The fetal classification dataset [19] is a publicly available dataset obtained from Kaggle with 22 columns & 2126 records extracted from cardiotocography examinations. Features such as “uterine contractions,” “light decelerations,” “prolonged decelerations,” “abnormal short-term variability,” and many more are included in the dataset. The dataset contains 3 different classes—“normal,” “suspect,” & “pathological”. The records reflect thorough assessments of various physiological parameters obtained from the cardiotocogram examinations. During prenatal therapy, it is possible to predict the fetal health state based on the assessment of these traits, which serve as the foundation for classification. The dataset’s attributes are summarized in Fig. 2, including information about the data type and possible values for each attribute.

A complete understanding of the distribution and uncertainty of the dataset is achieved via the study, which comprises predicting probability density and computing variances across several data points. Looking at the probability density and variances may give you a better idea of the underlying probabilistic features and how they change across different data types. In Fig. 3, a grid of Kernel Density Estimation (KDE) plots for each feature in the dataset depicts the expected probability density and variances across several ‘fetal_health’ classes (Blue-normal, orange-suspect, green-pathological). The differences in feature distributions among health classes in the color-coded charts. The exploratory data analysis (EDA) tool indispensable for this project is these KDE charts, which help with feature distribution insights, data pre-processing, feature selection, and model interpretation.

Improving data quality, and reducing noise to improve the efficacy of the classification models are all responsibilities of data pre-processing. Important tasks in this phase include feature selection, cleaning, missing values management, scaling, categorical variable encoding & handling data anomalies [20]. The efficacy of data pre-processing determines the success or failure of training & evaluating models.

Using the seaborn package and matplotlib, Fig. 4 generates a heatmap to display the dataset’s correlation matrix. By looking at the size and direction of the correlations among the divergent variables in the dataset, this visualization is a great tool for understanding their relationships.

By building a network of decision trees, the ensemble learning algorithm, Random Forest is useful for fetal health classification. A voting method is used during prediction to aggregate individual tree outputs and improve overall model performance after these decision trees are trained on random subsets of the data. Bootstrapping and feature selection are made more random by the algorithm at each split, which helps to improve generalization and decrease overfitting. RF evaluates feature relevance based on the decrease in impurity achieved by each feature across all trees, whereas the Gini impurity measure directs the development of decision trees. It can detect important features for reliable predictions using this approach. Complex fetal health classification tasks are well-suited to Random Forest due to its ensemble nature and feature interpretability. First, we can write the Gini impurity for an FHD in Eq. (6). (6)Gini(FHD)=1−∑i=1cpi2

The notation c in the above equation designates the number of classes, and the notation p designates the probability of class in the FHD dataset.

The ensemble learning algorithm XGBoost is essential for fetal health prediction because it combines regularization with gradient boosting. A loss term assessing prediction accuracy and a regularization term prohibiting overfitting make up the objective function that XGB minimizes through the sequential construction of decision trees. Computing gradients & Hessians, which stand for derivatives of the loss function, is an integral part of the optimization process. We distribute weights to trees according to their contributions, and the structure of each tree is decided by minimizing the objective function. Regularization terms control model complexity. XGB is resilient against overfitting & suitable for varied datasets; it is particularly effective for fetal health classification due to its adaptability to complex interactions, automatic management of missing values, and feature importance score. In multiclass classification, the objective function for XGB comprises a regularization term and the aggregate of the individual loss functions for each class [23]. Class probabilities are typically calculated using the softmax algorithm. Expressing the global objective function using Eq. (7) is possible. (7)O(XGB)=∑i=1nL(Yi,Y^i)+∑k=1KΩ(∫↷k)where, n is the total instances. K is the total classes. L(Yi,Y^i) is the individual loss function measuring the difference among the true class Yi and the predicted probability Y^i for instance i. In multiclass classification, the cross-entropy loss is commonly used. Ω(fk) is the regularization term for the k^th tree in the ensemble.

The predicted probability for class k is calculated using the softmax function Eq. (8). (8)P(Y=K)=eηk∑j=1𝒦eηk

In the above equation, the notation ηk is the output of the kth tree.

A binary classification algorithm, logistic regression can be enhanced to manage jobs involving multiple classes. Regarding fetal health classification, LR models the likelihood that an instance belongs to a specific class. It is usual practice to convert the linear combination of input features into probabilities using the logistic function, often known as the sigmoid function. The softmax function allows logistic regression to be extended for multiclass classification. After calculating the probability of each class, the one with the highest probability is used to make a prediction. Solving this Eq. (9) gives the class k of the softmax function. (9)P(Y=k)=eβ0k+β1kX1+β2kX2+…+βnkXn∑j=1Keβ0j+β1jX1+β2jX2+βnjXnwhere,

K is the number of classes.

Training the LR model entails adjusting the coefficients while maximizing the likelihood of the observed data. The model uses the computed probabilities to forecast the class, and these coefficients define the decision boundary.

One flexible and non-parametric approach useful in fetal health classification is K-Nearest Neighbours. When applied to fetal health, KNN works on the premise that instances with comparable features should have comparable class labels. To determine a data point’s classification, the algorithm looks at the labels of its K-Nearest Neighbors, where k is a number that the user specifies [24]. Data points are usually classified using majority voting, which assigns them to the class that appears most often among the K-Nearest Neighbors. The prediction for a new data point can be mathematically represented in Eq. (10). (10)Y^=argmax(∑n=1mI(Yn=k))

Here, Y^ is the predicted class label, Yn represents the class label of the n^th nearest neighbor, and I is the indicator function. The technique considers the distances among the new data point and every other data point across the training set, chooses the K-Nearest Neighbors, and subsequently majority vote to assign the class label. When the decision boundaries are complex or non-linear, KNN is a good fit for fetal health prediction since it is both simple and effective. It may, however, be performance-dependent on the distance metric and k values. Thus, fine-tuning is likely to be necessary.

Using decision trees, which are robust and easily interpretable models, allows for the classification of fetal health in circumstances involving many classes. Using feature values as a basis, recursively partitioning the feature space into pieces. Each partition in a decision tree represents a node, and the last partitions, known as leaf nodes, serve as labels for the classes [25]. As part of the decision-making process, the feature space is partitioned at each node based on a given feature and a splitting criterion, such as Gini impurity or entropy. Maximizing the homogeneity of class labels inside each partition is the goal of the splitting criterion, which aims to produce as pure of divisions as feasible. With each branch in the decision tree, minimizing the impurity measure identified by the notation IG is the mathematical objective over a node m with n classes are measured as shown in Eq. (11).

(11)IG(m)=1−∑i=1np(im)2where, p(im) is the proportion of samples in node m belonging to class i. The decision tree keeps recursively splitting the feature space until it reaches a stopping condition, like a maximum depth, minimum samples per leaf, or when there is no more improvement in impurity reduction. For fetal health classification, decision trees excel due to their interpretability and ability to manage non-linear decision boundaries. Clinicians can better understand the decision-making process in this way. On the other hand, overfitting is more common in deep learning. So, to prevent them, we might need to use regularization techniques.

Predicting the fetal health status from various input features is the domain’s specialty, and the Dynamic Multi-Layer Perceptron is the customizable neural network design used for this study. Layers that reflect relevant features obtained from diagnostic tests, layers that analyze inputs using weighted connections and activation functions, and final layers that generate predictions on fetal health issues are all part of a Multi-Layer Perceptron (MLP). The DMLP design’s adaptability allows for customizing the total hidden layers and neurons according to the classification task’s complexity. Upgrading the MLP model from a one-neuron to seven neurons in a multi-feedforward layered model is the logical next step for fetal health classification. With three neurons per class to seven hidden layers, we intend to train and assess networks with diverse topologies. In the bottom hidden layer, there are 24 neurons; subsequent layers increase this amount. To avoid overfitting, use the Early Stopping call-back and choose your model and parameters based on their performance on the validation set for accuracy. The use of dropout and batch normalization significantly enhances generalizability. Also, an F1-score can be used to monitor how well the model is doing [26]. This comprehensive strategy is implemented to improve deep learning models for accurate fetal health categorization. This comprehensive strategy is being implemented to decrease concerns about overfitting. The calculation determines the weighted total for each class k in the output layer in Eq. (12).

(12)zc=∑n=1Nwkn.afn+bcwhere,

zc is the weighted sum for class c.

The softmax activation function is applied to the logits to obtain class probabilities, as illustrated in Eq. (13).

(13)P(Y=k)=ezk∑i=1Ceziwhere, P(Y=k) is the probability of class k, and C is the total classes.

It is possible to represent the objective function of a MLP network, frequently utilized for supervised learning tasks, as the reduction of a cost or loss function [27]. In the case of regression tasks, the objective function O(θ) is typically expressed as the mean squared error (MSE), while in the case of classification tasks, it is expressed as the cross-entropy in Eq. (14).

(14)O(θ)=−1n∑i=1n∑k=1Kzk(i)log⁡(hθ(p(i))k)where, O(θ) is objective function. θ represents the parameters, i.e., weights and biases hθ(p(i)) is the predicted output at input p(i) and zk(i) is the actual output at input p(i), n is the number of training samples in a dataset, and K is the number of classes in fetal classification.

The weights and biases are updated using the gradient descent update rule. The gradient for the weights ∂wkj is given in Eq. (15).

(15)∂J∂wkj=−1c∑a=1c(xb(a).(ya(k)−P(Y(a)=K)))where, xb(a) is the bth feature of the ath training sample.

The complexity of the problem and the data determine the optimal number of hidden layers and neurons to increase the performance of DMLP. The following are a few possible benefits:

Increasing the amount of hidden layers and neurons improves the model’s capacity to interpret complex patterns and correlations in the data. This might improve performance, especially for fetal datasets with complicated structures or non-linear relationships.

Machine learning model performance is greatly influenced by hyperparameters, which is why they are so important. The training procedure and the final model are affected by these external configurations, which are not learned from data. Reaching peak model performance, warding off problems like overfitting and underfitting, and improving generalization to novel, unknown data depend on correct hyperparameter tuning. Examples of common hyperparameters are parameters about the model’s architecture, regularization, and learning rate. Machine learning weakening entails experimenting with different values to discover the optimal ones. The model parameters used in the current study are shown in Table 1.

Various evaluation indicators were plotted across different models and setups to provide a comprehensive perspective of model performance. As a result, the study evaluated each method for fetal health classification and drew comparisons between them. In addition, we analyzed the significance of traits to learn how they contributed to classification. By providing a comprehensive assessment of the performance and interpretability of both conventional classifications, the current study seeks to shed light on their efficacy for fetal health classification. A confusion matrix is a way to summarize a machine learning model’s performance in fetal health categorization. It does this by comparing the model’s predictions with the actual outcomes. The confusion matrix appears when evaluating the results of classifications across many classes, which often happens in fetal health categorization.

Usually, there are four entries in the confusion matrix:

First, there are cases where the model accurately predicts a positive class, such as the presence of a specific fetal health issue, these cases are called True Positives.

Fig. 9 shows how the confusion matrix may be used to obtain several metrics for evaluating the model’s performance, including recall, accuracy, precision, and F1-score. These indicators provide insights into the model’s accuracy in identifying fetal health issues and areas for improvement.

In fetal health categorization, feature importance describes how each feature or variable in a dataset affects the model’s prediction ability. It helps to determine which features are crucial for the model to accurately classify cases of fetal health [28]. Understanding which traits are most crucial for model interpretation is essential, as it exposes which factors play a pivotal role in determining the outcome. Analyzing how much weight the model gave to each input feature is what feature importance analysis is all about in the context of fetal health categorization. During training, popular algorithms like DT, RF, and XGB make it easy to determine which features are most important, are illustrated in Figs. 10 and 11. Bar graphs that show feature dependency highlight which features most influence a model’s predictions by displaying their importance scores. These graphs help identify key features, simplify the model, and improve performance by focusing on the most impactful data [29].

A feature’s impact on the accuracy of predictions is proportional to its significance score. When making decisions, the model gives more weight to features considered more important and less weight to features considered less important. Academics and practitioners can better understand the most important elements impacting fetal health outcomes through feature-importance visualizations or analyses. Improving the categorization model’s interpretability, directing more studies, and prioritizing characteristics for data gathering or enhancement can all be achieved with this information.

In the current study we have used the LOOCV based cross validation alongside the K-Fold validating the proposed model. A comparative analysis is also promised, encompassing three scenarios: the algorithm without any pre-processing, the algorithm with scaling, and the algorithm with the data balancing technique SMOTE. The final scenario involves applying both data balancing and scaling simultaneously. This comprehensive comparison of accuracy in Table 2 aims to provide insights into the impact of different pre-processing techniques on the algorithm’s performance, shedding light on the effectiveness of balancing and scaling individually and in combination. Analysis of various classifiers for fetal health classification is illustrated in Table 3 and Fig. 12.

Training and validation set performance metrics are the primary emphasis of the DMLP model, which aims to analyze the connection between the model’s capacity as indicated by its layers and neurons. The dataset has been partitioned into 80-20 for training and testing. The validation set used for fine-tuning the model is the integral part of the training sample, allotted 10% of the training size. According to graphical representations, a more noticeable disparity between training and validation measures is associated with enhanced model capacity, which may indicate overperformance. With an astounding F1-score of 91.33% on the validation set and 89.95% on the testing set, the model with five hidden layers and 51,000 parameters beats others.

Nevertheless, a more robust hyperparameter search is needed because worries regarding the significance of the results arise due to the lack of statistical tests and cross-validation. Future studies should utilize comprehensive cross-validation procedures and statistical testing to reliably choose models, as larger models show lower metrics, which could indicate overfitting or randomness. Neural network analysis with emphasis on layers and neurons is illustrated in Table 4, and training accuracy, validation accuracy, training loss, and validation losses are illustrated in Fig. 13.

As the number of hidden layers grows, there is a general trend towards improved accuracy, recall, precision, and F1-score across training, validation, and test sets. With 6 hidden layers, the model performs at its peak, with a training accuracy of 98.90%, a validation accuracy of 94.82%, and a test accuracy of 92.96%. The performance indicators appear to reach a plateau or decline at 7 hidden layers, suggesting that overfitting is possible as the model complexity increases.

It can be observed that DMLP model had attained 97% accuracy when utilizing pre-processing techniques, whereas without pre-processing, the model attained 96% accuracy. Removing the resampling, feature scaling, and feature selection phases allowed us to better grasp the significance of data preparation in our Dynamic Multi-Layer Perceptron (DMLP) model. After removing this pre-processing, we tested the model (Table 5) on the fetal health classification task. As a result, we can gauge how much these methods improve the model’s predictive power and, maybe, learn which features were most crucial.

The practical implications associated with the current study outline the proposed model’s limitations. We used a CTG dataset from medical centre and a specific population. The demographic, genetic, and environmental differences between populations restrict the generalizability of our findings. The dataset size with limited processing records is one of the potential limitations. The feature scaling that is performed in the current study yields better accuracy, but at the same time, smoothing values would miss out on some of the significant data for analysis. The other limitation in the evaluation process is that the k-value in multi-fold validation is confined to 10, where the model yields a better accuracy compared to k=5, but needs considerable computation efforts to yield the accuracy. The tradeoff between accuracy and computational efforts may be further analyzed to improve the comprehensibility of the model. The hyperparameters are considered fixed and can be fine-tuned by further evaluating the model in different settings. The feature weights and dependencies can be further evaluated using explainable models. The ablation study can be performed by discarding some of the less significant features from the dataset for better analysis of the robustness of the model.