Research and application of constructing football training linear programming based on multiple linear regression equation

The research object is the core winning factors and winning formulas of football matches.

The independent variable X₁ − X₂₇ in the regression model corresponds to each secondary index, and the detailed definition is shown in Table 1. The dependent variable Y corresponds to the total score of the school's education informatisation level. The preliminary analysis of the data shows that the dependent variable and the independent variable are roughly linear, so the following multiple linear regression model is established: (1) Y^=b0+b1X1+b2X2+⋅⋅⋅+bkXk \hat Y = {b_0} + {b_1}{X_1} + {b_2}{X_2} + \cdot \cdot \cdot + {b_k}{X_k} where Ŷ is the estimated value of the mean of the dependent variable under the condition of the respective variable taking a certain value, X₁, X₂, . . . , X_k are the independent variables, k is the number of independent variables, b₀ is the constant term of the regression equation, also called the intercept, b₁, b₂, . . . , b_k are called partial regression coefficients and b_j represents the average change of Y after X_j changes by one unit under the fixed condition of independent variables other than X_j.

In multiple linear regression analysis, it is necessary to study how many independent variables should be introduced into the model. If fewer independent variables are introduced, the regression equation will not be able to explain the changes of dependent variables well; but it is not that the more independent variables are the better, so it is necessary to adopt some strategies to control and filter the independent variables into the regression equation. We adopt the stepwise regression method, which is to test the import threshold of the P value according to the set regression coefficient significance, introduce independent variables into the model one by one, recalculate the P values of all coefficients in the model and screen variables according to the set elimination threshold. When selecting independent variables, we first select the variable with the highest linear correlation coefficient with the dependent variable into the equation, and perform various tests of the regression equation; then, we find the variable with the highest partial correlation coefficient with the dependent variable and pass the test among the remaining variables [3]. Regression equation, and various tests, are performed on the newly established regression equation; this process is repeated until there are no more variables that can enter the equation. Data processing is performed in SAS 8.01 for Windows software, the significance level of entering the model is set to 0.05 and the significance level of the excluded or retained variables is also set to 0.05. The analysis process of selecting variables and performing regression is shown in Table 1, the statistical results of model regression are shown in Table 2 and the regression coefficients are shown in Table 3.

The regression equation is: (2) Y=0.074+0.131X2+0.277X6+0.129X11+0.092X13+0.069X19+0.074X20+0.047X21+0.063X23 Y = 0.074 + 0.131{X_2} + 0.277{X_6} + 0.129{X_{11}} + 0.092{X_{13}} + 0.069{X_{19}} + 0.074{X_{20}} + 0.047{X_{21}} + 0.063{X_{23}}

Front court 30 m area and side road area, FIFA's standard for field division has been adopted (Figure 1). For running intensity, we are adopting the China Super League running intensity level standard, that is, extreme intensity running for speeds >21 km per hour; high intensity running for speeds of 17–21 km per hour; medium intensity running for speeds of 14–17 km per hour; and 11–14 km per hour run for low intensity. Other indicators refer to the 2016/2017 FIFA competition rules.

Fig. 1

Correlation analysis is used to obtain the correlation coefficients between 52 factor indicators and the results of the game (Table 4): total possession rate, number of goals, number of shots, number of shots, number of free kicks, number of successful attempts into the penalty area and number of counterattacks; forward 10 factors that reflect offensive skills and tactics, including: the number of straight passes, the number of cross passes and the number of breakthroughs; as well as 5 factors that reflect defensive skills and tactics, including: the number of goals conceded, the number of shots, the number of shots and the number of sieges and saves. The indicators have a significant relationship with the result of the game. This shows that these 15 technical and tactical factors have a significant impact on the results of football matches, and it also supports the theory that the level of skills and tactics is the core winning factor in determining the results of football matches.

Owning the ball is one of the fundamental ways to score goals in a football game. Therefore, long-term possession of the ball indicates an expansion of the chance of scoring goals. In addition, long-term possession of the ball can not only reduce the player's physical consumption but also increase the opponent's psychological pressure, which creates good conditions for winning the final game. The number of goals scored, the number of shots and the number of shots on target are all indicators that reflect a team's shooting ability and shooting efficiency. The only way to win a football game is to score more goals than the opponent. Therefore, if the team wants to win the game, it must improve the accuracy of the shot while strengthening the shot. Only in this way can the chance of scoring be increased and the game won. Therefore, these three indicators are the fundamental factors that determine the outcome of the game [4].

Under the guidance of the concept of ‘defines first, win first’, the coach puts the improvement of players and the team's defensive ability at the top of training. Through daily targeted training, the players’ personal defensive ability and the team's overall defensive ability have undergone significant improvement, resulting in the time and space available for the use of offensive players to become narrower and more difficult. In this case, as a method of set-kick offence, free-kick offence can use the advantageous conditions of set-kick to complete the shot and get the chance to score. Therefore, the implication is that free kick is also important for the team to win the game in modern football matches.

According to the sports classification method, football belongs to the hit category. Only by approaching the opponent's goal in the game can the goal rate be increased. Therefore, the restricted area has become key for both the offensive players and the defenders to compete in. Successfully scoring the opponent's penalty area means the threat of shooting, shooting accuracy and the improvement of the scoring rate, which largely determines the probability of the team winning. Therefore, the number of instances of successfully scoring the opponent's penalty area has become one of the important indicators that affect the outcome of football matches.

As the coaches of each team attach great importance to defensive capabilities, the individual defensive capabilities of the players of each team have been continuously improved, the team's partial and overall defensive organisation has become more rigorous, and coupled with the qualitative improvement of the players’ physical level, the offensive team is fighting in position. The time and space available in China is narrower and more difficult. In this case, taking advantage of the defensive team's loose defensive formation and the unformed defensive organisation, at the moment of the offensive and defensive transition, to launch a quick counterattack has become one of the important methods of each team's offensive, and it has also become the main attack method for scoring goals. Therefore, counterattack tactics are the main offensive means for teams to win games in modern football matches.

One of the outstanding features of modern football games is the high level of integrity. Therefore, the continuous improvement of the overall defensive ability of each team has made it more and more difficult for individual offences. Relying on the overall offence of the team is an important means to win the game. Therefore, passing has become the main means of connecting players and completing team offences. In the passing direction, a forward pass can break through the opponent's defines line and create a chance to score a goal; while a cross pass can shift the offensive direction, mobilise the opponent's defines line, create vertical penetration space and form a local number advantage. Therefore, these two types of passes greatly affect the outcome of the game.

Under the situation that each team attaches great importance to defines, the players’ personal defensive ability and the team's local and overall defensive organisation ability have been greatly improved. In this case, only the player's personal breakthrough ability can be continuously improved. Only in the confrontation with the defensive players can the victory be won, by breaking through the opponent's defines, thereby forming an advantage in the number of offensives in a local area, creating favourable conditions for finally breaking through the opponent's defines line, facilitating availability of optimal shooting opportunities and completing the goal. Therefore, personal breakthrough ability also has an important impact on the outcome of the game [5].

The number of goals conceded, the number of shots and the number of shots taken together reflect the final effect of the team's defines. In the course of the game, the team can only rely on the excellent personal defensive ability of the players and the local and overall tight defensive formation to make it difficult for the offensive team to complete the shot. In particular, to ensure that the goal is not lost, the team can win the game on this basis. Therefore, the number of goals conceded, the number of shots and the number of shots are important indicators that affect the outcome of the game.

Encirclement is the main indicator reflecting the local defines organisation. In the game, through the first defender's frontal delay or close pressing against the team player, combined with the partial coordination, orderly cooperation and rapid movement of the second and third defenders, they form an organisation with the first defender. Strict and well-defined local defensive formations form an encirclement to the team members, thereby improving the team's defensive quality and ensuring that the team concedes fewer or no goals. Therefore, the level of siege directly affects the quality of the team's defines, and to a large extent determines the outcome of the game.

As the last defender of the team, ‘a good goalkeeper is equal to half a team’ has become a consensus in football. A save is an important indicator that reflects the goalkeeper's defensive ability. The strength of this ability to save plays a very important role in ensuring that the team does not concede a goal. Therefore, improving the goalkeeper's ability to save has an important impact on the result of the game.

The high correlation between the above 15 offensive and defensive technical and tactical factors and the results of the game enlighten us to the following inference: deep understanding of modern football concepts, following modern football development trends, strengthening the players’ ball control and breakthrough capabilities, improving the quality of the team's direct and cross passes, making full use of the advantages of counterattacks and free kicks to launch an offence and a good grasp of shooting opportunities is where coaches need to focus on improving offensive training in the future. At the same time, it is necessary to improve the team's partial and overall defensive organisation ability, reduce the offensive team's shots and try to avoid the team's conceding. In addition, the importance of goalkeepers has become increasingly prominent, and goal-keeper training should be strengthened [6].

Through correlation analysis, we only understand the correlation between each factor index and the game result, but it cannot reflect the relationship between each factor index and the game result as a whole. In order to grasp and understand the impact of various factors on the results of the game as a whole, and to explore the quantitative relationship between the two, a multiple linear regression analysis was carried out with the game results as the dependent variable and each factor index as the independent variable.

The regression analysis adopts the full entry method. The correlation coefficient value (R) is 0.882, the determination coefficient R² (Square) is 0.778 and the adjusted R² (Adjusted Square) is 0.578, indicating that the regression model has a high degree of fit, and 10 selected independent variables: the number of goals, the number of free kicks, the number of successful attempts into the penalty area, the number of counterattacks, the number of forward passes, the number of cross passes, the number of breakthroughs, the number of goals conceded, the number of sieges and the number of saves explain 57.8 of the variance in the total score percentage, and the Durbin–Watson test value is 2.270, indicating that the residuals are independent of each other. The Durbin arson test is passed, and the regression equation can be used. The results of the analysis of variance calculated by the multiple linear regression analysis show (Table 5) that, in this multiple linear regression model, F is 3.888, P < 0.01 and there is a very significant difference, indicating that the regression analysis is meaningful [7].

Given the significance level α = 0.05, the P value of a goal is 0.029, the P value of a free kick is 0.016, the P value of a successful penalty area is 0.021, the P value of a counterattack is 0.008 and the pass is forward. The P value for cross passes is 0.003, the P value for cross passes is 0.038, the P value for breakthroughs is 0.033, the P value for conceded goals is 0.000, the P value for surrounds is 0.045 and the P value for saves is 0.048. It shows that the 10 variables introduced by the regression model have a significant impact on the results of the game. They passed the significance test, and the constant P value was 0.654, which failed the significance test. The following can be observed from the results of the regression equation: the result of the football match and the number of goals scored by the team, the number of free kicks, the number of successful attempts into the opponent's penalty area, the number of counterattacks, the number of forward passes, the number of cross passes, the number of breakthroughs and the number of sieges. There is a positive correlation among the eight technical and tactical indicators, namely, the number of goals scored, the successful entry into the penalty area, the free kick, the counterattack, the forward pass, the cross pass, the breakthrough and the siege; the higher the value of these indicators, the higher the probability of the team winning. The result of the game is negatively correlated with the number of goals conceded and the number of saves, that is, the fewer goals conceded and the fewer saves, the better the team's performance may be.

What needs further explanation is that of the 15 factor indicators that are highly correlated with the game results through correlation analysis, only 10 indicators in the multiple linear regression model have high correlation with the game results. The reason is the unselected shots. There is a high correlation between the number of shots and the number of goals scored, so the most representative number of goals is selected; the number of goals shot and the number of shots are highly correlated with the number of goals conceded, so the most representative conceded goals were incorporated in the number selected; and the ball possession rate was not selected, mainly because the ball control is mainly reflected by the passing of the team's offence and the breakthrough of the individual offence, so there is a high correlation with the passing and the breakthrough, forward passes and cross passes. The selection of the ball and breakthrough largely reflects and represents the ball possession rate.

In order to verify whether the winning formula has a certain degree of scientific and objectivity, we take the 26 games in which the win–loss relationship is determined within 90 min of the group stage of the 2016 European Cup finals as an example, the 26 games and 52 team goals etc. The core winning factor indicators are substituted into the winning formula. Among the 22 games, the team with the larger Y value calculated by the winning formula won, and the winning rate reached 85%, which shows that the winning formula has a certain degree of scientific temperament and objectivity [8].

In terms of specific competition cases, the two matches of Wales vs. Slovakia (the result of the two sides’ match is 2:1) and England vs. Wales (the result of the two sides’ match is 2:1) are randomly selected as examples. The relative ratios of the core winning factors of Wales and Slovakia, England and Wales are replaced by the equation. The calculation result is: the Y value of the Wales team is 0.22638, the Y value of the Slovak team is 0.15862, the sum of the two Y values is 0.38500, and the calculated Welsh relative winning percentage of the team is 58.80%, while the relative winning percentage of Slovakia is 41.20%; the Y value of England is 0.32222, the Y value of Wales is 0.06218 and the sum of the two Y values is 0.38440 [9, 10]. The calculated relative winning percentage of England is 83.82%., and the relative winning percentage of the Wales team was 16.18%. From the calculation results of the winning formula, it can be observed that Wales and England have a higher winning percentage than their respective opponents Slovakia and Wales, and the results of the game also prove that the team with a relatively high winning percentage has won the game, which further proves that the winning formula has a certain degree. The scientific and objectivity of It is important to note that the regression formula model obtained by this research only reflects the objective quantitative law between the football game and the core winning factors to a certain extent. Strictly speaking, the football game result is affected by multiple subjective and objective factors. There is no absolute quantitative relationship between the result of the game and the factors. This formula model is only a discussion. In the future, more in-depth discussions are required from football researchers.