# Predicting Putting Development Using Machine Learning Methods Based on RIOCHTrack Data
## 2 Methodology  
### Machine Learning Methods  
ANN is a computational model designed to simulate the neural network of the human brain. It consists of a series of tightly packed neurons, and each neuron has a certain number of inputs and a unique output [27]. It simulates biological neural networks, has strong adaptability and learning ability, establishes highly nonlinear mapping relationships, and solves both classification and regression problems very well. Hence, it has been widely applied in engineering fields. Comprising an input layer, a hidden layer, and an output layer, it is a type of one-way propagation neural network. A neuron is the basic unit of a neural network. The neurons in each layer are fully connected to the neurons in the next layer through weights and activation functions. After the input signal obtains the output value through the operation of neuron nodes in each layer, the weight of the network parameters is corrected by continuously reducing the error between the real and predicted values. The output variable of the next layer can be obtained by multiplying the input vector by a weight and bias, and an activation function. The calculation formulas are as follows. \[\mathrm{z}=\mathrm{W}^{\mathrm{T}}\mathrm{x}+\mathrm{b} \tag{1}\]  
\[\mathrm{h}=\mathrm{g}(\mathrm{z}) \tag{2}\]  
where x is the input vector, W is the weight matrix, b is the bias vector, z is the linear output function, g(.) is the activation function, and h is the output vector of the activation function. Support Vector Regression (SVR) is a commonly used supervised machine learning algorithm that makes predictions through the optimization of structural risk [28]. The SVR model was proposed by [PERSON] in 1992 [29]. Its popularity and applicability once exceeded those of ANNs. An SVR uses a kernel function to project the original dataset into a higher dimensional space, making it linearly separable. The objective function of SVR is transformed into a quadratic linear programming problem based on the idea of optimal plane classification:  
\[\mathrm{min}\left[\frac{1}{2}||w||^{2}+C\left(\sum_{i=1}^{m}\bar{\xi}_{i}+\sum_ {i=1}^{m}\bar{\xi}_{i}^{*}\right)\right] \tag{3}\]  
The constraint is:  
\[\varepsilon+\bar{\xi}_{i}\leq|y_{i}-f(x_{i})|\leq\varepsilon+\bar{\xi}_{i}^{*}, \ \bar{\xi}_{i}\geq 0,\bar{\xi}_{i}^{*}\geq 0,\ i=1,\ldots,m\]  
In the above equation, \(C\) is the penalty coefficient, representing the penalty for exceeding the data points \(\varepsilon\). The larger the \(C\), the greater the penalty for exceeding the data points \(\varepsilon\). \(\bar{\xi}_{i}\), \(\bar{\xi}_{i}^{*}\) are relaxation variables. Therefore, using \(\alpha_{i},\ \alpha_{i}^{*},f(\mathrm{x})\) can be expressed as  
\[f(x)=\sum_{i=1}^{m}(\alpha_{i}-\alpha_{i}^{*})K(x_{i},x)+b \tag{4}\]  
The commonly used kernel functions include the polynomial kernel function, the radial basis kernel function, and the sigmoid kernel function. Ensemble learning completes learning tasks by constructing and combining multiple base learners. Boosting and bagging are the most important ensemble learning algorithm models. Bagging randomly samples the original samples and trains the learning model based on the extracted samples. Repeat N times to obtain N base learners, and then obtain the final model by calculating the average error. Boosting is learn multiple weak classifiers by changing the weight distribution of training data and then linearly combining these classifiers to improve overall classification performance. RF and GB are representative models of ensemble learning [30]. Both models are supervised machine learning methods based on decision trees and constructed through integration. In the model training stage, RF, as a typical bagging algorithm, applies random sampling with dropout to the training set for each decision tree. When training a single decision tree, randomly select k features and select the optimal feature from these k features to split the nodes.