Leveraging Classifier Performance Using Heuristic Optimization for Detecting Cardiovascular Disease from PPG Signals

2. Materials and Methods

CVD analysis using PPG signals is crucial due to the rising prevalence of cardiovascular diseases globally. PPG technology offers a non-invasive, cost-effective, and easily deployable method for continuous heart health monitoring, facilitating early detection and timely intervention. This can reduce the burden on healthcare systems and improve patient outcomes. The integration of PPG analysis with wearable devices and advanced machine learning algorithms enhances diagnostic accuracy and efficiency. As healthcare shifts towards personalized and preventive models, PPG-based CVD analysis becomes a pivotal tool in the fight against CVDs. Therefore, CVD detection from PPG signals is considered in this research work. The main objective of this work is to enhance CVD diagnosis through more precise classification systems. Effectively categorizing CVD data not only ensures patients receive appropriate care at reduced costs but also lowers their risk of developing the disease. Classifier accuracy tends to decrease when unimportant and noisy signals are present in recorded signals. To enhance the quality of the recorded PPG signals, an efficient filtering technique is implemented to eliminate unwanted noise and artifacts. PPG data recordings with diverse wave shapes from the CapnoBase database have been used in this work. This database is a publicly available online resource that adheres to the IEEE TMBE pulse oximeter standard [25]. This article explores the CapnoBase dataset, utilizing the complete IEEE benchmark (41 records) for the experiment, with 20 records representing CVD and 21 representing normal conditions. The PPG signals are digitized at a rate of 200 samples per second. For analysis purposes, each one-second interval of the PPG signal is defined as a segment. Therefore, each patient has 720 individual segments for further examination. Consequently, each patient has 144,000 samples (720 segments × 200 samples per segment). The total number of CVD segments is [20 x 720 = 14400], and in normal cases, the total number of segments is [21 x 720 = 15120]. Therefore, in total, 29,520 one-second segments are available for analysis from the 41 cases. The PPG signals are analyzed based on signal segments across the patients. Beat-to-beat analysis is not included in this study. Noise components from the PPG signals were removed by utilizing Independent Component Analysis (ICA). Figure 1 shows a normal PPG signal, and Figure 2 depicts the PPG signal obtained from a CVD person.

This research utilizes heuristic dimensionality reduction (DR) techniques as the initial step to reduce the dimensionality of the PPG data. Specifically, the research incorporates methods such as ABC-PSO (Artificial Bee Colony-Particle Swarm Optimization), the cuckoo search algorithm, and the dragonfly algorithm. In the second stage, the optimized PPG data were input into ten different classifiers to detect Cardiovascular disease from PPG signals. The classifiers' performance is evaluated and differentiated using parameter metrics for heuristically optimized PPG values. Figure 3 provides a detailed illustration of the workflow.

3. Dimensionality Reduction Techniques:

Now, each person has 144,000 samples of PPG signals (720 × 200). The objective of dimensionality reduction in PPG signals is to decrease the number of variables, thereby improving computational efficiency and reducing the risk of overfitting. High-dimensional data can be noisy and redundant, making it difficult for classifiers to accurately identify patterns associated with CVD. By extracting the most relevant features, dimensionality reduction techniques such as ABC-PSO, Cuckoo Search, and Dragonfly are employed to enhance the performance of machine learning models.

3.1. ABC-PSO (Artificial Bee Colony-Particle Swarm Optimization)

ABC-PSO is a hybrid dimensionality reduction technique that combines the global search capabilities of the ABC algorithm with the local exploitation abilities of Particle Swarm Optimization (PSO). ABC emphasizes exploration through employed and onlooker bees, where solutions are iteratively improved. PSO, inspired by social behavior, optimizes by updating particle velocities based on personal best and global best solutions. By integrating these approaches, the hybrid algorithm aims to enhance exploration-exploitation balance, leveraging ABC's local search capability and PSO's global search efficiency.This synergy enhances the efficiency of handling large datasets and improving the performance of data analysis[28].

ABC-PSO Algorithm Steps:

• Initialization: Initialize bee and particle populations with random solutions. Set the number of employed bees, onlooker bees, and scout bees, as well as particle positions ${\mathit{x}}_i$ and velocities ${\mathit{v}}_i$
• Employed Bee Phase: Each employed bee explores new food sources (solutions) using:

  $${\mathit{V}}_{ij}={\mathit{X}}_{ij}+{\varnothing }_{ij}\left({\mathit{X}}_{ij}-{\mathit{X}}_{kj}\right)$$

  (1)

  where ${\varnothing }_{ij}$ is a random number and ${\mathit{X}}_{kj}$ is a neighboring solution.
• Onlooker Bee Phase: Onlooker bees in the algorithm choose their food sources probabilistically:

  $$P_{i }={\displaystyle \frac{f_i}{\sum _{i=1}^N{f}_i}}$$

  (2)

  where $f_i$ is the fitness of solution $x_i$
• Scout Bee Phase: Abandon poor solutions and have scout bees search for new random solutions.
• PSO Update: Update particle velocities and positions using:

  $${\mathit{v}}_i\left(t+1\right)=\omega {\mathit{v}}_i\left(t\right)+c_1{r}_1\left({\mathit{p}}_i-{\mathit{x}}_i\right)+c_2{r}_2\left(\mathit{g}-{\mathit{x}}_i\right)$$

  (3)

  $${\mathit{x}}_i\left(t+1\right)={\mathit{x}}_i\left(t\right)+{\mathit{v}}_i\left(t+1\right)$$

  (4)

  where ${\mathit{p}}_i$ is the personal best position, $\mathit{g}$ is the global best position,$\omega$ is the inertia weight, $r_1$​ and $r_2$ are random numbers, $c_1$ and $c_2$ are acceleration coefficients.
• Evaluation and Selection: Evaluate new solutions and select the best ones based on fitness.
• Convergence Check: Repeat the steps until stopping criteria, such as maximum iterations or a convergence threshold are met.

3.2. Cuckoo Search Algorithm (CSA)

It is a nature-inspired metaheuristic optimization technique established by Xin-She Yang and Suash Deb in 2009[29]. Figure 4 illustrates the methodology of the CSA. It's effective for dimensionality reduction by finding optimal feature subsets. The feature subset is chosen that optimizes a performance metric, specifically aiming for the lowest MSE. This algorithm draws inspiration from the reproductive strategy of cuckoo birds. It simulates the unique behavior of cuckoos, which lay their eggs in the nests of other bird species. The algorithm employs both randomization and local search techniques to guide the cuckoos towards the optimal solution. A notable characteristic of CSA is its utilization of Levy flights, which are random walks characterized by a heavy-tailed distribution. This method regulates the movement of cuckoos, enabling efficient exploration of the feature space while swiftly navigating towards promising regions. Therefore, CSA is selected as a DR technique for its capability to balance exploration and exploitation and effectively manage high-dimensional data. The CSA commences its process by generating an initial population of cuckoos with random positions in the feature space. The position of each cuckoo in the population is then updated using the following equation.

$$z_k^{new}=z_k^{old}+\alpha \ast \mathrm{L}\mathrm{e}\mathrm{v}\mathrm{y}\left(\mathsf{\lambda }\right)$$

(5)

Where, the cuckoo's previous position is represented by $z_k^{old}$, while its updated position is denoted as $z_k^{new}$.where α is a scaling factor, and Levy(λ) represents a step drawn from a Levy distribution.

The cuckoos' random search patterns are modeled by a concept called the Lévy flight, described by the following equation.

$$s=\tau \ast r^{{\displaystyle \raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$\mu $}\right.}}$$

(6)

Here, '$s$' represents the step size, $\tau$ is a parameter that dictates the scale of the Levy flight, and shape of the Levy flight is determined by the parameter μ. A random number $\prime r\prime$ chosen uniformly between 0 and 1. The optimal nest in the population is chosen based on the fitness value. MSE is used to derive the fitness value, and the best nest serves as the starting point for the next iteration of the algorithm. Also, abandon a fraction ${\prime P}_a\prime$ of the worst nests and generate new solutions to replace them. This process helps keep the search space varied, allowing the algorithm to explore new potential solutions. By using ${\prime P}_a\prime$ to periodically introduce new random solutions, the algorithm maintains a balance between exploration and exploitation, which is crucial for effective dimensionality reduction and optimization.

3.3. Dragonfly Algorithm

The Dragonfly Algorithm (DFA) is a heuristic optimization technique inspired by the fascinating static and dynamic swarming behaviors of dragonflies. DFA is developed by Seyedali Mirjalili in 2015[30], this algorithm mimics the way dragonflies search for food and avoid predators. Figure 5 presents the DFA flowchart. Dragonflies display two main behaviors: static, where they gather in small groups and hover around a target, and dynamic, where they collectively move towards a distant target. These behaviors are translated into mathematical models that help in exploring and exploiting the search space efficiently. It is effective for dimensionality reduction by finding optimal feature subsets through exploration and exploitation of the search space. As represented by Rahman et al. [31], separation, alignment, cohesion, attraction to food and distraction from the enemy are the important features of the DFA. In the following equations, $q$ represents the current positionof an individual dragonfly, $q_j$​ denotes the position of the $j^{th}$ dragonfly, and ‘M’ indicates the total number of neighboring dragonflies.

Separation: It prevents dragonflies from crowding together by maintaining a minimum distance from each other.

$$S_k=-{\sum }_{j=1}^{M}q-q_j$$

(7)

The symbol $S_k$ represents the motion of separation exhibited by the $k^{th}$ individual.

Alignment: It refers to the tendency of dragonflies to match their velocity with that of their neighbors.

$$A_k={\displaystyle \frac{\sum _{j=1}^M{V}_j}{M}}$$

(8)

where $A_k$ represents the alignment motion for the $k^{th}$ individual, $V_j$ indicates $j^{th}$ neighboring individual dragon velocity.

Cohesion: is the behavior that drives dragonflies to move towards the center of their neighboring individuals.

$$C_k={\displaystyle \frac{\sum _{j=1}^M{q}_j}{M}}-q$$

(9)

Attraction: It is the tendency of dragonflies to move towards food sources

$$F_k=q^{+}-q$$

(10)

where $F_k$ represents the attraction of nutrition source for the $k^{th}$ individual flyand the position of the food source is denoted as $q^{+}$.

Distraction: It is the tendency of dragonflies to move away from enemies.

$$E_k=q^{-}+q$$

(11)

where $E_k$ represents the distraction motion caused by the enemy for the $k^{th}$ individual and the location of the enemy is denoted as $q^{-}$.

The positions of artificial dragonflies inside the designated search area are revised using the current position vector (q) and the step vector (∆q). The direction of their movement is determined by the step vector (∆q), and it is calculated as,

$$∆q_{t+1}=\left(s{S}_k+a{A}_k+c{C}_k+f{F}_k+e{E}_k\right)+\omega q_{t }$$

(12)

where iteration number is denoted as ‘t’, 'ω' denotes the inertia weight and weights assigned to separation, alignment, cohesion, attraction, and enemy, are denoted as s, a, c, f, and e respectively. The exploitation and exploration phases can be achieved by modifying the weights, once the step vector calculation is completed, calculation of the position vectors commences as below:

$$q_{t+1}=q_t+∆q_{t+1}$$

(13)

Statistical metrics such as Pearson correlation Coefficient (PCC), Sample Entropy, and Canonical Correlation Analysis (CCA), mean, variance, kurtosis, and skewness were employed to identify distinct characteristics in the PPG signals between the different classes, allowing for faster analysis as presented in Table 2.

It is perceived from the table 2 that the calculated mean values are notably lower for both normal and CVD cases under ABC PSO DR technique and also for cuckoo search normal category. For cuckoo search CVD class, a higher mean value is attained. Under dragon fly optimization, negative mean values are achieved for both Normal and CVD. Skewness and kurtosis values are considerably skewed for both normal and CVD for all three DR techniques. Table 2 indicates that the sample entropy values are same across classes with the exception of the cuckoo search DR technique in the CVD case. Table 2 further illustrates that the low PCC values suggest that the optimized features exhibit nonlinearity and lack correlation across different classes. If CCA values exceed 0.5, significant correlation between the classes can be expected. Table 2 shows that the dragon fly optimization is highly related to all of the classes compare to other DR techniques. It also shows that the ABC PSO DR technique has lower correlation with the other classes.

Figure 6 dipicts the histogram plot of ABC PSO DR values for normal person and it gives that the histogram with higher peaks in the middle and lower at either end. Figure 7 displays the histogram plot of ABC PSO DR values for CVD patient and it provides that the histogram isnormal and Gaussian nature.The scatter plot of ABC PSO DR values of normal and CVD data is depicted in Figure 8, which clearly indicates that the DR features are highly merged among the classes at the center and lesser overlapping at the end.The scatter plot of Dragon Fly DR values is portrayed in Figure 9. It shows that the lesser overlapping is present among the normal and CVD classes.

4. Classifiers for Classification of CVD from Dimensionality Reduced Values

4.1. Linear Regression as a Classifier

Linear regression(LR) falls under the category of supervised learning techniques. It is primarily employed to forecast continuous numerical outcomes. This algorithm establishes a linear relationship between input features and the target variable, allowing it to make predictions on new, unseen data based on the patterns learned from the training set. The linear equation is defined by a set of coefficients that are estimated using the training data. Linear regression is is primarily designed for predicting continuous numerical outcomes, it demonstrates versatility in its application. Through appropriate modifications, this algorithm can be effectively repurposed to address classification problems. The basic linear regression model is defined as[32]:

$$q=b_0+b_1{x}_1+b_2{x}_2+\dots +b_n{x}_n+ϵ$$

(14)

Where $x_1$,$x_2$,…,$x_n$ are independent variables, $q$ is the dependent variable, $b_0$​ is the intercept ,the coefficients are $b_1$,$b_2$,…,$b_n$ and the error term is $ϵ$. For classification, the predicted $y$ values can be thresholded to determine class membership. In this research, we set a threshold of 0.5 to classify PPG data as either normal or CVD. Predictions greater than 0.5 are classified as CVD, while those less than 0.5 are considered normal. This approach is simple but can be limited by linear regression's assumptions and the nature of classification problems.

4.2. Linear Regression with BLDC

LR and LR-BLDC models use the same linear relationship, however LR is used for regression tasks with continuous outputs, whereas LR-BLDC adapts this relationship for binary classification by applying a threshold to the predicted values. The Bayesian Linear Discriminant Classifier operates as a probabilistic generative model specifically tailored for classification challenges. It involves estimating the class-conditional probability distribution of the input variables for each class and using Bayes' rule to calculate the posterior Probability of each class based on the input variables [33]. To combine linear regression and BLDC, the predicted output of the LR model as input to the BLDC model. The linear regression classifier output is used to estimate the mean of the class-conditional probability distribution for each class in the BLDC model.

4.3. K-Nearest Neighbor as a Classifier

The KNN algorithm compares the input point's distance to that of the K-nearest points in the training set. After identifying the K number of nearest neighbors, it allots the input point to the class with the maximum frequency among its K-nearest neighbors. It doesn't need a different phase for training. Instead, the algorithm retains the complete training dataset and employs it to classify new data points. Weighted KNN is a classification algorithm that assigns weights to the neighbors based on their distance to the query point. In the KNN algorithm, for a given point $z$, the algorithm finds the K data points closest to a new point by measuring their distance using a metric like Euclidean distance. [34]:

$$d\left(z,z_i\right)=\sqrt{{\sum }_{j=1}^m{\left(z_j-z_{ij}\right)}^2}$$

(15)

In weighted KNN, each neighbor $z_i$​ is assigned a weight $w_i$ inversely proportional to its distance from $z$:

$$w_i={\displaystyle \frac{1}{d\left(z,z_i\right)+\epsilon }}$$

(16)

where $\epsilon$ is a small constant to avoid division by zero. The algorithm predicts the class of the query point by considering the most frequent class among its K closest neighbors:

$$\widehat{b}=\arg {max}_c{\sum }_{i\in M_{K\left(z\right)}}{w}_i.I(b_i=c)$$

(17)

where $M_{K\left(z\right)}$ denotes the set of K nearest neighbors, the class label of neighbor $z_i$ is $b_i$ and the indicator function is I(⋅). This method improves the classification accuracy by giving more influence to closer neighbors.

4.4. PCA-Firefly

The hybrid PCA firefly technique is used to choose the most relevant features while removing irrelevant ones, thereby optimizing accuracy to its fullest extent. The PPG dataset undergoes dimensionality reduction using the PCA algorithm, which effectively reduces the quantity of stochastic variables to a concise group of primary variables. This approach greatly enhances the accuracy of the prediction results [35]. To further optimize the process, the firefly optimization algorithm is utilized to select the most appropriate attributes from this refined reduced dataset.

PCA simplifies data by finding a new set of features that are not related to each other by solving the eigenvalue problem:

$$Y^T{Y}_v=\mathsf{\lambda }v$$

(18)

where $Y$ is the data matrix, $v$ are the eigenvectors (principal components), and λ are the eigenvalues.

The Firefly Algorithm is a clever problem-solving method that mimics how fireflies communicate with their flashes, optimizes the selection of principal components. Each firefly represents a potential solution with an intensity I proportional to its fitness. The attractiveness $\alpha$ between fireflies $i$ and $j$ is given by:

$${\alpha }_{ij}={\alpha }_0{e}^{-\gamma p_{ij}^2}$$

(19)

where ${\alpha }_0$​is the attractiveness at $p$=0, $\gamma$ is the light absorption coefficient, and ​$p_{ij}$ is the distance between fireflies $i$ and $j$. Fireflies are drawn to brighter ones, constantly adjusting their location:

$$x_i=x_i+{\alpha }_{ij}\left(x_j-x_i\right)+\beta {ϵ}_i$$

(20)

where $\beta$ is a randomization parameter, and ${ϵ}_i$ is a random vector. This process iteratively refines the feature selection, enhancing classification performance [36].

4.5. Linear Discriminant Analysis as a Classifier

Linear Discriminant Analysis (LDA) is a dimensionality reduction technique used for classification, aiming to find the best way to combine features to distinguish between different groups. LDA assumes that each class follows a Gaussian distribution with a shared covariance matrix. The steps of LDA involve [37]:

• Calculate the mean vector for each class

  $${\mu }_z={\displaystyle \frac{1}{M_z}}{\sum }_{i\in C_z}{x}_i$$

  (21)
• Calculate the within-class scatter matrix

  $$S_w={\sum }_{z=1}^M{\sum }_{i\in C_z}\left(x_i-{\mu }_z\right){\left(x_i-{\mu }_z\right)}^T$$

  (22)
• Calculate the between-class scatter matrix

  $$S_b={\sum }_{z=1}^M{M}_z\left({\mu }_z-\mu \right){\left({\mu }_z-\mu \right)}^T$$

  (23)

  Where $\mu$ is the overall mean of the data set.
• Solve the generalized eigenvalue problem

  $$S_w^{-1}{S}_b\mathit{v}=\mathsf{\lambda }\mathit{v}$$

  (24)

The eigenvectors $\mathit{v}$ corresponding to the largest eigenvalues λ form the transformation matrix. Project the data onto this lower-dimensional space to maximize class separability. In classification, the new data point $x$ is projected and assigned to the class with the closest mean in this reduced space.

4.6. Kernel LDA as a Classifier

Kernel Linear Discriminant Analysis(KLDA) is an extension of LDA that incorporates the kernel trick to handle nonlinear data. The objective of this technique is to identify an optimal projection of the data in a high-dimensional feature space, which is created through the application of a kernel function. This projection is designed to simultaneously maximize the separation between different classes (between-class scatter) and minimize the spread within each individual class (within-class scatter).

The objective function of KLDA is formulated as:

$${max}_v{\displaystyle \frac{v^T{S}_{b}v}{v^T{S}_{w}v}}$$

(25)

where $S_b$ is the between-class scatter matrix and $S_w$ is the within-class scatter matrix in the kernel space. The solution $v$ corresponds to the eigenvector associated with the largest eigenvalue of the generalized eigenvalue problem $S_{w}v={\mathsf{\lambda }S}_{b}v$.KLDA is effective for nonlinear data leveraging a kernel function to implicitly project the original input space into a higher-dimensional feature space, where data separation is potentially easier to achieve. It has found applications in various fields where nonlinear relationships among data features are prevalent [38].

4.7. Probabilistic LDA as a Classifier

Probabilistic Linear Discriminant Analysis (ProbLDA) is an extension of Linear Discriminant Analysis (LDA) that models class distributions probabilistically. ProbLDA assumes that each class follows a Gaussian distribution and utilizes Bayesian principles for classification [39]. The class mean, within-class scatter matrix and between-class scatter matrix are computed as per the equations 21,22 and 23.Then for each class$\prime z\prime$ Gaussian distributions can be assumed as follows:

$$p\left(x|C_z\right)={\displaystyle \frac{1}{{\left(2\pi \right)}^{\raisebox{1ex}{$d$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\left|\mathbf{Ʃ}\right|}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{1}{2}}{\left(\mathit{x}-{\mu }_z\right)}^T{\mathbf{Ʃ}}^{-1}\left(\mathit{x}-{\mu }_z\right)\right)$$

(26)

Where$Ʃ$ is the Shared covariance matrix estimated as $S_w$.

To calculate the posterior probabilities for classification, Bayes’ theorem can be used as per below equation.

$$p\left(C_z|x\right)={\displaystyle \frac{p\left(x|C_z\right) p\left(C_z\right)}{p\left(x\right)}}$$

(27)

Here $p(C_z$) prior probability of class $z$ and $p\left(x\right)$ is the marginal probability of $x$.Assign the new data point $x$ to the class with the highest posterior probability.

$$\widehat{g}=arg {max}_z p\left(C_z|x\right)$$

(28)

ProbLDA integrates the strengths of LDA with probabilistic modeling, providing a robust framework for classification.

4.8. Support Vector Machine as a Classifier

The Support Vector Machine (SVM) is a sophisticated supervised learning technique primarily employed for classification purposes. Its core principle involves identifying the most effective hyperplane that creates the widest possible margin between distinct classes within the feature space. It is used in the hyperplane to establish decision boundaries, distinguish between relevant and irrelevant vectors, and categorize data points into different classifications [40]. In SVM, input data is mapped to a higher-dimensional feature space using kernel functions to address multiclass classification problems. SVM aims to achieve high generalization by effectively separating classes based on the training data. Consider a binary classifier where a feature $x$ is associated with label‘h’ that belong to the set (−1,1). The classifier's decision is defined using parameters $z$ and b instead of the vector θ, the classification expression is,

$$g_{z,b}\left(x\right)=K(z^{T}x+b)$$

(29)

Here K(y) =1 if y ≥ 0 and k(y)= -1 otherwise. The term ‘b’ is separated from other parameters using ‘z,b’. In this research, SVM with three different kernels are used: Radial Basis Function (RBF), linear, and polynomial [41].

SVM-Linear:

$$\ker \left(z_i,z_j\right)={z_i}^T{z}_j$$

(30)

SVM-Polynomial:

$$\mathrm{Ker}\left(z_i,z_j\right)={\left(1+{\Upsilon z}_i^T.z_j\right)}^q, \Upsilon >0$$

(31)

where the degree of polynomial kernel is denoted as q and ‘ϒ’ denotes the gamma term in the kernel function.

SVM-RBF:

$$\ker \left(z_i,z_j\right)=\mathrm{e}\mathrm{x}\mathrm{p}(-\Upsilon {‖z_i-z_j‖}^2), \Upsilon >0$$

(32)

where,$‖z_i-z_j‖$ is Euclidean distance between two input vectors$z_i$ and $z_j$.

These kernel functions operate by projecting the input data into a space of increased dimensionality. This transformation facilitates the identification of an optimal hyperplane that effectively delineates between different classes.

5. Results and Discussion

This research work employed a data partitioning strategy wherein 90% of the available dataset was dedicated to the model's training process, while the remaining 10% was set aside for subsequent testing and validation. To assess the classifier's efficacy and reliability, we implemented a ten-fold cross-validation approach. A True Positive (TP) occurs when the classifier correctly identifies a positive sample, while a True Negative (TN) is when it accurately labels a negative sample. A False Positive (FP) occurs when the model erroneously classifies a negative instance as positive, whereas a False Negative (FN) happens when the model incorrectly identifies a positive instance as negative. The Mean Squared Error (MSE) is defined by the following mathematical formula:

$$MSE={\displaystyle \frac{1}{\mathrm{M}}}{\sum }_{k=1}^M{\left(O_k-T_i\right)}^2$$

(33)

where M denotes the complete set of data points within the PPG dataset and it is assumed as 1000. The target value of model $\mathrm{\text{'}}i\mathrm{\text{'}}$ is designated by $T_i$,where the range of $\mathrm{\text{'}}i\mathrm{\text{'}}$ varies from 1 to 15; at a specific time, $O_k$ represents the observed value. The training was conducted in a manner that significantly reduced the classifier's Mean square error to a minimal value.

Table 3 presents the testing and training MSE for classifiers using three different DR techniques. The training MSE ranges from${10}^{-5}$ to ${10}^{-10}$, in contrast, the testing phase yielded MSE values ranging from${10}^{-3}$ to ${10}^{-9}$.The SVM-RBF classifier under the ABC-PSO dimensionality reduction method achieved the lowest training of 1.92 X${10}^{-10}$ and testing MSE value of 2.45 X ${10}^{-9}$. The DFA DR method yields somewhat reduced MSE values for both training and testing phases across the classifiers, when compared to the alternative DR approaches evaluated in this work. Similarly, for Cuckoo Search DR features the LR classifier yields the minimal testing MSE value of 1.37E-08, due the correct labelling of PPG signal for both CVD and normal subjects. The PCA Firefly classifier is plugged with a high number of FP and FN cases, resulting in the highest testing MSE of 6.08E-03. Similarly, for Dragon Fly DR values, the SVM-RBF classifier achieves the minimum testing Mean square error of 3.62E-09. This superior performance is attributed to its accurate labeling of PPG signals for both CVD and normal subjects. Conversely, the SVM-Linear classifier shows poorer performance, resulting in a much higher testing MSE of 9.03E-03. This higher error rate is due to the SVM-Linear misclassifying a significant number of cases, producing both false negatives and false positives.

5.1. Optimal Parameters Selection for Classifiers

When selecting the target values for the binary classification of PPG dataset (CVD and Normal class), a deliberate choice is made to assign the target ($T_{CVD})$ values towards the upper end of the 0 to 1 range. The criteria used to select $T_{CVD}$ is as follows:

$${\displaystyle \frac{1}{Y}}{\sum }_{p=1}^Y{\mu }_p\le T_{CVD}$$

(34)

The complete set of CVD PPG data features, denoted as (Y), underwent a normalization process with mean ${\mu }_p$. For the target $T_{Normal}$ values, a deliberate choice is made to assign them towards the lower end of the 0 to 1 range. The selection criteria for determining the value of $T_{Normal}$ are governed by the following parameters:

$${\displaystyle \frac{1}{X}}{\sum }_{q=1}^X{\mu }_q\le T_{Normal}$$

(35)

The complete set of normal PPG data features, denoted as (Y), underwent a normalization process with mean ${\mu }_p$. For optimal categorization,

$$‖T_{CVD}-T_{Normal}\ge 0.5‖$$

(36)

Based on the condition provided in (3), in this research work, the targets have been set at 0.1 for normal and 0.85 for CVD. Table 4 details the iterative process of selecting optimal parameters for the classifier during its training phase. A maximum of 1000 iterations is allowed to control the convergence criteria.

5.2. Performance Analysis of the Classifier

The classifiers' effectiveness is assessed using a comprehensive set of metrics, such as Accuracy, Good Detection Rate (GDR), F1 Score, Kappa, Matthews Correlation Coefficient (MCC), Error rate, and Jaccard Index. The following formula used for evaluating the overall effectiveness of the classification method.

$$\mathrm{A}\mathrm{c}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{y}={\displaystyle \frac{\mathrm{T}\mathrm{N}+\mathrm{T}\mathrm{P}}{\mathrm{T}\mathrm{P}+\mathrm{T}\mathrm{N}+\mathrm{F}\mathrm{P}+\mathrm{F}\mathrm{N}}} X 100\%$$

(37)

$$\mathrm{G}\mathrm{o}\mathrm{o}\mathrm{d}\mathrm{d}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}\left(\mathrm{G}\mathrm{D}\mathrm{R}\right)={\displaystyle \frac{\left(\mathrm{T}\mathrm{N}+\mathrm{T}\mathrm{P}\right)-\mathrm{F}\mathrm{N}}{(\mathrm{T}\mathrm{N}+\mathrm{T}\mathrm{P})+\mathrm{F}\mathrm{P}}} X 100\%$$

(38)

$$\mathrm{E}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}\mathrm{R}\mathrm{a}\mathrm{t}\mathrm{e}={\displaystyle \frac{FP+FN}{TP+TN+FP+FN}}\times 100\mathrm{\%}$$

(39)

$$\mathrm{where}, E_{acc}=\left(\right((TP+FP)/100)\ast (TP+FN)/100+(\left(\right(TN+FP)/100)\ast \left(\right(TN+FN)/100)$$

$$\mathrm{K}\mathrm{a}\mathrm{p}\mathrm{p}\mathrm{a}={\displaystyle \frac{\left(\frac{TP+TN}{100}-E_{acc}\right)}{\left(1-E_{acc}\right)}}$$

(40)

$$\mathrm{M}\mathrm{C}\mathrm{C}={\displaystyle \frac{\left(\mathrm{T}\mathrm{N}.\mathrm{T}\mathrm{P}\right)-(\mathrm{F}\mathrm{N}.\mathrm{F}\mathrm{P})}{\sqrt{\left(TN+FN\right).\left(TN+FP\right)\left(TP+FN\right).\left(TP+FP\right)}}} X 100\%$$

(41)

$$\mathrm{F}1\mathrm{S}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}={\displaystyle \frac{2\mathrm{T}\mathrm{P}}{2\mathrm{T}\mathrm{P}+\mathrm{F}\mathrm{P}+\mathrm{F}\mathrm{N}}} X 100\%$$

(42)

$$\mathrm{J}\mathrm{a}\mathrm{c}\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}\mathrm{I}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}={\displaystyle \frac{\mathrm{T}\mathrm{P}}{\mathrm{T}\mathrm{P}+\mathrm{F}\mathrm{P}+\mathrm{F}\mathrm{N}}} X 100\%$$

(43)

Table 5 presents the results of the performance analysis for classifiers with three dimensionality reduction techniques. The results in Table 5 reveal that the classifier SVM-RBF for ABC PSO dimensionality reduction technique achieved very high scores on all benchmark metrics, including a superior accuracy of 95.12%, a F1 score and GDR of 95%, and a lowest error rate of 4.88%. Furthermore, the high Kappa and MCC are 0.90, with a Jaccard Index of 98.48%. The result demonstrates that the SVM-RBF classifier is outperforming for the ABC-PSO dimensionality reduction technique. Conversely, the KLDA classifier for the ABC-PSO dimensionality reduction method and the PCA-Firefly classifier for the Cuckoo Search dimensionality reduction technique demonstrated poor performance across all parameter values. This was evident from the lowest accuracy and F1 score of 53.66%, a GDR of 38.71%, and the highest error rate of 46.34%. Also, the low Kappa and MCC are 0.07, with a Jaccard Index of 36.67%. Upon analyzing individual dimensionality reduction techniques, the SVM (RBF) classifier performs better than other classifiers for the ABC-PSO DR technique, with a F1 score and GDR of 95% and a Jaccard Index of 98.48%. On the other hand, KLDA is the lowest performing classifier due to a higher error rate of 46.34% and reduced Kappa and MCC values of 0.07. Similarly, for the CSA DR method, the linear regression classifier achieved better accuracy at 90.24%, with a strong F1 score of 90% and a lower error rate of 9.76%. Meanwhile, for the Dragon Fly DR technique, the SVM-RBF classifier attained a higher accuracy of 92.68%, a GDR of 92.50%, and a high Matthews correlation coefficient of 0.85 due to a lower error rate of 7.32%.

Figure 10 illustrates the comparison of accuracy performance among classifiers using ABC PSO, Cuckoo Search, and Dragon Fly DR techniques.

Figure 10 presents the performance evaluation of classifiers across various DR techniques based on accuracy. Figure 10 reveals that the SVM-RBF classifier reigns supreme with an accuracy of 95.12%. Conversely, the KLDA classifier yields the lowest accuracy, at 53.66% for the ABC PSO DR method. Similarly, with CSA dimensionality reduced values, the LR classifier achieves the highest accuracy of 90.24%, while PCA-Firefly exhibits the lowest accuracy at 53.66%. The SVM-RBF classifier exhibited robust performance, achieving a notable accuracy of 92.68% when paired with the Dragonfly DR method. In contrast, the SVM with linear kernel struggled comparatively, yielding a considerably lower accuracy of 58.54%. Additionally, Figure 10 shows that the LR classifier performed well across all three DR techniques, achieving an accuracy of 90.24%.

Figure 11 shows that all classifiers using the three different DR techniques achieve an F1 score of around 60% or higher, indicating they are performing significantly better than random guessing. This indicates a notable alignment between the model's forecasts and the true class labels. It is also perceived that the maximum error rate observed is 46%, while the highest F1 score is 95%.

Figure 12 presents the histogram of accuracy and the Jaccard index for the classifiers. It is perceived that the maximum accuracy reaches 95%, while the highest Jaccard index is 90%. The accuracy histogram is left-skewed, indicating that the classifier's accuracy does not drop below 50% for any of the DR techniques. The Jaccard index histogram spans the entire spectrum, with values below 50% attributed to the classifiers' high false-positive rates.

5.3. Analysis of the Computational Complexity of Classifiers

Computational complexity is an important performance metric for classifiers. It is evaluated by considering the input size, denoted as $m$. The computational complexity remains very low when the input size is $O\left(1\right)$. However, the computational complexity increases with the number of inputs. In this research, the computational complexity is independent of input size, which is a highly desirable characteristic for any algorithm. The term $O(\log m)$ denotes the logarithmic rise in computational complexity with respect to $m$.

The computational complexity of classifiers across different DR techniques is presented in Table 6. From table 6, the results clearly demonstrate that the SVM-RBF classifier has the highest computational complexity when used with the ABC-PSO and dragonfly DR techniques. Consequently, SVM-RBF achieves the highest accuracy, with 95.12% for the ABC-PSO dimensionality reduction technique and 92.68% for the dragonfly DR technique. Conversely, the KLDA classifier for the ABC-PSO dimensionality reduction technique and the PCA-Firefly classifier for the cuckoo search DR values produce the lowest accuracy of 53.66% while also having high computational complexity. This is attributed to their high false-positive rate and low Jaccard index.

As seen in Table 7, various machine learning classifiers, such as LR, NB, RBF NN, DCNN, KNN, DNN, ELM, ANN and SVM (RBF) have been utilized for classification of CVD from clinical database. The performance spectrum of these classifiers spans from a moderate 65% to an impressive 95% in terms of accuracy. However, this study specifically targets CVD detection using Capnobase dataset, with SVM (RBF) achieving the highest accuracy of 95.12%.

6. Conclusion

The main objective of this work is to classify PPG signals as either normal or indicative of cardiovascular disease. High-quality PPG samples were obtained by extracting useful features using heuristic-based DR methods such as ABC-PSO, Cuckoo Search, and Dragon Fly techniques. Ten classifiers were used for this purpose: LR, LR-BLDC, KNN, PCA-Firefly, LDA, KLDA, ProbLDA, SVM-Linear, SVM-Polynomial, and SVM-RBF. The SVM-RBF classifier, combined with the hybrid ABC-PSO dimensionality reduction technique, demonstrated superior performance. This approach achieved a remarkable accuracy of 95.12% while maintaining a minimal error rate of just 4.88%. Furthermore, it exhibited robust reliability, as evidenced by its high Matthews correlation coefficient and Kappa values, both reaching 0.90. Again, the second highest accuracy of 92.62% was achieved by the SVM-RBF classifier for Dragon Fly optimized values. The third highest accuracy of 90.24% was obtained by the LR classifier across all three DR techniques. Future research is focusing on convolutional neural networks (CNNs) and deep neural networks (DNNs) to swiftly detect cardiovascular disease.