## 2 Materials and Methods
### Orange Data Mining Application and Image Management
\[Gini\left(t\right)=1-\sum_{k=1}^{K}p_{k}^{2} \tag{6}\]  
where \(p_{k}\) is the proportion of samples belonging to class \(k\) in node \(t\), and \(K\) is the number of classes. The goal is to choose splits that minimize the weighted sum of Gini impurities of the child nodes. For regression tasks, the decision tree algorithm typically uses the mean squared error (MSE) as the splitting criterion. The MSE for a node \(t\) is given by Equation (7). \[MSE\left(t\right)=\frac{1}{N_{t}}\sum_{i\in t}(y_{i}-\overline{y_{t}})^{2} \tag{7}\]  
where \(y_{i}\) is the actual value of the target variable for observation \(i\), \(y_{t}\) is the mean of the target values in node \(t\), and \(N_{t}\) is the number of observations in node \(t\). The algorithm selects the split that minimizes the sum of the MSEs of the resulting child nodes. The splitting process continues until a stopping criterion is met, such as a maximum tree depth or a minimum number of samples per leaf. The resulting tree model can be visualized as a hierarchical structure where each internal node represents a decision based on a feature, and each leaf node represents a predicted outcome. One of the key strengths of tree-based models is their ability to capture non-linear relationships between the input features and the target variable without requiring feature scaling. Moreover, tree models are naturally capable of handling both numerical and categorical data. However, they tend to be prone to overfitting, which can be mitigated through techniques like pruning or by using ensemble methods such as random forests or gradient boosting; naive Bayes is a simple yet effective probabilistic classifier based on [PERSON] theorem, with the strong assumption that features are conditionally independent given the class label ([PERSON] et al., 2018). Despite its simplicity, naive Bayes often performs surprisingly well in a variety of classification tasks, particularly in text classification and other domainswhere this independence assumption holds approximately true. [PERSON] theorem forms the foundation of naive Bayes and is expressed as Equation (8). \[P\left(\left.C\middle|X\right)=\frac{P\left(\left.X\middle|C\right)\cdot P(C)}{P(X)}\right. \tag{8}\]  
where:  
\(P\left(\left.C\middle|X\right)\right.\) is the posterior probability of the class \(C\) given the feature vector \(X=\left\{x_{1},\ x_{2},\ \ldots,\ x_{n}\right\}\)  
\(P\left(\left.X\middle|C\right)\right.\) is the likelihood of observing the feature vector \(X\) given the class \(C\). \(P(C)\) is the prior probability of the class \(C\). \(P(X)\) is the marginal likelihood, which acts as a normalization constant. Finally, neural networks are a class of machine learning models inspired by the structure and function of biological neurons. These models are particularly powerful in capturing complex, non-linear relationships between inputs and outputs, making them highly suitable for a wide range of predictive tasks [49]. The architecture of a neural network consists of interconnected layers of artificial neurons, typically organized into an input layer, one or more hidden layers, and an output layer. The output of each neuron in a layer is computed as a weighted sum of its inputs, followed by the application of an activation function. Mathematically, for a given neuron \(j\) in layer \(l\), the output \(a_{j}^{(l)}\) is given by Equation (9). \[a_{j}^{(l)}=f\left(\sum_{i=1}^{n}w_{ij}^{(l)}a_{i}^{(l-1)}+b_{j}^{(l)}\right) \tag{9}\]  
where \(w_{ij}^{(l)}\) represents the weights connecting neuron \(i\) in layer \(l-1\) to neuron \(j\) in layer \(l\); \(a_{i}^{(l-1)}\) is the output of neuron \(i\) in the previous layer; \(b_{j}^{(l)}\) is the bias term for neuron \(j\); and \(f(\cdot)\) is the activation function, such as the ReLU (Rectified Linear Unit), sigmoid, or tanh function. The network's final output is produced by the neurons in the output layer, and the overall network function can be represented as Equation (10).