and the operation is performed according to the crossover probability Pc. The two paired chromosomes exchange part of their genes at the crossover position by a single point crossover, and a new generation of groups is generated through exchange. Figure 3 is a schematic diagram of a single point crossover operation.

Fig. 3

The specific implementation process of single-point crossover: 1. Randomly pair individuals in pairs. If the group size is M, there are [M/2] pairs of paired individual groups; 2. For each pair of paired individuals, randomly Set the position after a certain locus as the crossover point. If the length of the chromosome is N, there are N-1 possible crossover point positions; 3. For each pair of individuals, the crossover probability Pc is Part of the chromosomes of two individuals are exchanged at the intersection point, resulting in two new individuals [5].

To compare the influence of the number of college sports performance levels on the multiple regression model, the GA optimised multiple regression nonlinear model is used to calculate all the best fit goodness R2 within the range of the number of college sports performance levels C. The calculation results are shown in Figure 4. The data of Jiangxi province, Henan province, Heilongjiang province and Jiangsu province are the 4 sub-maps in Figure 4 respectively.

Fig. 4

It can be seen from Figure 4 that for the prediction of merit number, when the number of college sports performance grades is C = 7, the goodness of fit R2 is the largest, that is, the best college sports performance grade for the student (region) to obtain the merit number should be divided into 7 categories; for the prediction of excellent performance, when the college sports performance level C = 4, the goodness of fit R2 is the largest, that is, the best college sports performance level for students (regions) to obtain excellent results should be divided into 4 categories [6].

According to the above analysis, the number of college sports performance grades of the student (region) merit number prediction model is set to 7; the number of college sports performance grades of the excellent performance prediction model is set to 4, and the sample data is subjected to regression analysis (Table 2).

According to the results in Table 2, the excellent results in 2018 can be predicted (Table 3). Finally, respectively calculate the prediction results of the literature and the prediction ability evaluation indicators of the prediction results proposed in this study (Table 4). It can be seen from Table 4 that the prediction model proposed in this study has obvious advantages in predicting excellent performance; in the prediction of excellent performance, except for the slightly smaller MAE index, other indicators are better than the former.

From Table 4, it can be found that for the FCM-regression model, because the university sports performance evaluation based on unsupervised fuzzy C-means clustering is difficult to objectively describe, it has limited ability to effectively optimise the combination of student (regional) university sports performance and its predictive ability Naturally, there is no guarantee, making the prediction accuracy relatively low [7].

The GA-regression model proposed by this research can realise the supervised calculation of the student (regional) college sports performance grade through GA, and can dynamically mine the best college sports performance evaluation [8] so that the prediction model based on college sports performance can be optimised. At the same time, the subjectivity of the prediction model is reduced, and the accuracy and stability of the superior and prediction are higher [9].