A novel variant of social spider optimization using single centroid representation and enhanced mating for data clustering

Contributions

In the last decade, nature-inspired algorithms have been successfully applied for solving NP-hard clustering problems. In the state-of-the-art nature inspired algorithms for solving clustering problems, each agent in the population is taken as a collection of K clusters. Therefore, the memory requirements and CPU times of these algorithms are very high. These algorithms return the best agents in which SICD is minimized or the average of ICD is minimized. In other words, the fitness of the agent is measured by the consideration that all K clusters present in it as a whole. This, however, does not mean that all clusters should have low ICD individually in order to get a low SICD, as even the globally best agent with the best fitness may contain some clusters that have very high ICD.

Suppose DS = {dv1, dv2, dv3, dv4, dv5, dv6, dv7, dv8, dv9, dv10}, K = 3, number of the agents = 4, and the contents of the agents are as shown in Figs. 2–5. According to all state-of-the-art nature-inspired algorithms, the best agent will be agent1 as it has the lowest SICD value. However, these algorithms will not give any assurance that the three clusters have the lowest individual ICD. Table 1 specifies the best three agents (spiders) returned by our proposed algorithm. The SICD value of the globally best solution is 40 + 30 + 65 = 135, which is less than the SICD value of the globally best solution in the K-cluster representation of the agent. Therefore, the clustering results produced by the start-of-the art algorithms that use K-centroid representation for agents may not be highly accurate. The proposed approach not only focuses on SICD but also on the individual ICD of the clusters.

In our proposed algorithm, social spider optimization for data clustering using single centroid (SSODCSC), each spider is represented by a single centroid and the list of data instances close to it. This representation requires K times less memory requirements than the representation used by the other state-of-the-art nature-inspired algorithms like SSO, as shown below. Each data instance in the dataset is given an identification number. Instead of storing data instances, we stored their identification numbers (which are integer values) in spiders.

• For SSODCSC:

• Number of spiders used = 50

• Number of iterations for best clustering results = 300

• Total number of spiders to be computed = 300 * 50 = 15,000

• Memory required for storing a double value = 8 bytes

• Memory required for storing a spider’s centroid (that consists of m dimension values) = 8 * m bytes, where m is the number of dimensions present in the dataset.

• Memory required for storing an integer value representing identification number of a data instance = 4 bytes.

• Maximum memory required for storing the list of identification numbers of data instances closer to the centroid = 4 * n bytes, where n is number of data instances present in the dataset.

• Maximum memory required for a spider = 8 * m + 4 * n bytes

• Therefore, total computational memory of SSODCSC = 15,000 * (8 * m + 4 * n) bytes.

• For SSO:

• Number of spiders used = 50

• Number of iterations for best clustering results = 300

• Total number of spiders to be computed = 300 * 50 = 15,000

• Memory required for storing a double value = 8 bytes

• Memory required for storing K centroids of a spider = 8 * m * K bytes, where m is the number of dimensions present in the dataset.

• Memory required for storing an integer value representing identification number of a data instance = 4 bytes

• Maximum memory required for storing K lists of identification numbers of data instances (where each list is associated with a centroid) = 4 * n * K bytes, where n is the number of data instances present in the dataset, and K is the number of centroids present in each spider.

• Maximum memory required for a spider = K*(8 * m + 4 * n) bytes

• Therefore, total computational memory of SSO = 15,000 * K* (8 * m +4 * n) bytes.

The time required for initiating the spiders will be less in this representation. The average CPU time per iteration depends on the time required for computing fitness values and the time required for computing the next positions of the spiders in the solution space. The fitness values and next positions of spiders can be computed in less time with single centroid representation, so that the average CPU time per iteration reduces gradually. The proposed algorithm returns best K spiders such that the union of the lists of data instances present in them will produce exactly all of the data instances in DS.

In the basic SSO algorithm, non-dominant males are not allowed in the mating operation because of their low weight values. They do not receive any vibrations from other spiders and have no communication in the web, as the communication is established through vibration only (Shukla & Nanda, 2016). Therefore, their presence in the solution space is questionable. Moreover, their next positions are dependent on the existing positions of the dominant male spiders (Cuevas et al., 2013). They cannot be part of the selected solution when dominant male spiders are in the solution space. In our proposed algorithm, SSODCSC, we convert them into dominant male spiders by increasing their weight values and then allowing them to participate in the mating operation to produce a new spider, better than the worst spider in the population. In SSO, each dominant male mates with a set of females and produces a new spider. The weight of the new spider may or may not be greater than that of the worst spider. But as we make the weight of each non-dominant male spider greater than the average weight of the dominant male spiders in SSODCSC, the new spider produced by the non-dominant male spider is surely better than the worst spider. In other words, not only did we convert non-dominant male spiders into dominant male spiders, but also we made them more effective than the dominant male spiders. Therefore, each spider receives vibrations from other spiders and has a chance at becoming a part of the selected solution, unlike in SSO. At each iteration of SSODCSC, the population size is the same but the spiders with greater weight values are introduced in place of the worst spiders. As a result, the current solution given by SSODCSC moves toward the globally best solution as the number of iterations is increased. We applied SSODCSC on feature-based datasets and text datasets.

This paper is organized as follows: “Related Work” describes the recent related work on solving clustering problems using nature-inspired algorithms, “Proposed Algorithm: SSODCSC” describes SSODCSC, “Results” includes experimental results, and we conclude the paper with future work in the section “Discussion.”

Initialization

Social spider optimization for data clustering using single centroid starts with the initialization of spiders in the solution space. Initially all spiders are empty. The fitness of each spider is set to 0, and the weight is set to 1. Each spider s is initialized with a random centroid using Eq. (6).

where spider [s, d] is dth dimension of the centroid of spider s, lowerbound (d) and upperbound (d) are the smallest and largest values of the dth dimension of the dataset, respectively.

Assignment of data instances

The distances of each data instance from the centroids of all spiders are calculated using the Euclidean distance function. A data instance is assigned to the spider that contains its nearest centroid.

Next positions of spiders

The spiders are moved across the solution space in each iteration of SSODCLC based on their gender. The movement of a spider in the solution space depends on the vibrations received from other spiders. The intensity of the vibrations originated from spider s_j to spider s_i can be found using Eq. (7) and depends on the distance between the two spiders and the weight of spider s_j.

Next positions of female spiders

The movement of a female spider s_f depends on the vibrations from the globally best spider s_gbs and its nearest better spider s_nbs as shown in Fig. 9. To generate the next position of a female spider s_f, a random number is generated and if it is less than the threshold probability (TP), the female spider attracts other spiders and the position of it is calculated according to Eq. (8). If not, it repulses other spiders and the position of it calculated according to Eq. (9). In Eqs. (8) and (9), α, β, γ, and δ are random numbers from the interval [0, 1].

(8) $$\begin{array}{l}\text{spider}[s_f,d]=\text{spider}[s_f,d]+\alpha *\left(\text{spider}[s_f,d]-\text{spider}[s_{gbs},d]\right)*\text{weight}[s_{gbs}]\\ *{\text{e}}^{-\text{distance}{(s_f,s_{gbs})}^2}+\beta *\left(\text{spider}[s_f,d]-\text{spider}[s_{nbs},d]\right)\\ *\text{weight}[s_{nbs}]*{\text{e}}^{-\text{distance}{(s_f,s_{nbs})}^2}+\gamma *(\delta -0.5)\end{array}$$ (9) $$\begin{array}{l}\text{spider}[s_f,d]=\text{spider}[s_f,d]+\alpha *\left(\text{spider}[s_f,d]-\text{spider}[s_{gbs},d]\right)\\ *\text{weight}[s_{gbs}]*e^{-\text{distance}{(s_f,s_{gbs})}^2}-\beta *\left(\text{spider}[s_f,d]-\text{spider}[s_{nbs},d]\right)\\ *\text{weight}[s_{nbs}]*e^{-\text{distance}{(s_f,s_{nbs})}^2}+\gamma *(\delta -0.5)\end{array}$$

Next position of male spiders

The solution space consists of female spiders and male spiders. When data instances are added or removed from them, their fitness values and weight values will change. If the current weight of a male spider is greater than or equal to the median weight of dominant male spiders, it will be considered to be a dominant male spider. The male spiders that are not dominant male spiders are called non-dominant male spiders. The next position of a dominant male s_dm can be calculated using Eq. (10).

(10) $$\begin{array}{l}\text{spider}[s_{dm},d]=\text{spider}[s_{dm},d]+\alpha *\left(\text{spider}[s_{dm},d]-\text{spider}[s_{nfs},d]\right)\\ *\text{weight}[s_{nfs}]*{\text{e}}^{-\text{distance}{(s_{dm},s_{nfs})}^2}+\gamma *(\delta -0.5)\end{array}$$

The position of the spider depended only on the vibrations received from its nearest female spider s_nfs. The pictorial representation of this is specified in Fig. 10. The weighted mean of the male population, W, can be obtained using Eq. (11). Let N_f be the total number of female spiders in the spider colony and N_m be the total number of male spiders. Then the female spiders can be named as $s_{f_1},s_{f_2},s_{f_3},\dots \dots \mathrm{..},s_{f_{N_f}}$ and the male spiders can be named as $s_{m_1},s_{m_2},s_{m_3},\dots \dots \mathrm{..},s_{m_{Nm}}$.

(11) $$W=\frac{{\displaystyle {\sum }_{i=1}^{N_m}\text{spider}\left[s_{m_i},d\right]}*\text{weight}\left(s_{m_i}\right)}{{\displaystyle {\sum }_{i=1}^{N_m}*\text{weight}\left(s_{m_i}\right)}}$$

The next position of the non-dominant male spider s_ndm can be calculated using Eq. (12) and depends on the weighted mean of the male population. (12) $$\text{spider\hspace{0.17em}}[s_{ndm},d]=\text{spider\hspace{0.17em}}[s_{ndm},d]+\alpha *W$$

Mating operation

Each dominant male spider mates with a set of female spiders within the specified range of mating to produce a new spider, as shown in Fig. 11. The new spider will be generated using the Roulette wheel method (Chandran, Reddy & Janet, 2018). If the weight of the new spider is better than the weight of worst spider, then the worst spider would be replaced by the new spider. The range of mating r is calculated using Eq. (13).

where diff is the sum of differences of the upper bound and lower bound of each dimension, and n is the number of dimensions of the dataset DS.

In SSO, the non-dominant male spiders are not allowed to mate with female spiders, as they would produce new spiders having low weights. In SSODCSC, a non-dominant male spider is converted into dominant male spider by making sure that its weight becomes greater than or equal to the average weight of dominant male spiders so that it participates in the mating process and produces a new spider whose weight is better than that of at least one other spider. The theoretical proof for the possibility of converting a non-dominant male spider into a dominant male spider is provided in Theorem 1. Thus, non-dominant male spiders become more powerful than dominant male spiders as they are made to produce new spiders that surely replace worst spiders in the population. The theoretical proof for the possibility of obtaining a new spider that is better than the worst spider, after a non-dominant male spider mates with the female spiders is provided in Theorem 2. The following steps are used to convert a non-dominant male spider into a dominant male spider:

• Step 1: Create a list consisting of data instances of the non-dominant male spider s_ndm in the decreasing order of their distances from its centroid.

• Step 2: Delete the top-most data instance (i.e., the data instance which is the greatest distance from the centroid) from the list.

• Step 3: Find the weight of the non-dominant male spider s_ndm.

• Step 4: If the weight of non-dominant male is less than the average weight of dominant male spiders, go to Step 2.

• The flowchart for SSODCSC is specified in Fig. 12.

• Theorem 1. A non-dominant male spider can be converted into a dominant male spider in single centroid representation of SSO.

• Proof: Let s_ndm be the non-dominant male spider whose weight is w_ndm.

• Let medwgt be the median weight of male spiders (which is always less than or equal to 1).

• But according to definition of the non-dominant male spider, (14) $$w_{ndm}<medwgt$$

• Assume that the theorem is false.

• ⇒ s_ndm can not be converted into a dominant male spider

• ⇒ During the movement of s_ndm in the solution space, (15) $$w_{ndm}<1$$

• Let Sum be the sum of distances of data instances from the centroid of s_ndm.

• If the data instance that is the furthest distance from the centroid of s_ndm is removed from s_ndm, then

• ⇒ sum of distances of data instances from the centroid of s_ndm will decrease, as

• Sum = Sum-distance of removed data instance from the centroid of s_ndm.

• ⇒ fitness of s_ndm will decrease, as

•  fitness of s_ndm is proportional to Sum

• ⇒ the weight of s_ndm will increase as

•  the weight of s_ndm is inversely proportional to fitness of s_ndm

• Similarly,

  If a data instance is added to s_ndm, then

• ⇒ sum of distances of data instances from centroid of s_ndm will increase.

• ⇒ fitness of s_ndm will increase.

• ⇒ the weight of s_ndm will decrease.

• Therefore, (16) $$w_{ndm}=1-{\displaystyle \sum _{i=1}^n{w}_{d_i}}$$where 1 is the initial weight of s_ndm, n is the total number of data instances added to s_ndm, and $w_{d_i}$ is the decrease in the weight of s_ndm when ith data instance was added to s_ndm.

• When all the data instances are removed from s_ndm, (17) $$w_{ndm}=w_{ndm}+{\displaystyle \sum _{i=1}^n{w}_{d_i}=1-}{\displaystyle \sum _{i=1}^n{w}_{d_i}+{\displaystyle \sum _{i=1}^n{w}_{d_i}}}$$

• But according to Eq. (15), w_ndm can never be 1 during the movement of s_ndm in the solution space.

• Hence, our assumption is wrong.

• So, we can conclude that a non-dominant male spider can be converted into dominant male spider in single centroid representation of SSO.

• Theorem 2. The weight of the new spider resulting from the mating of a non-dominant male spider with a weight greater than or equal to the average weight of dominant male spiders will be better than at least one spider in the population.

• Proof: Let s_ndm be the non-dominant male spider whose weight became greater than or equal to the average weight of dominant male spiders.

• Let $s_{f_1},s_{f_2},s_{f_3},\dots \dots \mathrm{..},s_{f_m}$ be female spiders that participated in the mating.

• Let s_new be the resulting new spider of the mating operation.

• Let N be the total number of spiders in the colony.

• Assume that the theorem is false.

• It implies: (18) $${\displaystyle \sum _{i=1}^N\text{weight}\left(s_i\right)\le \text{weight}\left(s_{\text{new}}\right)?1:0=0}$$

• In other words, the total number of spiders whose weight is less than or equal to that of s_new is zero.

• But according to the Roulette wheel method: (19) $$s_{\text{new}}=\frac{\left({\displaystyle {\sum }_{i=1}^m{s}_{f_i}*\text{weight}\left(s_{f_i}\right)}\right)+s_{ndm}*\text{weight}\left(s_{ndm}\right)}{\left({\displaystyle {\sum }_{i=1}^m\text{weight}\left(s_{f_i}\right)}\right)+\text{weight}\left(s_{ndm}\right)}$$

  ⇒ $\underset{\text{weight}\left(s_{f_i}\right)\to 0\wedge \text{weight}\left(s_{ndm}\right)\to 1}{\lim }{s}_{\text{new}}$

  = s_ndm with weight equal to 1 = s_gbs (since any spider whose weight is 1 is always s_gbs)

• And $$\begin{array}{l}\underset{\text{weight}\left(s_{f_i}\right)\to 1\wedge \text{weight}\left(s_{ndm}\right)\to 1}{\lim }{s}_{\text{new}}\\ =\frac{\left(m\text{female spiders whose weight is1}\right)+\left(s_{ndm}\text{whose weight is }1\right)}{m+1}\\ =\frac{\left(m+1\right)*s_{gbs}}{m+1}\\ =s_{gbs}\end{array}$$

• So, when weight $\left(s_{f_i}\right)$ tends to 0, and weight (s_ndm) tends to 1, s_new becomes s_gbs.

• When weight $\left(s_{f_i}\right)$ tends to 1, and weight (s_ndm) (s_ndm) tends to 1, s_new becomes s_gbs.

• Similarly,

  When weight $\left(s_{f_i}\right)$ tends to 1, and weight (s_ndm) tends to 0, s_new becomes s_gbs.

• Substituting s_gbs in place of s_new in Eq. (18), (20) $${\displaystyle \sum _{i=1}^N\text{weight}\left(s_i\right)\le \text{weight}\left(s_{gbs}\right)?1:0=0}$$

According to Eq. (20), the number of spiders whose weight is less than or equal to the weight of s_gbs is zero. But according to the definition of s_gbs, its weight is greater than or equal to the weights of all remaining spiders. So, there are spiders whose weights are less than or equal to the weight of s_gbs. Therefore Eq. (20) is false.

Hence, our assumption is wrong. Therefore, we can conclude that the weight of s_new produced by s_ndm is greater than that of at least one spider in the population.

Applying SSODCSC on patent datasets

At first, we applied SSODCSC on six Patent corpus datasets. The description of the data sets is given in Table 2. Patent corpus5000 contains 5,000 text documents with technical descriptions of the patents that belong to 50 different classes. Each class has exactly 100 text documents. Each text document contains only a technical description of the patent. All text documents were prepared using the ASCII format.

As SSODCSC returned K best spiders, the SICD of clusters in those spiders was calculated. Table 3 specifies the clustering results of SSODCSC when applied on Patent corpus datasets. For each dataset the SICD value, Cosine similarity value, F-measure value and accuracy obtained were specified.

Table 4 specifies the relationship between the SICD values and number of iterations. Lower SICD value indicate a higher clustering quality. It was found that as we increased the number of iterations, the SICD decreased and thereby, the clustering quality increased.

To find the distance between data instances, we used the Euclidean distance function and Manhattan distance function. Data instances having small differences were placed in same cluster by the Euclidean distance function, as it ignores the small differences. It was found that SSODCSC produced a slightly better clustering result with the Euclidean distance function as shown in Table 5.

Table 6 specifies the comparison between clustering algorithms with respect to SICD values. Table 7 specifies the comparison between clustering algorithms with respect to accuracy. SSODCSC produces better accuracy for all datasets. The overall percentage increase in the accuracy is approximately 13%.

The silhouette coefficient SC of a data instance d_i can be calculated using Eq. (21).

where a is the average of distances between data instance d_i and other data instances present in its containing cluster, and b is the minimum of distances between data instance d_i and data instances present in other clusters. The range of the silhouette coefficient is [−1, 1]. When it is closer to 1, better clustering results will be produced. Table 8 specifies the comparison between clustering algorithms with respect to the average silhouette coefficient values of datasets. SSODCSC produces better average silhouette coefficient values for all Patent corpus datasets.

From Figs. 13 to 14, it is obvious that SSODCSC produces the largest inter-cluster distances and smallest ICD for Patent corpus datasets.

Applying SSODCSC on UCI datasets

We applied SSODCSC on UCI data sets as well. The description of the data sets is given in Table 9. Table 10 specifies the relationship between SICD values and the number of iterations. As we increase the number of iterations, the SICD is also reduced. For the Iris dataset, the SICD value is 125.7045 at 100 iterations but as we increase the number of iterations, the SICD value of the clustering result also decreases until it reaches 95.2579 at 300 iterations. However, it remains at 95.2579, after 300 iterations and it becomes obvious that SSODCSC converges in 300 iterations.

Table 11 specifies the best three spiders for the Iris dataset. We initialized a solution space with 50 spiders, among which, the first 30 spiders were females and the remaining were males. Our proposed algorithm returned spider 21, spider 35, and spider 16. Spider 21 and spider 16 were females and spider 35 was a male spider. The centroids of these spiders were (6.7026, 3.0001, 5.4820, 2.018), (5.193, 3.5821, 1.4802, 0.2402), and (5.8849, 2.8009, 4.4045, 1.4152), respectively. These centroids have four values as Iris dataset consists of four attributes. The sum of the distances between the 150 data instances present in the Iris dataset and their nearest centroids in Table 11 was found to be 95.2579, as shown in Table 10.

Table 12 specifies the best six spiders for vowel dataset. Our proposed algorithm returned spider 10, spider 25, spider 42, spider 22, spider 48, and spider 5. Spider 10, spider 25, spider 22, and spider 5 were females, and spiders 42 and spider 48 were male spiders. The sum of the distance between the 871 data instances present in the Vowel data set and their nearest centroids in Table 12 was found to be 146,859.1084, as shown in Table 10.

Table 13 specifies the best three spiders for the CMC dataset. Our proposed algorithm returned spider 23, spider 38, and spider 16 among which spider 23 and spider 16 were females and spider 38 was a male spider. The centroids of these spiders are specified. These centroids had nine values as the CMC dataset consists of nine attributes. The sum of the distance between the 1,473 data instances present in the CMC data set and their nearest centroids in Table 13 was found to be 5,501.2642, as shown in Table 10.

Tables 14 and 15 specify the best spiders and their centroids for the Glass and Wine datasets, respectively. The sum of the distances between the 214 data instances present in the Glass dataset and their nearest centroids in Table 14 was found to be equal to 207.2091, as shown in Table 10. If we find the sum of the distances between the 178 data instances present in the Wine dataset and their nearest centroids in Table 15, it would be equal to 16,270.1427, as shown in Table 10.

Table 16 specifies the average CPU time per iteration (in seconds), when clustering algorithms were applied on the CMC dataset. It was found that SSODCSC produces clustering results with the shortest average CPU time per iteration.

Table 17 specifies the average CPU time per iteration (in seconds), when clustering algorithms were applied on the Vowel dataset. It was found that K-means produces clustering results with the shortest average CPU time per iteration. The SSODCSC has the second shortest average CPU time per iteration.

Table 18 specifies F-measure values obtained by the clustering algorithms when they are applied on the Iris, Glass, Vowel, Wine, Cancer, and CMC datasets, respectively. It is evident that SSODCSC produces the best F-measure values. The overall percentage increase in the F-measure value is approximately 10%.

We computed the ICD and inter-cluster distances of the resultant clusters of the clustering algorithms when applied on UCI datasets. From Figs. 15 to 17 it is obvious that SSODCSC produces the smallest ICD for UCI datasets. From Figs. 18 to 19 it can be concluded that SSODCSC produces the largest inter-cluster distances for UCI datasets, as compared with other clustering algorithms.

Table 19 specifies the comparison between clustering algorithms with respect to the average silhouette coefficient values of UCI datasets. SSODCSC produces better average silhouette coefficient values for UCI datasets also. The average silhouette coefficient values produced by SSODCSC are 0.7505, 0.6966, 0.7889, 0.7148, 0.8833, and 0.6264 for Wine, Cancer, CMC, Vowel, Iris, and Glass datasets, respectively.

Statistical analysis: Patent corpus datasets

To show the significance of the proposed algorithm, we applied a one-way ANOVA test on the accuracy values shown in Table 7. Sum, Sum squared, Mean, and Variance of the clustering algorithms are specified in Table 20.

• Degrees of freedom: df1 = 10, df2 = 55

• Sum of squares for treatment (SSTR) = 2,761.313

• Sum of squares for error (SSE) = 1,625.378

• Total sum of squares (SST = SSE + SSTR) = 4,386.691

• Mean square treatment (MSTR = SSTR/df1) = 276.131

• Mean square error (MSE = SSE/df2) = 29.552

• F (= MSTR/MSE) = 9.344

• Probability of calculated F = 0.0000000080

• F critical (5% one tailed) = 2.008

• We can reject the null hypothesis as calculated F (9.344) is greater than F critical (2.008).

• 1) Post-hoc analysis using Tukeys’ honestly significant difference method:

• Assuming significance level of 5%.

• Studentized range for df1 = 10 and df2 = 55 is 4.663.

• Tukey honestly significant difference = 10.349.

• Mean of K-means and SSODCSC differs as 16.14667 is greater than 10.349.

• Mean of PSO and SSODCSC differs as 20.03833 is greater than 10.349.

• Mean of genetic algorithms (GA) and SSODCSC differs as 21.85000 is greater than 10.349.

• Mean of artificial bee colony (ABC) and SSODCSC differs as 21.13333 is greater than 10.349.

• Mean of improved bee colony optimization (IBCO) and SSODCSC differs as 24.12333 is greater than 10.349.

• Mean of ACO and SSODCSC differs as 16.38333 is greater than 10.349.

• Mean of SMSSO and SSODCSC differs as 21.73167 is greater than 10.349.

• Mean of BFGSA and SSODCSC differs as 12.71000 is greater than 10.349.

• Mean of SOS and SSODCSC differs as 19.54333 is greater than 10.349.

• Mean of SSO and SSODCSC differs as 12.27000 is greater than 10.349.

• Therefore, it may be concluded that SSODCSC significantly differs from other clustering algorithms.

Statistical analysis: UCI datasets

To show the significance of the proposed algorithm, we applied a one-way ANOVA test on the F-measure values shown in Table 17. Sum, Sum squared, Mean, and Variance of the clustering algorithms are specified in Table 21.

• Degrees of freedom: df1 = 10, df2 = 55.

• Sum of squares for treatment (SSTR) = 4,113.431.

• Sum of squares for error (SSE) = 7,901.638.

• Total sum of squares (SST = SSE + SSTR) = 12,015.069.

• Mean square treatment (MSTR = SSTR/df1) = 411.343.

• Mean square error (MSE = SSE/df2) = 143.666.

• F (= MSTR/MSE) = 2.863.

• Probability of calculated F = 0.0060723031.

• F critical (5% one tailed) = 2.008.

• So, we can reject the null hypothesis as calculated F (2.863) is greater than F critical (2.008).

• 1) Post-hoc analysis using Tukeys’ honestly significant difference method

• Assuming significance level of 5%.

• Studentized range for df1 = 10 and df2 = 55 is 4.663.

• Tukey honestly significant difference = 22.819.

• Means of GA and SSODCSC differ as 24.32000 is greater than 22.819.

• Means of ABC and SSODCSC differ as 23.15833 is greater than 22.819.

• Means of IBCO and SSODCSC differ as 25.89333 is greater than 22.819.

• Means of ACO and SSODCSC differ as 25.27667 is greater than 22.819.

• Means of SMSSO and SSODCSC differ as 27.14500 is greater than 22.819.

• Means of BFGSA and SSODCSC differ as 25.77000 is greater than 22.819.

• Means of SOS and SSODCSC differ as 26.29667 is greater than 22.819.

Therefore, it is obvious that SSODCSC significantly differs from most of the other clustering algorithms when applied on UCI datasets.