Detecting Parkinson’s disease from shoe-mounted accelerometer sensors using convolutional neural networks optimized with modified metaheuristics

Convolutional neural networks

Convolutional neural networks (CNNs or ConvNets) represent a class of deep neural networks specifically crafted for tasks related to visual data, such as images and videos. Renowned for their exceptional efficacy in computer vision applications, CNNs have consistently demonstrated cutting-edge performance across diverse challenges, like image classification, object detection, segmentation, and beyond (Li et al., 2021; Gu et al., 2018; Yamashita et al., 2018).

The core of the CNNs consists of the convolutional layers, which perform convolution operations over input data by applying filters (kernels) to identify patterns and features. Afterward, pooling layers reduce the spatial dimensions while retaining important features. Activation functions, like ReLU, are used for introducing the non-linear element to the network, which allows it to acquire intricate relations within the data. Fully connected layers, connecting each neuron in one layer to every neuron in the following layer, are employed in the final stages to perform classification tasks. Flattening is utilized to convert the output of the convolutional and pooling layers, which is fed to the fully connected layers. Batch normalization is commonly applied to standardize the input of every layer, aiding in the stabilization and acceleration of the training process. Finally, dropout serves as a regularization method wherein randomly selected cells are removed during the training phase to mitigate the risk of overfitting.

The accuracy of the model is heavily influenced by hyperparameters, making them a crucial aspect of optimization (Wang, Zhang & Zhang, 2019). Examples of hyperparameters include the number of kernels and their size in each convolutional layer, the learning rate, batch size, the architecture involving the count of convolutional and fully-connected (dense) layers, weight regularization in dense layers, the choice of activation function, the dropout rate, and more. Hyperparameter optimization is not a universally solvable process for all problems, necessitating a “trial and error” approach. However, these approaches are time-consuming and offer no guarantee of outcomes, contributing to their classification as NP-hard. Metaheuristics algorithms have shown promising outcomes in handling such challenges (Yamasaki, Honma & Aizawa, 2017; Qolomany et al., 2017; Bochinski, Senst & Sikora, 2017). For an in-depth mathematical formulation of CNN, refer to Albawi, Mohammed & Al-Zawi (2017), and a more recent exploration on the same topic is provided in Gu et al. (2018).

CNNs are frequently used for image and video recognition, medical image analysis, face recognition, and more (Krizhevsky, Sutskever & Hinton, 2012; Ranjan et al., 2017; Balaban, 2015; Spetlík, Franc & Matas, 2018; Cai, Gao & Zhao, 2020; Ting, Tan & Sim, 2019). A particularly important application of CNNs is the field of medical images, where they are successfully applied for the classification of brain tumors (Bíngol & Alatas, 2021; Bezdan et al., 2021b), breast cancer (Zuluaga-Gomez et al., 2021) and thoracic diseases (Abiyev & Ma’aitaH, 2018). Renowned pre-trained CNN architectures such as AlexNet (Krizhevsky, Sutskever & Hinton, 2017), VGGNet (Simonyan & Zisserman, 2014), ResNet (Szegedy et al., 2017), Inception (Soria Poma, Riba & Sappa, 2020), and MobileNet (Wang et al., 2020) have seen broad adoption across diverse tasks. It is noteworthy that, although CNNs are predominantly linked with computer vision tasks, their application extends beyond. They have been successfully employed in processing other types of data, such as one-dimensional signals in speech and audio processing.

XGBoost

The XGBoost (Chen & Guestrin, 2016) technique leverages a decision tree-based ensemble learning strategy to combine forecasts from numerous weak learners. Each tree, using a gradient-boosting framework, corrects faults caused by its ancestors. The efficacy of XGBoost is based on its regularization methods and parallel processing efficiency. Aside from optimization, regularization and gradient boosting can improve performance. The XGBoost model predicts using intricate relationships between input and target patterns. To improve the objective function, the XGBoost method employs an incremental training strategy. Due to an immense count of parameters requiring adjustment when tuning XGBoost, the trial and error technique is impractical. Given the intricacy of some situations, a strong model is required. The primary characteristics of a good model are speed, generalization, and accuracy.

To produce the best outcomes, the model should be trained iteratively. The objective function of the XGBoost is described by the Eq. (1)

(1) $$\mathrm{o}\mathrm{b}\mathrm{j}(\mathrm{\Theta })=L(\theta )+\mathrm{\Omega }(\mathrm{\Theta }),$$where $Theta$ is the collection of XGBoost hyperparameters, $L(Theta)$ is the loss function, and $Omega(Theta)$ is the regularization term. The last parameter controls the model’s complexity. The loss function depends on the specific problem being addressed.

(2) $$\mathrm{L}(\mathrm{\Theta })={\sum }_{\mathrm{i}}({\mathrm{y}}_{\mathrm{i}}-{\hat{\mathrm{y}}}_{\mathrm{i}}{)}^2,$$in which the $y_i$ is the predicted value, while the predicted target for each iteration $i$ is ${\hat{y}}_i$.

(3) $$L(\mathrm{\Theta })=\sum _i[y_i\ln (1+e^{-{\hat{y}}_i})+(1-y_i)\ln (1+e^{{\hat{y}}_i})].$$

The purpose of this procedure is to distinguish between real and expected values. The total loss function is minimized to enhance classification.

Metaheuristics optimization

The realm of metaheuristics optimizers gained popularity due to their efficiency in resolving NP-hard tasks. The biggest hurdle is discovering solutions within an acceptable time frame and upholding manageable hardware demands. These methods may be separated into distinctive groups, but there is no strict definition. One classification commonly acknowledged by the majority of scientists is distinguishing them concerning the phenomena that inspire these algorithms. Thus, these distinctive families comprise light, swarm, genetic, physics, human, and the most recent addition in the form of mathematically inspired approaches. For example light-based methods are inspired by light propagation properties (Alatas & Bingol, 2020), and the notable techniques include ray optimization algorithm (RO) (Kaveh & Khayatazad, 2012) and optics-inspired optimization (OIO) (Kashan, 2015; Bingol & Alatas, 2020).

Drawing motivation from breeds that thrive in huge swarms and benefit from collective behavior, swarm-inspired algorithms are particularly effective when a sole individual is insufficient to accomplish a task. The swarm family of algorithms was established as highly effective in solving NP-hard problems, but to optimize their performance, it is recommended to hybridize them with similar algorithms. The challenge with these stochastic population-based methods lies in their tendency to favor either exploration or exploitation. This can be addressed by incorporating mechanisms from different solutions. Notable approaches include particle swarm optimization (PSO) (Eberhart & Kennedy, 1995), genetic algorithm (GA) (Mirjalili & Mirjalili, 2019), sine cosine algorithm (SCA) (Mirjalili, 2016), firefly algorithm (FA) (Yang & Slowik, 2020), grey wolf optimizer (GWO) (Faris et al., 2018), reptile search algorithm (RSA) (Abualigah et al., 2022), red fox algorithm (Połap & Woźniak, 2021), polar bear algorithm (Polap & Woźniak, 2017), and the COLSHADE algorithm (Gurrola-Ramos, Hernàndez-Aguirre & Dalmau-Cedeño, 2020).

Swarm algorithms find practical applications across a diverse array of real-world challenges. These applications span various domains, including glioma MRI classification (Bezdan et al., 2020), credit card fraud detection (Jovanovic et al., 2022a; Petrovic et al., 2022), global optimization problems (Strumberger et al., 2019; Zamani, Nadimi-Shahraki & Gandomi, 2022; Nadimi-Shahraki & Zamani, 2022). Additionally, swarm metaheuristics are successfully employed in cloud computing (Predić et al., 2023; Bacanin et al., 2019), enhancing the audit opinion forecasting (Todorovic et al., 2023) predicting the number of COVID-19 cases (Zivkovic et al., 2021), software engineering (Zivkovic et al., 2023), feature selection (Bezdan et al., 2021a; Jovanovic et al., 2022b; Stankovic et al., 2022), security and intrusion detection (Savanović et al., 2023; Jovanovic et al., 2022c; Salb et al., 2023), and enhancing wireless sensor networks (Zivkovic et al., 2020a, 2020b).

The authors in Ahmadpour, Ghadiri & Hajian (2021) devised a genetic algorithm-grounded method for monitoring patients’ blood pressure, leading to a significant enhancement in their overall quality of life and enabling the early diagnosis of some preventable illnesses. Khan & Algarni (2020) investigated an IoT environment that facilitates continuous monitoring of patients’ conditions, resulting in substantial improvements in their cardiovascular health. Illustrative instances of AI-assisted medical diagnosis encompass the detection of diabetic retinopathy (Gupta & Chhikara, 2018), classification of skin lesions (Mahbod et al., 2019), categorization of lung cancer (Ren, Zhang & Wang, 2022), and applications such as magnetic resonance imaging (MRI) and X-ray imaging within the medical field (Zivkovic et al., 2022; Budimirovic et al., 2022).

The original sinh cosh optimizer

SCHO is a recent metaheuristics algorithm developed by Bai et al. (2023). It is a mathematically inspired method, as it relies on the properties of sinh and cosh. Hyperbolic functions encompass common trigonometric functions, with sinh and cosh being fundamental examples. Metaheuristic algorithms may benefit from two key characteristics of cosh and sinh. Firstly, cosh values consistently exceed one, serving as a crucial threshold between exploration and exploitation. Secondly, sinh values fall within the interval [−1, 1], approaching zero, thereby enhancing both exploration and exploitation aspects.

As a metaheuristic relying on population-based methods, the algorithm sets an initial population characterized by a considerable degree of randomness, as illustrated in Eq. (4).

(4) $$A=[\begin{array}{c}{a}_{1,1}...a_{1,j}...a_{1,D}\\ a_{1,2}...a_{2,j}...a_{2,D}\\ a_{N,1}...a_{N,j}...a_{N,D}\end{array}]$$

In this context, P represents a group of solutions, where the position $A_{i,j}$ of each agent is calculated according to Eq. (5). Here, the variables D and N signify the dimensional space of the solution and the number of solutions, respectively.

(5) $$a=rnd(N,D)\times (ub,lb)+lb$$where $rnd$ represents an arbitrary value, while $ub$ and $lb$ mark the upper and lower boundaries of the search domain.

After the initialization phase, the algorithm must strike a balance between exploration and exploitation, directing solutions toward promising regions within the search space. Exploration is divided into two strategies, and the equilibrium is controlled by Eq. (6):

(6) $$S=floor(\frac{T}{ct})$$here T represents the maximum count of rounds, and $ct$ denotes a control parameter with value empirically determined to be $3.6$.

In the exploration phase, solutions are updated as defined by Eq. (7):

(7) $$A_{(i,j)}^{t+1}=\{\begin{array}{cc}{A}_{best}^{(j)}+r_1\times W_1\times A_{(i,j)}^t\hfill & r_2>0.5\hfill \\ A_{best}^{(j)}-r_1\times W_1\times A_{(i,j)}^t\hfill & r_2\mathrm{\setminus }\mathrm{l}\mathrm{t}0.5\hfill \end{array}$$

In this context, $t$ represents the iteration number, $A^{t+1}(i,j)$ describes the $j$-th dimension of the $i$-th agent, and $A^{(j)}best$ denotes the best agent in the $j$ dimension. Random values within the range of $[0,1]$ are chosen for $r_1$ and $r_2$. The coefficient $W_1$ signifies a weighted coefficient for the specific agent and can be calculated as follows:

(8) $$W_1=r_3\times b_1\times (cosh{r}_4+\mu \times sinh{r}_4-1)$$

The value of $b_1$ is progressively reduced throughout the iterations, while $r_3$ and $r_4$ are randomly chosen within limits $[0,1]$. Additionally, a sensitivity parameter is defined and denoted as $\mu$.

During exploration, the second strategy involves the application of Eq. (9)

(9) $$A_{(i,j)}^{t+1}=\{\begin{array}{ll}{A}_{best}^{(j)}+|ϵ\times W_2\times A_{best}^{(j)}-A_{i,j}^{(t)}|& r_5>0.5\\ A_{best}^{(j)}-|ϵ\times W_2\times A_{best}^{(j)}-A_{i,j}^{(t)}|& r_5<0.5\end{array}$$

Within this equation, $ϵ$ is configured to the recommended value of $0.003$ from the original publication. The weight coefficient $W_2$ is calculated as follows:

(10) $$W_2=r_6\times b_2$$where $r_6$ represents a random number drawn from $[0,1]$, while $b_2$ represents a slowly decreasing value.

Another significant phase of the optimization process is exploitation, when solutions concentrate on promising regions of the search realm, taking more refined steps in the direction of the optima. One more time, the metaheuristic employs two techniques. The initial stage utilizes Eq. (11).

(11) $$A_{(i,j)}^{t+1}=\{\begin{array}{cc}{A}_{best}^{(j)}+r_7\times W_3\times A_{(i,j)}^t\hfill & r_8>0.5\hfill \\ A_{best}^{(j)}-r_7\times W_3\times A_{(i,j)}^t\hfill & r_8\mathrm{\setminus }\mathrm{l}\mathrm{t}0.5\hfill \end{array}$$the parameters $r_7$ and $r_8$ are selected within limits $[0,1]$, while $W_3$ is established as follows:

(12) $$W_3=r_9\times b_1\times (cosh{r}_{10}+\mu \times sinh{r}_{10})$$where $r_9$ and $r_{10}$ denote arbitrarily chosen values within $[0,1]$.

The second technique relies on the Eq. (13):

(13) $$A_{(i,j)}^{t+1}=A_{(i,j)}^t+r_{11}\times \frac{sing{r}_{12}}{cosh{r}_{1}2}|W_2\times A_{best}^t-A_{i,j}^t|$$where $r_{11}$ and $r_{12}$ are arbitrarily chosen within $[0,1]$.

The modified SCHO algorithm

While the basic SCHO algorithm performs well, being a relatively new method, it has plenty of space for improvement. Exploration inside this algorithm is lacking, according to evaluation utilizing CEC (Jiang et al., 2018) standard assessment methodologies. The improved version seeks to address this restriction by incorporating two more strategies.

The first technique is based on the artificial bee colony (ABC) (Karaboga & Basturk, 2008) algorithm. When tired solutions fail to improve, they are discarded and replaced with newly created solutions. Because there are only two iterations in this experiment, solutions that fail to demonstrate improvement are discarded after two iterations if no progress is seen. This method has been shown to improve exploration.

The second approach mentioned is quasi-reflective learning (QRL) (Fan, Chen & Xia, 2020), which is used to generate unique solutions and advance research. This approach is also used to produce potential solutions during the algorithm’s early stages. A given solution X’s quasi-reflected counterpart $z$ is calculated as follows:

(14) $$X_z^{qr}=rand(\frac{l{b}_z+u{b}_z}{2},x_z)$$here lb and ub represent the lower and upper limitations of the search realm and rand marks an arbitrary value inside the observed interval. The suggested method is labeled simply modified SCHO (MSCHO). The pseudocode of the proposed optimizer is exhibited in Algorithm 1.

Introduced framework

In this work, a two-layer framework is introduced to reduce computational demands while still maintaining a holistic observation of patient gait. Each patient’s walking gait recordings consist of several sensor readings over time. It can be difficult to observe this data using a simple approach without the use of feature engineering. This can in turn reduce the amount of information available to the algorithm when making a decision. An additional drawback of this approach is that different studies may require the application of different techniques depending on the sensors used and procedures followed during the study.

A flowchart of the framework can be observed in Fig. 1. The framework is capable of automatically adapting to the given task. The two layers in the framework form a collaborative unit that seeks to improve overall accuracy through feature selection and parameter optimization. The first layer of the framework leverages a CNN for feature selection. The second layer of the framework applies the XGBoost algorithm optimized by several contemporary metastatic tasked with demonstrating the best objective function outcomes.

The CNN parameters of the first layer of the framework are optimized by the introduced MSCHO algorithm. Respective optimization ranges empirically determined and for this experiment are as follows; learning rate $[0.01,0.0001]$, count of training epochs $[5,25]$, count of convolutions layers $[1,3]$, units in each the convolutional layer $[8,64]$ number of output features $[8,16]$. Models are tuned and trained until a predetermined accuracy is surpassed. In this work, an accuracy threshold of 90% is used. The final constructed model architecture is comprised of one convolutional layer with $16$ neurons. A learning rate of $0.001225$, $20$ training epochs are selected by the MSCHO algorithm to provide the best outcomes. Pooling layers are not optimized in this study. The optimization aims to attain a lighter CNN architecture suitable for feature selection while maintaining low resource demands. A depiction of the best architecture is provided in Fig. 2.

The reduced feature set is used by the second layer of the framework to further enhance performance. An interpretation of the best constructed CNN utilizing the SHAP (Lundberg & Lee, 2017) model on a selection of 10 random samples is provided in Fig. 3. Input images were zoomed in to improve readability. Once a reduced feature space is determined several metaheuristics are tasked with constructive well-tuned XGBoost models. The specifics of the setup for XGBoost are provided in detail in the following section.

Yogev et al. (2005) dataset simulations

Comparisons in terms of best, worst mean and median executions for the objective function are shown in Table 1 and in terms of indicator (Cohen’s kappa) are similarly shown in Table 2. Stability comparisons are shown in terms of Std and Var.

In Table 1, CNN-XG-MSCHO exhibits the lowest Best value, indicating strong potential in optimal scenarios. CNN-XG-BSO stands out for its consistency, having the lowest standard deviation and variance, suggesting predictable performance.

In Table 2, the CNN-XG-MSCHO, CNN-XG-SCA, and CNN-XG-WOA methods consistently show high Best, Mean, and Median Kappa values, suggesting they are highly effective. The CNN-XG-GA and CNN-XG-PSO methods have wider ranges in performance (noted in their higher standard deviation and variance), indicating less consistent but potentially high-quality outcomes. This suggests a trade-off between achieving peak performance and maintaining consistency across different evaluations.

A graphical comparison for the objective and indicator functions outcome distributions is provided in Fig. 6.

Distribution plots in Fig. 6 indicate that a high stability is demonstrated by the introduced algorithm in terms of objective and indicator function outcomes suggesting a strong reliability of the tested algorithms. An interesting observation can be made on the violin plots of the objective function. Multiple peaks in the objective function suggest that two local minimums might be present in the optimization space for this problem. However the SCA algorithm, as well as the introduced modified version managed to overcome this local optima attaining better overall outcomes, something the original SCHO algorithm did not manage to overcome. The improvements introduced by the modified mechanisms are therefore evident. A strong convergence hindered several algorithms in the simulation. Algorithms such as the original SCHO, BSOA, and COLSHADE showcased a limited exploration ability. Despite the BSO algorithm attaining the highest stability it also fails to locate the best solution within the search space relative to those located by other algorithms.

The final execution outcomes of each algorithm are shown in swarm plots for the objective and indicator evaluations in Fig. 7. As shown in Fig. 7 solution clustering for each algorithm does happen around the previously mentioned local minimum however, the original algorithm fails to field a solution that meets the performance of the solutions located by other algorithms. Nevertheless, the introduced modified algorithm manages to outperform the original and meet the performance of the best-performing competing metaheuristics.

Convergence rate comparisons between optimizers are provided in Fig. 8. Convergence demonstrated during optimizations through the interactions depicted in Fig. 8 brings to light several factors associated with each optimizer. Notably the exploration and exploitation rations between algorithms. Algorithms that converge too quickly often fail to find an optimal solution and get stuck in a local optimal such as the the case with the BSO. Similarly, algorithms with very slow convergence, fail to locate a solution within the allocated optimization period, yielding poor results such as the case with the COLSHADE algorithm. The modified algorithm showcases a good balance between these mechanisms locating an optimal solution after around 50% of the total allocated iterations.

Detailed metrics comparisons in terms of precision, sensitivity, F1-measure, and accuracy between the best-performing algorithms are provided in Table 3. The table presents detailed metrics for the best-performing models in detecting Parkinson’s disease using CNN-XGBoost methods. Notably, several methods (CNN-XG-MSCHO, SCA, GA, PSO, FA, WOA, RSA, COA) exhibit remarkable consistency across precision, sensitivity, and F1-measure for both control and Parkinson’s disease (PD) patients, with high accuracy. These models demonstrate excellent precision (above 0.94 for control and above 0.98 for PD patients) and sensitivity (above 0.96 for both groups), leading to high F1-measures (above 0.95), indicating their robustness and reliability in classification tasks. The results reflect the effectiveness of these CNN-XGBoost methods in accurately diagnosing Parkinson’s disease.

A comparison between the best-constructed models as well as the base XGBoost method in terms of error rate is provided in Table 4. The table compares the error rates of various CNN-XGBoost models in detecting Parkinson’s disease. Most models, including CNN-XG-MSCHO, SCA, GA, PSO, FA, WOA, RSA, and COA, exhibit remarkably low error rates of 0.029499. In contrast, CNN-XG-SCHO, BSO, and COLSHADE show slightly higher rates at 0.032448. Notably, the base XGBoost method (CNN-XG) has a significantly higher error rate of 0.056047. This comparison highlights the enhanced accuracy of the CNN-XGBoost methods over the base model, demonstrating the effectiveness of integrating CNN with XGBoost for this application.

Finally, parameter selections made by each algorithm for the best-performing models are shown in Table 5. The table showcases the parameter selections for the best-performing CNN-XGBoost models in Parkinson’s disease detection. A key observation is the preference for a high learning rate (0.9) in several methods like CNN-XG-MSCHO, SCA, GA, PSO, FA, WOA, indicating a faster learning process. Another notable aspect is the variation in ‘Min Child Weight’, ‘Max Depth’, and ‘Gamma’ across methods, reflecting the different strategies for handling overfitting and model complexity. For example, CNN-XG-GA and PSO opt for a ‘Gamma’ of 0, suggesting less regularization, while others like SCA and FA choose a higher ‘Gamma’ for more. This diversity in parameters underlines the customization and optimization involved in each method to achieve the best results.

Hausdorff et al. (2007) dataset simulations

Comparisons in terms of best, worst mean and median executions for the objective function are shown in Table 6 and in terms of indicator (Cohen’s kappa) are similarly shown in Table 7. Stability comparisons are shown in terms of Std and Var. Notably, CNN-XG-GA and CNN-XG-BSO exhibit remarkable consistency (zero standard deviation and variance) with identical best, worst, mean, and median values, suggesting highly stable performance. In contrast, CNN-XG-MSCHO, PSO, and FA show a wider range of outcomes but still maintain low error rates, indicating effectiveness albeit with less consistency. This comparison highlights differences in stability and performance range among the methods, underscoring the balance between achieving low error rates and maintaining consistent outcomes.

Table 7 displays the Cohen’s Kappa outcomes for various CNN-XGBoost methods, with CNN-XG-GA and CNN-XG-BSO showing remarkable stability (zero standard deviation and variance). CNN-XG-MSCHO, PSO, and FA exhibit higher Kappa values, suggesting superior agreement in certain scenarios, but with greater variability in results. The range of best-to-worst values across methods is narrow, indicating overall good and consistent performance in classification accuracy. This balance between high performance and stability is crucial in applications like medical diagnostics.

Additionally, a graphical comparison for the objective and indicator functions outcome distributions is provided in Fig. 9. Distribution plots in Fig. 9 indicate a strong presence of a local optimum within the search space. As evident, many algorithms fail to locate a relative best solution due to this local optima. The original SCHO, SCA, GA, WOA, and BSO all overly focus on local best solutions suggesting a lack of diversification within these methods. The introduced algorithm effectively covers to wards an optimal matching of the performance of the best algorithms. While high stability is demonstrated by several algorithms, these fail to locate a relatively best solution in this simulation. The improvements introduced by the modified mechanisms are therefore an effective way of overcoming the exploration limitations observed in the original.

The final execution outcomes of each algorithm are shown in swarm plots for the objective and indicator evaluations in Fig. 10. As shown in Fig. 10 solution clustering for each algorithm does happen around the previously mentioned local minimum however, the original algorithm fails to field a solution that meets the performance of the solutions located by other algorithms. Nevertheless, the introduced modified algorithm manages to outperform the original and meet the performance of the best-performing competing metaheuristics. Additionally, many of the solutions are located closed to the true optima indicating a strong reliability.

Convergence rate comparisons between optimizers are provided in Fig. 11. Convergence demonstrated during optimizations through the interactions depicted in Fig. 11 brings to light several factors associated with each optimizer. Notably the exploration and exploitation rations between algorithms. Algorithms that converge too quickly often fail to find an optimal solution and get stuck in a local optimal such as the case with the GA. Similarly, algorithms with very slow convergence, fail to locate a solution within the allocated optimization period, yielding poor results such as the case with the BSO algorithm. The modified algorithm showcases a good balance between these mechanisms in this simulataion as well. While a slower converges is evident it remains steady though the iterations with a local optima located in the final few iterations.

Detailed metrics comparisons in terms of precision, sensitivity, F1-measure, and accuracy between the best-performing algorithms are provided in Table 8. The table reveals that most CNN-XGBoost methods demonstrate exceptionally high performance in precision, sensitivity, and F1-measure for both control and PD patients, with accuracy consistently above 97%. Notably, methods like CNN-XG-MSCHO, PSO, FA, RSA, COA, and COLSHADE show remarkable precision and sensitivity values above 94% and 96% respectively, leading to F1-measures above 95%. This uniformity in high performance across different metrics indicates the robustness and reliability of these methods in accurately classifying PD patients.

A comparison between the best-constructed models as well as the base XGBoost method in terms of error rate is provided in Table 9. The table compares the error rates of various CNN-XGBoost methods with the base XGBoost (CNN-XG) method in detecting Parkinson’s disease. The CNN-XG-MSCHO, PSO, FA, RSA, COA, and COLSHADE methods exhibit notably low error rates of 0.018088. In contrast, CNN-XG-SCHO, SCA, GA, WOA, and BSO show slightly higher rates at 0.020672. The base XGBoost model has a higher error rate of 0.025840. This comparison highlights the superior accuracy of the CNN-XGBoost methods over the base model, indicating the effectiveness of integrating CNN with XGBoost in this application.

Finally, parameter selections made by each algorithm for the best-performing models using the (Hausdorff et al., 2007) dataset are shown in Table 10. There is a noticeable diversity in parameter choices. For instance, several models (CNN-XG-MSCHO, WOA, RSA) opt for a high learning rate of 0.9, indicating a preference for faster learning, while others like CNN-XG-SCHO, SCA, and GA choose lower rates for gradual learning. The ‘Min Child Weight’ and ‘Max Depth’ parameters vary significantly, reflecting different strategies to control model complexity and prevent overfitting. The ‘Gamma’ values also differ, suggesting varying degrees of regularization among the models. This diversity underscores the tailored approach each method takes to optimize performance.

Frenkel-Toledo et al. (2005) dataset simulations

Comparisons in terms of best, worst mean and median executions for the objective function are shown in Table 11 and in terms of indicator (Cohen’s kappa) are similarly shown in Table 12. Stability comparisons are shown in terms of Std and Var. CNN-XG-MSCHO and CNN-XG-SCA stand out with the lowest ‘Best’ values, indicating potential for optimal performance. However, CNN-XG-FA shows the highest variability and worst outcomes, suggesting less stability. This contrast in performance and consistency across methods highlights the importance of selecting the right algorithm and parameters for specific applications to achieve the most effective results.

Table 12 shows the Cohen’s Kappa outcomes for various CNN-XGBoost methods from the Frenkel-Toledo et al. (2005) dataset. CNN-XG-MSCHO, SCA, and WOA exhibit higher ‘Best’ Kappa values, indicating strong agreement in certain scenarios. However, CNN-XG-FA shows the largest range in outcomes, suggesting variability in its performance. The relatively narrow spread between the best and worst values across all methods indicates consistent performance, with a balance between achieving high agreement and maintaining consistency in different evaluations.

Additionally, a graphical comparison for the objective and indicator functions outcome distributions is provided in Fig. 12. Distribution plots in Fig. 12 indicate a strong stability of several algorithms such as the GA, PS, WOA, BSO, and RSA, however, these algorithms fail to locate a true optima. The interesting observation is that the introduced algorithm also demonstrates high stability. However, the introduced modified algorithm heavily converges on the best-located solution relative to the other algorithms.

The final execution outcomes of each algorithm are shown in swarm plots for the objective and indicator evaluations in Fig. 13. Swarm diagrams in Fig. 13 further enforce the previous observations. Many of the solutions provided by the modified algorithm fall near the located optimal. An improvement over the original version of the algorithm is evident, as many of the solutions of the original algorithm focus on a local optimal as opposed to the best solution located by competing algorithms.

Convergence rate comparisons between optimizers are provided in Fig. 14. The introduced algorithm showcases a high rate of convergence in comparison to competing algorithms as shown in Fig. 14. Nevertheless, the convergence is justified as an optimum is located that matches solutions located by other algorithms. It is important to emphasize that the NFL states that no single solution works equally well for all presented problems. While the quick convergence may be beneficial for this specific problem, other applications may prefer a stronger focus on exploration.

Detailed metrics comparisons in terms of precision, sensitivity, F1-measure, and accuracy between the best-performing algorithms are provided in Table 13. The table reveals that most CNN-XGBoost methods achieve high precision, sensitivity, and F1-measures for both control and PD patients, with overall accuracy consistently above 94%. Notably, methods like CNN-XG-MSCHO, SCA, and COA demonstrate exceptional precision and sensitivity, leading to high F1-measures, indicating their robustness and reliability in classification. This uniformity in high performance across different metrics underscores the effectiveness of these methods in accurately diagnosing PD.

A comparison between the best-constructed models as well as the base XGBoost method in terms of error rate is provided in Table 14. The table shows the error rates for various CNN-XGBoost models compared to the base XGBoost model in the Frenkel-Toledo et al. (2005) dataset. CNN-XG-MSCHO, SCA, WOA, RSA, COA, and COLSHADE show notably low error rates of 0.046875. CNN-XG-SCHO, PSO, FA, and BSO have slightly higher rates at 0.052083, while CNN-XG-GA is at 0.057292. The base XGBoost model (CNN-XG) has the highest error rate of 0.067708. This highlights the enhanced accuracy of the CNN-XGBoost methods over the base model, demonstrating the effectiveness of integrating CNN with XGBoost.

Finally, parameter selections made by each algorithm for the best-performing models are shown in Table 15. The table shows varied parameter selections for the best-performing models in the Frenkel-Toledo et al. (2005) dataset. Most methods opt for a high ‘Min Child Weight’, indicating a preference for more complex models. Learning rates vary significantly, with some models choosing moderate rates (around 0.5–0.7), while others like CNN-XG-SCHO and FA go for a high rate of 0.9. The ‘Max Depth’ is consistently low (mostly around 3), except for CNN-XG-WOA and RSA, suggesting an overall preference for shallower trees. ‘Gamma’ values also vary, indicating different degrees of regularization across methods.

Statistical outcome validation

As experimental data is often insufficient to establish the superiority of one algorithm over its competitors, scientists in current computer research must assess the statistical significance of proposed advancements. According to Eftimov, Korošec & Seljak (2016), literature recommendations suggest that statistical tests in such scenarios should involve creating a representative collection of outcomes for each method. However, when dealing with outliers from a non-normal distribution, this strategy may be ineffective, potentially leading to misleading results. The unresolved dispute highlighted by Eftimov, Korošec & Seljak (2016) revolves around whether employing the mean objective function value in statistical tests is suitable for comparing stochastic techniques. Notwithstanding these potential drawbacks, the objective function of the classification error rate was averaged over 30 distinct runs to evaluate the performance of the optimizer.

Following the execution of the Shapiro-Wilk test (Shapiro & Francia, 1972) for single-problem analysis employing the specified procedure, a determination was reached. A data sample was compiled for each algorithm and each problem by aggregating the results of each run, and the corresponding $p$-values were computed for all method-problem combinations. The resulting $p$-values are presented in Table 16.

These findings are further supported by Fig. 15, illustrating the distributions of objective function outcomes for each optimizer across 30 independent runs.

The rejection of the null hypothesis is warranted as the $p$-values in Table 16 are all below the predetermined significance threshold, denoted as $\alpha$, set at $0.05$. Consequently, the data samples for solutions do not all conform to a Gaussian distribution, indicating that usage of the average objective value in future statistical tests is inappropriate. Hence, the best results were selected for further statistical analysis in this study. Due to the non-met normalcy assumption, parametric tests were deemed to be unsuitable.

Next, the non-parametric Wilcoxon signed-rank test (Wilcoxon, 1992) has been employed. It can be applied to the identical data series containing the best scores achieved in each metaheuristic run. During the test, the proposed approach was employed as the control method, and the Wilcoxon signed-rank test was conducted on the provided data series. For each of the three instances observed, the calculated p-values were lower than $0.05$. Utilizing the significance threshold of $0.1$ ( $\alpha =0.1$), the results indicate that the new algorithm statistically surpassed all contending techniques considerably. The comprehensive outcomes of the Wilcoxon signed-rank test are presented in Table 17.

The Wilcoxon signed-rank test results indicate that the developed algorithm (MSCHO) statistically significantly outperformed the other algorithms in all three datasets (Yogev et al., 2005; Hausdorff et al., 2007; Frenkel-Toledo et al., 2005). The p-values for all comparisons are below 0.05, well within the significance threshold of 0.1. This consistency across different datasets suggests that the MSCHO algorithm consistently offers supreme performance in comparison to the contending methods evaluated in this study.

Summarizing, and determining the best algorithm based on the provided results involves considering multiple factors like error rates, Cohen’s kappa, precision, sensitivity, F1-measure, and Shapiro-Wilk test results. Algorithms like CNN-XG-MSCHO, SCA, and WOA consistently showed low error rates across different datasets, indicating high accuracy. Similarly, in terms of Cohen’s kappa, precision, sensitivity, and F1-measures, these algorithms demonstrated robust performance, suggesting effective classification capability. However, the Shapiro-Wilk testing outcomes suggest a non-normal distribution in performance metrics, suggesting variability in algorithm performance across different problems. Therefore, while CNN-XG-MSCHO, SCA, and WOA generally exhibit strong performance, the best choice may depend on specific dataset characteristics and application requirements.