Keywords

1 Introduction

Data clustering aims to discover natural existing structures in a dataset. For this purpose, data are partitioned into groups (clusters) of objects. Objects within a cluster are similar, whereas they are dissimilar in different clusters. Since there is a large variety of datasets different clustering algorithms and their configurations are still created, e.g. [9, 11, 12, 31]. Note that among clustering methods two major categories are distinguished: partitioning and hierarchical clustering. For example, the well-known partitioning algorithms are, e.g. K-means, Partitioning Around Medoids (PAM) [5, 24] and Expectation Maximization (EM) [21]. Whereas the agglomerative hierarchical clustering includes such methods as, e.g. the Single-linkage, Complete-linkage or Average-linkage [16, 22, 25]. Data clustering is applied in many areas, such as biology, spatial data analysis, business and so on. It can be noted that there is no a clustering algorithm, which creates the right data partition for all datasets. Moreover, the same algorithm can also give different results depending on the input parameters. Therefore, cluster validation should be used to assess the results of data clustering. Generally, it is a very difficult task and is the most frequently realized by validity indices. Techniques of the cluster validation are usually classified into three groups, i.e. external, internal and relative validation [16, 30]. The external validation is based on a comparison of partitions of a dataset obtained by a clustering algorithm with the correct partition of this data. In turn, the internal approach uses only the intrinsic properties of the dataset. On the other hand, the relative validation method compares the data partitions obtained by chaining input parameters of a clustering algorithm. It should be noted that the number of clusters is the key parameter for many clustering algorithms. So far, a number of authors have proposed different validity indices or modifications of existing indices, e.g., [1, 10, 18, 29, 32, 33, 36, 38]. Among internal cluster validity indices, the Silhouette (SIL) [26] and Simplified Silhouette (SimSIL) [15] indices are frequently used to evaluate the efficacy of the clustering algorithms in detecting the right data partitioning. It is important to note that clustering methods in conjunction with cluster validity indices can be used during a process of designing various neural networks [2,3,4, 6, 17], neuro-fuzzy structures [7, 8, 20, 27, 28] and creating some algorithms for identification of classes [13, 14].

In this paper, new cluster validity indices called the SimSILA and the SimSILAv1, are presented. These new indices modify the Simplified Silhouette (SimSIL) index. The proposed approach is based on an additional component and it is a detailed explanation in Sect. 3. In order to present effectiveness of the validity indices, several experiments were performed for various datasets. This paper is organized as follows: Sect. 2 presents a detailed description of the Silhouette, SILA and SILAv1 indices. In Sect. 3 the Simplified Silhouette, SimSILA and SimSILAv1 indices are outlined. Section 4 illustrates experimental results on datasets. Finally, Sect. 5 presents conclusions.

2 Modification of the Silhouette Index

In this section modification of the Silhouette (SIL) index is described. This approach was proposed and discussed in papers [34, 35]. Let us denote K-partition scheme of a dataset X by C = \({\{C_1, C, ..., C_K\}}\), where \({C_k}\) indicates \(k_{th}\) cluster, \({k=1,..,K}\). The original SIL index is presented as follows:

$$\begin{aligned} SIL = \frac{1}{K}\sum \limits _{k = 1}^K {SIL(C_k)} \end{aligned}$$
(1)

where \(SIL(C_k)\) is the Silhouette width for the given cluster \(C_k\) and is defined as:

$$\begin{aligned} SIL(C_k) = \frac{1}{{n_k }}\sum \limits _{\mathbf{x} \in C_k } { {\frac{{b(\mathbf{x}) - a(\mathbf{x})}}{{max\left( {a(\mathbf{x}),b(\mathbf{x})} \right) }}}} \end{aligned}$$
(2)

\(n_k\) is a number of elements in \(C_k\), and \(a(\mathbf{x})\) is the within-cluster mean distance, i.e. it is the average distance between \(\mathbf x\) and the rest of the patterns belonging to the same cluster, \(b(\mathbf{x})\) is the smallest of the mean distances of \(\mathbf x\) to the elements belonging to the other clusters. The values of the index are from the range −1 to 1 and a maximum value (close to 1) provides the best partitioning of the dataset.

Now let us turn to the modification of this index [34]. This approach is based on using an additional component, which improves a performance of the index. The new index is called SILA index and it is defined as follows:

$$\begin{aligned} SILA = \frac{1}{n}\left( {\sum \limits _{\mathbf{x} \in X} {\left( {\frac{{b(\mathbf{x}) - a(\mathbf{x})}}{{\max \left( {a(\mathbf{x}),b(\mathbf{x})} \right) }} \cdot \frac{1}{{\left( {1 + a(x)} \right) ^q }} } \right) } } \right) \end{aligned}$$
(3)

where the exponent q is equal 1 and n is the number of elements in a dataset. A maximum value of the new index indicates the right partition scheme. Noted that the choice of the value of the q is very important and \(q=1\) can be too small for the very large differences of distances between data points. Hence, the new concept was proposed in paper [35]. This new index, called SILAv1, can be presented by Eq. (3), where q is defined as below:

$$\begin{aligned} q = 2 + \frac{{K^2 }}{n} \end{aligned}$$
(4)

Generally, the SILA and SILAv1 indices ensure a better performance compared to the original Silhouette index. In the next section, a detailed explanation of modification of the Simplified Silhouette index is presented.

3 Modification of the Simplified Silhouette Index

It can be noted that the Silhouette index depends on of the computation of all the distances between data elements and it can lead to a computational cost \(O(mn^2)\) [37], where m is the number of features. On the other hand, the Simplified Silhouette index is much less computationally expensive, and the overall complexity of the computation of the index is estimated as O(kmn) [37]. Although the Simplified Silhouette index is similar to the Silhouette index, there are very significant differences. First, the distance of \(\mathbf x\) to the cluster is not the average distance between \(\mathbf x\) and the rest of the elements belonging to the same cluster. It is calculated as the distance between \(\mathbf x\) and the centroid of the cluster and can be written as follows:

$$\begin{aligned} \hat{a}(\mathbf{x}) = {d\left( {\mathbf{x},\bar{C}_k } \right) } \end{aligned}$$
(5)

where \({\bar{C}_k}\) is the centroid of the cluster \({C_k}\) and \({d\left( {\mathbf{x},\bar{C}_k}\right) }\) is a function of the distance between \(\mathbf{x}\) and \({\bar{C}}_k\). Next, the distance of \(\mathbf x\) to the other cluster is defined as follows:

$$\begin{aligned} \hat{b}(\mathbf{x}) = \mathop {\min }\limits _{\scriptstyle l = 1 \atop \scriptstyle {} l \ne k }^K \ d(\mathbf{x},{\bar{C}}_l ) \end{aligned}$$
(6)

where \({\bar{C}_l}\) is the centroid of the cluster \({C_l}\) and \(l \ne k \). Finally, the \({ Simplified \ Silhouette}\) (SimSIL) index is defined as:

$$\begin{aligned} SimSIL = \frac{\mathrm{1}}{\mathrm{n}}\sum \limits _{\mathbf{x} \in X} {\frac{{\hat{b}(\mathbf{x}) - \hat{a}(\mathbf{x})}}{{\max (\hat{a}(\mathbf{x}),\hat{b}(\mathbf{x}))}}} \end{aligned}$$
(7)

where n is the number of elements in the dataset X. The value of the index is also from the range −1 to 1 and a maximum value indicates the right partition scheme.

As in the previous index, the modification of the Simplified Silhouette index is based on using the additional component, which is expressed as:

$$\begin{aligned} \hat{A}(\mathbf{x}) = \frac{1}{{\left( {1 + \hat{a}(\mathbf{x})} \right) ^{q} }} \end{aligned}$$
(8)

For the exponent \(q=1\), the newly proposed index is called SimSILA and can be written as:

$$\begin{aligned} SimSILA = \frac{1}{n}\left( {\sum \limits _{\mathbf{x} \in X} {\left( {\frac{{\hat{b}(\mathbf{x}) - \hat{a}(\mathbf{x})}}{{\max \left( {\hat{a}(\mathbf{x}),\hat{b}(\mathbf{x})} \right) }} \cdot \frac{1}{{\left( {1 + \hat{a}(x)} \right) ^q }} } \right) } } \right) \end{aligned}$$
(9)

It can be noted that the additional component \(\hat{A}(\mathbf x)\) corrects the value of the index. When a clustering algorithm greatly increases sizes of clusters, the ratio of \(1/(1+\hat{a}(x))^q\) decreases significantly and the value of the index is also decreased. However, the value \(q=1\) can be too small to appropriately correct the SimSILA index. Hence, the issue of the choice of the exponent q for \(\hat{A}(\mathbf x)\) is a very significant problem. As with the previous index, the new index called SimSILAv1 is proposed and contains a formula of the change of the exponent q depending on the number of clusters. This formula is expressed by (4). Thus, the SimSILAv1 index can be presented by Eq. (9), where q is calculated by (4). It should be noted that the new indices can take values between \(1/(1+\hat{a}(x))^q\) and \(-1/(1+\hat{a}(x))^q\). A maximum value of the index selects the right data partitioning for a dataset. In the next section, the results of the experimental studies are presented to confirm the effectiveness of these new indices.

4 Experimental Results

In this section, several experiments have been conducted on artificial and real datasets using the Partitioning Around Medoids (PAM) clustering algorithm. This algorithm is a realisation of K-medoid clustering, which is a more robust version of K-means method. Both K-medoids and the K-means algorithm are partitional, but the first method searches K representative data elements (medoids) among all elements of a dataset. After finding K medoids, K clusters are created by assigning each data point to the nearest medoid. In contrast to the K-means, the K-medoids algorithm chooses data elements as centers (medoids). Moreover, the Manhattan Norm is used to define distances between elements of the dataset. These make that the PAM algorithm is robust to noise and outliers. As mentioned in Sect. 1, the different parameter configurations of clustering algorithms can lead to different results. Thus, the choice of these input parameters is a key issue. Furthermore, one of the essential configuration parameters is a number of clusters. This parameter should be set before the start of the algorithm, but it is not usually known in advance. The common way to resolve this problem is to run the clustering algorithm multiple times with a different number of clusters and select the best result. For the clustering analyze, the range of the different number of clusters should be varied from \({K_{min}=}\) 2 to \({K_{max}=}\) \({\sqrt{n}}\), [23]. Whereas, the evaluation of results is usually realized by cluster validity indices. In experiments conducted on artificial and real datasets, the six indices, i.e. the Silhouette (SIL), SILA, SILAv1, Simplified Silhouette (SimSil), SimSILA and SimSILAv1 are used to determine the right number of clusters. To show the efficacy of the new validity indices, the results are also presented on the plots. It is assumed that the value of the validity indices equals 0 for \(K=1\). Furthermore, the min-max normalization of data has been applied to all the datasets used in the experiments. In order to better compare of the new indices, the maximum value of all the indices is modified and it is equal to 1.

4.1 Datasets

In the conducted experiments four artificial and six real datasets are used. The artificial data was called Data 1, Data 2, Data 3 and Data 4 and they were 2-dimensional with 3, 5, 8 and 11 clusters, respectively. Note that they consist of various cluster structure and densities. The scatter plot of these data is presented in Fig. 1. As it can be observed on the plot the distances between clusters are very different and some clusters are quite close. Generally, clusters are located in groups and some of the clusters are very close and others quite far. Moreover, the sizes of the clusters are different and they contain the various number of elements. Hence, many clusters validity indices can provide incorrect partitioning schemes. The real datasets are numeric data from the UCE Irvine Machine Learning Repository [19]: Diabetes, Ecoli, Glass, Iris, Spectf, Wine. The Diabetes dataset includes results of studies relating to the signs of diabetes in patients. This set includes 768 instances belonging to 2 classes and each item is described by 8 features. The second set is Ecoli dataset consisting of 336 instances, and the number of attributes equals 7. It has 8 classes, which represent the protein localization sites. Next comes the Glass dataset, which contains information about 6 types of glass defined in terms of their oxide content. The set has 214 instances and each of them is described by 9 attributes. The well-known Iris data are extensively used in many comparisons of classifiers. This set has three classes, which contain 50 instances per class. Moreover, each item is represented by four features. The Spectf dataset describes diagnosing of cardiac Single Proton Emission Computed Tomography images. This set includes 267 instances and each of them is described by 44 features. It has 2 classes. Finally, the Wine dataset shows the results of a chemical analysis of wines. It comprises three classes of wines. Altogether, the dataset contains 178 patterns, where each of them is described by 13 features.

Fig. 1.
figure 1

2-dimensional artificial datasets: (a) \({Data\;\mathrm 1}\), (b) \({Data\;\mathrm 2}\), (c) \({Data\;\mathrm 3}\), and (d) \({Data\;\mathrm 4}\)

Full size image
Table 1. A detailed description of the artificial datasets
Full size table

Additionally, Table 1 shows a detailed description of these datasets used in experiments.

Fig. 2.
figure 2

Variations of the Silhouette, SILA and SILAv1 indices with respect to the number of clusters for 2-dimensional datasets: (a) \({ Data\; \mathrm 1}\), (b) \({ Data\; \mathrm 2}\), (c) \({ Data\; \mathrm 3}\), and (d) \({ Data\; \mathrm 4}\) partitioned by the PAM method.

Full size image
Fig. 3.
figure 3

Variations of the \(Simplified \ Silhouette\), SimSILA and SimSILAv1 indices with respect to the number of clusters for 2-dimensional datasets: (a) \({ Data\; \mathrm 1}\), (b) \({ Data\; \mathrm 2}\), (c) \({ Data\; \mathrm 3}\), and (d) \({ Data\; \mathrm 4}\) partitioned by the PAM method.

Full size image

4.2 Experiments

The experimental analysis is designed to evaluate the performance of the new indices. In these studies, the partitional PAM method as the underlying clustering method was adopted to clustering of the datasets. First of all, the Silhouette, SILA and SILAv1 indices are analyzed. For this purpose, the 2-dimensional \({ Data\; \mathrm 1}\), \({ Data\; \mathrm 2}\), \({ Data\; \mathrm 3}\) and \({ Data\; \mathrm 4}\) datasets have been clustered by the PAM algorithm. As shown in Fig. 1, these datasets create groups of clusters, which are far away from each other and their sizes are very different. As mentioned above, the number of clusters is the key configuration parameter of clustering methods and it is usually varied from \(K_{min}=2\) to \(K_{max}= {\sqrt{n}}\). It is assumed that the value of the validity indices is equal 0 for \(K=1\). In Fig. 2 the comparison of the variations of the Silhouette, SILA and SILAv1 indices with respect to the number of clusters is presented for the artificial datasets. It is also noticeable that the SILA and SILAv1 indices provide the correct number of clusters for all the artificial datasets. In addition, the value of the SILAv1 index more decreases than the value of SILA for the small number of clusters, i.e. when the number \(K<c^*\) (where \(c^*\) is the right number of clusters). This means that the additional component \(A(\mathbf{x})\) used in the SILAv1 index more reduces the value of the index than the value of the SILA index. On the other hand, when the number of clusters \(K>c^*\) the component \(A(\mathbf{x})\) can increase the values of these indices slightly (see Fig. 2). On the contrary, the Silhouette index incorrectly selects all partitioning schemes and mainly provides the greatest values when the number of clusters \(K=2\). Next, the Simplified Silhouette, SimSILA, and SimSILAv1 indices are analyzed. As with the previous studies, four artificial datasets, i.e. \({ Data\; \mathrm 1}\), \({ Data\; \mathrm 2}\), \({ Data\; \mathrm 3}\) and \({ Data\; \mathrm 4}\) have been clustered by the PAM algorithm. The comparison of the variations of the Simplified Silhouette, SimSILA and SimSILAv1 indices with respect to the number of clusters is presented in Fig. 3. Despite the fact that the differences of distances between clusters are large, the SimSILA and SimSILAv1 indices provide the correct partitioning for all these data. It can be noted that the component \(\hat{A}(\mathbf{x})\) strongly reduces values of the SimSILAv1 index when the number of clusters \(K<c^*\). This is due to the fact that the exponent q in \(\hat{A}(\mathbf{x})\) is calculated by the formula (4). Generally, the component \(\hat{A}(\mathbf{x})\) improvements the results especially when the clustering algorithm combines clusters into larger ones and differences of distances between clusters are large. Then the influence of the separability measure is significant and consequently, it can strongly affect the value the index. On the other hand, when \(K>c^*\) the values of these new indices are increased slightly. It can be noted that the Simplified Silhouette and the Silhouette indices incorrectly select the number of clusters, whereas the new indices provide the right results for all the artificial datasets. The next experiments are related to the real datasets. As outlined above, the real datasets are numeric data: Diabetes, Ecoli, Glass, Iris, Spectf, Wine. In the experimental process, these datasets have been clustered by the PAM algorithm. Moreover, for the evaluation of the clustering validity, the six indices have been used. Table 2 shows the comparison of these indices taking into account the number of clusters, which is the configuration parameter for the clustering algorithm. In addition, the Table also includes results from previous experiments related to the artificial data. From the Table 2, it can be noted that for the real datasets the best results are achieved by the SILA, SILAv1, SimSILA and SimSILAv1 indices. Moreover, for the Glass and Iris data, the results of the SimSILAv1 index are better in comparison with other indices. Based on these results, it can be concluded that for all the experiments carried out on artificial and real data the best clustering results are selected by using these new indices.

Table 2. Comparison of the number of clusters obtained when using the PAM algorithm in conjunction with the SIL, SILA, SILAv1, SimSil, SimSILA and SimSILAv1 indices. N denotes the actual number of clusters in the datasets.
Full size table

5 Conclusions

In this paper new indices called SimSILA and SimSILAv1 are proposed, which are the modification of the Simplified Silhouette index. As mentioned above, neither the Simplified Silhouette index nor the Silhouette index performs well when there are large differences of distances between clusters in a dataset. Similarly to the modification of the Silhouette index, the change of the Simplified Silhouette relies on the application of the additional component, which improves the performance of the index. This additional component contains a measure of cluster compactness and reduces the high values of the index caused by large differences between clusters. In these conducted experiments, several datasets were used, where the number of clusters varied within a wide range. Moreover, the PAM clustering algorithm was selected for clustering of all the artificial and the real datasets. It has been noticeable that the SILA, SILAv1, SimSILA and SimSILAv1 indices have provided the best results. However, the Simplified Silhouette index is much less computationally expensive than the Silhouette index. From this perspective, the SimSILA and SimSILAv1 indices have the competitive performance to the SILA and SILAv1 indices in the selection of the right clustering results. All the presented results confirm the very high efficiency of the newly proposed indices.