630 Maria Halkidi and Michalis Vazirgiannis that increase (or decrease) as the number of clusters increase we search for the values of nc at which a significant local change in value of the index occurs. This change appears as a ‘knee’ in the plot and it is an indication of the number of clusters underlying the data set. Moreover, the absence of a knee may be an indication that the data set possesses no clustering structure. Below, some representative validity indices for crisp and fuzzy clustering are presented. Crisp Clustering Crisp clustering considers non overlapping partitions meaning that a data point either belongs to a class or not. In this section we discuss validity indices suitable for crisp clustering. The modified Hubert Γ statistic The definition of the modified Hubert Γ (Theodoridis and Koutroubas, 1999) statistic is given by the equation Γ =(1/M) N−1 ∑ i=1 N ∑ j=i+1 P(i, j) ·Q(i, j) (31.16) where N is the number of objects in a data set, M = N ·(N −1)/2, P is the proximity matrix of the data set and Q is an N ×N matrix whose (i, j) element is equal to the distance between the representative points (v c i ,v c j ) of the clusters where the objects x i and x j belong. Similarly, we can define the normalized Hubert σ statistic, given by equation ˆ Γ = (1/M) ∑ N−1 i=1 ∑ N j=i+1 (P(i, j) − μ P )(Q(i, j) − μ Q ) σ P · σ Q (31.17) where μ P , μ Q , σ P , σ Q are the respective means and variances of P, Q matrices. If the d(v c i ,v c j ) is close to d(x i ,x j ) for i, j = 1,2, ,N, P and Q will be in close agreement and the values of Γ and ˆ Γ (normalized Γ ) will be high. Conversely, a high value of Γ ( ˆ Γ ) indicates the existence of compact clusters. Thus, in the plot of normalized Γ versus n c , we seek a significant knee that corresponds to a significant increase of normalized G. The number of clusters at which the knee occurs is an indication of the number of clusters that occurs in the data. We note, that for n c = 1 and n c = N the index is not defined. Dunn family of indices A cluster validity index for crisp clustering proposed in (Dunn, 1974), aims at the identifica- tion of ‘compact and well separated clusters’. The index is defined in the following equation for a specific number of clusters D n c = min i=1, ,n c min j=i+1, ,n c d(c i ,c j ) max k=1, ,n c (diam(c k )) (31.18) where d(c i ,c j ) is the dissimilarity function between two clusters c i and c j defined as d(c i ,c j )=min x∈C i ,y∈C j d(x, y) , and diam(c) is the diameter of a cluster, which may be con- sidered as a measure of clusters’ dispersion. The diameter of a cluster C can be defined as follows: diam(C)=max x,y∈C { d(x, y) } (31.19) If the data set contains compact and well-separated clusters, the distance between the clusters is expected to be large and the diameter of the clusters is expected to be small. Thus, 31 Quality Assessment Approaches in Data Mining 631 based on the Dunn’s index definition, we may conclude that large values of the index indicate the presence of compact and well-separated clusters. Index D n c does not exhibit any trend with respect to number of clusters. Then the maxi- mum in the plot of D n c versus the number of clusters can be an indication of the number of clusters that fits the data. The problems of the Dunn index are: i) its considerable time complexity, and ii) its sen- sitivity to the presence of noise in data sets, since these are likely to increase the values of diam(c) (i.e. dominator of equation 31.18). Three indices, are proposed in (Pal and Biswas, 1997) that are more robust to the presence of noise. They are known as Dunn-like indices since they are based on the Dunn index. More- over, the three indices use for their definition the concepts of Minimum Spanning Tree (MST), the relative neighborhood graph (RNG) and the Gabriel graph respectively (Theodoridis and Koutroubas, 1999). Consider the index based on MST. Let a cluster ci and the complete graph G i whose vertices correspond to the vectors of c i . The weight, we, of an edge, e, of this graph equals the distance between its two end points, x,y. Let E MST i be the set of edges of the MST of the graph G i , and e MST i the edge in E MST i with the maximum weight. Then the diameter of C i is defined as the weight of e MST i . Dunn-like index based on the concept of the MST is given by equation D n c = min i=1, ,n c min j=i+1, ,n c d(c i ,c j ) max k=1, ,n c (diam MST k ) (31.20) The number of clusters at which D MST m takes its maximum value indicates the number of clusters in the underlying data. Based on similar arguments we may define the Dunn-like indices for GG and RGN graphs. The Davies-Bouldin (DB) index A similarity measure R ij between the clusters C i and C j is defined based on a measure of dispersion of a cluster C i , denoted by s i , and a dissimilarity measure between two clusters, d ij . The R ij index is defined to satisfy the following conditions (Davies and Bouldin, 1979): 1. R ij = 0 2. R ij = R ji 3. if s i = 0 and s j = 0 then R ij = 0 4. if s j > s k and d ij = d ik then R ij > R ik 5. if s j = s k and d ij < d ik then R ij > R ik . These conditions state that R ij is non-negative and symmetric. A simple choice for R ij that satisfies the above conditions is (Davies and Bouldin, 1979): R ij =(s i + s j )/d ij (31.21) Then the DB index is defined as DB n c = 1 n c · n c ∑ i=1 R i ,andR i = max j=1, ,n c , j=i {R ij },i = 1, ,n c (31.22) It is clear for the above definition that DB n c is the average similarity between each cluster c i , i = 1, ,n c and its most similar one. It is desirable for the clusters to have the minimum possible similarity to each other; therefore we seek partitionings that minimize DB n c . The DB n c index exhibits no trends with respect to the number of clusters and thus we seek the minimum value of DB n c in its plot versus the number of clusters. 632 Maria Halkidi and Michalis Vazirgiannis Some alternative definitions of the dissimilarity between two clusters as well as the dis- persion of a cluster, c i , is defined in (Davies and Bouldin, 1979). Three variants of the DB n c index are proposed in (Pal and Biswas, 1997). They are based on the MST, RNG and GG concepts, similar to the cases of the Dunn-like indices. Other validity indices for crisp clustering have been proposed in (Dave, 1996) and (Mil- ligan and Cooper, 1985). The implementation of most of these indices is computationally very expensive, especially when the number of clusters and objects in the data set grows very large (Xie and Beni, 1991). In (Milligan and Cooper, 1985), an evaluation study of 30 validity indices proposed in literature is presented. It is based on tiny data sets (about 50 points each) with well-separated clusters. The results of this study (Milligan and Cooper, 1985) place Caliski and Harabasz (1974), Je(2)/Je(1) (1984), C-index (1976), Gamma and Beale among the six best indices. However, it is noted that although the results concerning these methods are encouraging they are likely to be data dependent. Thus, the behavior of indices may change if different data structures are used. Also, some indices are based on a sample of clustering results. A representative example is Je(2)/Je(1) whose computations based only on the information provided by the items involved in the last cluster merge. RMSSDT, SPR, RS, CD This family of validity indices is applicable in the cases that hierarchical algorithms are used to cluster the data sets. Below we present the definitions of four validity indices, which have to be used simultaneously to determine the number of clusters existing in the data set. These four indices are applied to each step of a hierarchical clustering algorithm (Sharma, 1996). • RMSST D (root mean square standard deviation) of a new clustering scheme defined at a level of a clustering hierarchy is the square root of the variance of all the variables (attributes used in the clustering process). This index measures the homogeneity of the formed clusters at each step of the hierarchical algorithm. Since the objective of cluster analysis is to form homogeneous groups the RMSST D of a cluster should be as small as possible. Where the values of RMSSTD are higher than the ones of the previous step, we have an indication that the new clustering scheme is worse. In the following definitions we shall use the term SS, which means Sum of Squares and refers to the equation: SS = n ∑ i=1 (X i − ¯ X) 2 (31.23) Along with this we shall use some additional symbolism like: 1. SSw referring to the sum of squares within group, 2. SSb referring to the sum of squares between groups, 3. SSt referring to the total sum of squares, of the whole data set. • SPR (Semi-Partial R-squared) for the new cluster is the difference between SSw of the new cluster and the sum of the SSw’s values of clusters joined to obtain the new cluster (loss of homogeneity), divided by the SSt for the whole data set. This index measures the loss of homogeneity after merging the two clusters of a single algorithm step. If the index value is zero then the new cluster is obtained by merging two perfectly homogeneous clusters. If its value is high then the new cluster is obtained by merging two heterogeneous clusters. • RS (R-Squared) of the new cluster is the ratio of SSb over SSt. SSb is a measure of differ- ence between groups. Since SSt = SSb + SSw, the greater the SSb the smaller the SSw and vice versa. As a result, the greater the differences between groups, the more homogenous 31 Quality Assessment Approaches in Data Mining 633 each group is and vice versa. Thus, RS may be considered as a measure of dissimilarity between clusters. Furthermore, it measures the degree of homogeneity between groups. The values of RS range between 0 and 1. Where the value of RS is zero, there is an indi- cation that no difference exists among groups. On the other hand, when RS equals 1 there is an indication of significant difference among groups. • The CD index measures the distance between the two clusters that are merged in a given step of the hierarchical clustering. This distance depends on the selected representatives for the hierarchical clustering we perform. For instance, in the case of Centroid hierarchi- cal clustering the representatives of the formed clusters are the centers of each cluster, so CD is the distance between the centers of the clusters. Where we use single linkage, CD measures the minimum Euclidean distance between all possible pairs of points. In case of complete linkage, CD is the maximum Euclidean distance between all pairs of data points, and so on. Using these four indices we determine the number of clusters that exist in a data set, plotting a graph of all these indices values for a number of different stages of the clustering algorithm. In this graph we search for the steepest knee, or in other words, the greatest jump of these indices’ values from the higher to the smaller number of clusters. The SD validity index Another clustering validity approach is proposed in (Halkidi et al., 2000). The SD validity index definition is based on the concepts of average scattering for clusters and total separation between clusters. Below, we give the fundamental definition for this index. Average scattering for clusters. It evaluates scattering of the points in the clusters comparing the variance of the considered clustering scheme with the variance of the whole data set. The average scattering for clusters is defined as follows: Scat(n c )= 1 n c · ∑ n c i=1 σ (v i ) σ (S) (31.24) The term σ (S) is the variance of a data set; and its p-th dimension is defined as follows: σ p = 1 n · n ∑ k=1 x p k − ¯x p (31.25) where ¯x p is the p-th dimension of ¯ X = 1 n · ∑ n k=1 x k ,∀x k ∈ S. The term σ (v i ) is the variance of cluster c i and its p-th dimension is given by the equation σ (v i ) p = n i ∑ k=1 (x p k −v p i ) 2 /n i (31.26) Further the term Y is defined as: Y = Y T Y 1/2 , where Y =(y 1 , ,y k ) is a vector (e.g. σ (v i )). Total separation between clusters. The definition of total scattering (separation) between clusters is given in the following equation: Dis(n c )= D max D min · n c ∑ k=1 n c ∑ z=1 v k −v z −1 (31.27) 634 Maria Halkidi and Michalis Vazirgiannis where D max = max( v i −v j ) i, j ∈ 1,2,3,,n c is the maximum distance between cluster cen- ters. The D min = min( v i −v j )), ∀i, j ∈{1, 2, ,n c } is the minimum distance between clus- ter centers. Now, we can define a validity index based on 31.24 and 31.27 as follows SD(n c )=a ·Scat(n c )+Dis(n c ) (31.28) where a is a weighting factor equal to Dis(c max ) and where c max is the maximum number of input clusters. The first term (i.e. Scat(n c )) is defined in Eq. 24, indicating the average compactness of clusters (i.e. intra-cluster distance). A small value for this term indicates compact clusters and as the scattering within clusters increases (i.e. they become less compact) the value of Scat(n c ) also increases. The second term Dis(n c ) indicates the total separation between the nc clusters (i.e. an indication of inter-cluster distance). Contrary to the first term the second one, Dis(n c ), is influenced by the geometry of the clusters and increases with the number of clusters. The two terms of SD are of the different range, thus a weighting factor is needed in order to incorporate both terms in a balanced way. The number of clusters, n c , that minimizes the above index is an optimal value. Also, the influence of the maximum number of clusters c max , related to the weighting factor, in the selection of the optimal clustering scheme, is discussed in (Halkidi et al., 2000). It is proved that SD proposes an optimal number of clusters almost irrespectively of the c max value. The S Dbw validity index A recent validity index is proposed in (Halkidi and Vazirgiannis, 2001a). It exploits the inherent features of clusters to assess the validity of results and select the optimal partitioning for the data under concern. Similarly with the SD index, its definition is based on the compact- ness and separation of clusters. The average scattering for clusters is defined as above in 31.24. Inter-cluster Density (ID) - It evaluates the average density in the region among clus- ters in relation with the density of the clusters. The goal is the density among clusters to be significantly low in comparison with the density in the considered clusters. Then, considering a partitioning of the data set into more than two clusters (i.e. n c > 1) the inter-cluster density is defined as follows: Dens bw(c)= 1 n c ·(n c −1) n c ∑ i=1 n c ∑ j=1, j=i density(u ij ) max{density(v i ),density(v j )} ,c > 1 (31.29) where v i , v j are the centers of clusters c i , c j respectively, and u ij the middle point of the line segment defined by the clusters’ centers v i , v j . The term density(u) is defined in the following equation: density(u)= n ij ∑ l=1 f (x l ,u) (31.30) where x l is a point of data set S, n ij is the number of points (tuples) that belong to the clusters c i and c j , i.e. x l ∈ c i ∪c j ⊆ S. It represents the number of points in the neighborhood of u.In our work, the neighborhood of a data point, u, is defined to be a hyper-sphere with center u and radius the average standard deviation of the clusters, stdev. The standard deviation of the clusters is given by the following equation: 31 Quality Assessment Approaches in Data Mining 635 stdev = 1 n c n c ∑ i=1 σ (v i ) where c is the number of clusters and s(v i ) is the variance of cluster C i . More specifically, the function f (x, u) is defined as: Y = 0ifd(x,u) > stdev , 1 otherwise (31.31) It is obvious that a point belongs to the neighborhood of u if its distance from u is smaller than the average standard deviation of clusters. Here we assume that the data has been scaled to consider all dimensions (bringing them into comparable ranges), as is equally important during the process of finding the neighbors of a multidimensional point (Berry and Linoff, 1996). Then the validity index S Dbw is defined as: S Dbw(n c )=Scat(n c )+Dens bw(n c ) (31.32) The above definitions refer to the case that a cluster presents clustering tendency, i.e. it can be partitioned into at least two clusters. The index is not defined for n c = 1. The definition of S Dbw indicates that both criteria of ‘good’ clustering (i.e. compactness and separation) are properly combined, enabling reliable evaluation of clustering results. Also, the density variations among clusters are taken into account to achieve more reliable results. The number of clusters, n c , that minimizes the above index is an optimal value indicating the number of clusters present in the data set. Moreover, an approach based on the S Dbw index is proposed in (Halkidi and Vazirgian- nis, 2001b). It evaluates the clustering schemes of a data set as defined by different clustering algorithms and selects the algorithm resulting in optimal partitioning of the data. In general terms, S Dbw enables the selection both of the algorithm and its parameter values for which the optimal partitioning of a data set is defined (assuming that the data set presents clustering tendency). However, the index cannot properly handle arbitrarily shaped clusters. The same applies to all the aforementioned indices. There are a number of applications where it is important to identify non-convex clusters such as medical or spatial data applications. An approach to handle arbitrarily shaped clusters in the cluster validity process is presented in (Halkidi and Vazirgiannis, 2002). 31.4.5 Fuzzy Clustering In this section, we present validity indices suitable for fuzzy clustering. The objective is to seek clustering schemes where most of the vectors of the data set exhibit a high degree of mem- bership in one cluster. Fuzzy clustering is defined by a matrix U = u ij , where u ij denotes the degree of membership of the vector x i in cluster j. Also, a set of cluster representatives is defined. Similar to a crisp clustering case a validity index, q! is defined and we search for the minimum or maximum in the plot of q versus n c . Also, where q exhibits a trend with respect to the number of clusters, we seek a significant knee of decrease (or increase) in the plot of q. Below two categories of fuzzy validity indices are discussed. The first category uses only the membership values, u ij , of a fuzzy partition of data. The second involves both the U matrix and the data set itself. 636 Maria Halkidi and Michalis Vazirgiannis Validity Indices involving only the membership values Bezdek proposed in (Bezdeck et al., 1984) the partition coefficient, which is defined as PC = 1 N N ∑ i=1 n c ∑ j=1 u 2 ij (31.33) The PC index values range in [1/n c ,1], where n c is the number of clusters. The closer the index is to unity the ”crisper” the clustering is. In case that all membership values to a fuzzy partition are equal, that is, u ij = 1/n c , the PC obtains its lower value. Thus, the closer the value of PC is to 1/n c , the fuzzier the clustering is. Furthermore, a value close to 1/n c indicates that there is no clustering tendency in the considered data set or the clustering algorithm failed to reveal it. The partition entropy coefficient is another index of this category. It is defined as follows PE = − 1 N N ∑ i=1 n c ∑ j=1 u ij ·log a (u ij ) (31.34) where a is the base of the logarithm. The index is computed for values of n c greater than 1 and its value ranges in [0,log a n c ]. The closer the value of PE to 0, the ‘crisper’ the clustering is. As in the previous case, index values close to the upper bound (i.e. log a n c ), indicate absence of any clustering structure in the data set or inability of the algorithm to extract it. The drawbacks of these indices are: • their monotonous dependency on the number of clusters. Thus, we seek significant knees of increase (for PC) or decrease (for PE) in the plots of the indices versus the number of clusters, • their sensitivity to the fuzzifier, m. More specifically, as m → 1 the indices give the same values for all values of n c . On the other hand when m → ∞ , both PC and PE exhibit significant knee at n c = 2, • the lack of direct connection to the geometry of the data (Dave, 1996), since they do not use the data itself. Indices involving the membership values and the data set The Xie-Beni index (Xie and Beni, 1991), XB, also called the compactness and separation validity function, is a representative index of this category. Consider a fuzzy partition of the data set X = {x j ; j = 1, ,n} with v i ,(i = 1, ,n c ) the centers of each cluster and u ij the membership of the jth data point belonging to the ith cluster. The fuzzy deviation of x j form cluster i, d ij , is defined as the distance between x j and the center of cluster weighted by the fuzzy membership of data point j with regards to cluster i. It is given by the following equation: d ij = u ij x j −v i (31.35) Also, for a cluster i, the sum of the squares of fuzzy deviation of the data point in X , denoted σ i , is called variation of cluster i. The term π i =( σ i /n i ), is called compactness of cluster i. Since n i is the number of point in cluster belonging to cluster i, π i is the average variation in cluster i. Then the compactness of a partitioning of n c clusters is defined as the average compactness of the defined clusters, given by the equation: π = ∑ n c i=1 π i n c (31.36) Also, the separation of the fuzzy partitions is defined as the minimum distance between cluster centers, that is 31 Quality Assessment Approaches in Data Mining 637 d min = min v i −v j (31.37) Then XB index is defined as XB = π N ·(d min ) 2 (31.38) where N is the number of points in the data set. It is clear that small values of XB are expected for compact and well-separated clusters. We note, however, that XB is monotonically decreasing when the number of clusters n c gets very large and close to n. One way to eliminate this decreasing tendency of the index is to determine a starting point, c max , of the monotonic behavior and to search for the minimum value of XB in the range [2, c max ]. Moreover, the values of the index XB depend on the fuzzifier values, so as if m →∞ then XB → ∞. Another index of this category is the Fukuyama-Sugeno index, which is defined as FS m = N ∑ i=1 n c ∑ j=1 u m ij x i −v j 2 A − v j −v 2 A (31.39) where v is the mean vector of X and A is an lxl positive definite, symmetric matrix. When A = I, the above distance becomes the squared Euclidean distance. It is clear that for compact and well-separated clusters we expect small values for FS m . The first term in brackets measures the compactness of the clusters while the second one measures the distances of the clusters representatives. Also some other fuzzy validity indices are proposed in (Gath and Geva, 1989), which are based on the concepts of hyper volume and density. 31.4.6 Other Approaches for Cluster Validity Another approach for finding the optimal number of clusters of a data set was proposed in (Smyth, 1996). It introduces a practical clustering algorithm based on Monte Carlo cross- validation. More specifically, the algorithm consists of M cross-validation runs over M chosen train/test partitions of a data set, D. For each partition u, the EM algorithm is used to define n c clusters to the training data, while n c is varied from 1 to c max . Then, the log-likelihood L u c (D) is calculated for each model with n c clusters. It is defined using the probability density function of the data as L k (D)= N ∑ i=1 logf k (x i / Φ k ) (31.40) where f k is the probability density function for the data and Φ k denotes parameters that have been estimated from data. 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