Application of Fuzzy Mathematics Calculation in Quantitative Evaluation of Students’ Performance of Basketball Jump Shot

With the rapid development of modern basketball, the skills and level that basketball players need to master are of high standard. Jump shot technology is the most important for a basketball player and good jump shot technology can make the player in the field of play to have a certain advantage. The jump shot first originated from the NBA in the United States. In the early days of the establishment of the NBA, players used to shoot from standing in a place. In the early 1950s, some players began to use the jump shot. Due to the obvious improvement in the efficiency of blocking the jump shot, it has gradually become popular in the whole NBA and even the international basketball world, and has become the main means of scoring in today’s basketball games. In modern basketball, athletes basically master a variety of jump shots, such as the jump shot, the fadeaway jump shot, the step-back jump shot, and so on. Each jump shot has its own characteristics and skills. However, most basketball jump shot technical evaluation is only limited to subjective evaluation due to lack of certain quantitative verification [3] and the use of fuzzy mathematics method combined with basketball jump shot technical subjective evaluation can make up for its defects.

In this paper, 40 basketball students majoring in 2015 in the Physical Education Institute of a university were selected as the research objects. First, the application of fuzzy mathematics method in the field of sports and basketball was understood through literature review, and then the professional investigation of basketball teaching and training in the studied universities was conducted by questionnaire survey, so as to obtain part of the weight. Finally, the fuzzy mathematics method is used to quantitatively evaluate the basketball jump shot.

The work of basketball referee is an indispensable part of basketball competition, which plays an important role in promotion of the basketball game. With the development of the competitive level of modern basketball, high requirements are needed for the referee’s work. Basketball referees’ on-the-spot work performance is the integrated embodiment of its theory and the practice ability, and accurate, comprehensive and scientific evaluation is highly beneficial to the competent authorities at all levels of basketball referees, and by evaluation one can find the problem, find out gaps or flaws and thus can improve the level of the referee. At present, the evaluation of basketball referees is mainly based on experience, which has limitations and also one-sidedness. The job of basketball referee on the spot is a complicated work with many factors, which is dynamic and fuzzy. In this paper, fuzzy mathematics method is used to evaluate the on-the-spot refereeing work of basketball referees, making the evaluation of the on-the-spot refereeing work more scientific, standardised and procedural [4].

In production practice, scientific experiments and daily life, people often encounter fuzzy concepts (or phenomena). For example, big and small, light and heavy, fast and slow, dynamic and static, deep and shallow, beautiful and ugly and so on all contain certain vague concepts. With the development of science and technology, practical problems related to these fuzzy concepts often need to be analysed quantitatively in various fields, which requires the use of the tool of fuzzy mathematics to solve.

The fuzzy comprehensive evaluation method belongs to the category of fuzzy mathematics. The membership degree [5] generally refers to A function. For example, for any element x in the field U, there is A number A(x)∈ [0,1] corresponding to it, then A is called the fuzzy set on U, A(x) is called the membership of x to A. Based on the membership principle of fuzzy mathematics, the qualitative problem is transformed into a quantitative one, that is, the problem of overall evaluation of complex system is solved by using fuzzy mathematics.

Improving the free throw shooting rate is achieved through a variety of training means. To determine the degree of correlation between these training methods and free throw shooting rate, we denoted this correlation as the fuzzy correlation coefficient R; At the same time, think of these drills as set X, such as: return free throw running drills, free throws with eyes closed, Freehand imitation exercises, two-team basketball games, imaginary shooting contests … And so on [6].

Given two finite domains, let the domain U = [U₁, U₂,.., U_a] be the set of evaluation indexes or evaluation factors, each of these factors is the ‘focus point’ of evaluation. Let V = [V₁, V₂, …, V_a] be the collection of evaluation grades, let the evaluation of the I factor be R˜1=[r11,r12,…,rim]$_1 = \left[ {r_{11} ,r_{12} , \ldots ,r_{im} } \right]$. It can be thought of as a Fuzzy subset on U, and r_ik represents the membership degree of the ith factor to grade K, thus, the evaluation matrix of n factors can be obtained. Each evaluated factor determines the fuzzy relation R˜$\matrix{ R \hfill \cr \~ \hfill \cr } $ from U to V, which is a matrix: R~=rijn*m{r11r12…r1imr21r22…r2imrn1rn2…raim} $\matrix{ R \hfill \cr \~ \hfill \cr } = {\rm{r}}_{{\rm{ij}}} {\rm{n*m}}\left\{ {\matrix{ {r_{{\rm{11}}} } \hfill & {r_{{\rm{12}}} } \hfill & \ldots \hfill & {r_{{\rm{1}}im} } \hfill \cr {r_{{\rm{21}}} } \hfill & {r_{{\rm{22}}} } \hfill & \ldots \hfill & {r_{{\rm{2}}im} } \hfill \cr {r_{n{\rm{1}}} } \hfill & {r_{n{\rm{2}}} } \hfill & \ldots \hfill & {r_{{\rm{a}}im} } \hfill \cr } } \right\}$

Where r_ij represents the following from factor V, this evaluation object can be rated as V_i membership [i = 1,2 … n; j = 1,2 … m]. The evaluation of objects is based on the factors in the index set U. This fuzzy relation matrix is not enough to evaluate things. A fuzzy subset A~$\matrix{ A \hfill \cr \~ \hfill \cr } $ on U is proposed to be introduced and it is called the weight or weighting allocation set, A~=[a1,a2…,aa]$\matrix{ A \hfill \cr \~ \hfill \cr } = \left[ {a_1 ,a_2 \ldots ,a_a } \right]$, among them a₁ > 0, and ∑i=1mai=1\sum_{i=1}^{m} a i=1. Its reflection is a trade-off between factors. At the same time, introduce a fuzzy subset B~$\matrix{ B \hfill \cr \~ \hfill \cr } $ on V, it’s called the evaluation set, and its fuzzy evaluation, B~=[b1,b2…,bm]$\matrix{ B \hfill \cr \~ \hfill \cr } = \left[ {{\rm{b}}_1 ,b_2 \ldots ,{\rm{b}}_{\rm{m}} } \right]$. Find the solution of B~$\matrix{ B \hfill \cr \~ \hfill \cr } $ from R~$\matrix{ R \hfill \cr \~ \hfill \cr } $ and A~$\matrix{ A \hfill \cr \~ \hfill \cr } $, general R~=A~⋅R~=[b1,b2,…,bm]$\matrix{ R \hfill \cr \~ \hfill \cr } = \matrix{ A \hfill \cr \~ \hfill \cr } \cdot \matrix{ R \hfill \cr \~ \hfill \cr } = \left[ {{\rm{b}}_{\rm{1}} ,b_2 , \ldots ,b_m } \right]$, (“.” Is operator symbol) is called fuzzy transformation, A~$\matrix{ A \hfill \cr \~ \hfill \cr } $ is the input and B~$\matrix{ B \hfill \cr \~ \hfill \cr } $ is the output, R~$\matrix{ R \hfill \cr \~ \hfill \cr } $ is used as a fuzzy converter from U to V, for each set of weights A~$\matrix{ A \hfill \cr \~ \hfill \cr } $, the corresponding comprehensive evaluation result B~$\matrix{ B \hfill \cr \~ \hfill \cr } $ can be obtained.

Set U and X are both finite sets, and a fuzzy correlation matrix table [7] can be established. First, set U and five elements are filled into the horizontal species of the table, and then eight elements (training methods) of set X are filled into the vertical column of the table, as shown in Table 1. In order to objectively reflect the degree of correlation between things, we adopt a five-level method; that is, a complete correlation score is 1 point, a major correlation score is 0.75 points, a general correlation score is 0.50 points, a major correlation score is 0.25 points and a complete correlation score is 0 points. Similarly, the author invited 10 experts with rich experience in the basketball teaching and research department to make an evaluation, and finally took the average value for application, the coefficient of correlation with X₁U₁, U₂, U₃, U₄, U₅: was determined as 0.225, L, 0.625, 0.875, 0.45. Then, the correlation coefficient between R₁ and R₂is calculated (Table 1), the result of R₁ is the correlation value between the training means and the field. The result of R₂ is the degree of correlation between each element in U and the training method.

From Table 1, we can find the correlation coefficient of various training means to various purposes and tasks. The correlation level between each training means and free throw shooting rate is shown through R₂ and through R₁, indicating the extent to which the factors that affect the free throw percentage may be developed. X₁ and X₇ from R, have biggest effect in improving the free throw. But X₁ and X₇ are weak points to U₁’s development, if A and B can be combined to train together and learn from each other, better training effect will be obtained.

After processing the data obtained from the interview survey according to Table 2, the measurement value of the relative importance of each representing factor of the technology assessment factors in the study was obtained. A~=[0.095,0.092,0.083,0.092,0.089,0.122,0.141,0.131,0.15] $\matrix{ A \hfill \cr \~ \hfill \cr } = \left[ {{\rm{0}}{\rm{.095,0}}{\rm{.092,0}}{\rm{.083,0}}{\rm{.092,0}}{\rm{.089,0}}{\rm{.122,0}}{\rm{.141,0}}{\rm{.131,0}}{\rm{.15}}} \right]$

Weighted average operator model selection: B∼=A∼⋅=[b1,b2,…,bm]b1=∑i=1m(ai⋅r1j)=(a1⋅r1j)⊕(az⋅rzj)⊕…⊕(anrmj) \underset{\sim}{B}=\underset{\sim}{A} \cdot=\left[\mathrm{b}_{1}, b_{2, \ldots}, b_{m}\right] b_{1}=\sum_{i=1}^{m}\left(a_{i} \cdot r_{1 j}\right)=\left(a_{1} \cdot r_{1 j}\right) \oplus\left(a_{z} \cdot r_{z j}\right) \oplus \ldots \oplus\left(a_{n} r_{m j}\right)

Fuzzy evaluation evaluates the weight vector of the meta factor W = {W₁, W₂, …, W_n} representing the set of weights corresponding to (R|x₁), which indicates the relative importance of each element, for example, how important the former factor is to the latter. Its weight value can be obtained by the analytic hierarchy process, or can be determined by combining with the data obtained from the questionnaire, thus determining the weight coefficient.

{r11r12…r1mr21r22…r2m…………ra1ra2…rnm}(W1W2…,Wa)=(b1b2…,ba)=B \left\{\begin{array}{cccc} r_{11} & r_{12} & \ldots & r_{1 m} \\ r_{21} & r_{22} & \ldots & r_{2 m} \\ \ldots & \ldots & \ldots & \ldots \\ r_{a 1} & r_{a 2} & \ldots & r_{n m} \end{array}\right\} \begin{aligned} &\left(W_{1} W_{2} \ldots, W_{a}\right)= \\ &\left(b_{1} b_{2} \ldots, b_{a}\right)=B \end{aligned}

According to the fuzzy comprehensive evaluation method, this paper establishes the evaluation model of students’ basketball jump skills, designs the concrete steps and constructs the factor set of the evaluation model of students’ basketball jump skills according to the evaluation index system.

Q = {Q1,Q2,Q3,Q4,Q5} = {excellent, good, fair, pass, fail}

E = {E1,E2,E3,E4} = {lift the ball position, the smoothness of lifting the ball, Cast dancing posture} Set up a review collection: P={P1,P2,P3,P4,P5}={ excellent, good, fair, pass, fail } \mathrm{P}=\{\mathrm{P} 1, \mathrm{P} 2, \mathrm{P} 3, \mathrm{P} 4, \mathrm{P} 5\}=\{\text { excellent, good, fair, pass, fail }\}

Assign the values for the above five levels as shown in Table 3.

The weight of the factor set obtained from the questionnaire data is: W={W1,W2,W3}=(0.558,0.320,0.122) W=\left\{W_{1}, W_{2}, W_{3}\right\}=(0.558,0.320,0.122)

This paper evaluates the basketball jump shot skills of the basketball major students in the grade of 2015 in a physical education college of a university, and takes the evaluation results of two of them, as shown in Tables 3, 4 and 5.

Student 1 Evaluation vector of basketball jump shot skills: B1=W⋅R1=(0.558,0.320,0.122)(0.250.270.220.2600.170.210.310.3100.210.340.220.230)=(0.225,0.283,0.234,0.258,0) B_{1}=W \cdot R_{1}=(0.558,0.320,0.122)\left(\begin{array}{llll} 0.25 & 0.27 & 0.22 & 0.26 & 0 \\ 0.17 & 0.21 & 0.31 & 0.31 & 0 \\ 0.21 & 0.34 & 0.22 & 0.23 & 0 \end{array}\right)=(0.225,0.283,0.234,0.258,0)

Student 2 Assessment vector of basketball jump shot skills: B1=W⋅R2=(0.558,0.320,0.122)(0.280.210.240.2700.220.240.270.2700.270.310.190.270)=(0.259,0.231,0.243,0.270,0) B_{1}=W \cdot R_{2}=(0.558,0.320,0.122)\left(\begin{array}{lllll} 0.28 & 0.21 & 0.24 & 0.27 & 0 \\ 0.22 & 0.24 & 0.27 & 0.27 & 0 \\ 0.27 & 0.31 & 0.19 & 0.27 & 0 \end{array}\right)=(0.259,0.231,0.243,0.270,0)

Take the middle score value of each evaluation grade as the column vector: P={9585756530} P=\left\{\begin{array}{l} 95 \\ 85 \\ 75 \\ 65 \\ 30 \end{array}\right\}

Then the evaluation results of student 1’s basketball jump shot skills are as follows: F1=B1⋅P=(0.255,0.283,0.234,0.258,0){9585756530}=79.75 F_{1}=B_{1} \cdot P=(0.255,0.283,0.234,0.258,0)\left\{\begin{array}{l} 95 \\ 85 \\ 75 \\ 65 \\ 30 \end{array}\right\}=79.75

The evaluation results of student 2’s basketball jump shot skills are as follows: F2=B2⋅P=(0.259,0.231,0.243,0.270,0){9585756530}=80.1 F_{2}=B_{2} \cdot P=(0.259,0.231,0.243,0.270,0)\left\{\begin{array}{l} 95 \\ 85 \\ 75 \\ 65 \\ 30 \end{array}\right\}=80.1

Combined with the above calculation results and the values in the assignment table, it can be seen that the basketball jump ball and jump shot skills of the evaluated students are generally at a ‘good’ level, indicating that the basketball teaching system of this college is relatively complete. Student 1 is at a ‘good’ level and Student 2 is at a ‘good’ level. Student 2 should focus on improving the skill level of basketball jump shot, further should strengthen the training in lifting the ball, and also ensure the training of the landing point of jump shot.

From this article, you can see, the method of fuzzy evaluation analysis can be effectively used for basketball jumper grade evaluation needed for selection, evaluation index including presence and decision effect, psychological control ability, the pre-match preparations and on-the-spot control competition ability from four aspects, and to lift the ball position, smooth lift degrees, jumper position from three aspects to throw and get scores. The method of fuzzy mathematics evaluation can accurately and quantitatively evaluate the level of basketball players, which has high feasibility and can provide objective basis for the evaluation and management of referees.

You’ve all heard the term home court advantage, or home court effect. In one statistic, teams have a significantly higher winning percentage at home than on the road in most competitive games. In this variable, audience support is an important aspect, but it is not the only one, nor is it the most important one. The most important one is players’ habit or familiarity. The cheering of the audience is more positive for the team, especially when the team is facing adversity, the encouragement of the fans is a very good heart booster. Listening to others call your name when you are playing games is a very powerful motivating factor and also a kind of encouragement [9].

After the target and key point of the league marketing work are clear, the countermeasures corresponding to the influence factor are extracted from the ‘countermeasure database’ and sent to the basketball association and basketball club in the form of ‘countermeasure report on increasing the number of spectators in the basketball match scene.’ In addition, due to the concerted efforts of all aspects, as well as the comprehensive use of multilevel, all-around countermeasures, finally the prosperity is achieved as the number of spectators in basketball matches has increased greatly and jumped from 12,500 to 28,000 spectators, and note that some matches have never seen the full audience in many years [10].