An Efficient K-means Clustering Algorithm for Analysing COVID-19

Many machine learning algorithms, including both supervised and unsupervised methods have been applied to the COVID-19 dataset [15]. Each attribute of the dataset must contain numerical values for approaching the k-means algorithm [5]. If not then, some pre-processing might be required for handling missing values and non-numeric values. Before going to further methodology, it needs to be ensured that each attributes values in numeric.

For creating our model, we have used a few datasets for selecting the features, required for analyzing the health care quality of the countries. The selected datasets are owid-covid-data [19], covid-19-testing-policy [20], public-events-covid [20], covid-containment-and-health-index [20], inform-covid-indicators[21]. It is worth mentioning that we used the data up to 11^st August 2020. For instance, some of the attributes of the owid-covid-data [19] are shown in Table 1.

Covid-19-testing-policy [20] dataset contains the categorical values of the testing policy of the countries shown in table 2.

Other datasets also contains such types of features which is required for ensuring the health care quality of a country. These real-world datasets help us to analyze our proposed method for real-world scenarios.

Principal component analysis (PCA) is a method, that utilizes an orthogonal transformation to translate a set of observations of potentially correlated variables into a set of values of linearly uncorrelated variables that are called principal components. PCA is widely used in data analysis and for making a predictive model considering dimensionality reduction [22] [18], that has been taken into account in our approach.

On the other hand, the percentile method [23] used in our model divides the whole dataset into 100 different parts. Each part contains 1 percent data of the total dataset. For example, the 25th percentile means this part contains 25 percent data of the total dataset. That implies, using the percentile method, we can split our dataset into different distributions according to our given values. The percentile formula is given below [23]:

Here, $\rho$ = The Percentile wants to find, $\eta$ = Total Number of values, R = Percentile at $\rho$.

After splitting the reduced dimensional dataset through percentile, we then extract the split data from the primary dataset by indexing for each percentile. In this process, we get back the original data. After retrieving the original data for each percentile, we have calculated the mean of each attribute. These means from the split dataset is the initial centroids of our proposed method.

After splitting the dataset and calculating the mean according to the subsection 3.3, we have selected the means of each split dataset as a centroid. These centroids are taken into account as initial centroids for the efficient k-means clustering algorithm.

At the last step, we have executed our modified k-means algorithm until the centroids converge. Passing our proposed centroids instead of random or kmeans++ centroids through the k-means algorithm we have generated the final clusters [24]. The proposed method always considers the same centroids for each test. The pseudocode of our whole proposed methodology is given in the algorithm 1. In the next section, evaluation and experimental results are discussed.

We have merged the datasets described in subsection 3.1 according to the country name. With the regular expression, some pre-processing and data cleaning have been conducted in the case of merging the data. We have also handled some missing data consciously. There are so many attributes regarding COVID-19, among them 25 attributes had been selected finally, as these attributes closely signify the health care quality of a country. The attributes represent categorical and numerical values. These are country name, cancellation of public events (due to public health awareness), stringency index¹¹1It is one of the matrices used by Oxford COVID-19 Government Response Tracker [25]. It delivers a picture of the country’s enforced strongest measures., testing policy ( category of testing facility available to the mass people), total positive case per million, new cases per million, total death per million, new deaths per million, cardiovascular death rate, hospital beds available per thousand, life expectancy, inform the COVID-19 risk (rate), hazard and exposure dimension rate, people using at least basic sanitation services (rate), inform vulnerability(rate), inform health conditions (rate), inform epidemic vulnerability (rate), mortality rate, prevalence of undernourishment, lack of coping capacity, access to healthcare, physicians density, current health expenditure per capita, maternal mortality ratio. All of the attributes are closely investigated before feeding the model.

As we are going to make clusters for the countries with similar types of healthcare quality, the optimum number of clusters is 4, defined by the elbow method [26].

We have applied Principal Component Analysis (PCA) to convert a high dimensional dataset into two dimensions. The number of clusters is 4 for the COVID-19 dataset. After applying PCA, we have used percentile concepts to divide the whole dataset into K equal parts. For K=4, each portion contains 25% of the dataset. For this purpose, we have calculated the percentile for 25%, 50%, 75%, and 99.9% data. We have divided the data horizontally because it provides a good intuition to the cluster. Figure 2 depicts plotting the data with two dimensional PCA where horizontal lines represents the splitting according to percentile.

As described in subsection 3.3, after splitting the dataset and mean calculation we have got the proposed initial centroids of k-means. Efficient centroids of COVID-19 for 4 centroids are given in table 3. Here only 7 attributes from 25 attributes are shown for demonstration purposes. In table 3, c1,c2,c3 and c4 represent initial clusters centroids consecutively.

In machine learning and data science, computational power is one of the main issues. Because at a time, a computer needs to process a large amount of data. So, reducing computational cost is a big deal. As discussed in the methodology in section 3, we implemented our proposed method. Though many researchers proposed many ideas those are discussed in the related work section. We have compared our method with the best existing kmeans++ method [24]. We have measured the effectiveness of the model with

• $\bullet$

  Number of iterations needed for finding the final clusters

• $\bullet$

  Execution time for reaching out to the final clusters

These two things are compared in the upcoming subsections.

In figure 3, we show the experimental result of the COVID-19 dataset for 10 tests. In the graph, the green and the red line represents results for the proposed and kmeans++ algorithms consecutively. As our centroid is pre-defined, the number of iterations is constant in every test case. On the contrary, kmeans++ method works with random iterations. For getting insight into the clusters, we have plotted the cluster map in Figure 4 showing similar types of countries in terms of health care quality. The 2D plot also shows the cluster distributions in a two-dimensional space. Before jumping to the demonstration, it is worth mentioning that the model have been tested on the Intel^® Core^TM i7-8750H processor.

• $\bullet$

  Figure 3(a) provides the iteration comparison for existing kmeans++ and our proposed method. When our proposed method have been applied, we find a constant number of iterations in each test. But the existing kmeans++ method’s iteration is random. It might be varied for different test cases.

• $\bullet$

  Figure 3(b) provides the execution time comparison for existing kmeans++ and our proposed method. For each test case, our model have executed the k-means clustering algorithm with the shortest time.

Based on the experimental results, we claim that our model outperforms in the case of real-world application and reduces the computational power for the k-mean clustering algorithm.