Quantum-inspired seagull optimised deep belief network approach for cardiovascular disease prediction

Research questions

The present research is carried out to find answers to the following questions,

1. How successful is the quantum inspired seagull optimisation algorithm (QISOA) at optimising the deep belief network’s parameters?

2. Does the QISOA algorithm enhance the DBN’s ultimate performance and rate of convergence?

3. What are the possible outcomes of using the QISOA-DBN model for the early detection of cardiovascular diseases?

Research contributions

The major contributions of this research are,

• 1.

  To propose a quantum-inspired seagull optimization algorithm that incorporates quantum principles into the conventional seagull optimization algorithm to improve the balance between exploration and exploitation, which leads to better convergence towards optimal solutions.

• 2.

  To implement deep learning-based methods such as deep belief networks algorithm optimized by QISOA technique to perform cardiovascular disease prediction with enhanced efficiency and reliability.

• 3.

  To conduct a comprehensive performance analysis of the proposed QISOA-DBN model to demonstrate its effectiveness in diagnosing cardiovascular disorders by comparing it to the state-of-the-art methods described in the literature.

Paper organization

The remainder of the article is organized as follows: “Related Works” describes the state-of-the-art work on cardiovascular disease identification using ML/DL techniques and optimization algorithms. “Preliminaries” provides a theoretical background on quantum computing concepts and conventional seagull optimisation techniques. “Proposed Methodology” discusses the proposed methodology addressing the core concepts of the suggested QISOA-DBN approach. “Results and Discussion” presents the results obtained by applying various ML/DL techniques on the Cleveland Heart Disease prediction dataset and compares the existing and proposed systems. “Conclusion” concludes the present research.

Motivations of current research

The cardiovascular disease diagnosis techniques currently in operation frequently have low diagnosis rates, and thus, improvement in detection rates usually necessitates several iterations with poor convergence. This shortcoming emphasizes the need for a quicker and more precise diagnosis technique that might improve the general efficacy of heart disease prediction. Using sophisticated machine learning models—especially those influenced by quantum algorithms—offers a viable way to improve heart disease diagnostic performance. By increasing the precision of disease prediction, utilizing approaches like QISOA and DBN may be able to advance beyond the drawbacks of the present methods. Machine learning models such as DBNs must fine-tune their parameters for better classification accuracy. QISOA offers a valuable mechanism to strike a balance between exploration and exploitation in the search space, thus it is implemented to improve the efficiency of the DBN model.

Basics of quantum computing

Quantum computing’s foundations are quantum physics, with the Schrodinger equation as the underlying principle. Unlike the traditional approach, quantum computing demonstrates its potential to operate more quickly, particularly when tackling complicated, challenging problems requiring a wide range of possible solutions. This section describes some of the basic concepts in quantum computing.

Conventional seagull optimization

An innovative swarm optimization algorithm mimics seagulls’ relocating and assaulting activity is called the seagull optimization algorithm (SOA) (Asif et al., 2023). For multi-objective optimisation problems, an SOA extension has been created that incorporates variable network, leader-selection, and conserving techniques. When migrating, they move in clusters. To avoid crashes, seagulls adjust their initial points of interest. The gulls will flock together and fly towards the fittest gull, who has the most chance of surviving. The other seagulls will adjust their starting places based on the most fitting gull. When birds travel from one spot to another over the sea, gulls attack them. When they attack, they move in a spiral pattern. The migration operation uses the SOA’s exploratory power to replicate how a flock of seagulls moves from one location to another. The attack operation uses the algorithm’s exploitative power to mimic how flocks of seagulls pursue their target.

Data preprocessing

The data preprocessing in this work involves three techniques: cleaning, transformation, and standardization. Data cleaning is essential to identify and eliminate redundant, erroneous, or unavailable values. Data transformation is needed to remove compatibility issues based on the nature of the data. Data standardization employs statistical approaches to normalize the data values within a particular range.

QISOA

The QISOA approach incorporates quantum computing superposition concepts, layout ranking, seagull mobility and assault behaviour, and opposition-driven learning (ODL). The ODL process initialises the seagull group to produce an improved setup population. The real-coded quantum representation of the existing ideal approach and quantum rotation gate have been adjusted to more effectively preserve an equilibrium between exploration and exploitation to attain the best possible solutions. A layout mechanism that combined the layout density ordering and the global layout ordering additionally offered an indicator for the maintenance and selection of leaders.

Step 1: Initializing population using ODL mechanism

The likelihood of sampling better regions in population-based algorithms is decreased by initialisation using random roulette in conventional evolutionary algorithms when beforehand data is not available. On the other hand, without previously acquired knowledge, ODL can identify more appropriate initial candidate regions, raising the likelihood of identifying superior regions and holding out hope for enhancing fitness.

Step 2: Choosing the best possible solution

This step employs layout density ordering and global layout ordering mechanisms to characterize the most optimal solutions. This technique can display the differences in objective values between optimal solutions and alternative solutions and highlight the superiority and inferiority of each option. This work uses layout density ordering (LDO) to increase the spread of the solutions and global layout ordering (GLO) to improve the algorithm’s efficiency.

The total number of unique layout positions in each objective that are better than those of other points can be calculated by the GLO technique as represented in Eq. (8):

(8) GLO(ak)=∑h=1Pkcount(Lm(ak)<Lm(al)).

In Eq. (8), akandal are two individuals chosen from the population created using Algorithm 1 in which k and l are different. Lm(ak) denotes the layout positions of ak for one objective from the total set of objectives represented as Pk. Furthermore, the value of LDO is computed as given in Eq. (9):

(9) LDO(ak)=count(∑h=1Pk|Lm(ak)−Lm(al)|<G).

In the above equation, G represents the present set of objectives in the population. This mechanism prefers the candidate solution with a higher GLO value among those in the collection. Furthermore, when multiple solutions have identical GLO values, the solution with the lowest value is favored as the best option. A roulette wheel chooses a solution as the optimal solution among several solutions with the same GLO values.

Step 3: Quantum illustration of best solutions

This step is comprised of the quantum illustration of the best possible solution and the quantum rotation gate, which updates the likelihood magnitudes of two states. A combined function of dimensions and duration, known as the wave function, is needed to fully characterize a quantum state in the theory of quantum mechanics. The magnitude coefficient denotes the likelihood that the quantum state will occur at a suitable place and time. To compute the observed values of the quantum illustration, a typical wave function is implemented as shown in Eq. (10):

(10) |W(ak)|2=12πθkexp[−(ak−μk)22θk2].

In the above equation, μk denotes the average location of the individual and θk represents the dispersal spectrum of the individuals with respect to the value of μk. The quantum illustration of the best possible solution ( Rci) is determined using Eq. (11) as:

(11) Rci=△[ac,1ac,2…ac,kγ1γ2…γk].

In Eq. (11), i represents the present iteration for a set of values, γk represents the coefficient of magnitude for each optimal solution.

Step 4: Seagull exploration

During this stage, the seagulls relocate themselves from one position to another so they do not collide. The orientation of relocation without interference is determined using the movement nature of the seagull in the search space (F) as shown in Eq. (12):

(12) Dz=F×Cz(i).

In the above equation, Cz represents the present location of individual and Dz denotes the relocated position such that it does not collide with any other individual in the search space. The value of F is computed in a non-linear fashion in order to improve the efficiency of the algorithm using Eq. (13):

(13) F=ms(2τ−1)

(14) τ=e1−imaximax−i.

The relocation direction of each individual in quantum representation towards the best possible solution is given using Eq. (15) as:

(15) Ez=G×(Rc(i)−Cz(i))

(16) G=2×F2×rand(0,1).

The modified position of the relocated seagull is calculated by combining the values of the orientation of relocation without interference and relocation direction towards the best possible solution as shown in Eq. (17):

(17) Pz=|Dz+Ez|.

Step 5: Assault behaviour of seagull

During the process of relocating from one position to another, seagulls implement an assault against each other in a 3-D space. The movement of seagulls in 3-D space is described in Eqs. (18) to (21):

(18) u=rd×cos(l)

(19) v=rd×sin(l)

(20) w=rd×l

(21) rd=m×eln.

In the above equations, l is an arbitrary value between 0 and 2π and rd denotes the radius with which the seagull makes a spiral motion in the orientation of m and n. The updated position of the seagull is an aggregation of the relocation and assault behaviour, and the final most optimal position is determined as shown in Eq. (22):

(22) Cz(i)=(Pz×u×v×w)+Rc(i).

QISOA-DBN

One neural network that is stacked with numerous supervised back propagation networks as well as restricted Boltzmann machine networks is the DBN. The DBN model is extensively used in the healthcare domain for disease prediction and classification, due to its great approximation capability to the nonlinear function. The arbitrary initialisation of the DBN connectivity weights as well as biases results in a lack of global optimisation potential and final descent into the local optimum. As a result, the DBN’s connection weights and biases have a significant impact on the reliability of disease classification.

The QISOA algorithm handles the functions of quick convergence, population variability, and global search potential. To maximise each node’s connection weights, the QISOA with global optimization potential is incorporated into the DBN. To achieve an optimised DBN such as the QISOA-DBN model, a parametric optimization approach based on QISOA is proposed. This method helps to prevent a lack of clarity in parameter selection, minimise its effect on modelling precision, and improve the efficiency of the model for forecasting and classifying cardiovascular diseases.

A transparent layer and a concealed layer constitute every restricted Boltzmann machine. For any neuron x in the transparent layer and y in the concealed layer denoted as (t,c), the energy parameter is represented as in Eq. (23):

(23) EG(t,c|δ)=−∑x=1Xmxtx−∑j=1Ynycy−∑x=1X∑y=1Ytxwxycy.

In the above equation, δ=(wxy,mx,ny) in which wxy denotes the weight parameter between the nodes in the transparent and concealed layers t and c, respectively. mxandny are the bias values for t and c layers. The joint probability distribution is computed based on the energy parameter calculated using Eq. (23) as given in Eq. (24):

(24) JP(t,c|δ)=e−EG(t,c|δ)/[R(δ)].

In Eq. (24), the standardization parameter is determined as R(δ)=∑t∑ce−EG(t,c|δ). The actuation states of the nodes in the concealed layer are independent since there is no connection between the layers in the restricted Boltzamnn when the state of the nodes in the transparent layer is determined. The likelihood of actuating a node from the concealed layer y is given by the Eq. (25):

(25) JP(cy=1|t,δ)=σ(ny+∑x=1Xtxwyx).

In the above equation, the sigmoid function σ(a) is given by σ(a)=11+e−a. Similarly, the likelihood of actuating neurons from the transparent layer is given by the Eq. (26) as:

(26) JP(tx=1|c,δ)=σ(mx+∑y=1Ycxwyx).

The sensitivity ( φx) in the final output layer ( fx) is calculated from the actual value ( vx) as in Eq. (27):

(27) φx=fx(1−fx)(vx−fx).

Similarly, the sensitivity for i^th layer can be expressed as shown in Eq. (28):

(28) φxi=gxi(1−gxi)∑ywxyiφyi+1.

The modified mathematical formulations for the weight as well as bias values for every layer of the deep belief network as the rate of learning ( γ) for every epoch is given by Eqs. (29) to (31):

(29) wxyi=wxyi+γ∗gxiφyi+1

(30) mxyi=myi+γ∗φyi+1

(31) nxyi=nyi+γ∗φyi+1.

Dataset description

The Cleveland benchmark dataset from the UCI machine learning repository is used in this study. The Cleveland dataset has fourteen characteristics and 303 records. The classification of the target attribute is represented with two class values labeled 1 and 0, denoting patients with and without heart disease, respectively. The assortment is approximately balanced because 165 records represent individuals having heart disease and 138 records represent individuals without heart disease. The dataset used in this research can be accessed using the link: https://archive.ics.uci.edu/dataset/45/heart+disease. This dataset is a combination of numerical and categorical features. The numerical features in the dataset include age, cholesterol, resting blood pressure, maximum heart rate, ST depression, and number of major vessels. The categorical features in this dataset are chest pain types, sex, fasting blood sugar, resting electrocardiogram, exercise-induced angina, ST slope, and Thalassemia. The data distribution of numerical and categorical features is analyzed and presented in Figs. 2 and 3.

Experimental setup

The experimental setup used in this research includes an NVIDIA GTX 1080 Ti as a dedicated GPU with CUDA compatibility with 16GB RAM and Windows 11 Operating System. Python 3.7 version is utilized with the Jupyter Notebook environment for code execution. Libraries such as TensorFlow, pandas, numpy, scikit-learn and matplotlib were used for the algorithm implementation.

The parameter settings for the QISOA-DBN model are described in this section. The population size of QISOA, which specifies the number of seagull agents, is 50, with maximum optimization iterations set as 200. The quantum parameter, which balances exploration and exploitation, is assigned a value of 0.5. The inertia weight, which controls the impact of previous positions on the current iteration, is set to decrease from 0.9 to 0.4. The adaptation speed of seagull agents is 0.7. The probability of mutation to maintain diversity is assigned as 0.05. The DBN model comprises five layers with four stacked Restricted Boltzmann Machine layers. The model is executed for 300 iterations with a maximum number of nodes set as 20 and a minimum number of nodes set as 2,000. The learning rate for the first, second, and third layers is 0.015. The learning rate for the final fourth layer of RBM is set to be 0.001. The value of the initial momentum is initialized with 0.5, and the regularization parameter, which prevents overfitting by penalizing large weights, is assigned as 0.001.

Experimental evaluation

The correlation matrix representing the relationship between each of the variables in the dataset is presented in Fig. 4. The inferences interpreted from the details in Fig. 5 are presented in Table 2.

From the correlation matrix obtained it can be inferred that chest pain type, maximum heart rate, ST slope, number of major vessels and thalassemia are the most important features that can predict the presence of heart diseases. These features have strong positive correlations with the target indicating that the individual is affected by heart disease. The features such as exercise-induced angina and oldpeak can also be considered as significant contributors as they are correlated with several other features. The performance of the existing and proposed techniques is assessed using metrics such as accuracy, precision, recall and F1-score.

For this research, accuracy, precision, recall, and F1-score are critical performance indicators since they offer a thorough assessment of the suggested QISOA-DBN model’s efficacy in classifying heart diseases. While precision concentrates on the consistency of positive diagnoses while reducing false positives, accuracy assesses the overall correctness of forecasts. By minimising false negatives, recall evaluates the model’s sensitivity in detecting actual cases of heart disease. Because false positives and false negatives can both have serious repercussions in medical applications, the F1-score, which is the harmonic mean of precision and recall, offers a balanced measurement.

The confusion matrix is built using the predictions made by various models based on true positive/negative and false positive/negative values.

Initially, the performance of the DBN model is compared with the baseline machine learning and deep learning models such as support vector machines (SVM), Random Forest (RF), decision trees (DT), multilayer perceptrons (MLP), and fully convolutional networks (FCN).

Support vector machines (SVM): SVM is a supervised learning approach that divides observations into different categories by determining the best hyperplane. SVM is useful in heart disease research for binary classification and performs well with high-dimensional data, assisting in the discovery of correlations that point to the presence of heart disease.

Random Forest (RF): Several decision trees are constructed using the Random Forest ensemble learning technique, combining them to increase the model’s predictability and reduce overfitting. RF handles large datasets with numerous attributes in heart disease research, yielding reliable insights into the significance of various risk indicators for heart disease.

Decision trees (DT): A decision tree is a straightforward yet effective method that divides data into branches depending on feature values to make predictions. Because of its interpretability, DTs are employed in heart disease research to determine which features—such as age and blood pressure—are most important in predicting the risk of heart disease.

Multilayer perceptrons (MLP): This model is a composition of multiple layers of neurons that are capable of learning intricate, non-linear correlations from data. This technique is effective in forecasting the existence or severity of heart disease, complex patterns, and interactions between different clinical and demographic data related to cardiovascular research.

Fully convolutional networks (FCN): This is a deep learning architecture mainly used to analyze sequential data in heart disease research. It is a valuable tool for identifying patterns in heart disease data because it can automatically extract features from raw data without requiring manual feature extraction.

The confusion matrix obtained based on the predictions made by each model is presented in Fig. 5.

Table 3 presents the performance metric values obtained according to the efficiency of the models in classifying heart diseases. It can be observed that the SVM model produces an accuracy of 92.35%, precision of 91.32%, recall of 90.53% and F1-score of 91.75%. RF model produces a lower accuracy of 83.21% compared to the SVM model. The precision, recall and F1 score values are also less, with 82.36%, 81.65% and 82.95%, respectively. The DT model shows the least performance with poor accuracy of 81.52% compared to the other baseline models. MLP model shows a better performance compared to SVM, RF and DT models with an average accuracy of 93.25%, precision of 92.45%, recall of 91.36% and F1 score of 92.86%. The FCN model is 94.51% accurate in classifying cardiovascular disease with 93.31%, 91.64% and 92.98% of precision, recall and F1 score correspondingly. The DBN model is more precise and reliable than other traditional ML/DL models, with an accuracy level of 96.78% in predicting individuals with and without heart diseases. Figure 6 depicts the performance comparison of the baseline ML/DL models.

Furthermore, the performance of the baseline models, along with various optimization algorithms, is analyzed, and the corresponding confusion matrix is presented in Fig. 7. The optimization algorithms such as particle swarn optimization (PSO), genetic algorithm (GA), grey wolf optimization (GWO), cuckoo search optimization (CSO), crow search algorithm (CSA) and seagull optimization algorithm (SOA) are employed to optimize the parameters of the traditional ML/DL algorithms. The obtained values of various performance metrics are shown in Table 4. It can be seen that the performance of the SVM model optimized by PSO, the RF model optimized by GA and the DT model optimized by GWO algorithm are quite close to each other with accuracy levels of 93.89%, 94.12% and 94.36% respectively. This is because the total true positives and true negatives predicted by these models are the same, with slight variations in the number of false positives as well as false negatives forecasted by these models.

It is also evident that the FCN and CSA combination produces the lowest accuracy of 92.78% with a precision of 91.23%, recall of 90.45%, and F1 score of 91.15%. MLP model optimized by the CSA technique produced a better accuracy of 95.24% with precision, recall, and F1 score values of 94.52%, 93.14%, and 94.16%, respectively.

DBN model optimized by QISOA produces an accuracy of 98.6%, precision of 97.6%, recall of 96.8% and F1 score of 97.1%. It has been observed that the optimized DBN model produces performance superior to other baseline models that are optimized using several optimization algorithms. The graphical depictions of the achieved outcomes are presented in Fig. 8.

Table 5 shows the performance of the proposed QISOA-DBN model, which is compared against a few existing works in the literature that focus on cardiovascular disease classification using quantum-based ML/DL or optimization methods. The work proposed in Nandy et al. (2023) uses a swarm artificial neural network to predict heart diseases with an accuracy of 91.6%, precision of 90.9%, recall of 90.1% and F1 score of 90.4%. The researchers in Ullah et al. (2022) have proposed a fully connected quantum convolutional neural network to classify individuals with and without heart disease and have achieved an accuracy of 95.3%. Quantum K-means clustering was suggested as a quantum machine learning algorithm for heart disease classification in Kavitha & Kaulgud (2023), and the outcomes were found to be 93.5% accurate and precise.

The SVM model was optimized by a quantum-based PSO algorithm in Tharwat & Hassanien (2019) to produce a better accuracy of 97.6%, which is higher than the DBN model used separately and optimized by the SOA model. However, it was clear that the QISOA-DBN model produced the highest accuracy of 98.6% in identifying and categorizing individuals with cardiovascular issues. The proposed model’s precision, recall and F1-scores were also higher, with values such as 97.6%, 96.8% and 97.1%, respectively. The performance comparison existing and proposed methods is shown in Fig. 9.

Limitations of present research

The major limitation of the proposed research is that it can be challenging to deploy resource-intensive models for real-time prediction, particularly in settings with limited resources. Also, elevated computational capabilities result in amplified energy usage, which may constrain extensive or instantaneous applications. In addition to this, DBNs can grow quite complex and prone to overfitting, especially when optimized for high performance. This is the case if the training data is insufficiently vast or heterogeneous. Depending on the particular disease under investigation, medical data can differ substantially. The patterns and aspects associated with cardiovascular diseases may not be appropriate for diseases like cancer or neurological problems. To obtain comparable performance, the QISOA-DBN model could, therefore, require significant retraining and feature modification. Highly complicated data types, including multi-omics data and histopathology images, are frequently used in cancer diagnosis. Compared to specialized models created especially for multi-modal medical data, the QISOA-DBN technique may find it challenging to integrate and handle these various data kinds, resulting in inadequate performance. Genomic and medical imaging data can include extremely high-dimensional characteristics. Quantum-inspired optimization may not scale well in such cases, resulting in slower processing times and higher computing costs, even though DBNs can manage such data to some extent.