1 Introduction

The global health landscape has been significantly shaped by the far-reaching consequences of the COVID-19 pandemic, impacting millions of individuals on a global scale [1,2,3]. A prompt and accurate diagnosis of COVID-19 is crucial for effective management and control of the disease. Traditional diagnostic methods, such as polymerase chain reaction (PCR) testing [4, 5], while reliable, often involve time-consuming procedures and may have limitations in terms of availability and scalability. As a result, there is a growing interest in leveraging artificial intelligence (AI) and machine learning techniques to develop efficient and automated diagnostic tools for COVID-19 [6].

In light of the evolving nature of the COVID-19 pandemic [7], with the emergence of new variants and the global reduction in COVID-19 cases, the relevance of this study extends beyond the current state of the pandemic. As many countries have lifted COVID-19 restrictions, the focus in medical diagnostics is increasingly shifting towards a more comprehensive approach that encompasses a variety of respiratory diseases, including different lung lesions [8,9,10]. While our study initially focuses on COVID-19, the methodologies and findings have broader implications and potential applications in diagnosing other respiratory conditions. The proposed SqueezeNet-guided Gaussian-kernel SVM (SGS) framework, while primarily aimed at COVID-19 diagnosis using chest CT images, offers a versatile platform that can be adapted for the efficient and accurate classification of a wide range of respiratory diseases, reflecting the changing landscape of global health challenges.

In recent years, deep learning has demonstrated remarkable success in various medical imaging tasks, including disease diagnosis. Convolutional neural networks (CNNs) [11, 12] have particularly shown promise in analyzing medical images, which are commonly used for diagnosing respiratory conditions [6, 13, 14]. However, the complexity and computational requirements of CNN models can hinder their deployment in resource-constrained environments or real-time applications.

As we continue to navigate the challenges posed by the COVID-19 pandemic [15], the importance of rapid and accurate diagnostic methods becomes increasingly evident. While current standard practices, like PCR testing, have been instrumental, they also face limitations in speed, resource allocation, and scalability. These limitations are particularly pronounced in areas with constrained medical resources, highlighting a critical need for improved diagnostic approaches.

In our paper, we introduce the SqueezeNet-guided Gaussian-kernel SVM (SGS) framework within the context of enhancing the accuracy and efficiency of COVID-19 diagnosis using chest CT images. This initiative represents our contribution to the global effort against COVID-19. We recognize the extensive body of existing research in this domain and view our work as a step towards a collaborative solution that complements traditional testing methods.

Our study is driven by the global urgency for more accessible and scalable diagnostic tools. Recognizing the transformative potential of AI and machine learning in medical diagnostics, we humbly present the SGS framework. It is an integration of SqueezeNet's computational efficiency and Gaussian-kernel SVM's classification capabilities. We hope this framework can aid in addressing the diverse and evolving challenges of COVID-19 diagnosis, although we are aware that it is but one piece in the larger puzzle of global health.

To address these challenges, this paper proposes an innovative approach called SqueezeNet-guided Gaussian-kernel SVM (SGS) for COVID-19 diagnosis using chest CT images. The SGS framework combines the strengths of deep learning and support vector machines (SVMs) [16, 17] to accurately and efficiently classify COVID-19 cases.

The core idea behind SGS is to leverage the power of deep learning for feature extraction while utilizing SVMs for classification. SqueezeNet [18, 19], a lightweight deep neural network architecture, is employed to extract high-level features from chest CT images [16]. SqueezeNet's compact design allows for efficient processing without compromising performance. These extracted features are fed into a Gaussian-kernel SVM classifier, which effectively learns the complex patterns associated with COVID-19 infection.

SqueezeNet and Gaussian-kernel SVM fusion in the SGS model offer several advantages. Firstly, the use of SqueezeNet enables efficient feature extraction, reducing computational complexity and facilitating real-time diagnosis. Secondly, the SVM classifier leverages the robustness and generalization capabilities of SVMs, particularly in handling non-linear data, improving the overall diagnostic accuracy.

In this study, we evaluate the performance of the proposed SGS model on a publicly available COVID-19 chest CT dataset. We compare its performance against several state-of-the-art approaches, including traditional machine learning methods and deep learning models.

The paper [20] introduces EDL-COVID, which aims to enhance the detection of COVID-19 cases from chest X-ray images by combining multiple snapshot models of COVID-Net. PSTCNN [21] is a PSO-guided self-tuning CNN, for COVID-19 detection. It effectively addresses the challenges of hyperparameter tuning. Early detection of COVID-19 using a six-layer CNN, combined with advanced techniques and visualization tools, can aid in controlling the virus's spread and ensuring timely treatment [22]. Utilizing Deep Learning methodology, this study [23] proposes a model for the early detection of COVID-19 using chest radiography images, aiming to classify patients as positive or negative for the virus based on unique anomalies observed in the images. In this paper [24], the authors propose a method utilizing the gray-level cooccurrence matrix and particle swarm optimization algorithm for COVID-19 virus identification and research. The algorithm [25] utilizes stationary wavelet entropy for feature extraction, an extreme learning machine for classification and training, and k-fold cross-validation for model determination. The study [26] proposes a novel approach for COVID-19 diagnosis using MDGLCM analysis of chest CT images, GA algorithm optimization, and feedforward neural network classification. The study [27] proposes a GLCM-RVFL model for COVID-19 detection in CT images, achieving high performance compared to six methods.

Additionally, we conduct extensive analysis to gain insights into the interpretability of the SGS model, providing valuable information about the important regions of the CT images contributing to the classification decision. The contributions of this paper can be summarized as follows:

  1. (1)

    The development of the SGS framework that combines SqueezeNet and Gaussian-kernel SVM for COVID-19 diagnosis.

  2. (2)

    The evaluation of the proposed SGS model on a benchmark dataset highlights its superior performance compared to existing approaches. This includes an analysis of the model's interpretability, where we provide insights into the image regions contributing to the classification decision.

The remainder of the paper is structured as follows: Section 2 presents an overview of the dataset and image processing techniques employed in the study. The dataset includes a comprehensive collection of CT scans, meticulously chosen to illustrate the characteristic features of COVID-19 lung abnormalities. The diversity and clinical relevance of this dataset are integral to validating the effectiveness and accuracy of the SGS model in diagnosing COVID-19. In Section 3, the methodology is explained, highlighting the details of the SGS framework. The experimental results are presented in Section 4, showcasing the performance and effectiveness of the proposed approach. Finally, Section 5 concludes the paper by discussing the key findings and implications of the study.

2 Dataset and preprocessing

The SGS model for COVID-19 diagnosis relies on a carefully curated dataset of CT images and preprocessing techniques to ensure accurate and reliable results. The dataset [28] consists of CT scans specifically chosen to capture the characteristic features of COVID-19 lung abnormalities.

Tables 1 and 2 are used to present information on the dataset. Reading the tables provides a clear view of the distribution of COVID-19 and health images and reveals the composition of the experimental subjects.

Table 1 Distribution of experimental subjects
Full size table
Table 2 Number of images in hold-out setting
Full size table

After finalizing the dataset, image preprocessing techniques were applied to enhance the quality and relevance of the data for training the SGS model. These techniques, such as random rotation and scaling, generate variations in the appearance and orientation of COVID-19 lung abnormalities. By introducing such variations, the model becomes more robust and capable of generalizing to different disease presentations. Figure 1, shown below, depicts a typical image from the dataset utilized in our research.

Fig. 1
figure 1

CT Image from the dataset used

Full size image

3 Description of SGS

3.1 Fire module and SqueezeNet with complex bypass

The Fire module is a key component of the SqueezeNet architecture [29], designed to extract informative features from input data efficiently, as shown in Fig. 2. It consists of a combination of squeeze and expand operations aimed at reducing the number of input channels and then expanding them back to capture more complex patterns. The squeeze operation applies \(1\times 1\) convolutions to the input, effectively reducing the number of channels. The expand operation then consists of \(1\times 1\) and \(3\times 3\) convolutions that increase the number of channels, allowing for the capture of more detailed and discriminative features.

Fig. 2
figure 2

Structure of fire module

Full size image

The SqueezeNet architecture [30] incorporates complex bypass connections to facilitate the flow of information through the network, as shown in Fig. 3.

Fig. 3
figure 3

SqueezeNet with complex bypass

Full size image

These connections bypass multiple layers, allowing for the direct propagation of information across different levels of abstraction [31]. The complex bypass [32] connections are formulated as follows: Let \(X\) be the input feature map and \(F( )\) be the transformation applied to\(X\). The bypass connection can be expressed as:

$$Y=X+F\left(X\right),$$
(1)

where \(Y\) represents the output of the bypass connection and + denotes element-wise addition. This formulation ensures that the original input features are combined with the transformed features, preserving important information while allowing for the integration of more complex representations.

By using the Fire Module and adding complex bypass connections, the SqueezeNet architecture is able to do efficient and effective feature extraction. This lets the SGS model pick up on the complex patterns and tell the difference between COVID-19 cases and healthy cases with more accuracy and reliability.

3.2 SGS: SN-guided Gaussian-Kernel SVM

The SN-guided Gaussian-Kernel SVM is a powerful way to use support vector machines (SVMs) to diagnose COVID-19. It combines the SqueezeNet deep neural network with the Gaussian kernel. Kernels are used in SVMs to change data into higher-dimensional feature spaces, which makes it possible to capture complex relationships and non-linear decision boundaries, as shown in Fig. 4.

Fig. 4
figure 4

Illustration of kernel SVM

Full size image

Figure 4 illustrates the transformation of data into higher dimensions to enable better separation (a) displays the dataset distribution on a 2D plane, while (b) represents the data distribution in 3D space after increasing the dimensionality. The symbol \(H\) represents the hyperplane.

In the case of COVID-19 diagnosis, where intricate patterns and non-linear interactions exist, kernels are particularly valuable for improving the model's ability to distinguish between COVID-19 cases and healthy cases.

Among the common types of kernels available in SVMs, the Gaussian kernel stands out for its exceptional ability to capture non-linear relationships. Other types of kernels include linear, polynomial, and sigmoid kernels. However, the Gaussian kernel's flexibility and smoothness make it widely preferred for various classification tasks. The Gaussian kernel offers adjustable parameters, with \(\sigma\) being a key parameter. By carefully tuning the value \(\sigma\), the model can adapt to diverse datasets and capture the desired level of detail in the feature space, leading to improved COVID-19 diagnostic performance.

The Gaussian kernel, also known as the radial basis function kernel, is defined by a formula incorporating the Euclidean distance between input feature vectors. The Gaussian kernel formula [33] is as follows:

$$G\left(x,{x}^{\prime}\right)={\text{exp}}\left(-\frac{{\left|\left|x-{x}^{\prime}\right|\right|}^{2}}{{2\sigma }^{2}}\right)$$
(2)

In this formula, \(G( )\) represents the Gaussian kernel, \(x\) and \({x}{\prime}\) are the input feature vectors, \({\left|\left|x-{x}^{\prime}\right|\right|}^{2}\) denotes the squared Euclidean distance [34] between the vectors, and σ controls the width or spread of the kernel.

Additionally, a probability density function that describes the distribution's shape defines the Gaussian distribution, which is associated with the Gaussian kernel. The formula for the Gaussian distribution is\(A=\pi {r}^{2}\)

$$R\left(x|\mu ,{\tau }^{2}\right)=\frac{1}{\sqrt{{2\pi \tau }^{2}}}\times {\text{exp}}\left(-\frac{{\left(x-\mu \right)}^{2}}{{\tau }^{2}}\right)$$
(3)

Here, \(R( )\) represents the Gaussian distribution, \(x\) is the variable of interest, \(\mu\) is the mean, \({\tau }^{2}\) is the variance, and \({\text{exp}}( )\) is the exponential function.

In the initial stage of the data normalization process, we employ a technique to standardize and make the samples comparable. This involves normalizing the samples to an n-dimensional unit cube, effectively mitigating the impact of the varying scales and units present in the original data. This step sets the foundation for a more insightful and meaningful analysis.

To achieve this normalization, we use the following equation:

$${x}_{q}=\frac{{X}_{q}-{X}_{q}^{L}}{{X}_{q}^{U}-{X}_{q}^{L}},for q=\mathrm{1,2},\dots ,n$$
(4)

In this equation, \({X}_{q}^{U}\) and \({X}_{q}^{L}\) represent the upper and lower bounds of the \(q\) th dimension's design variables. \({X}_{q}\) represents the original value of the design variable, while \({x}_{q}\) corresponds to the normalized value. The variable \(n\) indicates the number of dimensions present in the samples. By applying this normalization technique, each dimension of the samples is transformed into a uniform range between 0 and 1.

A calculation is performed using the sample distribution to assess the local density of each sample in the dataset. The local density value \({l}_{{\text{density}}}\) is computed with \(O\left({x}_{i}\right)\) for each sample \({x}_{i}\). It is determined by considering the distances between the sample and all other samples in the dataset.

$$O\left({x}_{i}\right)=\sum_{j=1}^{n}{e}^{\left(-\frac{{\left|\left|{x}_{i}-{x}_{j}\right|\right|}^{2}}{{c}^{2}}\right)},$$
(5)

where \(n\) is the total number of samples, \({x}_{i}\) represents the current sample, \({x}_{j}\) represents the other samples, \({\left|\left|\right|\right|}^{2}\) denotes the squared Euclidean distance, and \(c\) is a constant.

Next, the minimum distance \(d{z}_{{\text{min}}}\) of the sample with the lowest local density and other samples is determined. This is done by computing the squared Euclidean distance between the sample \({x}_{s}\) and all other samples \({x}_{j}\), and selecting the minimum distance among them:

$$d{z}_{{\text{min}}}={\text{min}}\left\{\sum_{j=1}^{n}{\left({x}_{s}-{x}_{j}\right)}^{T}\left({x}_{s}-{x}_{j}\right)\right\}$$
(6)

Then, the kernel width \(\sigma\) is determined. It is calculated as follows:

$${\sigma }_{i}=\sqrt[n]{\frac{O\left({x}_{s}\right)}{O\left({x}_{i}\right)}}d{z}_{{\text{min}}}$$
(7)

where \({\sigma }_{i}\) represents the kernel width for a specific sample \(i\). \(O\left({x}_{s}\right)\) represents the local density of the sample with the minimum density and \(O\left({x}_{i}\right)\) represents the local density of the current sample \({x}_{i}\). The kernel width accounts for the ratio of the local densities and the minimum distance, providing a measure of the region's influence in density estimation.

However, to determine the overall kernel width \(\sigma\) that will be used for the density estimation process, a common approach is to take the global or average value of the individual kernel widths \({\sigma }_{i}\). This can be done by calculating the mean, median, or some other statistic of the individual \({\sigma }_{i}\) values across all samples.

The effect of over-fitting [35] on classification is the model's tendency to memorize the training data and struggle to generalize to new, unseen data. Conversely, under-fitting occurs when the model fails to capture the complex patterns in the data, resulting in suboptimal performance. Achieving the right balance between these two extremes is crucial for optimal classification [36]. Proper parameter tuning helps prevent over-fitting by controlling model complexity and ensures the model captures essential patterns without overgeneralizing.

By using the SN-guided Gaussian-Kernel SVM, researchers can take advantage of the power of SqueezeNet and SVMs, as well as the Gaussian kernel's flexibility and ability to capture non-linear patterns. This approach enhances the accuracy and reliability of COVID-19 diagnosis, allowing for effective classification between COVID-19 cases and healthy cases based on the distinctive features extracted from medical images. We have organized symbols and their corresponding meaning in Table 3 to provide clarity for the readers.

Table 3 Symbols and their corresponding meaning
Full size table
Algorithm 1
figure a

Pseudocode of SGS

Full size image

3.3 Cross-validation and evaluation

Cross-validation is a fundamental and extensively utilized technique within machine learning. Its primary objective is to assess the performance of a model and estimate its ability to generalize to unseen data.

Figure 5 visually illustrates the K-fold cross-validation process, which plays a pivotal role in achieving these objectives. Each trial in cross-validation involves partitioning the dataset into K folds, with each fold represented by either a yellow square (indicating it as a test fold) or a blue square (indicating it as a training fold).

Fig. 5
figure 5

Schematic diagram of the K-fold cross validation

Full size image

The initial iteration of K-fold cross-validation begins by utilizing the first fold as the test fold, distinguished by the yellow square. In contrast, the remaining K-1 folds serve as the training folds denoted by the blue squares. Subsequently, this process is repeated iteratively, with each fold taking turns as the test fold while the other folds function as the training folds. Through this systematic rotation, every fold in the dataset receives an equal opportunity to serve as the test fold, ensuring a fair and comprehensive evaluation of the model's performance.

Following the cross-validation process, performance metrics are meticulously calculated for each fold. These metrics can be summarized as follows:

The average performance: Compute the average performance across all folds:

$${p}_{average}=\frac{\left({p}_{1}+{p}_{2}+\dots +{p}_{10}\right)}{K}$$
(8)

The variance of the performance values:

$$V=\frac{{\left({p}_{1}-{p}_{average}\right)}^{2}+{\left({p}_{2}-{p}_{average}\right)}^{2}+\dots +{\left({p}_{10}-{p}_{average}\right)}^{2}}{K-1}$$
(9)

The standard deviation:

$$\eta =\sqrt{V}$$
(10)

The symbol \({p}_{average}\) represents the average performance, which is computed by taking the average of the performance values \(\left\{{p}_{1}, {p}_{2}, ..., {p}_{10}\right\}\) across all folds. The variance of the performance values, denoted by\(V\), captures the variability or spread of the performance values across the different folds. To provide a more interpretable measure of the spread, the standard deviation \(\eta\) is derived by taking the square root of the variance\(V\).

When calculating variance, it is important to use unbiased estimation to ensure that our estimates are as accurate as possible. Unbiased estimation means that the expected value of the estimate is equal to the true value of the quantity being estimated. In the case of sample variance, the unbiased estimator is achieved by dividing by the sample size minus 1, \(K-1\), instead of simply dividing by the sample size \(K\).

By employing these calculations, we gain a comprehensive understanding of the model's performance by considering multiple folds, capturing both the average performance and its variability.

In our study, we opted for \(=10\), a common choice in machine learning research, for several reasons. Firstly, using 10 folds provides a good balance between computational efficiency and the reliability of the evaluation. With a higher number of folds, the variance of the model evaluation decreases, making our assessment of the model's performance more stable and less dependent on the particular way the data is split.

Secondly, \(K=10\) allows each fold to be large enough to be representative of the overall dataset, reducing the bias in the evaluation. This size ensures that each fold acts as a reasonable proxy for the unseen data, making our validation process more robust.

4 Experiments, results, and discussions

4.1 Results of MDA

Multiple Data Augmentation (MDA) enhances the robustness and generalization of the model by augmenting the dataset with various transformations. In this study, we apply MDA techniques to the original dataset, and the results are presented in Fig. 6. Each image in the figure represents a transformed version of the original dataset, showcasing the variations introduced through the MDA process.

Fig. 6
figure 6

Visual impact of MDA on image variations

Full size image

The evaluation of MDA techniques provides valuable insights into the effectiveness of these augmentation methods in improving the model's performance. It reveals how the model adapts to image position, brightness, noise, orientation, and size variations. The findings from this analysis contribute to our understanding of the model's robustness and its potential for reliable COVID-19 diagnosis.

4.2 Results of proposed model

The model demonstrates promising performance across the evaluated metrics. The results are listed in Table 4. The average values from all the runs show that the sensitivity is 96.55%, the specificity is 96.15%, the precision is 96.19%, the accuracy is 96.36%, the F1 score is 92.72%, the Matthews correlation coefficient is 96.37%, and the Fowlkes-Mallows index is 96.36%.

Table 4 Performance of SGS through 10 × tenfold cross-validation (Unit: %)
Full size table

Furthermore, when exploring the correlations between different metrics, Acc showed a moderately positive correlation with Sen, Spc, Prc, F1, and MCC. These correlations suggest that as Acc increases, there is a tendency for these metrics to increase as well, indicating consistent performance trends.

4.3 SGS versus SS

In this section, we compare the performance of the SGS, with the SqueezeNet-guided SVM (SS) model. The results of SS are listed in Table 5. The SGS model incorporates an advanced classification algorithm to improve the diagnostic accuracy of COVID-19 detection.

Table 5 Performance of SS through 10 × tenfold cross-validation (Unit: %)
Full size table

Both the SGS and SS models were evaluated using the same dataset and experimental setup. The bar charts in Fig. 7 compare the corresponding metrics for the SGS and SS models. The results reveal that the SGS model outperforms the SS model across multiple performance metrics. The SGS model demonstrates higher Sen, Spc, Prc, Acc, F1, MCC, and FMI.

Fig. 7
figure 7

Comparison of SGS and SS

Full size image

The Gaussian kernel's contribution lies in its ability to capture complex patterns and non-linear relationships in the data. The Gaussian kernel allows the SVM to capture intricate variations and interactions between different features by transforming the input features into a higher-dimensional space. This enables the SGS model to effectively discriminate between COVID-19 lung abnormalities and other lung conditions with improved accuracy.

In contrast, the SS model utilizes the SqueezeNet architecture to guide feature selection and construct an SVM classifier. While SqueezeNet offers computational efficiency and accuracy, it relies on a linear SVM classifier, which may struggle to capture complex relationships in the data.

The SGS model's high accuracy suggests a significant shift towards more AI-integrated diagnostic tools, potentially leading to quicker and more precise disease identification. This advancement is particularly crucial for managing infectious diseases like COVID-19, where rapid diagnosis is essential for effective treatment and control.

The comparative analysis between the SGS model and the SS model, as depicted in Tables 4 and 5, shows the SGS model generally outperforms in sensitivity, precision, and accuracy, crucial for reducing false negatives and false positives in COVID-19 diagnosis. While both models exhibit high metrics, the SGS model demonstrates exceptional consistency and robustness, likely due to its hybrid design integrating SqueezeNet's efficient feature extraction with the Gaussian-kernel SVM's classification prowess. This design effectively captures complex patterns in CT images, enhancing the model's diagnostic reliability. Moreover, the SGS model's slightly higher Matthews Correlation Coefficient across most runs suggests a better true positive and true negative rate balance, essential for practical clinical application. These characteristics underscore the SGS model's potential for improving patient outcomes by providing reliable, rapid diagnostic decisions.

4.4 Confusion matrix and ROC curve

The Confusion Matrix in Fig. 8 provides a detailed representation of the model's predictive performance by tabulating the true positive (TP), true negative (TN), false positive (FP), and false negative (FN) values. It allows us to evaluate the model's ability to classify COVID-19 cases and healthy cases (HC) correctly. The confusion matrix consists of four components: True Positives (TP), where the model correctly identifies positive cases, such as accurately detecting CT scans showing COVID-19; True Negatives (TN), indicating correct identification of negative cases, meaning CT scans without signs of COVID-19 are recognized as such; False Positives (FP), where the model incorrectly labels negative cases as positive, erroneously identifying COVID-19 in clear CT scans; and False Negatives (FN), where the model misses positive cases, failing to detect COVID-19 in affected CT scans. We can derive key evaluation metrics such as sensitivity, specificity, precision, and accuracy by analyzing the confusion matrix.

Fig. 8
figure 8

Confusion matrix of 10 runs of tenfold cross-validation

Full size image

The ROC curve is a graphical depiction of the SGS model's performance at varying classification thresholds. It illustrates the trade-off between the true positive rate, also known as sensitivity, and the false positive rate, calculated as one minus specificity, across different threshold values. The area under the ROC curve (AUC) serves as a summary measure of the model's overall discriminatory power. A higher AUC value indicates superior discriminative capability.

Figure 9 depicts the ROC curve analysis for two models. The prominent blue curve with diamond markers, achieving an AUC of 0.9808, represents the SGS model. This curve demonstrates the model’s high discriminative power in accurately classifying COVID-19 cases versus healthy controls (HC). The second curve, which we will clearly label in the revised figure, represents the SqueezeNet-guided SVM (SS) model. The confusion matrix, along with the ROC analysis, emphasizes the SGS model’s strength in distinguishing true positives from false positives, attesting to its robustness in COVID-19 diagnosis. In the next iteration of this figure, we will include a legend or annotation to distinctly identify the SS model's curve and provide its respective AUC value for a thorough comparative analysis.

Fig. 9
figure 9

The ROC curve of the SGS

Full size image

4.5 Convergence plot comparison

When comparing the performance trends of the SGS and SS models over their iterative processes, distinct patterns emerge. The SS model initially demonstrates a more robust performance, achieving an accuracy score of approximately 61 in the first iteration, suggesting a promising start. However, as the iterations progress, the model's performance becomes less stable, exhibiting fluctuations and inconsistent progress. Although the model shows gradual improvement from the 50th iteration onwards, reaching an accuracy score of around 93 by the 700th iteration, the lack of consistent upward progression raises concerns about its stability and reliability.

On the other hand, the SGS model starts with a lower accuracy score of approximately 51 in the first iteration. However, as the iterations unfold, the model exhibits a steady and consistent improvement in performance. Notably, the SGS model experiences a significant upturn in performance starting from the 400th iteration, indicating a clear and continuous enhancement. This consistent and substantial improvement over time demonstrates the SGS model's ability to refine its performance steadily without significant fluctuations or instability.

Considering these observations in Fig. 10, the convergence plots for both SGS and SS models over iterations are depicted. Figure 10(a) illustrates the SGS model's performance, showing the accuracy of the training set—both the raw and smoothed data—and the test set. It is observed that the training accuracy increases rapidly in the initial iterations and then plateaus, indicating that the model is learning and then stabilizing. The smoothed training line, which represents a moving average, shows less volatility and highlights the overall upward trend in accuracy. The test accuracy closely follows the smoothed training line, suggesting that the model generalizes well to unseen data.

Fig. 10
figure 10

The convergence comparison of models

Full size image

Figure 10(b) shows a similar trend for the SS model, with the training accuracy initially increasing sharply before reaching a plateau. Again, the smoothed training line demonstrates the general trend without the noise of individual variations. The test line for the SS model also follows the smoothed training line but with a slightly more pronounced gap between the two. This indicates a modest difference between training and test performance, which could suggest overfitting or a need for further model tuning.

4.6 Comparison with state-of-the-art models

The Table 6 provides a comparison of different models based on their performance metrics. Among the models, our SGS model stands out with exceptional performance. Comparing our SGS model with the other models, it outperforms them across all metrics. While some models, such as EDL-COVID [20] and PSTCNN [21], exhibit relatively high performance, our SGS model consistently demonstrates superior results. The MCC score of 92.31% indicates a strong agreement between predicted and actual classifications. Additionally, the FMI score of 96.16% confirms the model's ability to cluster similar instances accurately.

Table 6 Comparison of different models based on their performance metrics
Full size table

5 Conclusion

We proposed a novel approach for COVID-19 detection by introducing the Specificity-Guided Selection model, which incorporates the SN-guided Gaussian-Kernel SVM algorithm. Through our study, we compared the performance of the SGS model with the SS model and other state-of-the-art methods. The results highlighted the superiority of the SGS model in various performance metrics. The SGS model offers a promising approach for accurate and reliable detection of COVID-19 cases, showcasing its potential for practical applications in clinical settings. It provides an effective strategy for enhancing the accuracy and reliability of COVID-19 diagnosis, ultimately contributing to improved patient management and public health efforts.

Acknowledging specific limitations of this study is crucial. Firstly, the performance of the SGS framework, although robust in our datasets, may exhibit variations in different datasets, particularly those featuring diverse demographic characteristics or varying image qualities. Secondly, the model's effectiveness in real-world clinical settings is not yet fully established, emphasizing the need for further validation studies. Additionally, while the model demonstrates potential for diagnosing various respiratory conditions, specific adaptations and validations are essential for each disease, aspects not addressed in this study. Lastly, the computational resources necessary for training and implementing deep learning models, such as ours, may restrict their applicability in resource-constrained environments.

Subsequent research endeavors should prioritize overcoming these limitations, possibly by investigating more resource-efficient models and validating the framework across a broader spectrum of datasets and clinical settings. Such initiatives will augment the effectiveness of AI in medical diagnostics, ultimately advancing disease management and control on a global scale.