Random forest is an integrated machine learning algorithm that improves the accuracy and robustness of a model by combining the predictions of multiple decision trees. For the classification task, the prediction result of random forest is usually obtained through the majority voting mechanism, that is, each tree gives a prediction result, and the majority category is finally selected as the prediction result of random forest. This process can be expressed as: for the input feature vector x, each tree $t_i$ in the random forest gives a prediction result $y_i$ (for classification tasks, $y_i$ is a category label). The final prediction class y* is the class that maximizes the following expression:

$$\mathrm{y}*=\arg \underset{y}{\max }\sum _{i=1}^{k}I(y_i=y)$$

Where, $I()$is the indicator function, which takes the value 1 when the parenthesis condition is true, otherwise it is 0; k is the number of trees in the random forest; y is the possible category label.

LR belongs to probabilistic nonlinear regression, which is mainly used to study the relationship between the outcome index of binary classification (dependent variable) and some influencing factors (independent variable) (can be extended to multiple categories). It is commonly used in epidemiology to analyze quantitative relationships between diseases and associated risk factors. The LR model can be expressed as:

$$P=\frac{1}{1+\exp \left[-\left({\beta }_0+{\beta }_1{X}_1+{\beta }_2{X}_2+\mathrm{\cdots }+{\beta }_m{X}_m\right)\right]}$$

In the formula, P is the probability when a positive result occurs, ${\beta }_0$ is the constant term, ${\beta }_1$, ${\beta }_2$…, ${\beta }_m$ is the independent variable regression coefficient of $X_1$, $X_2$, …, $X_m$. Logarithmic conversion of the formula can be expressed in linear form:

$$\mathrm{logit}P=\ln \frac{P}{1-P}={\beta }_0+{\beta }_1{X}_1+{\beta }_2{X}_2+\mathrm{\cdots }+{\beta }_m{X}_m$$

The $logitP$for positive results with negative results occur when probability of the natural logarithm and the value of$logitP$ ranges have no numerical bounds.

XGBoost is an ensemble learning algorithm based on gradient-raising decision trees that optimizes the loss function by adding prediction trees, each attempting to correct the error of the previous tree‌. The core idea of XGBoost is to combine multiple weak classifiers (decision trees) into one strong classifier. Its mathematical formula mainly involves the definition and optimization of the objective function. The objective function of XGBoost can be expressed as:

$$\mathrm{obj}(\theta )=\sum _{i=1}nL(y_i,\widehat{y})+\sum _{k=1}K\mathrm{\Omega }\left(f_k\right)$$

L ($y_i$, $\widehat{y}$) represents a loss function that measures the difference between the model’s predicted value $\widehat{y}$ and the actual value $y_i$. $\mathrm{\Omega }\left(f_k\right)$ represents the complexity of the $k$-th tree and is used to control the complexity of the model to prevent overfitting. $\theta$ represents the parameters of the model. XGBoost performs a second-order expansion of the loss function using Taylor’s formula to better approximate and optimize the loss function. By adding prediction trees, XGBoost gradually reduces residuals and improves the predictive performance of the model. Because of its efficiency, flexibility and powerful performance, XGBoost has been widely used in a variety of machine learning tasks such as classification, regression, and sequencing.

SVM is a two-class classification model, its basic model is defined as the linear classifier with the largest interval on the feature space, its learning strategy is to maximize the interval, and finally can be transformed into a convex quadratic programming problem. The goal of SVM is to find a hyperplane:

$$w^{T}x+b=0$$

$w$ is the weight vector and $b$ is the bias term. The sample points of different classes are separated, and the distance from the nearest point (i.e., support vector) to the line is maximized as far as possible. This distance is called the margin, and the SVM attempts to maximize this gap.

KNN algorithm is a simple and intuitive classification and regression method, namely the K nearest neighbor algorithm. The core idea is that a sample belongs to a class if most of the K nearest neighbors of the sample in the feature space belong to that class. The general flow of the KNN algorithm is as follows:

First determine the size of the K, that is, how many neighbors to choose to participate in the decision. The choice of K value has great influence on the performance of the algorithm. Then calculate the distance between the test object and all objects in the training set: Euclidean distance is generally adopted, and the formula is:

$$d(x,y)=\sqrt{{\sum }_{i=1}^n{\left(x_i-y_i\right)}^2}$$

Where x and y are two points in n-dimensional space, and $x_i$ and $y_i$ are their coordinates on the i-th dimension, respectively. According to the calculated distance, the K training samples closest to the test sample are found, which are the K nearest neighbors of the test sample. Then take a vote or weighted average and finally use the prediction category or predicted value as the output of the algorithm.

First, it is necessary to sample N times from the original data set (Bootstrap sampling method) to form a training set with the same size N (the comparison with the original data set is not completely consistent). If each sample in the data set has T attributes, t (t $\le$ T) attributes will be randomly selected when the RF internal decision tree is split, and then the split attributes of the node will be selected according to some strategy, and finally a decision tree will be grown on the training set with size N. This is repeated m times, and a random forest of m decision trees is trained.

After completing the model parameter tuning, the predictive ability of each model was verified using a test machine, and the receiver operating characteristic curve (ROC) of each model was plotted, and precision, accuracy, and recall were selected, and the F1 score and area under curve (AUC) were selected as the evaluation indexes of model effectiveness. The model with the largest AUC was selected as the best model, and the Hosmer-Lemeshow (HL) was further performed to assess the degree of correspondence between the predicted probabilities and observations using the Hosmer-Lemeshow goodness-of-fit test [55]. p-values less than 0.05 indicate that there may be a model fitting problem, such as overfitting or underfitting [56]. We used the SHapley Additive exPlanation (SHAP) algorithm to explain the prediction model, which provides a globally consistent explanation of the model from the theory of game theory, can explain each feature output of the machine learning model at the group level as well as at the individual level, and visualize the output results to study the relative importance of each feature, in which the SHAP value of the edible oil bar graphs and scatter plots composed of summary graphs in a graphical representation, are used to illustrate the importance of individual features and their overall impact on model predictions [57].

Accuracy is a measure of the percentage of all predicted samples that the model correctly predicts. In this study, accuracy provides an intuitive evaluation criterion to help us understand the predictive power of the model as a whole. By evaluating the accuracy, it is possible to determine whether the model’s performance on the test data meets expectations, thus providing a basis for further optimization and the calculation formula is:

$$\text{Accuracy}=\frac{TP+TN}{TP+TN+FP+FN}$$

TP (True Positive) is the number of samples correctly predicted as a positive class, TN (True Negative) is the number of samples correctly predicted as a negative class, FP (False Positive) is the number of samples incorrectly predicted as a positive class, FN (False Negative) is the number of samples incorrectly predicted as a negative class.

The accuracy rate measures the proportion of all samples predicted to be positive that are actually positive. In this study, the accuracy rate reflects the accuracy of the model when predicting positive classes, such as patients with cardiovascular disease. The higher accuracy indicates that the model can identify the real positive samples well, which is of great significance for avoiding false positive prediction and reducing misdiagnosis, and the formula is:

$$\text{Precision}=\frac{TP}{TP+FP}$$

Where TP is the number of samples correctly predicted as positive, and FP is the number of samples incorrectly predicted as positive.

The recall represents the percentage of all samples that are actually positive that are correctly predicted to be positive. Recall rates in this study were used to assess the model’s ability to identify positive samples, especially in high-risk patients. The higher recall rate means that the model can capture more actual positive samples, which is crucial for early detection of diseases and reducing missed diagnoses and is calculated as:

$$\text{Recall}=\frac{TP}{TP+FN}$$

Where TP is the number of samples correctly predicted as a positive class, FN is the number of samples incorrectly predicted as a negative class.

The F1 score is the harmonic average of the accuracy rate and the recall rate, which takes into account the accuracy and completeness of the model in the positive prediction. In this study, F1 scores provide a way to balance accuracy and recall, especially when dealing with data imbalances. With F1 scores, we are able to evaluate the classification performance of the model more comprehensively and its calculation formula is:

$$\text{F1 Score}=\frac{2\times TP}{2\times TP+FP+FN}$$

The AUC value is calculated by plotting the area under the ROC curve, and the calculation formula is usually obtained by numerical integration. AUC is an important index to evaluate the performance of binary classification models, which measures the ability of models to distinguish between positive and negative classes. In this study, the AUC values reflect the comprehensive performance of the model under different thresholds. A higher value of AUC means that the model can distinguish positive and negative samples more effectively, and has a strong classification ability. Especially in the case of unbalanced categories, AUC is a very useful performance evaluation standard.

MCC is a comprehensive index considering all classification results, which can fully reflect the classification performance of the model. In this study, MCC is used to evaluate the performance of the model in the face of unbalanced data. The closer the value of MCC is to 1, the better the prediction results of the model are. Especially when dealing with small samples of positive or negative classes, MCC provides a more stable performance evaluation and its calculation formula is:

$\mathrm{MCC}={\displaystyle \frac{TP\times TN-FP\times FN}{\sqrt{(TP+FP)(TP+FN)(TN+FP)(TN+FN)}}}$

The Hosmer-Lemeshaw test is used to evaluate the goodness of fit of a model, and it tests the agreement between the predicted values of the model and the actual observed values. In this study, the test is used to determine whether the model can accurately fit the data and whether there are systematic errors. Through this test, we can confirm the model’s consistency across different data sets, thereby enhancing its reliability for clinical application and the statistics of the Hosmer-Lemeshaw test are usually calculated by the following formula:

$$\widehat{C}=\sum _{k=1}^G\frac{{\left(O_{1k}-E_{1k}\right)}^2}{N_k\times {\overline{\pi }}_k\times \left(1-{\overline{\pi }}_k\right)}$$

Here, G represents the number of groups (usually 10 groups), $O_{1k}$is the actual number of events observed in group kk (i.e., the number of samples with the dependent variable taking the value of 1), $E_{1k}$ is the number of events predicted by the model in group kk (that is, the sum of the predicted probabilities of all samples in this group), $N_k$is the total sample size of Group k, ${\overline{\pi }}_k$is the average of the predicted probabilities of the k group.

The p-value is calculated from the statistics of the Hosmer-Lemeshaw test, which is usually tested based on the Chi-square distribution. The p-value of Hosmer-Lemeshaw test is used to evaluate the fitting effect of the model, and a higher p-value indicates a better match between the model and the actual data. In this study, the p-value helped us judge the applicability of the model to clinical data, ensuring that its prediction results have a high degree of confidence in practical applications.

The confusion matrix intuitively reveals the classification results of the model by showing the comparison between the actual categories and the predicted categories of the model. In this study, the confusion matrix helps us to understand the predictive performance of the model for various samples, especially whether it can correctly identify positive and negative classes. With this tool, we are able to evaluate the specific performance of the model in each category and provide specific directions for subsequent improvement. The confusion matrix usually looks like this:

Where TP, TN, FP and FN represent true example, true counter example, false positive and false counter example respectively.

The calibration curve evaluates the calibrability of the model by calculating the difference between the actual incidence and the predicted probability for each predicted probability interval. Specific formulas usually involve calculating by ratios or differences. The calibration curve shows the agreement between the probability predicted by the model and the actual results. In this study, the accuracy of the model’s prediction probabilities was evaluated by calibration curves to ensure that the model was not only able to classify, but also to provide reliable probability predictions. The good calibration curve shows that the probabilistic prediction of the model is consistent with the actual incidence rate, which enhances its operability and reliability in the actual clinical environment.