An efficient approach on risk factor prediction related to cardiovascular disease around Kumbakonam, Tamil Nadu, India, using unsupervised machine learning techniques

Abstract

Nowadays, human beings suffer from varieties of diseases due to the environmental circumstances and their residing habits. Cardiovascular diseases (CVD) are the leading cause of mortality among all diseases. CVDs are heart-related diseases. In early days, the lack of technological advancements resulted in the loss of many human lives. That is, delay in diagnosis resulted in delay in treatments, which obviously becomes the reason for loss of human lives. Hence, the prediction of diseases in advance becomes an inevitability that subsequently supports in providing the necessary treatments. Thus, the present paper deals with the risk factor prediction based on unsupervised learning methods and also identifying the predominant parameters that are vital to risk factors by using principal component analysis. In this article, we have collected the patient data of size 130 × 12 from four different laboratories in and around Kumbakonam, Tamil Nadu, and India. Here, various clustering techniques like k-means clustering, partition around medoids (PAM) clustering, hierarchical clustering, and fuzzy clustering have been applied to the patient data, and the results show that data can be taken in clusters of “patients with risk” and “patients with no risk”. The optimal number of clusters is determined using elbow and silhouette methods. The efficiency of the clustering is evaluated using the Hopkins statistic, Dunn’s index, and average Silhouette widths. The agglomerative coefficients computed indicate that there is a strong cluster structure in the dataset. The stability of the clusters is tested using bootstrapping cluster analysis, and the result showed that the clusters are highly stable. We have applied feature selection using principal component analysis. Also, on applying PCA, out of 12 parameters, it is inferred that Total Cholesterol is the highly correlated factor which plays an important role in the identification of risk factors among patients.

Introduction

Human lives are a great boon out of all assets. There exist so many threats to human lives like food scarcity, climatic changes, accidents, social and economic problems, war and crimes, diseases, etc. Out of these factors, some threats cannot be prevented by humans. For example, threats created by natural calamities are out of the scope of humans which cannot be prevented or controlled. Out of all the factors, the major cause of threat to human lives is because of several types of diseases. Diseases can be categorized into two forms, namely communicable and non-communicable diseases.

Infectious diseases otherwise called communicable diseases get easily spread to a large number of people. This spread occurs within a short span of time, and the duration is even less than two weeks. Hence, preventing the spread becomes necessary, which otherwise would lead to unfortunate consequences. In order to stop the spread at an early stage, prediction of the cause of the same is important. This is useful in understanding how the disease should be treated further. Also, the prediction will be useful to plan and allocate resources that help in deciding what treatment should be followed as well as what preventive measures should be undertaken to control the spread of the disease. It is very common that every year, people suffer from various types of influenza. Moreover, the nature of each influenza is quite different. This nature leads to difficulty in identifying the symptoms of the same in advance so that the influenza can be predicted in advance. The duration of the spread, strength of the spread, rate of spread of the influenza, and the geographical locations where the influenza originates vary differently during each year. Hence, with the help of the historical data and various statistical models available, samples related to influenza are collected and explored precisely.

The major threat among NCDs is CVD. In 2021, 20.5 million people died from a cardiovascular condition. Even undergoing regular checkups or taking medications, the CVDs are still a major threat to human lives. Also, there may be so many reasons for a person to get affected by the same and this may be because of environmental and behavioral factors. The decline in death rates for CVDs has been much faster in High-Income countries (HICs) compared to low and middle-income countries (LMICs), where more than 80% of CVD deaths occur globally. A person may be affected by CVDs based on several factors like behavioral, environmental etc. Humans suffer from premature deaths because of CVDs. It is said that 80% of the premature deaths can be avoided if proper diagnosis and treatments are provided. Hence, to provide proper treatments to patients, it is necessary to identify the important risk factors related to CVDs.Therefore, in the current paper, we identity various patterns related to CVD dataset based on patients with risk and patient with no risk using various clustering algorithms and also PCA is applied to identify highly correlated factor in risk factor prediction.

The major contribution of the paper are as follows.

• Dataset collection

• Data Pre-processing

• Feature selection using PCA

• Implementing various clustering algorithms to identify the hidden patterns in the dataset

• Performance evaluation

Related works

Cardiovascular disease, otherwise called heart disorders, is the basic cause of mortality globally. Therefore, attention is focused on identifying the risk factors related to heart disorders at an early stage. Ripan2021¹ presented a data-driven prediction model that states that heart-related risk factors play an important role in diagnosing heart disease in advance. Here, the authors used k-means clustering for anomalies detection by identifying optimal value of k using silhouette method. Further, several research studies are available on early forecasting of heart disorders. Ripan et al.² used an unsupervised algorithm called K-means clustering on healthcare data and predicted the heart disease efficiently, which detects the irregularities called outliers. Ayon et al.³ conducted relative research to predict coronary artery heart disease. Multiple data mining techniques like Support Vector Machine (SVM), Logistic Regression (LR), Decision Tree (DT), Naive Bayes (NB), K-Nearest Neighbor (KNN), Random Forest (RF), and Deep Neural Network (DNN) techniques were applied to predict coronary artery heart disease. A new method was proposed by Mohan et al.⁴.The authors proposed a hybrid model combining random forest with linear method (HRFLM). The model’s novelty lies in identifying important features, which greatly improves the prediction accuracy. Hence, a new idea about the prediction model is introduced, which is based on the feature selection method, and the performance of the same is analyzed using various classifiers. A probabilistic neural network approach was used by Dessai et al.⁵ to predict heart disease efficiently. But the probabilistic neural network approach did not reflect the influence on anomaly cases when the model is evaluated.

A new hybrid approach was presented by Fatma Zahra Abdeldjouad et al. for the efficient prediction of cardiovascular disease (CVD). The approach used different machine learning techniques like Adaptive Boosting (AdaBoostM1), Logistic Regression (LR), Fuzzy Unordered Rule Induction (FURIA), Multi-Objective Evolutionary Fuzzy Classifier (MOEFC), Genetic-Based Machine Learning (FH-GBML), and Genetic Fuzzy System-LogitBoost (GFS-LB)⁶. The prediction accuracies of various classification models were compared, and the model with the highest accuracy was taken as the best model for efficient cardiovascular disease risk prediction. The most significant features related to heart disorders were presented by Devansh Shah et al.⁷, who used various supervised algorithms such as decision tree, K-nearest neighbor, random forest, and Naïve Bayes. The authors used the Cleveland database from the UCI repository, which consists of records of heart disease patients, and the dimension of the dataset is 303 × 76 (303 instances with 76 features). The subset of features is considered for efficient prediction, and this subset contains only 14 attributes out of 76. Out of various classifiers, K-nearest neighbor resulted in the highest prediction accuracy . Further, the Cleveland heart disease dataset was used by Kavitha et al., on which various data mining techniques such as regression and classification were applied. A hybrid model consisting of a decision tree and random forest was applied, and the model yielded 88.7% accuracy. A GUI was developed by them that takes the user’s input parameter and predicts the heart disease⁸.Shahadat Uddin et al.⁹analyzed the application of machine learning techniques in various disease predictions and it has been observed that the random forest methods resulted in highest accuracy in most of the studies. PCA is extensively used in finding eco-environmental geographical locations using remote sensing ecological index (RSEI)¹⁰. Amitha et al. used k-means and Artificial neural network for heart disease prediction to improve the efficiency of the disease prediction¹¹.

In 2007, Principal Component Analysis (PCA) and Artificial Immune Recognition System (AIRS) were used to predict hepatitis disease. The techniques yielded a classification accuracy of 94.12% using 10-fold cross-validation. This accuracy stood better than other classification applications in the literature and was presented by Polat¹². For certain clinical results, it is very important to identify the risk factors, and this is accomplished by the PCA–logistic regression model. This result promoted clinical research for other diseases. Huang¹³ used this research to identify the disease early and also to take effective decisions about whether to continue the current treatment or to take any preventive measures. The impact of complex cytokines on interacting with systemic lupus erythematosus (SLE) was described by principal component analysis in the literature. The increase in the levels of disease activity as there are changes in B cell–activating factor (BAFF) and TGF-$beta$1 levels was not considered as significant factors in PC1¹⁴. The integration of co-clustering and supervised learning methods was highlighted by Wang et al. to show the incomplete clinical data mining. Data-driven approaches were promoted by Wang¹⁵ to examine the predictors as well as the efficient algorithms that help in effective decision-making in the healthcare system. Abdul Razzaque et al.¹⁶ used Principal Component Analysis to extract the features for medical data classification to detect leukemia, colon tumors, and prostate cancer. The MPSO (modified particle swarm optimization) technique is used at the second stage to pick features from high-dimensional microarray medical datasets like prostate cancer, leukemia, and colon tumours.

Even though several works exist related to CVD diagnosis, some of the issues were not addressed clearly. For example, many works were involved using k-means clustering to detect anomalies, but continuous data were not considered in clustering. Also, in the literature we need labelled data for disease prediction. Hence, we intended to compare the classification of various patterns for heart disease risk factor prediction purely based on unsupervised machine learning algorithms with real data.

Methodology

The research aims to find the efficiency of various clustering algorithms like k-means clustering, PAM clustering, hierarchical clustering, and fuzzy clustering for the real data collected in and around Kumbakonam, Tamil Nadu, India. The application of various clustering algorithms is important to identify the risk patterns that could be used in the healthcare system for further diagnosis. Also, PCA is used for feature selection, which is used to determine the dominant parameters in risk prediction. Figure 1 presents the overall architecture diagram.

Fig. 1.

Architecture of the proposed work.

Dataset

The current work is to study the forecast of risk feature of patients using the unsupervised machine learning techniques (MLT) using R software. The patient information was obtained from the renowned labs (Vaishnavi Clinical Lab, Vignesh Clinical Lab, Karthik Lab, Keerthi X-Ray) in the city of Kumbakonam, Tamil Nadu, India). The dataset contains the features, namely, Gender (G), Blood Haemoglobin (BH), Sugar Fasting (SF), Sugar PP (SPP), Blood Urea (U), Serum Creatinine (SC), Total Cholesterol (TC), H.D.L. Cholesterol (HDLC), L.D.L. Cholesterol (LDLC), V.L.D.L. Cholesterol (VLDLC), Triglycerides (T), and Risk Factor-I (RF) of 130 patients.The study area map is given below in Fig. 2.

Fig. 2.

Location of Kumbakonam, Tamil Nadu, India.

K-means clustering

One of the prevalent and modest unsupervised machine learning procedures is K-means clustering¹⁷. It takes the centroids to be equal to k, which is identified by the algorithm. Also, the algorithm keeps the centroids to be very small, which then assigns the data points to the nearest cluster. Each group is formed by randomly selecting the centroids, which are the starting points for every cluster, and it calculates iteratively in order to boost the centroid positions. Data points are allotted to a cluster grounded on the minimum remoteness taken amid the sum of squares of two points and the centroid of the cluster. If the variation is very small, the data points are more identical within the same cluster. However, its efficiency is usually not as good as those of the other sophisticated clustering methods because a minor variation in the data could lead to a high discrepancy.

PAM Clustering

The expansion for PAM¹⁸ is “Partition around medoids.” The intention behind the algorithm is to look for a line of objects called medoids that are located centrally. The set of objects that are selected is taken as O, and U = O – S represents the set of unselected objects. The neutral of the algorithm is to reduce the mean divergence of objects to the objects in close proximity. Equivalently, it is possible to lower the total differences between objects and their nearest selected objects. On comparing PAM with k-means, it is less subtle to outliers. For large datasets, the extension of PAM called the CLARA algorithm (Clustering Large Applications) is adapted.

Hierarchical clustering

In hierarchical clustering¹⁹, objects that are similar are clustered into groups. Either the distance matrix or the raw data is used to do hierarchical clustering. If the distance matrix is not provided, the algorithm calculates the same from the raw data. The algorithm works as follows.

1. The first step involves the detection of the two clusters that are closest together.

2. In the second stage, the two most analogous groups are fused, and it is continued till all the clusters are combined.

The dendrogram, which is the important output, shows the categorized relationship between clusters. The dimension of the straight line between two groups indicates the distance between the same. This distance is usually Euclidean distance. Once the distance metric is finalized, it is important to identify from where the distance is calculated, which could be either between two similar parts of a cluster or the two least similar of a cluster (complete-linkage) or the center of the clusters (mean or average-linkage). Some other criteria could also be used. This is called linkage, and many linkage criteria have been developed.

Theoretically there is no justification on the type of linkage used; Ward’s method is used practically. In this method, the decision on which samples are to be grouped is taken based on minimizing the sum of squared distances of each sample from the average sample in a cluster. Regarding the types of hierarchical clustering, if the simple clusters are merged, then it is called agglomerative clustering, whereas another type, divisive clustering, involves successively splitting into clusters from the initial large cluster, which includes all the observations as a whole. Divisive clustering is rarely done in practice.

Fuzzy clustering

Fuzzy clustering methods²⁰were established by Dunn in 1973 and upgraded by J.C. Bezdek²¹ in 1981, discovering uncertain dividers where observations can be softly allotted to more than one group. The language R contains a package named “fclust,” which is used to implement fuzzy clustering. It not only outfits the broadly used fuzzy k-means (FkM) procedure but also has many Fuzzy k-Means deviations. Fuzzy cluster likeness measures, cluster legitimacy indices, and cluster picturing tools are also presented. Furthermore, novel fuzzy relational gathering procedures for dividing qualitative/mixed data are provided together with an enhanced form of the so-called Gustafson-Kessel procedure to circumvent individuality in the cluster covariance matrices. Lastly, it is now conceivable to routinely select the number of groups by means of the available fuzzy cluster validity indices. Fuzzy clustering is implemented as making items as members belonging to many groups. Every element has a set of associated constants agreeing to the degree of being in a given group. The efficient fuzzy clustering algorithm is the fuzzy c-means method.

Principal component analysis

It is often complex to infer the output of a large-sized data set, which has been common today. Hence, to diminish the dimensions of the dataset, the principal component analysis (PCA) technique is used. It increases the interpretations of the output but at the same time with the least loss in statistics²². It is realized by making new uncorrelated variables that maximize the variance. The process of discovering novel variables that are not a priori makes PCA an adaptive data analysis method that makes utilization of solving eigenvalue/eigenvector problems. On the other hand, it is tailored to dissimilar data structures and data types. This article is initiated by presenting the elementary thoughts of PCA, conferring what it can and cannot do. It will then define some variants of PCA and their application.

Results and discussion

Cluster analysis

As the first step, the Correlation coefficient matrix is evaluated. The following are the observations made.

• Sugar PP is positively correlated with Sugar Fasting with correlation coefficient as 0.847.

• The Creatinine and Urea are correlated (its correlation coefficient is 0.625).

• L.D.L. cholesterol is positively correlated with Total Cholesterol with correlation coefficient as 0.983.

• The V.L.D.L. cholesterol and Triglycerides are correlated (its correlation coefficient is 0.998).

Hence, for further analysis creatinine, Sugar PP, L.D.L. cholesterol and V.L.D.L. cholesterol are not considered.

The sample data related to BH, SF, U, TC, TGC, HDLC, G and RFI of 130 patients is presented in Table 1.

Table 1.

Sample data.

Sl. No.	BH (mg/dl)	SF (mg/dl)	U (mg/dl)	TC (mg/dl)	T (mg/dl)		HDLC (mg/dl)	Gender	RF
1	14	172	48	209	156		61	Male	No
2	11.6	104	27	171	136		54	Female	Yes
3	13.6	110	21	240	125		37	Male	Yes
4	11.5	80	19	175	86		49	Female	No
5	6	110	17	122	87		51	Male	Yes
6	11	228	23	191	117		53	Female	No
7	11.5	107	22	140	183		38	Male	No
8	12	206	28	261	288		32	Female	Yes
9	8	79	37	229	169		50	Female	No
10	11	130	30	140	64		37	Female	No

Identification of pertinent patient groups based on a similarity factor relating to the test parameters present in patient data was the first step of the analytical approach. In this case, we used cluster analysis theories and methods. In this study, the best number of natural clusters within the data set was estimated using the Silhouette approach²³. Principal component analysis was also used to forecast risk factors.

This section presents our study’s conclusions based on the data collected using clustering techniques and principal component analysis. Now, the following sections summarize and illustrate the experimental data and their conclusions.

Now for the remaining parameters, a correlation matrix is computed. From the correlation matrix,

it is known that there is a maximum positive correlation between triglycerides and total cholesterol with the correlation coefficient 0.35. As triglycerides increase, there is a possibility of an increase in total cholesterol and vice versa. But gender and blood hemoglobin are negatively correlated with the correlation coefficient − 0.48.

Next, the interquartile range statistic (IQR) is used to check whether the data set contains outliers. The equation for checking interquartile range is IQR = Q3 - Q1. According to Tukey’s rule, the outliers are values more than 1.5 times the interquartile range from the quartiles: either below Q1–1.5 (IQR) or above Q3 + 1.5 (IQR). It seems that there exist outliers in the particular features “Blood Haemoglobin,” “Total Cholesterol,” “Triglycerides, “Sugar Fasting,” and “Urea” which are 6, 5, 4, 2, and 2, respectively, whereas there are no outliers detected in H.D.L. Cholesterol and Gender. To check the cluster ability of the data set, the Hopkins statistic is executed. It is observed that the Hopkins statistic value of the data set is 0.5796. Hence, the data set can be taken as clusters.

Subsequently, the three flat clustering algorithms, namely k-means, PAM, and CLARA, hierarchical clustering, and fuzzy clustering techniques are applied to the dataset. In order to find the count of clusters, the silhouette statistic is applied to the three flat clustering algorithms. On applying the statistic, it is observed that the number of clusters obtained is 2 for k-means, PAM, CLARA, hierarchical, and fuzzy clustering. The optimal number of clusters is emphasized by the dashed line against the cluster numbers, respectively, which is shown in Fig. 3.

Fig. 3.

Optimal number of clusters plot.

Flat clustering (k-means)

The basic concept of k-means is that it takes cluster centers (means) to represent groups. Its objective is to reduce the square error of the intra-class distinction. It means that the algorithm aims to the situation in which clusters are consistent and different from each other. Under both Pearson’s coefficient and the Euclidean distance metric (optimal size k = 2), we have the same result for cluster size, average silhouette width, and efficiency, which are shown in Table 2.

Table 2.

K-means cluster size, average silhouette width and efficiency.

Cluster Number	Cluster Size	Average Silhouette Width	Risk Factor-I
			No	Yes
1	48	0.16	24	24
2	82	0.23	41	41

Figures 4 and 5 show the implementation of Pearson’s coefficient and Euclidean distance metric in k-means clustering.

Fig. 4.

K-means clustering with Pearson’s Correlation.

Fig. 5.

K-means clustering with Euclidean Distance.

Summarizing the above results, it occurs that there exist no difference between Pearson’s correlation and Euclidean distance. Silhouette statistic for both metrics is equal to 0.20. This indicates that clusters are not well separated.

Regarding accuracy, both methods seem to wrongly assign labels of No and Yes.

Flat Clustering (PAM)

PAM stands for Partitioning around Medoids. The algorithm finds clusters that have minimum average dissimilarity between objects that belong to the same cluster. Regarding PAM clustering, under both Pearson’s coefficient and Euclidean distance metric (optimal size k = 2), we have the same result for cluster size, average silhouette width of 0.2, and efficiency, which are shown in Table 3.

Table 3.

PAM Cluster size, average silhouette width and efficiency.

Cluster Number	Size	Average Silhouette Width	Risk Factor
			No	Yes
1	50	0.14	24	26
2	80	0.24	41	39

Figures 6 and 7 show the implementation of Pearson’s coefficient and Euclidean distance metric in PAM clustering.

Fig. 6.

PAM Clustering with Pearson’s Correlation.

Fig. 7.

PAM Clustering with Euclidean Distance.

Summarizing the above results, it is clearly shown that there exists no change between Pearson’s correlation and Euclidean distance. For both the metric, the Silhouette statistic is 0.20. Regarding accuracy, both methods seem to wrongly assign labels of No and Yes. In addition to testing the efficiency, the validity of the clusters also involves testing how stable the clusters are.

The stability of clusters is the circumstance in which clusters remain stable when the dataset is changed by adding noise to the data or making subsamples. Bootstrap is the method to test the cluster stability. The cluster stability values for the patient data with k = 2 are 0.9223 and 0.8860, which specify the cluster stability. The values equal to 1 indicate that the clusters are perfectly stable. The nine gained clusters are extremely stable, with the cluster-wise means of the boot result equal to 0.9223 and 0.8860.

The number of times each cluster was dissolved was given by the bootbrd component. No cluster was dissolved for out of 100 bootstrap iterations, and for our dataset, it is 2, 1, which means out of 100 iterations, only 2, 1 clusters were dissolved in clusters 1 and 2, respectively. The boot recover vector is 74, 88, which means that out of 100 bootstrap iterations, clusters 1 and 2 are recovered successfully with the success percentage equal to 74% and 88%, respectively.

Agglomerative hierarchical clustering

Agglomerative hierarchical clustering works in a bottom-up approach. A number of different cluster agglomeration methods (i.e., linkage methods) have been developed to identify the dissimilarity between two clusters of observations. They are “single” ,“complete”, “average” and “ward.D2”. Under the Euclidean distance metric (optimal size k = 10) with the methods “single”, “complete”, “average” and "ward.D2", we have obtained the efficiencies that are shown in Table 4.

Table 4.

Hierarchical clustering efficiency.

Cluster Number	Risk Factor-I Single		Risk Factor-I Complete		Risk Factor-I Average		Risk Factor-I Ward.D2
	No	Yes	No	Yes	No	Yes	No	Yes
1	64	64	10	17	7	15	6	16
2	1	1	55	48	58	50	59	49

The algorithm created two clusters with the sample counts equal to 22 and 108. The within-cluster sum of squares is equal to 840.4964. The average silhouette width is 0.2518. The average silhouette widths for each cluster are 0, –0.1037, and 0.3242. Further, the minimum separation and maximum diameter are 1.0713 and 8.5824, respectively. Dunn’s Index is equal to 0.12, and the agglomerative coefficient is equal to 0.95 and this value indicate the strong clustering nature of our dataset.

Divisive hierarchical clustering

In contrast to agglomerative clustering, divisive works in top-down manner. Now, the only parameter within the function that has to be set is the number of clusters, and here it is k = 2; the efficiency is shown in Table 5.

Table 5.

Fuzzy clustering efficiency.

Cluster	Risk Factor-I
	No	Yes
1	7	14
2	58	51

The algorithm created two clusters with the count of observations equal to 21 and 109. The within-cluster sum of squares is equal to 837.5834. The average silhouette width is 0.2591. The average silhouette widths for each cluster are − 0.0907 and 0.3265 respectively. Dunn Index is equal to 0.19, and the agglomerative coefficient is equal to 0.85. The cluster stability for the patient data are 0.5936 and 0.6924 for the value of k = 2. The above vector indicates the values that are more than 0.6 which indicates the clusters are stable. The bootstrap component indicates how many times each cluster was dissolved. It means that out of 100 bootstrap iterations, no cluster was ever dissolved, and for our data set, it is 49 and 5, which means out of 100 iterations, only 49 and 5 clusters were dissolved in the clusters 1 to 2.The boot recover vector is 27, which means that out of 100 bootstrap iterations, no clusters are recovered successfully.

Fuzzy clustering

On applying the fuzzy clustering, the algorithm produces the ideal number of clusters to be two clusters. To check the accuracy of the algorithm, the algorithm is executed on the assumption that the number of clusters is 2. The membership exponent controls the fuzziness between clusters in determining how much a data point can belong to multiple clusters simultaneously. Three levels of membership exponent (1.5, 1.7, and 2) are set in order to spot the difference between the outputs and decide which one to choose. Here, the Euclidean distance metric is used. Tables 6 and 7 represent the output, and they are visualized in Figs. 8 and 9.

Table 6.

Fuzzy clustering efficiency with k = 2.

Cluster Number	k = 2
	Membership Exponent
	1.5		1.7		2
	Cluster Size	Average Silhouette Width	Cluster Size	Average Silhouette Width	Cluster Size	Average Silhouette Width
1	63	0.06	62	0.07	61	0.07
2	67	0.28	68	0.27	69	0.27
2	67	0.28	68	0.27	69	0.27

Table 7.

Fuzzy clustering efficiency with k = 3.

Cluster Number	K = 3
	Membership Exponent
	1.5		1.7		2
	Cluster Size	Average Silhouette Width	Cluster Size	Average Silhouette Width	Cluster Size	Average Silhouette Width
1	63	0.06	63	0.06	62	0.07
2	67	0.28	67	0.28	68	0.27

Fig. 8.

Fuzzy Clustering with k = 2.

Fig. 9.

Fuzzy Clustering with k = 3.

Changes made in the membership exponent and number of clusters resulted in changes of silhouette statistic too. The highest silhouette is seen for two clusters for the membership exponent equal to 1.7 and 2 respectively. Thus, the next step will be testing for clustering accuracy which is given in the Table 8.

Table 8.

Fuzzy clustering efficiency.

Cluster Number	1.7		2
	Risk Factor-I		Risk Factor-I
	No	Yes	No	Yes
1	29	33	28	33
2	36	32	37	32

Principal component analysis

Here, all the eleven parameters are taken into account. The key factor examination can be accomplished on both the covariance matrix S and the correlation matrix R. First, we start with the covariance matrix S. The variables appear to be taken measurements in diverse units that may bring out the features with higher variances, leading to the prime factors of the covariance matrix. Now the covariance matrix S is S =.

In the above table, Hg represents blood hemoglobin, SF represents sugar fasting, SPP stands for sugar PP, SC stands for serum creatinine, and tot Ch. Stands for Total_Cholesterol, Tri stands for triglycerides, HDL stands for H.D.L. cholesterol, LDL stands for L.D.L. cholesterol, and VLDL stands for V.L.D.L. cholesterol.

Now, the total variance is, where k is the order of S. Further, the total variance is also equal to the sum of the eigenvalues of S, and it is 19039.64. Also, the eigenvalues are 8345.5900850, 5980.5908181, 3932.1240141, 542.9497781, 120.3226961, 113.3960246, 2.3157201, 1.3463986, 0.5708996, 0.3014173, and 0.1350137, and the corresponding eigenvectors represent the principal components of S. The eigenvalues of S are used to find the percentage of the total discrepancy explained by the 11 principal components, and they are 0.4383, 0.3141, 0.2065, 0.0285, 0.0063, 0.0060, 0.0001, 0.0001, 0, 0, and 0 respectively.

Hence, the first two principal components account for 75.24% of the total variance, and the first three principal components account for 95.90% of the total variance, whereas the first four principal components account for 98.75% of the total variance. A scree graph (Fig. 10) of the eigenvalues is drawn to picture the percentage of variance explained by each subsequent eigenvalue.

Fig. 10.

The Scree Graph.

The elements of the eigenvectors are the “coefficients” or “loadings” of the principal components. Thus, the five key factors are given by.

PC1 = −0.0001 (Blood Haemoglobin) −0.4608 (Sugar Fasting) −0.6821 (Sugar PP) −0.0008 (Urea) − 0.0002 (Serum Creatinine) −0.2746 (Total Cholesterol)−0.4474 (Triglycerides) + 0.0102 (H.D.L Cholesterol)−0.1966 (L.D.L Cholesterol)−0.0897 (V.L.D.L Cholesterol) + 0.0000 (Gender).

PC2 = 0.0050 (Blood Haemoglobin) −0.2990 (Sugar Fasting) −0.4774 (Sugar PP) −0.0238 (Urea) −0.0018 (Serum Creatinine) + 0.4652 (Total_Cholesterol) + 0.5714 (Triglycerides) −0.0098 (H.D.L Cholesterol) + 0.3548 (L.D.L Cholesterol) + 0.1152 (V.L.D.L Cholesterol)−0.0006 (Gender).

PC3 = −0.0061 (Blood Haemoglobin) −0.0473 (Sugar Fasting) −0.0542 (Sugar PP) + 0.0252 (Urea) + 0.0001 (Serum Creatinine) −0.4877 (Total Cholesterol) + 0.6495(Triglycerides)−0.0550 (H.D.L Cholesterol)−0.5610 (L.D.L Cholesterol) + 0.1294 (V.L.D.L Cholesterol) + 0.0004(Gender).

PC4 = −0.0156 (Blood Haemoglobin) + 0.8341 (Sugar Fasting) − 0.5508 (Sugar PP) −0.0090 (Urea) −0.0041 (Serum Creatinine) −0.0108 (Total_Cholesterol)−0.0057 (Triglycerides) + 0.0069 (H.D.L Cholesterol)−0.0151(L.D.L Cholesterol) + 0.0006 (V.L.D.L Cholesterol) + 0.0043 (Gender).

PC5 = −0.0518 (Blood Haemoglobin) −0.0033 (Sugar Fasting) −0.0194 (Sugar PP) + 0.9926 (Urea) + 0.0385 (Serum Creatinine) + 0.0534(Total_Cholesterol)−0.0065 (Triglycerides) + 0.0838 (H.D.L Cholesterol)−0.0164 (L.D.L Cholesterol)−0.0071 (V.L.D.L Cholesterol) + 0.0003 (Gender).

PC6 = −0.0062 (Blood Haemoglobin) + 0.0064 (Sugar Fasting) −0.0068 (Sugar PP) + 0.0954 (Urea) + 0.0068 (Serum Creatinine) −0.3743 (Total_Cholesterol) + 0.0220(Triglycerides)−0.8128 (H.D.L Cholesterol) + 0.4353(L.D.L Cholesterol) + 0.0007(V.L.D.L Cholesterol)−0.0069 (Gender).

PC7 = 0.9849 (Blood Haemoglobin) + 0.0144 (Sugar Fasting)−0.0078 (Sugar PP) + 0.0530 (Urea) −0.0380 (Serum Creatinine) −0.0088 (Total_Cholesterol) + 0.0014 (Triglycerides) + 0.0040 (H.D.L Cholesterol) + 0.0001(L.D.L Cholesterol) + 0.0005 (V.L.D.L Cholesterol)−0.1593 (Gender).

PC8 = 0.0144 (Blood Haemoglobin) + 0.0010 (Sugar Fasting) + 0.0017 (Sugar PP) + 0.0091 (Urea) + 0.0091 (Serum Creatinine) + 0.4747 (Total_Cholesterol)−0.2010 (Triglycerides) −0.4787 (H.D.L Cholesterol)−0.4778 (L.D.L Cholesterol) + 0.5230 (V.L.D.L Cholesterol) + 0.0467 (Gender).

PC9 = 0.0107 (Blood Haemoglobin) + 0.0019 (Sugar Fasting) + 0.0001 (Sugar PP) + 0.0022 (Urea) −0.0663 (Serum Creatinine) + 0.3161 (Total_Cholesterol) + 0.1017(Triglycerides)−0.3141 (H.D.L Cholesterol)−0.3170 (L.D.L Cholesterol)−0.8265 (V.L.D.L Cholesterol) + 0.0551(Gender).

PC10 = 0.0145 (Blood Haemoglobin) −0.0038 (Sugar Fasting) + 0.0029 (Sugar PP) + 0.0379 (Urea) −0.9403 (Serum Creatinine) −0.0225 (Total Cholesterol)−0.0112 (Triglycerides) + 0.0163 (H.D.L Cholesterol) + 0.0233 (L.D.L Cholesterol) + 0.0724 (V.L.D.L Cholesterol) + 0.3279(Gender).

PC11 = 0.1626 (Blood Haemoglobin) −0.0003 (Sugar Fasting) −0.0002 (Sugar PP) −0.0044 (Urea) + 0.3282 (Serum Creatinine) −0.0384 (Total_Cholesterol) + 0.0085(Triglycerides) + 0.0316(H.D.L Cholesterol) + 0.0384(L.D.L Cholesterol)−0.0027 (V.L.D.L Cholesterol) + 0.9283 (Gender).

The amount of discrepancy described by PC1 to PC11 components are summarized below (Tables 9 and 10).

Table 9.

Significance of components.

Significance of components	PC1	PC2	PC3	PC4	PC5
Standard deviation	91.3542	77.3343	62.7067	23.3013	10.9692
Proportion of Variance	0.4383	0.3141	0.2065	0.0285	0.0063
Cumulative Proportion	0.4383	0.7524	0.9590	0.9875	0.9938

Table 10.

Significance of components.

Significance of components	PC6	PC7	PC8	PC9	PC10	PC11
Standard deviation	10.6488	1.5218	1.1603	0.7556	0.5490	0.3674
Proportion of Variance	0.0060	0.0001	0.0001	0.0000	0.0000	0.0000
Cumulative Proportion	0.9998	0.9999	1.0000	0.99998	1.0000	1.0000

The scatter plot of the principal components is shown in the following Fig. 11.

Fig. 11.

The Scatterplot of Principal Components.

The remarkable characteristics of the data, such as leavings from normality, outliers, or non-linearity, may be communicated by the first and second principal components, which are frequently drawn as a scatter plot. The first dual prime factors are estimated for each observation vector and plotted (Fig. 9). As stated before, though prime factor examination is naturally done on the covariance matrix, it often makes more intuitive sense to apply PCA to the correlation matrix R. The correlation matrix R is R =.

The eigenvalues of R are used to find the proportion of the total variance explained by the components, and they are 0.2499, 0.2080, 0.1547, 0.1286, 0.1153, 0.0657, 0.0339, 0.0312, 0.01233, 0.0002, and 0. Hence, the first four principal components account for 74.12% of the total variance, whereas the first five principal components account for 85.65% of the total variance. Moreover, the last six components have little significance. Thus, one may want to keep the first four components rather than the first two or three with the matrix R.

Thus, the five principal components are given by.

PC1 = 0.2309 (Blood Haemoglobin) + 0.1615 (Sugar Fasting) + 0.1573 (Sugar PP) −0.1694 (Urea) − 0.1766 (Serum Creatinine) + 0.4931 (Total Cholesterol) + 0.4416 (Triglycerides) − 0.0813 (H.D.L Cholesterol) + 0.4272 (L.D.L Cholesterol) + 0.4431 (V.L.D.L Cholesterol) − 0.1110 (Gender).

PC2 = 0.3873 (Blood Haemoglobin) −0.4235 (Sugar Fasting) −0.4168 (Sugar PP) −0.3980 (Urea) −0.3235 (Serum Creatinine) + 0.1306 (Total_Cholesterol) − 0.2526 (Triglycerides) + 0.1896 (H.D.L Cholesterol) + 0.1800 (L.D.L Cholesterol) − 0.2502 (V.L.D.L Cholesterol)−0.1488 (Gender).

PC3 = 0.0458 (Blood Haemoglobin) + 0.3865 (Sugar Fasting) + 0.3872 (Sugar PP) + 0.0500 (Urea) + 0.1324 (Serum Creatinine) + 0.2732 (Total Cholesterol) − 0.3887(Triglycerides) + 0.4331 (H.D.L Cholesterol) + 0.3384 (L.D.L Cholesterol) − 0.3874 (V.L.D.L Cholesterol) + 0.0253(Gender).

PC4 = −0.1548 (Blood Haemoglobin) + 0.2683 (Sugar Fasting) + 0.1706 (Sugar PP) −0.4563 (Urea) −0.5699 (Serum Creatinine) −0.1515 (Total_Cholesterol) + 0.0050 (Triglycerides) + 0.0698 (H.D.L Cholesterol)−0.1834(L.D.L Cholesterol) + 0.0067 (V.L.D.L Cholesterol) + 0.5295 (Gender).

PC5 = 0.4437 (Blood Haemoglobin) + 0.2676 (Sugar Fasting) + 0.3655 (Sugar PP) − 0.2049 (Urea) − 0.0934 (Serum Creatinine) − 0.3103 (Total Cholesterol)−0.0810 (Triglycerides)−0.2761 (H.D.L Cholesterol)−0.2607 (L.D.L Cholesterol)−0.0831 (V.L.D.L Cholesterol)−0.5415 (Gender).

PC6 = 0.2131 (Blood Haemoglobin) − 0.0318 (Sugar Fasting) + 0.0274 (Sugar PP) + 0.1153 (Urea) − 0.0124 (Serum Creatinine) −0.1152 (Total Cholesterol) + 0.2629(Triglycerides) + 0.8033 (H.D.L Cholesterol) − 0.3603(L.D.L Cholesterol) + 0.2624(V.L.D.L Cholesterol)−0.1139 (Gender).

PC7 = 0.6339 (Blood Haemoglobin) −0.0199 (Sugar Fasting) + 0.0312 (Sugar PP) + 0.5560 (Urea) −0.1974 (Serum Creatinine) −0.0323 (Total_Cholesterol) − 0.0398 (Triglycerides) − 0.1632 (H.D.L Cholesterol) + 0.0037(L.D.L Cholesterol) − 0.0412 (V.L.D.L Cholesterol) + 0.4665 (Gender).

PC8 = −0.3331 (Blood Haemoglobin) + 0.1070 (Sugar Fasting) − 0.0959 (Sugar PP) + 0.4867 (Urea) − 0.6773 (Serum Creatinine) + 0.0381 (Total_Cholesterol)−0.0505 (Triglycerides) + 0.0281 (H.D.L Cholesterol) + 0.0610 (L.D.L Cholesterol) − 0.0612 (V.L.D.L Cholesterol)−0.4007 (Gender).

PC9 = −0.1243 (Blood Haemoglobin) − 0.6991 (Sugar Fasting) + 0.6914 (Sugar PP) + 0.0293 (Urea) −0.1236 (Serum Creatinine) + 0.0242 (Total Cholesterol) − 0.0039(Triglycerides)−0.0013 (H.D.L Cholesterol) + 0.0292 (L.D.L Cholesterol)−0.0141 (V.L.D.L Cholesterol) + 0.0050(Gender).

PC10 = 0.0031 (Blood Haemoglobin) + 0.0036 (Sugar Fasting) − 0.0050 (Sugar PP) − 0.0057 (Urea) + 0.0052 (Serum Creatinine) −0.0747 (Total Cholesterol) + 0.7139 (Triglycerides) + 0.0139 (H.D.L Cholesterol) + 0.0693 (L.D.L Cholesterol)−0.6926 (V.L.D.L Cholesterol) + 0.0041(Gender).

PC11 = −0.0024 (Blood Haemoglobin) + 0.0035 (Sugar Fasting) + 0.0014 (Sugar PP) + 0.0010 (Urea) − 0.0012 (Serum Creatinine) + 0.7239 (Total Cholesterol) − 0.0257(Triglycerides) − 0.1255(H.D.L Cholesterol) − 0.6556(L.D.L Cholesterol)−0.1727 (V.L.D.L Cholesterol)−0.0043 (Gender).

The proportion of variance explained by the components are summarized below (Tables 11 and 12).

Table 11.

Significance of components.

Significance of components	PC1	PC2	PC3	PC4	PC5
Standard deviation	1.6581	1.513	1.3044	1.1896	1.1260
Proportion of Variance	0.2499	0.208	0.1547	0.1286	0.1153
Cumulative Proportion	0.2499	0.458	0.6127	0.7413	0.8566

Table 12.

Significance of components.

Significance of components	PC6	PC7	PC8	PC9	PC10	PC11
Standard deviation	0.85032	0.61084	0.58617	0.36828	0.04438	0.02265
Proportion of Variance	0.06573	0.03392	0.03124	0.01233	0.00018	0.00005
Cumulative Proportion	0.92229	0.95621	0.98744	0.99977	0.99995	1.00000

The first two principal components are assessed for each observation vector and plotted (Fig. 12).

Fig. 12.

The Scatterplot of Principal Components.

One way of understanding the prime factors is to compute the correspondence between the real data and the component. First, we compute the associations amid the data and the computed factors of the covariance matrix S, and it is represented in Table 13.

Table 13.

Correlation table of PC for covariance matrix S.

Features	PC1	PC2	PC3
Blood Haemoglobin	0.005336622	−0.22407781	0.220187545
Sugar Fasting	0.811131078	0.44555827	0.057127764
Sugar PP	0.846252149	0.50133433	0.046124459
Urea	0.063334955	0.16389894	−0.140926440
Serum Creatinine	0.032077526	0.19642375	−0.008044836
Total Cholesterol	0.467859570	−0.67088100	0.570292753
Triglycerides	0.562346269	−0.60800779	−0.560416795
H.D.L Cholesterol	−0.098375553	0.07980492	0.364839872
L.D.L Cholesterol	0.371657434	−0.56789953	0.727966078
V.L.D.L Cholesterol	0.561439442	−0.61007995	−0.555871079
Gender	−0.004072955	0.09291156	−0.057337911

Now, the principal components can be deduced based on which variables they are most associated with in either a confident or adverse way. The first component is positively correlated with all the factors (maximum of 0.846252149 for Sugar PP) other than H.D.L. Cholesterol and Gender which could be considered as important factor. Further, the first component is negatively interrelated with H.D.L. cholesterol and gender. This association proposes these two variables differ together in opposite directions, and when one goes down, the other increases as well.

The next factor is best linked with blood haemoglobin, total cholesterol, triglycerides, L.D.L. cholesterol, and V.L.D.L. cholesterol (negatively correlated) and others with positive correlation. Sugar PP is associated with the subsequent factor in a confident way. The tertiary factor is mainly associated with L.D.L. cholesterol (correlation with 0.7280) and not much else. This factor could be observed as a degree of the L.D.L. cholesterol of the risk factor-I as “yes” or “no”. Now, we calculate the associations amid the data and the calculated factors of the correlation matrix R, and it is represented in Table 14.

Table 14.

Correlation table of PC for correlation matrix S.

Features	PC1	PC2	PC3
Blood Haemoglobin	0.3828588	0.5859225	0.05971563
Sugar Fasting	0.2677877	−0.6405980	0.50411769
Sugar PP	0.2607719	−0.6304329	0.50511574
Urea	−0.2809157	−0.6020384	0.06521248
Serum Creatinine	−0.2928303	−0.4893157	0.17275769
Total Cholesterol	0.8176796	0.1974885	0.35638688
Triglycerides	0.7322060	−0.3821836	−0.50696472
H.D.L Cholesterol	−0.1348354	0.2867844	0.56493427
L.D.L Cholesterol	0.7082968	0.2723171	0.44145224
V.L.D.L Cholesterol	0.7347949	−0.3784557	−0.50532444
Gender	−0.1841336	−0.2250481	0.03299325

Now, the chief components can then be inferred based on which variable quantities they are best linked to in either a confident or adverse way. The first component is positively correlated with total cholesterol (correlation coefficient 0.8176796), which could be measured as a principal parameter. Further, the first component is negatively associated with urea, serum creatinine, H.D.L. cholesterol, and gender. This association proposes these four variables differ together in opposite directions, and when one goes down, the others increase as well. The subsequent factor is best related to blood hemoglobin (positively with 0.5859225) and sugar fasting (negatively with − 0.6405980). The third component is most interrelated with triglycerides (negatively with − 0.50696472) and VLDL cholesterol (negatively with − 0.50532444). H.D.L. cholesterol is associated with the third component in a positive way (0.56493427).

Conclusion

The current work identified the clustering nature of the real data related to CVDs collected from different laboratories. On applying various clustering algorithms, we observed that both k-means and PAM clustering yielded the same cluster size, efficiency, and average silhouette widths for both Pearson and Euclidean distance metrics, provided the optimal number of clusters for the dataset is 2. The efficiency of clustering algorithms depends on various available metrics. Hence, we applied the Hopkins statistic to check the cluster tendency, and the statistic result showed that the data can be taken in clusters. Also, various statistical measures like Dunn’s index and agglomerative coefficients indicate that there exists a strong clustering structure for the dataset. The stability of the clusters is tested using bootstrapping techniques, and the bootstrap values obtained are 0.9223 and 0.8860, respectively. The values that are close to one indicate that clusters are highly stable on resampling. On applying fuzzy clustering, the highest silhouette values are obtained for membership exponents equal to 1.7 and 2. Also, we applied the PCA technique for feature selection, and it is inferred that total cholesterol is highly positively correlated with a correlation coefficient of 0.8176796, and this value indicates that total cholesterol plays a major role in predicting risks. In the future, model-based clustering methods, artificial neural networks, and structural equation modeling techniques could be applied for risk prediction. Moreover, deep learning techniques may also be considered further.

Author contributions

Dr.K.Kannan and A.Menaga have equally contributed to this article. All authors reviewed the manuscript.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Data availability

The data that support the findings of this study are not openly available due to reasons of sensitivity and are available from the corresponding author upon reasonable request.

Declarations

Competing interests

The authors declare no competing interests.

Research involving human participants and/or animals

All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional research committee.