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The application of the improved chaos algorithm in the football performance prediction.

1. Introduction

It is the current developing trend to combine the football with the prediction. The scholars studied the prediction of the football industry (Xia and Jiang, 2015) and the prediction of football talents (Chen and Zhao, 2006). The improvement of the football performance cannot be separated from scientific training. The prediction of the football performance can help the coach analyze the rules of the game. Then, that can formulate the scientific and reasonable training plan. In the modern football development, the professional trainers will find rules from training data and help the athletes for the football training. Under this background, it is very necessary to predict the football performance.

The Chaos theory is an important part of the nonlinear theory (Andr, 2016; Lino et al., 2016). The sensitive characteristics of the initial value for the chaotic system makes the changes of the system input can be reflected quickly in the output. The feature indicates that the chaotic model is closer to the reality of the world (Konrad and Rafa, 2015).

In the long development course, the Chaos theory unceasingly consummates. The research direction has included the chaotic dynamics (d'Aquino et al, 2016), the chaos prediction (Heydari et al, 2016), the chaos control (Chattopadhyay et al, 2015) etc. with the urgent demand of the social practice to the prediction theory and the limitations of the traditional forecasting models, the chaos prediction has become a focus of the chaos theory study (Wu et al, 2009). Therefore, in this paper, we choose the chaos prediction theory to study. It has the great theoretical value and the practical significance.

In the long development course, the Chaos theory unceasingly consummates. The research direction has included the chaotic dynamics, the chaos prediction, the chaos control etc.

In this paper, we study the prediction of the football performance. By using the thought of chaos prediction, we forecast the football performance. In order to achieve a good prediction effect, we put forward the improved chaos algorithm. Then, we use the algorithm to study and predict the athletes' football performance. Finally, the experiment shows that the algorithm which applies to the football performance prediction has achieved the good results.

2. The chaos algorithm

Li Yorke definition gives the first mathematical definition of the chaos. It is the first time that the scholar gives the chaos the strict scientific significance. It is the chaos mathematical definition which has the great influence. It is defined from the interval mapping. For the continuous self-mapping on the interval, if it satisfies the following conditions, it can have the chaos phenomenon.

1. For the periodic point f, the periodic has no the upper bound. f has the periodic points with arbitrary positive integer period. That is, for any natural number n, there is x e I making [f.sub.n](x) = x .

2. For the closed interval I, there exists S satisfying,

When [for all]x, y [member of] S , x [not equal to] y

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

When [for all]x, y [member of] S

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

When [for all]x [member of] S and the any periodic point of is y,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

sup[absolute value of ([f.sup.n](x) - [f.sup.n](y))] is the upper bound of the distance between [f.sup.n](x) and [f.sup.n](y). inf[absolute value of ([f.sup.n](x) -[f.sup.n](y))] is the lower bound of the distance between [f.sup.n](x) and [f.sup.n](y).

Li-Yorketheorem: we assume that f (x) is the mapping in [a,6]. If f (x) has three periodic points, for any positive integer n, f (x) has n periodic points.

Takenstheorem: M is d dimensional space, [phi]: M [right arrow] M, [phi] is a smooth diffeomorphism. y: M [right arrow] R, y has two order continuous derivative. [phi]([phi]y): M [right arrow] [R.sup.2d+1], where,

[phi]([phi], y) = (y(x), y([phi](x)), y([[phi].sup.2](x)), ..., y([[phi].sup.2d] (X))) (4)

Therefore, [phi]([phi], y) is a embed from M to [R.sup.2d+1].

There are many ways to choose the delay time and embedding dimension. According to the embedding theorem, if m [greater than or equal to] 2d +1, we can reveal the chaotic attractor. There are many ways to choose the embedding methods. We introduce the Cao method and the C-C method.

The Cao method was proposed by Cao Liangyue in 1997. Firstly, we define,

a(i, m) = [parallel][Y.sub.i](m+1)-[Y.sub.n(i,m)](m+1)[parallel]/[parallel][Y.sub.i](m)-[Y.sub.n(i,m)](m)[parallel] (5)

Where, [Y.sub.i](m+1) is the i vector in (m + 1) dimension reconstructed phase space. In m dimension reconstructed phase space, n(i, m) makes [Y.sub.n(i,m)](m) as the nearest neighbor integer of [Y.sub.i](m). n(i, m) is determined by i and m. [parallel]*[parallel] is Euclid records. It can also be calculated and achieved by the maximum norm. That is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

We define that all the average value of a(i,m) is,

E(m) = [1/N-m[tau]] [N-m[tau]].summation over (i=1)] a(i,m) (7)

Then, we define the change from m to m+1,

[E.sub.1](m) = E (m + 1)/E (m) (8)

In Cao method, at the same time, we use another parameter to distinguish the new good and random signal of the deterministic chaos. That is,

[E.sup.*](m) = [1/N-m[tau]] [N-m[tau].summation over (i=1)] [absolute value of ([x.sub.i+m[tau]] - [X.sub.n(i,m)+ m[tau]] (9)

[E.sub.1](m) = [E.sup.*](m + -)/[E.sup.*](m) (10)

For each time series of the C-C method, C-C method adopts the following to calculate.

[bar.S]([tau]) = [1/16] [7.summation over (m=2)](4.summation over (j=1))S(m, [r.sub.j], [tau]) (11)

[DELTA][[bar.S].sub.1]([tau]) = [1/4] [7.summation over (m=2)][DELTA]S (m, [tau]) (12)

[S.sub.cor] ([tau]) = [DELTA][[bar.S].sub.1] ([tau]) + [absolute value of ([bar.S]([tau]))] (13)

S(m, [r.sub.j], [tau]) = C(m, [r.sub.j], t) - [C.sup.m](1, [r.sub.j], [tau]) reflects the system's self- correlation characteristic. r is the size of the neighborhood radius.

C (m, r, [tau]) = 1/[M.sub.2] [summation over (1[less than or equal to]j[less than or equal to]k[less than or equal to]M)] [theta](r -[parallel][X.sub.j] - [X.sub.k][parallel]) is the correlation integral of the system. #0 is

the Heaviside function.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

For the estimation of the correlation dimension, we adopt G-P algorithm [22]. When r [right arrow] o, N [right arrow] [infinity], the associated integral function is approximated to the following function.

[log.sub.2] C(m, r, [tau]) [right arrow] v [log.sub.2] r

When m continues to increase, the limit value of v is the correlation dimension of the system.

We select the first zero point of [bar.S](t) or the first minimum point which goes through [DELTA]S(t) at the first time as the most appropriate time delay [tau]. The minimum point of [S.sub.cor] (r) is the optimal time delay window.

3. Improved chaos prediction method

During the prediction process of the chaotic time series, the selection of the embedding dimension and delay time is the basis of chaotic time series prediction. At present, there are a lot of computation methods (Li et al, 2011).The same chaotic time series can be embedded in different (or multiple) phase space. The predicted results are also different. In order to improve the reliability and practicability of the chaotic time series prediction, we propose a new multi-embedding dimension method. Near the obtained embedding dimension, the method takes several different dimensional numerical and calculate the load forecasting results under different embedding dimension. Finally, we weight and average the obtained results and they are the final prediction value.

For the known time series{x, i=i, 2, N}, firstly, according to the certain embedding dimension and the delay time calculation method, we calculate the embedding dimension m and the delay timer. Then, we reconstruct the phase space and establish the first order linear prediction model. In general, the different embedding dimension calculation method or the difference of the artificial subjective factor can make the different value of the embedding dimension. The predicted results are also different in different embedding dimensions. We assume that there are k different embedding dimensions. In each embedding dimension, the prediction result of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The prediction error is,

[DELTA][e.sub.i] = [[??].sub.i] (n + 1) - x(n + 1), i = 1,2, ...,k (15)

Where, x (n +1) is the actual value. [[??].sub.i](n +1) is the prediction value of the i embedding dimension. We assume that the prediction average value in k embedding dimension is [x.sub.p]. That is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

The error of each prediction is,

[r.sub.i] = [[??].sub.i] (n + 1) - [x.sub.p] (17)

We assume that the probability that we obtain [r.sub.i] is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

Because the prediction results are independent in different embedding dimensions, the occurrence probability for all k is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

If p is bigger, it shows that the prediction results are reliable. That is, when min [k.summation over (i=1)] [r.sup.2.sub.i], the prediction results are the most reliable. Therefore, according to the least square method, the final prediction result in different embedding dimension is,

[??](n + 1) = [k.summation over (i=1)] p([r.sub.i])[[??].sub.i](n + 1)/[k.summation over (i=1)]p([r.sub.i]) (20)

Where, [[??].sub.i] (n +1) is the prediction value of the I embedding dimension at n+1 time. p([r.sub.i]) is the probability of the prediction error for the i embedding dimension.

In the phase space, any two adjacent tracks diverges with the trend that largest lyapunov exponents [[lambda].sub.1]. That is, in normal circumstances, if the initial two state displacement d(o) is not entirely along the vertical [[lambda].sub.1], we can easily get,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

We assume that there is a time series. We use it to reconstruct the phase space. We select arbitrarily two adjacent initial positions. In general, the two points are the initial point and the nearest neighbor. Then, we calculate the initial distance d(o) of the two points. Next, we select appropriately the step size [DELTA]t and the time length t = k[DELTA]t. We calculate the distance d(k[DELTA]t) that elapsing the time t. Then, we can get,

[[lambda].sub.1] = [1/k[DELTA]t] In [d(k[DELTA]t)]/d(o) (22)

We select the different initial distance [d.sub.j](o) to calculate [d.sub.j](k[DELTA]t) and calculate the average value.

[[lambda].sub.1] = [1/k[DELTA]t] In [[d.sub.j](k[DELTA]t)/[d.sub.j](o)] (23)

4. Experiment

We take the football school 1500 meter race as the example. We apply the improved chaos method to predict the 1500 meter long-distance race performance. We select the performance of 4 athletes to predict. Among them, the first 20 data are as the training set and the last 10 data are as the prediction set. The training set data are shown in the following table.

We put the data in a figure.

Then, we predict the performances. The prediction value and the actual value are as follows.

From the above table, we can see that the actual values and the predicted values of the four athletes are very similar. The errors are very small. It shows that the improved chaos algorithm has achieved the good results for the football performance. The experimental results demonstrate furtherly the effectiveness of the improved algorithm.

5. Conclusion

The development of the football cannot have the science. Training the football by the scientific way can find out the rule, improve the training performance of the athletes and promote the development of the football. Chaos prediction is a mature forecasting method. In this paper, we use the chaos prediction method to study the prediction of the football performance. At the same time, we put forward the improved chaos prediction method to make the result more accurate. Firstly, this paper introduces the basic theory of the chaos prediction. Then, this paper puts forward the improved chaos algorithm. In the experimental part, we use the improved chaos algorithm to predict the performance of the athletes and achieve the good experimental results.

Recebido/Submission: 23/07/2016

Aceitacao/Acceptance: 01/10/2016

References

Andre, F. (2016). A polynomial chaos approach to narrow band modeling of radiative heat transfer in non-uniform gaseous media. Journal of Quantitative Spectroscopy & Radiative Transfer, 175, 17-29, DOI: 10.1016/j.jqsrt.2016.01.037

Chattopadhyay, J., Pal, N., Samanta, S., Venturino, E., Khan, Q. J. (2015). Chaos control via feeding switching in an omnivory system. Bio Systems, 138, 18-24, DOI: 10.1016/j.biosystems.2015.10.006

D'Aquino, M., Quercia, A., Serpico, C., Bertotti, G., Mayergoyz, I. D., Perna, S. (2015). Chaotic dynamics and basin erosion in nanomagnets subject to time-harmonic magnetic fields. Physica B Condensed Matter, 486, 121-125, DOI: 10.1016/j. physb.2015.09.032

Heydari, G., Vali, M. A., Gharaveisi, A. A. (2016). Chaotic time series prediction via artificial neural square fuzzy inference system. Expert Systems with Applications, 55, 461-468, DOI: 10.1016/j.eswa.2016.02.031

Jiang, T., Xia, M., Deptof, P. E. (2015). Development of Chinese sports industry under new norm. Journal of Wuhan Institute of Physical Education, DOI: 10.15930/j. cnki.wtxb.2015.05.008

Kolesko, K., Latala, R. (2010). Moment estimates for chaoses generated by symmetric random variables with logarithmically convex tails. Statistics & Probability Letters, 48(4), 1103-1136, DOI: 10.1016/j.spl.2015.08.019

Li, S., Li, Q., Li, J., Feng, J. (2011). Chaos prediction and control of good win's nonlinear accelerator model. Nonlinear Analysis Real World Applications, 12(4), 1950-1960, DOI: 10.1016/j.nonrwa.2010.12.011

Lino, A., Rocha, A., & Sizo, A. (2016). Virtual teaching and learning environments: Automatic evaluation with symbolic regression. Journal of Intelligent & Fuzzy Systems, 31(4), 2061-2072. DOI: 10.3233/JIFS-169045

Wu, J., Lu, J., Wang, J. (2009). Application of chaos and fractal models to water quality time series prediction. Environmental Modelling & Software, 24(5), 632-636, DOI: 10.1016/j.envsoft.2008.10.004

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Hui Song

songhui31939@126.com

Xi'an Physical Education University, Xi'an 710068, China

Table 1--Data of the training set

No    Play 1   Play 2   Play 3   Play 4

1      275      243      257      248
2      277      248      246      241
3      265      245      253      244
4      266      243      258      237
5      268      238      253      235
6      273      235      254      238
7      272      233      246      243
8      270      230      244      246
9      265      236      241      237
10     267      231      242      240
11     262      230      238      241
12     268      226      235      249
13     264      234      241      247
14     265      237      232      253
15     266      243      240      248
16     259      238      245      251
17     261      232      244      254
18     263      234      251      247
19     258      236      246      244
20     260      233      243      239

Table 2--The actual values and the predicted values

No         Play 1                Play 2

      Actual    Predicted   Actual    Predicted
      results    results    results    results

21      257        258        238        240
22      264        266        237        239
23      261        264        235        237
24      253        256        236        236
25      255        255        231        233
26      260        261        230        231
27      265        263        235        233
28      257        261        233        234
29      259        260        231        235
30      261        262        235        234

No          Play 3               Play 4

      Actual    Predicted   Actual    Predicted
      results    results    results    results

21      248        249        241        244
22      246        247        248        250
23      254        254        247        251
24      246        248        253        252
25      247        244        256        254
26      245        246        249        252
27      244        245        248        247
28      249        247        242        246
29      245        240        241        242
30      243        241        239        240
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Author:Song, Hui
Publication:RISTI (Revista Iberica de Sistemas e Tecnologias de Informacao)
Date:Nov 15, 2016
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