Introduction

Deep Sequencing is the process of DNA fractioning, which dramatically transformed the genomic research field. The advancement that this process witnessed during the last decade has led to the continuous generation of immense amounts of data, putting the genomic field among the top big data generating fields [1]. Although the captured sequence itself does not express ready to use information, it can be transformed throughout a complex process that deduces the proteins drawn from the sequence. In order to predict the expression of this protein and whether it is cancerous or not, the draft genome is compared to previously known cancer genome sequences [2].

The accumulation of genomic data has raised multiple challenges to produce a logical and coherent picture of the genomic basis of Cancer. Although cancer prognosis is very complicated due to the nature of the genomic datasets that contain thousands of features but relatively fewer samples. Early diagnosis is the key to increase the chances of healing, which makes the process highly crucial. Traditional machine learning techniques fall short in this area since they are used to dealing with datasets that have few features and multiple samples [3] leading to the necessity of novel technologies, hence, big data analytical techniques.

Mining genomic data is a challenging process due to the fact that this type of data meets the criteria, problems and challenges of big data. Big data refers to information with massive volume and high dimensional space [4], it is usually defined through its four characteristics: volume (physical size of data); variety (structure and diversity of data types); velocity (rate at which data is being generated and at which it needs to be processed) and value (the usefulness of analyzing big data). To overcome these challenges, frameworks such as the Hadoop’s implementation of MapReduce [5] were developed, which are designed not only to address these issues but also to work with low commodity hardware [6]. Other prominent big data technologies such as Apache Spark [7] support the same applications and share the same parallelization background as Hadoop, while retaining the scalability and fault tolerance of MapReduce with more flexibility. The different new frameworks offer numerous ecosystems that allow data scientists to conduct several operations among the data analysis process. One of the most important steps of this process, when it comes to genomic big data, is the preprocessing task, more precisely the feature selection step due to the complex nature of these datasets. Feature selection is the process of finding a new subset identifying relevant features in the original dataset and discarding irrelevant and redundant features in order to eventually build models in different analysis tasks [8]. Feature selection algorithms can be categorized into six groups (see Table 1): filters, wrappers, embedded, hybrid, ensemble and integrative, depending on the structuration nature of the selection algorithm and the clustering model building [9].

Table 1 Groups of feature selection algorithms
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Successfully dealing with big and complex data preferably requires parallel processing and cluster computing. Among the numerous solutions for parallel processing, lately, Apache Spark has proved to be more potent than other solutions when dealing with massive data [15]. “Apache Spark is an open source tool that can complete jobs considerably faster than previous Big Data tools, namely Apache Hadoop, by virtue of its in-memory caching, and optimized query execution” [16]. Spark’s capacity to efficiently and expeditiously process colossal datasets, led us to choose it as a platform for our approach.

The need for parallelizing the feature selection process is highly desired [17], which raises multiple issues due to the complexity of the dependencies between the different features [18]. The usage of efficient feature selection algorithms that ensure high accuracy with time optimization is the key to a successful analysis. Therefore, in this study, a three layered hybrid distributed approach using Apache Spark is proposed and the following contributions were attained:

  • A parallelized version of the BPSO algorithm for effective feature selection.

  • A parallelized combination of PSO and k-means algorithm in order to present a relevant clustering.

  • The approach was tested on five benchmark datasets, breast cancer tumor, Colon cancer, Leukemia, Lung cancer, Gene expression cancer RNA-Seq datasets to prove its efficiency and speed.

  • The SDPSO approach provides an average purity and F-measure scores that are significantly higher than four state of the art methods, namely, k-means, Genetic Algorithm (GA), the Density-Based Spatial Clustering of Applications with Noise (DBSCAN) and the hybrid PSO-GA [19].

The rest of this paper is organized as follows. In “Related works” section, a review of previous related works is presented. “Methods” section highlights the mathematical background, the methods and the components of the proposed approach. The detailed description of the proposed approach is explained in “Our proposed approach” section. “Results” section discusses the experimental design, evaluation metrics, configurations of experiments and a description of the used datasets for the purpose of experimentation, followed by the experimental results. The findings and results of the approach are discussed in “Discussion” section, followed by the limitations of the study in “Limitations of the study” section. In “Conclusion, perspectives and future work” section, the paper is concluded with a summary of the findings and future work directions.

Related works

This section is divided into two main subsections, the first one deals with previously proposed Binary Particle Swarm Optimization (BPSO) based feature selection methods while the second sub section reviews a number of Particle Swarm Optimization (PSO) based approaches for clustering problems (see Tables 2 and 3).

Table 2 BPSO based feature selection methods
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Table 3 PSO based clustering methods
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The main idea behind feature selection is choosing subsets of features from an original set. A subset that should necessarily and reasonably represent the original data along with being beneficial for analysis tasks. The feature selection task is centered on the search for an optimal solution in a usually large search space in order to assuage the clustering task. Although immensely critical, a good feature selection algorithm is not sufficient, the later should be supported by an appropriate clustering algorithm. Several researchers have chosen the PSO algorithm for data clustering.

The solutions described above generally focus on one aspect in the analytical process, the proposed approach is a conglomerate one that targets big and complex data through the parallelization of the computation using Apache Spark, as well as combining an integrative feature selection algorithm BPSO with the PSO for clustering. Along with the contribution in enhancing the computational time due to the parallelized implementation, the proposed approach provides the best in average, of purity and F-measure and the lowest entropy when tested with five complex multidimensional datasets compared to four state of the art algorithms.

Methods

Particle Swarm Optimization (PSO) has attracted significant attention as a technique that enhances the feature selection process due to its efficiency in solving optimization problems [30]. In addition for it being an “anytime algorithm” that produces solutions for any given computational time [31], its simple yet effective principle and solid global search capacity, leading to finding the optimal solution in relatively few iterations [28] were among the main motivations behind the choice of the algorithm as the main constituent of our approach. This section provides a description of the background algorithms behind the approach as well as the statistical preliminaries.

Feature selection algorithm

Carefully understanding the dataset along with dimensionality reduction issues before any data analysis process are crucial to the success of the analysis itself [32]. Data preprocessing involves transforming raw data into a format that is suitable for processing. Real-world data is often incomplete, inconsistent, and is likely to contain noise and errors. Data preprocessing is an endorsed method resolving such issues [33]. Feature selection is a preprocessing technique that automatically selects the features which contribute the most either to the clustering process or to the desired output. Having irrelevant features in the dataset can decrease the accuracy of clustering models and forces the clustering algorithm to process based on irrelevant features. Therefore, it is recommended to conduct a feature selection task before training a model. The feature selection algorithm in this work is a PSO-based algorithm termed the Binary PSO algorithm. The algorithm was originally introduced as an optimization technique for real-number spaces and has since then been successfully applied in many areas: function optimization, artificial neural network training, fuzzy system control, and other application problems [34]. Many optimization problems occur in a space featuring discrete, qualitative distinctions between variables and between levels of variables. Kennedy and Eberhart introduced BPSO [35], which can be applied to discrete binary variables.

In BPSO, a particle i is a pair \((X_i, V_i)\) of two n-dimensional vectors \(X_i\) and \(V_i\), where n is the total number of genes in a dataset. \(X_i\) is a binary vector that represents the position of the particle and \(V_i\) is the velocity of the particle. In the PSO algorithm, every solution of a given problem is considered as a particle which is able to move in a search landscape. In a binary space, a particle may move to near corners of a hypercube by flipping various numbers of bits, if a set of genes is selected, a bit takes the value 1, if not, it takes the value 0. An example of the process’ outcomes using an example section of the gene expression data is featured in Tables 4 and 5.

Table 4 Example of gene expression data with particle position
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Table 5 Example of selected gene subset
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The movement of the particle is expressed by the two following vectors, the particle vector, Eq. (1), and the velocity vector, Eq. (2):

$$\begin{aligned} X_i= & {} (x_i^1, x_i^2, \ldots, x_i^n) \end{aligned}$$
(1)
$$\begin{aligned} V_i= & {} (v_i^1, v_i^2, \ldots, v_i^n) \end{aligned}$$
(2)

where \(x_i^d \in {0,1}, i = 1,2,\ldots,m\) (m is the total number of particles), \(d = 1, 2,\ldots, n\) (n is the dimension of data).

To update the position vector of a particle, the movement direction and the speed of that particle are defined as follows:

$$\begin{aligned} v_i^{d+1} = w \times v_i^d + c_1r_1 \times (pbest_i^d - x_i^d) + c_2r_2 \times (gbest_i^d - x_i^d) \end{aligned}$$
(3)

where w is the inertial weight, \(v_i^d\) is the velocity of particle i at dimension d, \(c_1\) and \(c_2\) are acceleration constants, \(r_1\) and \(r_2\) are random values, \(x_i^d\) is the position of particle i at dimension d, \(pbest_i^d\) is the best previous position of the \(i{\text{th}}\) particle, \(gbest_i^d\) is the global best position of all particles.

The next velocity is defined by three components, the current velocity \(v_i^d\), the distance towards the personal best (pbest) and the distance towards the global best (gbest). For the second and third components, the distance to the personal best and the distance to the global best must be defined and each component is multiplied by an acceleration constant c and a random value r to increase or decrease its impact.

Equation (4) is applied to update the position of each particle. The velocity in BPSO indicates the probability of the corresponding element in the position vector taking value 1. A sigmoid function s(\(v_i^d\)) is introduced to transform \(v_i^d\) to the interval of (0, 1).

$$\begin{aligned} x_i^d = \left\{ \begin{array}{ll} 1, if \ r_3 \ < s(v_i^d) \\ 0, \ otherwise \end{array} \right. \end{aligned}$$

where \(r_3\) is a generated random value.

$$\begin{aligned} s(v_i^d) = \frac{1}{1+e^{-v_i^d}} \end{aligned}$$
(4)

The BPSO fitness function is:

$$\begin{aligned} fitness(X_i) = w_1 \times A(X_i) + (w_2 (M - R(X_i)) / M) \end{aligned}$$
(5)

where: \(A(X_i) \in [0,1]\) is the leave one cross validation accuracy on the training set using the only genes in \(X_i\), \(R(X_i)\) is the number of selected genes in \(X_i\), M is the total number of genes for each sample, \(w_1\) and \(w_2\) are two priority weights corresponding to the importance of accuracy and the number of selected genes, respectively, \(w_1 \in [0.1, 0.9]\), \(w_2 = 1 - w_1\).

Clustering algorithm

After the preprocessing step, a final step of analysis is conducted based on the PSO algorithm. PSO algorithm is a population based stochastic optimization technique developed by Kennedy and Eberhart [36]. As the name itself asserts, this method draws inspiration from natural life of swarms of birds. It uses the same principle to find the most optimal solution to a problem in the search space as birds do to find their most suitable place in a swarm [37]. PSO shares many similarities with evolutionary computation techniques such as Genetic Algorithms (GA) [38], but it is proved that the PSO algorithm provides faster convergence and finds better solutions when compared to GA. The implementation of PSO is also simple with a higher computational efficiency [39]. The many advantages within this algorithm have led us to choose it for the clustering model.

The clustering task is the main goal of the study. Therefore, a combination of PSO algorithm along with k-means algorithm is suggested in this work. PSO is a population-based stochastic optimization technique, it simulates the social behavior of organisms. This behavior can be described as an automatically and iteratively updated system. In PSO, each single candidate solution can be considered a particle in the search space. Each particle makes use of its own memory and the knowledge gained from the swarm as a whole to find the best solution. All of the particles have fitness values, which are evaluated using a fitness function in order to be optimized. During movement, each particle adjusts its position by changing its velocity according to its own experience and according to the experience of a neighboring particle, thus making use of the best position encountered by itself and its neighbor. Particles move in the problem space following a current of optimum particles. The PSO algorithm consists of three steps, which are repeated until a predefined stopping condition is met [40]:

  1. 1.

    Evaluate the fitness of each particle,

  2. 2.

    Update personal and global best fitness and positions,

  3. 3.

    Update velocity and position of each particle,

Fitness evaluation is conducted through supplying the candidate solution to the objective function. Personal and global best fitness and positions are updated through comparing the newly evaluated fitness against the previous personal and global best fitness, and replacing the best fitness and positions. The velocity of each particle i in the swarm is updated using Eq. (6):

$$\begin{aligned} v_i^{d+1} = w \times v_i^d + c_1r_1 \times (pbest_i^d - x_i^d) + c_2r_2 \times (gbest_i^d - x_i^d) \end{aligned}$$
(6)

In Eq. (6), \(v_i^d\) is the velocity of particle i at time d, and \(x_i^d\) is the position of particle i at time d. The parameters w, \(c_1\), and \(c_2\) (0 \(\le\) w \(\le\) 1.2, 0 \(\le\) \(c_1\) \(\le\) 2, and 0 \(\le\) \(c_2\) \(\le\) 2) are user-specified coefficients. The values \(r_1\) and \(r_2\) (0 \(\le\) \(r_1\) \(\le\) 1 and 0 \(\le\) \(r_2\) \(\le\) 1) are random values regenerated for each velocity update. The value \(pbest_i^d\) is the personal best candidate solution for particle i at time d, and \(gbest_i^d\) is the swarm’s global best candidate solution at time d.

Each of the three terms of the velocity update equation have different roles in the PSO algorithm. The first term \(w \times v_i^d\) is the inertia component, responsible for keeping the particle moving in the same direction it was originally heading. The value of the inertial coefficient w is typically between 0.8 and 1.2, which can either weaken the particle’s inertia or accelerate the particle in its original direction [41]. Generally, lower values of the inertial coefficient speed up the convergence of the swarm to optima, and higher values of the inertial coefficient encourage exploration of the entire search space.

The second term \(c_1r_1 \times (pbest_i^d - x_i^d)\), labeled the cognitive component, acts as the particle’s memory, causing it to tend to return to the regions of the search space in which it has experienced high personal fitness. The cognitive coefficient \(c_1\) is usually close to 2, and affects the size of the step the particle takes toward its personal best candidate solution \(pbest_i^d\). The third term \(c_2r_2 \times (gbest - x_i^d)\), labeled the social component, causes the particle to move to the best region the swarm has previously found. The social coefficient \(c_2\) is typically close to 2, and represents the size of the step the particle takes toward the global best candidate solution gbest the swarm has found so far.

The random values \(r_1\) in the cognitive component and \(r_2\) in the social component cause these components to have a stochastic influence on the velocity update. This stochastic influence causes each particle to move in a semi-random manner influenced in the directions of the personal best solution of the particle and the global best solution of the swarm. In order to keep the particles from moving too far beyond the search space, a technique called velocity clamping is used to limit the maximum velocity of each particle [40]. For a search space bounded by the range \([-x_{max}, x_{max}]\), velocity clamping limits the velocity to the range \([-v_{max}, v_{max}]\), where \(v_{max} = k \times x_{max}\).

The value k represents a user-specified velocity clamping factor, \(0.1 \le k \le 1.0\). In many optimization tasks, such as the ones proposed in the paper, the search space is not centered around 0 and thus the range \([-x_{max}, x_{max}]\) is not an adequate definition of the search space. In such a case where the search space is bounded by \([x_{min}, x_{max}]\), \(v_{max} = k \times (x_{max} - x_{min})/2\) are defined. Once the velocity for each particle is defined, each particle’s position is updated by applying the new velocity to the particle’s previous position:

$$\begin{aligned} x_i^{d+1} = x_i^d + v_i^{d+1} \ R^{n,n} \end{aligned}$$
(7)

This process is repeated until a predefined stopping condition is met. Common stopping conditions include a predefined number of iterations of the PSO algorithm or a predefined target fitness value.

Our proposed approach

Medical datasets tend to be characterized by missing values and noise, therefore, before engaging in the analysis step, datasets are cleaned as a first step. The list of available preprocessing functions includes instance selection. Approaches for instance selection can be applied for reducing the original dataset to a roughly manageable volume, leading to a reduction of the computational resources that are necessary for performing the analysis process. Algorithms of instance selection can also be applied for removing noisy instances, before applying learning algorithms, which is a step that can improve the accuracy in clustering problems [42].

Another important step within the preprocessing task is the feature selection process that consist mainly of three types of methods filter, wrapper and embedded methods. Filter methods act as preprocessors that rank the features wherein the highly ranked features are selected and applied to a predictor. In wrapper methods, the feature selection criterion is the performance of a predictor, which is wrapped in a search algorithm that finds a subset giving the highest predictor performance. Embedded methods include variable selection as part of the training process. Three other feature selection methods inspired from the previously mentioned ones, are the hybrid, ensemble and integrative methods, they also have shown good performance depending on the type of the problem in hand.

For this work, a BPSO based wrapper approach to feature selection is proposed along with mutual information of each pair of features, which determines the relevance and redundancy of the selected feature subset. Mutual information is a nonparametric, model-free method for scoring a set of features. It can be used to spot all relevant features, and to identify groups of features that allow building a valid clustering [43]. BPSO algorithm is used in binary discrete search spaces. BPSO along with the entropy of each group of features, evaluate the relevance and redundancy of the selected feature subset [44]. The BPSO-based wrapper is followed by a hybrid PSO-k-means algorithm to process the previously cleaned and preprocessed data in order to achieve a desired performance.

Our SDPSO approach consists of three main phases, (see Fig. 1). The datasets tend to contain null values, extra spaces and insignificant duplications. Therefore, as a first step, datasets are cleaned in order for them to only contain acceptable formats. The cleaned data is then used as an input to construct the BPSO model that provides a preprocessed dataset with a convenient number of features ready for analysis. The approach can be summarized in the following steps:

  • Step 1: Keeping in mind that a particle is considered as the centroid of the cluster, initialize the position and the velocity of the particles.

  • Step 2: Evaluate the fitness for each particle based on Eq. (5).

  • Step 3: If the maximum number of iterations (15 for our approach) is reached go to step 7, if not go to step 4.

  • Step 4: The pbest and the gbest are saved and used to update the particles’ position and velocity according to Eqs. (3) and (4).

  • Step 6: If the gbest is stable when the maximum iterations is reached, move to step 7, if not move back to step 3.

  • Step 7: BPSO results: recommended set of features.

  • Step 8: Reinitialize the swarms from the solution space.

  • Step 9: Evaluate the fitness for each particle.

  • Step 10: The pbest and the gbest are saved and used to update the particles’ position and velocity according to Eqs. (6) and (7). If a particle shows tendency to converge exceeding the boundary, each space is bounded by the range \([-x_{max}, x_{max}]\), velocity clamping limits the velocity to the range \([-v_{max}, v_{max}]\), where \(v_{max} = k \times x_{max}\).

  • Step 11: If the gbest is unvarying when the maximum iterations is reached go to Step 12, if not go back to Step 10.

  • Step 12: Finish the clustering task with the k-means algorithm to find the local optimum using as initial the centroids that resulted from the previous PSO steps.

Fig. 1
figure 1

SDPSO flowchart for feature selection and clustering

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The BPSO starts by the initialization of the population. First, the position and velocity of particles within the search space are randomly initialized. Second, the fitness values of particles is defined. The first fitness values and positions are the personal best values and the personal best positions (pbest). The global best value and the global best position (gbest) are set to the fitness value and position of the particle with the best fitness value in the entire population. Third, all particles are moved to their new positions using Eq. (3). All fitness values are evaluated again and personal best positions are updated for particles that have a new fitness value (ultimately better than the old personal best value). The global best position is updated if there is any particle with fitness value that is better than the old global best value. All particles are moved to their new positions. The algorithm continues evaluating the fitness values and updating the personal best values, the personal best positions, the global best value, and the global best position. It stops when a limit on the number of iterations is reached.

The clustering algorithm used in the SDPSO is a hybrid one. A combination of PSO with k-means is favored due to the fact that the convergence speed of PSO algorithm near the solution is relatively slow. The k-means algorithm, on the contrary, near the solution, converges fast to a local optimum result, but its ability to find the global solution is weak. Therefore, the combination of the two allows us to find the optimum more quickly. The hybrid algorithm starts with clustering data through PSO, which allows to search all space for a global solution. When the region of global optimum is found by the PSO, the clustering task is resumed using k-means. The hybrid algorithm accelerates the convergence speed as well as the accuracy. Thus, the k-means algorithm finalizes the clustering task. When the value of fitness function for a number of successive iterations changes negligibly the clustering algorithm switches to k-means. All the particles are updated and a new generation of particles is generated. The new particles are used to search the global best position in the solution space. The novelty is that the k-means algorithm is used to search around the global optimum.

Results

Experimental design

In order to test the performance of the SDPSO approach and its capacity to process highly dimensional datasets in low computational runtime, the experimentation is initiated using a non-distributed architecture, followed by a distributed one using PySpark on Spark 2.4. The non-distributed experiments are conducted on a single machine with Ubuntu 16.04 using 4 GB RAM, 4 CPU, and a stable version of Python 2.7 is used to implement the approach. The parallelized experiments are conducted using Apache Spark.

Our approach is implemented through the use of the many advantages presented by Apache Spark, namely, the parallel processing notion using Spark Resilient Distributed Datasets (RDD). In the master node, each particle is modelled through the use of a Python class. A swarm is then created using instances of the class and processed into the RDD. The updating of the particles’ position and velocity are conducted in the slave nodes. The updated best values are sent back to the master node to determine and update the new global best, which is again forwarded to the slave nodes for update. This process continues until the limited number of iterations is met.

Dataset description

Five benchmark datasets are chosen to test the performance of this approach, each dataset has two labels M = Malignant, B = Benign.

  • Breast cancer tumors: A dataset with examples labeled as either malignant or benign with 30 features [45].

  • Colon cancer: A colon cancer dataset which contains information on 62 samples for 2000 genes. The samples belong to tumor and normal colon tissues [46].

  • Leukemia: The total number of genes to be tested is 7129, and the number of samples to be tested is 72 [47].

  • Lung cancer: The lung dataset contains 181 tissue samples. Each sample is described by 12533 genes [48].

  • Gene expression cancer RNA-Seq dataSet: composed of 20531 features, this dataset is used in the experiments due to its complex nature and the diversification of its features [49].

The datasets were selected to have various numbers of features, classes and instances as representative samples of the problems that the proposed approaches can address (see Table 6).

Table 6 Dataset description
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Evaluation metrics

As evaluation metrics to test the performance of our approach, entropy, purity and F-Measure are used. Entropy and purity are widely used measures to determine the clustering efficiency. For each cluster, entropy uses external information class labels to test the performance. Lower entropy means better clustering. The Entropy is magnified when the members of the cluster are more diversified. So we aspire low entropy for every cluster in order to maintain the efficacy of the clustering task. For each cluster, purity determines the largest class and it attempts to capture how well the groups match with the reference on average [50]. To define the entropy E, we start by defining the probability Prob that a member of a cluster j belongs to class i.

$$\begin{aligned} Prob(i,j)=\frac{N_{ij}}{N_j} \end{aligned}$$
(8)

where \(N_{ij}\) is the number of members of a class i in a cluster j and \(N_j\) is the number of members in a cluster j. The entropy of a cluster is defined as follows:

$$\begin{aligned} E(j)=-\sum \limits _{i=1}^{k}Prob_{ij}\log _2Prob_{ij} \end{aligned}$$
(9)

where k is the number of classes. The total entropy is defined as follows:

$$\begin{aligned} E=\sum \limits _{j=1}^{n} \frac{N_j}{N}E_j \end{aligned}$$
(10)

where N is the total of members and n is the number of clusters.

The purity Pu of a cluster is defined as follows:

$$\begin{aligned} Pu_j=\max \limits _{i}Prob_{ij} \end{aligned}$$
(11)

The overall purity is defined as follows:

$$\begin{aligned} Pu=\sum \limits _{j=1}^{n} \frac{N_j}{N}Pu_j \end{aligned}$$
(12)

where N is the total of members and n is the number of clusters.

F-measure is a technique that combines the precision and the recall measurements from information retrieval literature [51]. The precision P and recall R of a cluster j (generated by the clustering algorithm) with respect to a class i (prior knowledge of the datasets) is defined as [52]:

$$\begin{aligned} P(i,j)= & {} \frac{N_{ij}}{N_i} \end{aligned}$$
(13)
$$\begin{aligned} R(i,j)= & {} \frac{N_{ij}}{N_j} \end{aligned}$$
(14)

where \(N_{ij}\) is the number of examples of class i within cluster j, \(N_j\) is the number of items of cluster j and \(N_{i}\) is the number of members of class i. The corresponding value of the F-measure is:

$$\begin{aligned} F(i)=\frac{2PR}{P+R} \end{aligned}$$
(15)

With respect to class i, members of i may be organized into different clusters. That will generate multiple F-measure value for class i. The cluster with the highest F-measure score is considered as the cluster for class i. The overall F-measure for the clustering result of one algorithm is computed as:

$$\begin{aligned} F=\frac{\sum \limits _{i=1}^{n}|i|*F(i)}{\sum \limits _{i}^{n}|i|} \end{aligned}$$
(16)

where n is the number of the clusters in the dataset and |i| is the number of data objects in class i. The F value is limited within the interval [0, 1]. The higher, the F-measure is, the better the clustering result are.

Experimental results

The experimentations are divided into two main parts. The first experiment tests the BPSO feature selection method against two prominent feature selection methods, which are implemented on the same datasets of interest. The second part of the experimentation is testing the runtime as well as the accuracy of the SDPSO approach. Both experiments are conducted on the five benchmark datasets previously described in “Dataset description” section.

Feature selection results

The preprocessing step using the BPSO-based feature selection algorithm is tested against two prominent feature selection algorithms in order to justify our choice:

  • CFS: Correlation-based feature selection is a filter method that ranks feature subsets according to an appropriate correlation measure and a heuristic search strategy [53].

  • ReliefF: An extension of the binary-classification Relief algorithm [54] which was limited to binary classification problems. ReliefF [55] can deal with multiclass problems, which makes of it an improved algorithm that is more likely to handle incomplete and noisy data [56].

Table 7 displays the number of features selected by each of the three feature selection methods, CFS, ReliefF and BPSO. It is noted that the number of features selected by BPSO is noticeably higher than the number of features selected by CFS and reliefF respectively.

Table 7 Number of selected features per algorithm
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SDPSO results

The accuracy of the SDPSO results, after it was tested on five benchmark datasets with different characteristics are expressed in Table 8 and Fig. 2. Five experimental trials are conducted with each dataset and the means of the five values was taken into consideration as the final result. For Breast cancer and Lung cancer the approach achieved considerably high accuracy. For the rest of the datasets, despite the lower purity and F-Measure scores, it is still in general relatively credited compared to results provided by the four state of the art methods, namely, k-means, GA, DBSCAN and hybrid PSO-GA algorithms, applied on the same datasets of interest. As can be observed, the effectiveness of the SDPSO approach is demonstrated as it yields the lowest in average of entropy 0.16 compared to the average entropy of 0.23, 0.22, 0.19 and 0.18 for the k-means, GA, DBSCAN and hybrid PSO-GA algorithms respectively. Our approach also shows a better purity average of 0.88 and F-measure average of 0.87 compared to a purity average of 0.79, 0.77, 0.82 and 0.85 and an F-score average of 0.82, 0.79, 0.84 and 0.85 for the k-means, GA, DBSCAN and hybrid PSO-GA algorithms respectively.

Table 8 Comparative results of SDPSO entropy and purity with k-means, GA, DBSCAN and hybrid PSO-GA for the five datasets
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Fig. 2
figure 2

Comparative results of SDPSO F-Measure with k-means, GA, DBSCAN and hybrid PSO-GA for the five datasets

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Another test was conducted to test computational cost as well as to emphasize on the importance of the parallelized approach. The parallelized interpretation of the approach is compared with the non-parallelized one. The average total runtime of the SDPSO approach is indicated in Table 9 for both the non-parallelized and the parallelized implementations. Similarly to the previous experimentation, all the experiments are run five times and the parallelized implementation shows faster runtime compared to the non-parallelized one along with a good accuracy average (see Figs. 3 and 4). Evidently, the number of the features influences the total runtime since it adds to the complexity of the computation. For the Gene Expression Cancer, the total runtime exceeded 8 days without showing any iteration result, which prompts us to consider it as an infinite runtime.

Table 9 SDPSO runtime for the five datasets (s)
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Fig. 3
figure 3

SDPSO non-parallelized implementation runtime for the five datasets (s)

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Fig. 4
figure 4

SDPSO parallelized implementation runtime for the five datasets (s)

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Discussion

Nowadays, it is pivotal in many fields to develop scalable approaches that can efficiently optimize processes and solve scalability issues. In the medical field, due to the critical nature of the data, it is crucial to obtain satisfying efficiency with lower time response. The use of horizontal scaling through Apache Spark combined with conglomerate and encompassing computational techniques allow organizations to access strong computational strength in approachable prices compared to the vertical scaling which demands high commodity hardware. The aim of this work is to provide an encompassing fast and accurate approach that can be afforded by medical organization through the implementation of a PSO-based approach in Apache Spark that is famously known for its capacity to optimize the computation cost in comparison with the vertically scalable solution that require the purchase of high commodity hardware.

Performance and accuracy

In the previous section, the experimental results demonstrate that SDPSO provides promising results both in terms of runtime and accuracy. High accuracy is pivotal, especially with medical data where very little room of error is allowed. SDPSO performance was assessed using five genomic datasets, the breast cancer, the colon cancer, the leukemia, the lung cancer and the genetic expression cancer datasets; it noticeably outperforms four state of the art algorithms, namely, k-means, GA, DBSCAN and hybrid PSO-GA yielding a purity that varies from 0.78 to 0.97, an F-measure that varies from 0.75 to 0.96 and an entropy that varies from 0.06 to 0.30 depending on the complexity of the dataset; taking into consideration that the higher the purity and F-measures are and the lower the entropy is, the higher the performance of the clustering task is.

Parallelization, cost and runtime

Moreover, the Apache Spark implementation of the SDPSO approach shows to be much faster than a single node non-parallelized implementation of the same approach. The BPSO selects more features compared to other feature selection methods, which allows it to be designated as a more inclusive method, eventually allowing the accuracy of the approach as a whole to increase. Owing to Apache Spark’s efficient data processing, the SDPSO approach gains computational power. SDPSO manages to provide a satisfying accuracy in a fairly lower computational time compared to non-parallelized approaches. The use of Apache Spark is remarkably cost friendly, it allows the analysis to run smoothly even with low commodity hardware, which is highly required in the medical field in order to satisfy the needs of the unfortunate medical organizations.

Limitations of the study

Our approach is mainly based on PSO due to the adaptability of the algorithm towards different problems simply by modifying or adding fitness functions. This flexibility, along with its efficiency, allow it to be used to operate a wide range of clustering processes. As highlighted through the article, PSO merits to be widely used due to its capability to adapt to large amounts of data, using fitness functions distinctively meeting the needs of the study in hand, and the fact that it preforms an effective global search of the solution space. Despite its numerous advantages, the PSO-based approaches fall short in many aspects. Due to its nature, PSO algorithm relies on a number of hyper parameters that are user-supplied and critical to determine in practice. Methods to test the cohesion and separation measures of the combination of k-means with PSO are important and ought to be addressed in future works, however, they are beyond the scope of this paper as we rely on the results of the entropy, the purity and the F-measure to validate our approach as a whole.

Conclusion, perspectives and future work

In this work a large scale study of the PSO algorithm is presented both as a feature selection and as a clustering method applied to cancer datasets. The performance of the SDPSO approach is systematically evaluated using five datasets, in order to analyze the influence of the multidimensional datasets on the outcomes of the analysis process. The proposed approach yields the best in average of purity ranging from 0.78 to 0.97 and in average of F-measure ranging from 0.75 to 0.96 compared to four state of the art methods, namely, k-means, GA, DBSCAN and hybrid PSO-GA. Although the results vary within the datasets, the general picture provided here helps emphasize on the importance of the combination of the PSO-based computational techniques with the parallelization’s touch that is added through Apache Spark usage for the prognosis tasks on cancer datasets.

Despite the multiple benefits of parallelism, it can be risky if not used properly. Over-parallelism can be malicious to the accuracy of the results; these issues grow in scale, when large and complex datasets are used, the distribution can result in disregarding certain meaningful relationships between features. In the light of the above, the proposed approach successfully contributes to the reduction of runtime while maintaining reasonable accuracy. In a near future, the hope is to contribute in making the literature on large scale data analysis be as mature as the one small scale data. Our main objective is to help ensure accessibility to strong computational creativity, with the minimum of expenses. Given the large dimensionality of the used datasets, further investigation is required regarding clustering in these spaces, which will be the leading perspective of our future work. The efforts towards elucidating this question will most probably involve the use and evaluation of even more elaborated feature selection and clustering algorithms.