The teaching of sports science of track and field-based on nonlinear mathematical equations

In writing this article, I consulted many academic papers, newspapers and magazines, and other materials related to prediction, grey system theory, and track and field performance analysis [3].

We logically summarize the statistical results. It analyzes the current situation and development trend of the overall athletic level of men and women in Chinese track and field events from 1999 to 2021.

We summarize and organize the collected data and use Spss 16.0 and Matlab software for statistical analysis and comparison. Statistical diagnosis is to diagnose the whole process of solving problems by statistical inference methods, and impact analysis is a very important branch of statistical diagnosis [4]. At present, the research on the statistical diagnosis of linear regression models has been very in-depth. There are also some research results on the diagnosis methods of nonlinear models and generalized linear models. This paper systematically studies its statistical diagnosis and impact analysis methods by aiming at the widely used Poisson regression model. Consider the following model. Suppose y_i obeys the Poisson distribution, and its probability function is: (1) P(yi,θi)=1yi!exp(yiθi−eθi) P\left( {{y_i},{\theta _i}} \right) = {1 \over {{y_i}!}}\exp \left( {{y_i}{\theta _i} - {e^{{\theta _i}}}} \right) Knowable (2) E(yi)=μi=eθi, Var(yi)=eθi μi=exp(xiTβ), i=1,2,⋯,n \matrix{ {E({y_i}) = {\mu _i} = {e^{{\theta _i}}},\quad Var({y_i}) = {e^{{\theta _i}}}} \hfill \cr {\;\;\;\;\;{\mu _i} = \exp \left( {x_i^T\beta } \right),\quad i = 1,2, \cdots ,n} \hfill \cr } (yi,xiT) \left( {{y_i},x_i^T} \right) represents the data point of the group i, xiT=(x1,x2,⋯,xp)∈Rp x_i^T = \left( {{x_1},{x_2}, \cdots ,{x_p}} \right) \in {R^p} is a list of known non-random design points. β = (β₁, β₂,⋯β_p)^T is an unknown parameter. Equation (2) is called the Poisson regression model. If we assume Y = (Y₁, Y₂,⋯Y_n)^T, X = (X₁, X₂,⋯X_n)^T, μ = E(Y |X) = (μ₁, μ₂,⋯ μ_n)^T, the vector form of the Poisson regression model is: μ = exp(Xβ). The log-likelihood function of Y concerning β is: (3) L(β)=∑i=1n[yixiTβ−exp(xiTβ)−ln(yi!)] L(\beta ) = \sum\limits_{i = 1}^n \left[ {{y_i}x_i^T\beta - \exp \left( {x_i^T\beta } \right) - \ln \left( {{y_i}!} \right)} \right] We mark the first and second derivatives of L(β) concerning β as L˙(β) \dot L(\beta ) and L¨(β) \ddot L(\beta ) respectively, then (4) L˙(β)=XTZ \dot L(\beta ) = {X^T}Z (5) L¨(β)=−XTVX \ddot L(\beta ) = - {X^T}VX Where Z=(z1,z2,⋯zn)Tzi=yi−exp(xiTβ) Z = {\left( {{z_1},{z_2}, \cdots {z_n}} \right)^T}{z_i} = {y_i} - \exp \left( {x_i^T\beta } \right) , V = diag(v₁, v₂,⋯v_n). We assume that the Poisson regression model (2) The maximum likelihood estimate of β is β, then from (4) we know that β satisfies the following likelihood equation: (6) XTzX=0 {X^T}zX = 0 β can be solved by the Gauss-Newton iteration method. From (4) and (5), the Gauss-Newton iteration formula can be obtained as: (7) βi+1=βi+(XTV(βi)X)−1XTZ(βi) {\beta ^{i + 1}} = {\beta ^i} + {\left( {{X^T}V({\beta ^i})X} \right)^{ - 1}}{X^T}Z\left( {{\beta ^i}} \right) This formula can be formally expressed in the form of weighted least squares estimation: (8) βi+1=(XTV(βi)X)−1XTV(βi)Wi Wi=Xβi+V1(βi)Z(βi) {\beta ^{i + 1}} = {\left( {{X^T}V({\beta ^i})X} \right)^{ - 1}}{X^T}V\left( {{\beta ^i}} \right){W^i}\quad {W^i} = X{\beta ^i} + {V^1}\left( {{\beta ^i}} \right)Z\left( {{\beta ^i}} \right) In the actual problem, we choose an appropriate initial value. The iteration of (7) can quickly converge. When the iteration converges (9) β=(XTVX)−1XTVW \beta = {\left( {{X^T}VX} \right)^{ - 1}}{X^T}VW (10) W=Xβ+V1Z W = X\beta + {V^1}Z (8) and (9) In the two formulas, Z, V is calculated at β.

The so-called nationwide system specifically refers to the guiding ideology, training management methods, and network system implemented in the field of competitive sports in China. The Chinese government attaches great importance to the strategic policy of the Olympic Games, while some coaches and athletes attach great importance to the strategic policy of the National Games [8]. From the rank-sum ratio of the individual best results of men and women in Chinese track and field from 1999 to 2005, we can see the best results in 2004 and 2005, namely the Olympic year and the National Games year. The ratio is 2:7 for men and 1:3 for women. At present, the population of China participating in basic track and field training does not exceed 100,000, and the population participating in systematic training is <100,000. A total of 52 athletes participated in the 2004 Olympic Games. Therefore, the jobs of most coaches and athletes are maintained in the National Games. If the National Games system is not resolved, it will affect the results of the Olympic Games.

Insufficient investment and difficulty in financing. Many necessary measures to improve the level of sports cannot be implemented in training competitions. Track and field events are more one-way (46 items), the team is large. Still, the level of exercise is relatively low, the social influence is low, and the attraction to enterprises is small, so it is more difficult to obtain social sponsorship. The insufficient investment makes track and field events, and although they are basic projects, they cannot be fully supported and developed [9]. Although we can make a difference in some projects, we cannot fully guarantee various teaching training methods in training and competitions. This is another reason why China's track and field performance is not high.

Scientific theoretical knowledge and technical methods are widely used in various fields of sports. Various new materials provided by material science are widely used in sports equipment, clothing, construction, and other utensils. Electronic computers, televisions, video recordings, and other audio-visual equipment directly lead people to the process of physical education and sports training as a means of education and training [10]. Various scientific research tools such as scientific research experiment equipment and testing tools are adopted by sports science research. Physiology, biology, biomechanics, biochemistry, psychology, and other disciplines have more deeply involved in the research and guidance of track and field sports from more aspects. These technologies increasingly play a role in improving technology, improving physiological functions, performing psychological adjustment, preventing and curing trauma, and selecting athletes.

Coaches play a leading role in sports training, and they are directly related to the improvement of scientific training levels and sports levels. The textbook ‘Sports Training Management’ proposes that coaches have moral character, knowledge quality, ability quality, conceptual quality, temperament quality, and physical quality [11]. After the mid-1990s, the Track and Field Sports Management Center of the State Sports General Administration was established. We have changed the training management system from the ‘centralized system’ to a ‘federal system.’ This has achieved certain results, but it also exposed many problems. Some coaches’ training plans, methods, and methods are kept secret, and even the phenomenon of mutual suspicion and unhealthy competition has arisen.

The competitive ability of track and field athletes refers to the ability of athletes to perform the highest exceptional results or defeat opponents in track and field competitions. In major competitions, the normal performance level of outstanding foreign athletes reaches about 60%, while that of outstanding Chinese athletes is only about 25%. In the 28th Olympic Games, 27.1% of Chinese track and field athletes performed normally, and <7% showed the highest level. 19.23% of the athletes performed well [12]. The level of 53.84% of athletes dropped more. This is significantly lower than the comprehensive international ratio.

Sports life refers to the period from good performance to the best performance for athletes until retirement. Chinese men's best results are between 20 years old and 26 years old, and women are between 18 years old and 24 years old. Compared with the best athletes globally, the best competitive age is <5 years old [13]. The entire sports life span is 8–10 years short, and the best competitive age is between 25 years old and 29 years old. They stay in the competitive age group for 8–10 years. The best competitive age range of Chinese elite track and field athletes is between 20 years old and 24 years old, and the length of the competitive age group is relatively short. Generally, 4–5 years, and the entire sports life span is relatively short.

The training management of grassroots sports schools deviates from the development stage of sports training. The selection of Chinese competitive sports consists of a three-level training system. The prefecture-level sports school manages the sports training process. The specific method is to link the various treatments of the coaches with the achievements of the athletes he trains. This ignores the physical and psychological development stages of children and adolescents. The evaluation system that only seeks results but not process leads to adultizing the training methods of children and adolescent athletes.