Research on Olympic medal prediction based on GA-BP and logistic regression model

2.1 Data preprocessing

The data of multiple data sets (data set references) are merged into an excel form, and descriptive statistical analysis is made. Taking the 2024 Paris Olympic Games as an example, as shown in Figure 1, and the missing values are replaced and filled. The descriptive statistical results are shown in Tables 1-2 and Table 3 model accuracy evaluation in the Table 4.

Figure 1. Medals for the 2024 Paris Olympic Games.

2.2 Correlation analysis

The variables were analyzed for correlation, and Pearson correlation test was used because the data passed the normality test.

The result is shown in the following Figure 2:

Figure 2. Correlation coefficient thermogram.

As shown in Figure 3, the axis of the icon lists the correlation coefficients of gold, silver, bronze and total medals. These correlation coefficients reflect the degree of correlation between the number of medals and other columns in the table. Among them, it is shown that there is a very close relationship between them. The correlation between the HOST variable and the total number of medals is 0.3, which shows that the athletes in the host country have the advantage of gold medal, but it is not significant. Through further analysis, we can find that gold medals occupy an important position in the total number of medals. In the Olympic Games, the gold medal is the symbol of the highest honor, and every participating country or region will strive for more gold medals. At the same time, although the number of silver medals and bronze medals is also included in the total number of medals, their value is relatively low. Therefore, the change of the number of gold medals will often affect the total number of medals. It can also be seen from the figure that the correlation between gold medals and silver medals and bronze medals is also high, which is 0.933 and 0.898 respectively, but it is still not as large as the correlation coefficient between gold medals and total medals, which further proves the important position of gold medals in the total number of medals. This also shows that in the Olympic Games, when countries compete for medals, they often pay attention to the number of gold medals, silver medals and bronze medals at the same time, but the gold medal is always the most important goal.

Figure 3. Proportion of awards in Olympic events.

2.3 Analysis of the number of medals won in sports events

In the provided grouping summary table, the data show Medals and Total of different Sport. Through the analysis of these data, we can draw the following rigorous, refined, accurate, complete, coherent and logical interpretations: from traditional Athletics and swimming to Cricket and Croquet. Medals, as a key index to measure the performance of each event, reveals the competitive situation and strength distribution among events. Total may reflect the participation, influence or overall scale of the events. In order to deeply analyze the competitive level and popularity of each event, swimming (83) and track and field (52, on average) are the two events with the largest number of medals, indicating that these two events have a very high level of competition and extensive participation. At the same time, basketball (11.5), boxing (18.5), gymnastics (if rhythmic gymnastics and competitive gymnastics medals are considered together, the total number is 50.5, but the analysis here is based on separate data) and other projects have also won more medals, revealing the popularity and competitiveness of these projects in the global scope.

2.4 Evaluation of indicators

3.1 Establishment of medal prediction model

3.1.1 BP neural network model

BP network is a nonlinear system, and its most remarkable feature is its high adaptability (Sanglin Zhao, 2024).⁵ The algorithm takes 100 samples of the research object as the input nodes of the neural network, and the expected results as the corresponding output nodes, calculates the weights and thresholds, and calculates the error between the actual results and the predicted results, as shown in Formula (1). Fitness function is the standard to measure whether the error value meets the requirements. For the calculation results that do not meet the requirements, the network will carry out error back propagation. Equation (2) is the correction of the hidden layer and the output layer. Through repeated iterations, if the error meets the expectations, the model is successfully established. $\mathit{nX}=(x_1,x_2,\cdots ,x_j,\cdots ,x_n)Y=(y_1,y_2,\cdots ,y_j,\cdots ,y_n)$

Where: the expected result is the prediction result, the learning rate is the hidden layer, the output value is the output layer, and the output value of this node is the correction. $y_i\left(n\right);y_i^{\prime }\left(n\right);\eta ;E;\Delta W_{\mathit{kj}};\Delta W_{\mathit{kj}}$ .

3.1.2 Genetic algorithm

The basic idea of genetic algorithm is a population that needs to be optimized. On the basis of the survival of the fittest, it evolves from generation to generation, leaving the evolved population with good fitness and inheriting it to the next generation. In this process, it needs to be crossed and mutated to produce a new population with better fitness. The optimization process is shown in Figures 4 and 5.

Figure 4. Schematic diagram of genetic algorithm.

Figure 5. GA-BP flow chart.

A. Chromosome coding

Floating-point coding is selected as the coding rule, which can obtain higher precision network weights and thresholds on the one hand, and a wider search range than before on the other hand, with the length s as shown in Formula (3). Where is the number of neurons in input layer, hidden layer and output layer.

B. Fitness function

During the training of a BP neural network, the network’s outputs and desired outputs are selected as training samples. After training, the resulting weights and thresholds are chosen as the optimal parameters. Genetic algorithm metrics are employed as indicators for the optimization process. Subsequently, the sum of squared errors between the predicted values and actual values is computed., as shown in the formula. $\overline{y}\left(k\right)y\left(k\right)$

In this context:

k represents the logarithmic transformation of both input and output sampling data.

n denotes the total count of network output nodes.

The kth node’s anticipated output is compared against its predicted output value.

In order to avoid the phenomenon of dividing by zero, a value close to zero is introduced here, and the fitness function is Formula (5). $\overline{y}\left(k\right)y\left(k\right)\mathrm{\delta }$

C. Choose

Let the population size be n and fi be the fitness, and this fitness is for the individual I in the population, then the probability of I being selected is shown in Formula (6).

D. Crossing

Arithmetic crossover method is widely used in individuals with floating-point crossover. Assuming that arithmetic crossover is performed between individuals, the new individuals generated after the operation are shown in the Formula (7). $x_A^{\mathrm{t}}\text{and}{x}_B^{\mathrm{t}}$

Where: α is a random number evenly distributed in the interval [0,1].

E. Variation

In order to prevent premature phenomenon, based on a small range of random numbers, genetic variation occurs in a small range of probability, and the local search ability of the algorithm will also be strengthened in this step. At this time, it is assumed that the maximum and minimum values of the initial individuals of the population are respectively summed, and then the mutated gene is cut as shown in Formula (8). x_min and x_max then, the mutated gene can be seen in Formula (8)

Where: β is a random number evenly distributed in the interval [0,1]. Figure 4 is a schematic diagram of genetic algorithm, and Table 5 is a flow chart of combinatorial optimization model.

3.2 The solution of medal prediction model

Figure 6 shows the fitting effect of the prediction model:

Figure 6. Comparison between predicted value and true value of GA-BP model.

Figure 6, Figure 7 and Table 5 show the training process of GA-BP algorithm model. According to the information analysis in the figure, the R value of BP algorithm model on the target data set is 0.9001, while the R value of GA-BP algorithm model on the target data set is 0.9881, which is closer to 1 than BP algorithm model, which indicates that the fitting degree of GA-BP model is higher than that of BP model. In addition, MSE can also be observed in Figure 7. The models of the two algorithms have different MSE values after different rounds of training. After three rounds of training, the MSE value of GA-BP algorithm reaches the minimum value of 0.629, which greatly reduces the mean square error compared with the BP algorithm model, which shows that the model is simpler and the prediction accuracy is higher. This paper shows the performance comparison results of GA-BP algorithm and BP algorithm on different data sets through several performance indicators. As can be seen from the figure, GA-BP algorithm is superior to BP algorithm in training error, R value, MSE value, Evar value and MAE value.

Figure 7. Model training process diagram.

The prediction results are shown in Table 6:

As shown in Figure 8, data collection and analysis based on GA-BP algorithm show that United States is expected to win 42 gold medals in the 2028 Los Angeles Olympic Games, ranking first in the gold medal list, followed by China with 37 gold medals and South Korea with 16 gold medals. The number of gold medals predicted by other countries is shown in the figure. As a global sports power, the United States has always performed well in international sports events such as the Olympic Games. The United States has strong competitiveness in various sports, especially track and field, swimming, gymnastics and other events, which are often the major events that produce gold medals. The number of gold medals predicted this time is as high as 42, which reflects the leading position of the United States in global sports competition. As a new sports power, China has performed better and better in international sports events such as the Olympic Games in recent years. The number of gold medals predicted this time is 37, although it is slightly less than United States, but it is still a very impressive number, which is far from the third place. China has traditional advantages in table tennis, badminton, diving, weightlifting, etc. At the same time, it is also striving to enhance the competitiveness of other new sports and strive to achieve the all-round development of national sports. Other countries such as Great Britain and Italy have relatively few gold medals. This does not mean that the sports strength of these countries is weak, but that the number of gold medals predicted may be low due to fierce competition or other factors. In addition, there are some countries behind, although the number of gold medals predicted is not much, but they may also make breakthroughs in some advantageous events.

Figure 8. American gold medal forecast chart.

3.3 Logistic Regression Prediction Model and Solution of First Gold Medal Countries

The sports events and winning trends are used as data training sets for logical probability regression, and the logical variables of winning gold medals or not are used as dependent variables for analysis.

The binary logistic regression results in Tables 7, 8 and 9 show that:

The significance p value of Total is 0.001***, which is significant horizontally, rejecting the original hypothesis. Therefore, Total will have a significant impact on Y (whether it wins the prize or not), which means that the probability of Y (whether it wins the prize or not) is 1.099% higher than that of 0.0 for every additional unit of Total. Sport has no significant influence on Y (whether winning or not) (p=0.718).

ROC curve

As can be seen from Table 10 and Figures 9-11, the true positive rate (TPR) increases with the increase of false positive rate (FPR). When the FPR is at a low level of 0-0.2, the TPR increases slowly, ranging from 0-0.4; When FPR starts to rise from 0.2, the rising speed of TPR is accelerated, and when FPR is about 0.4, TPR reaches about 0.8; After that, FPR continued to increase, and the rising trend of TPR slowed down, finally approaching 1.0. On the whole, the chart shows typical ROC curve characteristics, which reflects the classification performance of the model under different thresholds. With the increase of the allowable false positive rate, the true positive rate also increases, but the increase amplitude decreases when the FPR is high, indicating the performance of the model in balance misjudgment and correct recognition.

Figure 9. ROC curve.

Figure 10. Predicting the probability of winning gold medals in American events (part).

Figure 11. Probability map of winning gold by countries (some top countries).

Probability map of winning gold by countries (some top countries):

Based on the logistic regression prediction model, data collection and analysis show that United States will be the first in the gold medal list of the 2028 Los Angeles Olympic Games, and all its events have considerable winning probability, with the winning probability of Athletics and Badminton even as high as 1.0. The track and field event United States has always been in a dominant position in the Olympic history, and badminton is an Olympic event that the United States has paid more attention to in recent years. Although the winning probabilities of baseball, baseball/softball, basketball, beach volleyball and boxing are lower than 0.8, they are all higher than 0.6. The winning probability of the remaining Olympic events, such as Breaking, Canoe Slalom, Canoe Sprint, Canoeing and Cycling, is between 0.6 and 0.4, and the winning probability of Dividing, Equestrianism and Fencing is between 0.2 and 0.4. It can be seen that the advantages and disadvantages of the United States coexist in Olympic events. Its advantage lies in that track and field and badminton are traditional strengths, and there are many gold medals in these two events. With excellent physical fitness and advanced training system, the United States has strong competitiveness in many of them. The United States also has outstanding strength in collective big ball games, such as basketball, volleyball and rugby, with a mature professional league system and many high-level athletes. In addition, baseball/softball and other new events in the 2028 Los Angeles Olympic Games have a broad mass base and profound cultural heritage in the United States, and American players occupy the advantage of home court. However, there are also some inferior events in the United States. In table tennis, diving and other technical events, it is difficult for the United States to compete with traditional strong teams such as China, and the results are often poor. Although the United States may win gold medals in fencing, equestrian and other events, the overall probability is small, and it does not have obvious advantages compared with some countries that focus on these events.

In the statistics of predicted medals in some countries in 2028, Algeria (Algeria) and Argentina (Argentina) are outstanding, both of which are regarded as the countries most likely to win the championship for the first time. Driven by policies and GDP, it is predicted that Algeria and Argentina will win two gold medals, one silver medal and one bronze medal in 2028, totaling four medals.

In recent years, Algeria has invested a lot of policy support in sports development. The country has made great efforts to build sports infrastructure and built a number of modern stadiums nationwide, providing athletes with a high-quality training and competition environment. At the same time, the training plan for sports talents has been implemented, and potential young talents have been selected from the youth groups for systematic professional training, and excellent athletes have been provided with opportunities for overseas exchanges and competitions. On the economic level, global macro models and analysts of Trading Economics predict that Algeria’s GDP will reach 249.02 billion US dollars by the end of 2024; In the long run, it is estimated that it will be about $256.49 billion in 2025 and $262.90 billion in 2026, which provides a strong financial guarantee for the government in sports. Argentina also has a positive policy orientation in sports development. The government attaches great importance to the inheritance and development of traditional advantageous sports such as football, and at the same time, it is constantly tapping the potential of other sports and encouraging more young people to participate in various sports. In the field of football, Argentina has a perfect youth training system, which continuously transports talents for the national team. In addition, Argentina’s GDP ranks high in South America. Although the GDP has declined in recent years, Argentina’s economy is recovering strongly, and its developed agriculture, animal husbandry and industry provide economic support for sports development.

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