Comparison of hierarchical cluster analysis methods by cophenetic correlation

Abstract

This study proposes the best clustering method(s) for different distance measures under two different conditions using the cophenetic correlation coefficient.

In the first one, the data has multivariate standard normal distribution without outliers for [InlineEquation not available: see fulltext.] and the second one is with outliers (5%) for [InlineEquation not available: see fulltext.]. The proposed method is applied to simulated multivariate normal data via MATLAB software.

According the results of simulation the Average (especially for [InlineEquation not available: see fulltext.]) and Centroid (especially for [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext.]) methods are recommended at both conditions.

This study hopes to contribute to literature for making better decisions on selection of appropriate cluster methods by using subgroup sizes, variable numbers, subgroup means and variances.[PUBLICATION ABSTRACT]

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R E S E A R C H Open Access

Comparison of hierarchical cluster analysis methods by cophenetic correlation

Sinan Sarali1*, Nurhan Doan2 and Ismet Doan2

*Correspondence: [email protected]

1Department of Statistics, Faculty of Arts and Sciences, Afyon Kocatepe University, Afyonkarahisar, 03200, TurkeyFull list of author information is available at the end of the article

AbstractPurpose: This study proposes the best clustering method(s) for dierent distance measures under two dierent conditions using the cophenetic correlation coecient.

Methods: In the rst one, the data has multivariate standard normal distribution without outliers for n = 10, 50, 100 and the second one is with outliers (5%) forn = 10, 50, 100. The proposed method is applied to simulated multivariate normal data via MATLAB software.

Results: According the results of simulation the Average (especially for n = 10) and Centroid (especially for n = 50 and n = 100) methods are recommended at both conditions.

Conclusions: This study hopes to contribute to literature for making better decisions on selection of appropriate cluster methods by using subgroup sizes, variable numbers, subgroup means and variances.

Keywords: cophenetic correlation; hierarchical clustering methods; distance measures

1 Introduction

Classication, in its widest sense, has to do with forms of the relatedness and with the organization and display of the relations in a useful manner. The items to be studied could be anything: people, bacteria, religions, books, etc. The attributes in each case would be those features of the items that are of interest for the purpose of the study []. Classications are generally pictured in the form of hierarchical trees, also called a dendrogram. A dendrogram is the graphical representation of an ultrametric (= cophenetic) matrix; so dendrograms can be compared to one another by comparing their cophenetic matrices [].

Cluster Analysis (CA), Principal Components Analysis (PCA) and Discriminant Analysis (DA) are three of the primary methods of modern multivariate analysis. Because of its utility, clustering has emerged as one of the leading methods of multivariate analysis [].

Cluster analysis is a multivariate statistical technique which was originally developed for biological classication. Biologists Robert Soka and Peter Sneath published their seminal text Principles of Numerical Taxonomy in . Sokal and Sneath demonstrated that cluster analysis could be utilized to eciently classication a data set which contained all relevant characteristics of an organism. When the organisms had been classied based on these characteristics, it could be determined in which way they diered, and if they belonged to dierent species. In this way, Sokal and Sneath asserted, researchers could trace the path of evolution from one species to another [].

2013 Sarali et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0

Web End =http://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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In this study for clustering, two measures of cluster goodness or quality are used. One type of measure allows us to compare dierent sets of clusters without reference to external knowledge and is called an internal quality which is used as a measure of overall similarity based on the pairwise similarity of documents in a cluster. The other type of measures allows evaluating how well the clustering is working by comparing the groups produced by the clustering techniques to known classes. This type of measure is called an external quality measure, which is not scope of this study [].

The joining or tree clustering method uses the dissimilarities (similarities) or distances (Euclidean distance, squared Euclidean distance, city-block (Manhattan) distance, Chebychev distance, power distance, Mahalanobis distance, etc.) between objects when forming the clusters. Similarities are a set of rules that serve as criteria for grouping or separating items. These distances (similarities) can be based on a single dimension or multiple dimensions, with each dimension representing a rule or condition for grouping objects. The joining algorithm does not care whether the distances that are fed to it are actual real distances, or some other derived measure of distance that is more meaningful to the researcher; and it is up to the researcher to select the right method for his/her specic application [].

The next step is to identify how one can nd the natural clusters among items characterized by many attributes. A number of cluster analysis procedures (single linkage (nearest neighbor), Complete linkage (furthest neighbor), Unweighted pair-group average (UPGMA), Weighted pair-group average (WPGMA), Unweighted pair-group centroid (UPGMC), Weighted pair-group centroid (median), Wards method, etc.) are available; many of these begin with an n-dimensional space in which each entity is represented by a single point. The dimensions in the space represent the characteristics upon which the entities are to be compared. Similarity between entities can be measured by: () the correlation of entities scores on the dimensions (cophenetic correlation) or () the distance between points in the space (points closest to each other are most similar) [, ].

Suppose that the original data {Xi} have been modeled using a cluster method to pro

duce a dendrogram {Ti}; that is, a simplied model in which data that are close have been

grouped into a hierarchical tree. Dene the following distance measures. x(i, j) = |Xi Xj|, the ordinary Euclidean distance between the ith and jth observations. t(i, j) = the dendro-grammatic distance between the model points Ti and Tj. This distance is the height of the node at which these two points are rst joined together. Then, letting x be the average of the x(i, j), and letting t be the average of the t(i, j), the cophenetic correlation coecient c is dened as in () [].

c = [summationtext]

i<j(x(i, j) x)(t(i, j) t)

[[summationtext]i<j(x(i, j) x)][[summationtext]i<j(t(i, j) t)]. ()

Since its introduction by Sokal and Rohlf [], the cophenetic correlation coecient has been widely used in numerical phenetic studies, both as a measure of degree of t of a classication to a set of data and as a criterion for evaluating the eciency of various clustering techniques []. In statistics, and especially in biostatistics, cophenetic correlation (more precisely, the cophenetic correlation coecient) is a measure of how faithfully a dendrogram preserves the pairwise distances between the original unmodeled data points. Although it has been most widely applied in the eld of biostatistics (typically to

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assess cluster-based models of DNA sequences, or other taxonomic models), it can also

be used in other elds of inquiry where raw data tend to occur in clumps, or clusters. This

coecient has also been proposed for use as a test for nested clusters [].

The problem of comparing classications with numerical methods is not new; the rst eective numerical method known to us is the cophenetic correlation technique of Sokal and Rohlf []. Beginning with the development of cophenetic correlations methods for comparison of dendrograms have recently been the object of strong interest. Baker [] investigated the impact of observational errors on the dendrograms produced by the complete linkage and single linkage hierarchical grouping techniques. The goodness of t of the dendrograms was measured by means of the Goodman-Kruskal gamma coecient. The gamma coecients indicated that the single linkage grouping technique was more sensitive to the type of data errors employed than the complete linkage technique. Hubert [] compared two rank orderings of the object pairs. He tested hypothesis that the given set of proximity values have been assigned randomly by referring the Goodman-Kruskal rank correlation statistic to an approximate permutation distribution. Kuiper and Fisher [] compared six hierarchical clustering procedures (single linkage, complete linkage, median, average linkage, centroid and Wards method) for multivariate normal data, assuming that the true number of clusters was known. The authors used the Rand index, which gives a proportion of correct groupings, to compare the clustering methods. In their study for clusters of equal sizes, Wards method and complete linkage method, with very unequal cluster sizes centroid and average linkage method found best, respectively. Blasheld [] compared four types of hierarchical clustering methods (single linkage, complete linkage, average linkage and Wards method) for accuracy in recovery of original population clusters. He used Cohens statistic to measure the accuracy of the clustering methods. According to his results, Wards method performed signicantly better than the other clustering procedures and average linkage gave relatively poor results. According to Milligan [], complete linkage and Wards method reacted badly when outliers were introduced into the simulated data.

Hands and Everitt [] compared ve hierarchical clustering techniques (single linkage, complete linkage, average, centroid, and Wards method) on multivariate binary data. They found that Wards method was the best overall than other hierarchical methods. Yao [] discussed six classical clustering algorithms: k-means, SOM, EM-based clustering, classication EM clustering, fuzzy k-means, leader clustering and dierent combination scenarios of these algorithms. He used a count of cluster categories, classication accuracy and cluster entropy. Ferreira and Hitchcock [] compared the performance of four major hierarchical methods (single linkage, complete linkage, average linkage and Wards method) for clustering functional data. They used the Rand index to compare the performance of each clustering method. According to their study, Wards method was usually the best, while average linkage performed best in some special situations, in particular, when the number of clusters is over specied. Milligan and Cooper [] used four agglomerative hierarchical clustering methods to generate partition solutions and formed one factor in the overall design. These were the single link, complete link, group average (UPGMA) and Wards minimum variance methods. As a result, they found that the single link technique was least eective while the group average and Wards methods gave the best overall recovery.

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Consider the studies in the literature and the importance of using the most convenient cluster method under dierent conditions (sample size, variables number and distance measures), a detailed simulation study is undertaken. This study gives more insight into the functioning of the cluster method under dierent conditions. The purpose of this research is to investigate the best clustering method under dierent conditions.

2 Method

In this study, seven cluster analysis methods are compared by the cophenetic correlation coecient computed according to dierent clustering methods with a sample size (n = , n = and n = ), variables number (x = , x = and x = ) and distance measures via a simulation study. The simulation program is developed in a MATLAB software development environment by the authors. We have dierent simulation scenarios and ,/n replications for each scenario. The performance is monitored by two dierent conditions that are mentioned in Table and Table with cluster methods, distance measures by cophenetic correlation coecient in various settings of subgroup means, variances, sample size and variable numbers simultaneously.

For dierent simulation scenarios, the data was derived from multivariate normal distribution for = , = with and without outliers, respectively. The data set for outliers is obtained according to Dixons [] Outlier Model like (N r) N(, ) + r

N(, ). In this study, r = [, + , N] means that while % of the data set does not

include any outliers, % of the data set includes outliers.

3 Results and discussion

All numerical results, obtained by running the simulation program, are given in Table and Table . According to Table and Table , the average method gives the best results at all measures and at all variable numbers for both distributions with sample size n = . Moreover, increasing the sample size to n = and n = favors the complete, weighted, and centroid methods for all measures. However, the cophenetic correlation coecient for the Mahalanobis measure cannot be calculated in both distributions when there are variables with sample size n = , whereas there is not any meaningful explanation for this unexpected result, we still could not nd the main reason for this situation, but the same result is obtained for more than three times run of the simulation program.

4 Conclusion

In general, researchers especially nonstatisticians use cluster analysis methods and distance measures in dierent conditions. In addition, they choose to use the most famous cluster analysis methods and distance measures, which are available in statistical packages, without evaluating the validity of dierent conditions. When the dierent conditions are considered, drawn inferences are dubious, and may lead the decision-makers to incorrect decisions. It is noted that, with respect to the selection of a distance measures, the researcher must be aware that their choice can often signicantly aect the results of the clustering. For example, some distance measures are inappropriate when dierent conditions of the variables are not met. On this point, the determination of the correct distance measures to use under various cases is the main motivation of researchers working on this subject to determine which distance measures should be used in case of dierent conditions.

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Table 1 The cophenetic correlation coefcient values for = 0, 2 = 1 (without outliers)

Distance measure

Average 0.7381 0.6352 0.5978 0.7090 0.5974 0.5546 0.6792 0.5700 0.5315 Centroid 0.7285 0.6402 0.6056 0.6874 0.6085 0.5765 0.6254 0.5831 0.5632 Complete 0.6565 0.4826 0.4271 0.6128 0.4014 0.3325 0.5659 0.3318 0.2586 Median 0.7102 0.5538 0.4745 0.6679 0.5386 0.4553 0.6013 0.5344 0.5039 Single 0.6874 0.6038 0.5950 0.6453 0.5656 0.5578 0.6009 0.5324 0.5301 Ward 0.6533 0.4830 0.4350 0.6093 0.3995 0.3421 0.5661 0.3264 0.2589 Weighted 0.7191 0.5351 0.4615 0.6904 0.4978 0.4054 0.6623 0.4768 0.3969

Mahalanobis Average 0.6957 0.6276 0.5950 0.6176 0.5865 0.5469 NaN 0.5591 0.5284

Centroid 0.6749 0.6325 0.6010 0.5864 0.5993 0.5723 NaN 0.5789 0.5630 Complete 0.5627 0.4578 0.4121 0.3958 0.3625 0.3097 NaN 0.2711 0.2270 Median 0.6587 0.5426 0.4719 0.5762 0.5320 0.4538 NaN 0.5385 0.5106 Single 0.6565 0.6032 0.5956 0.5642 0.5655 0.5589 NaN 0.5293 0.5317 Ward 0.5541 0.4621 0.4235 0.3876 0.3539 0.3147 NaN 0.2495 0.2155 Weighted 0.6760 0.5202 0.4549 0.6078 0.4843 0.3963 NaN 0.4869 0.4013

Cityblock Average 0.7427 0.6228 0.5844 0.7120 0.5810 0.5349 0.6787 0.5484 0.5076

Centroid 0.7372 0.6280 0.5944 0.6983 0.5902 0.5557 0.6343 0.5579 0.5368 Complete 0.6716 0.4780 0.4229 0.6247 0.3976 0.3272 0.5771 0.3285 0.2502 Median 0.7194 0.5460 0.4710 0.6799 0.5233 0.4455 0.6129 0.5117 0.4774 Single 0.6876 0.5839 0.5756 0.6443 0.5410 0.5324 0.5975 0.5037 0.5009 Ward 0.6728 0.4869 0.4348 0.6281 0.4052 0.3427 0.5823 0.3322 0.2596 Weighted 0.7244 0.5290 0.4546 0.6936 0.4878 0.3969 0.6615 0.4601 0.3824

Minkowski Average 0.7552 0.6393 0.6009 0.7255 0.6017 0.5557 0.6922 0.5728 0.5339

Centroid 0.7497 0.6447 0.6087 0.7099 0.6124 0.5791 0.6450 0.5833 0.5629 Complete 0.6858 0.4940 0.4313 0.6397 0.4135 0.3386 0.5920 0.3439 0.2640 Median 0.7309 0.5600 0.4797 0.6897 0.5412 0.4605 0.6220 0.5338 0.4989 Single 0.7034 0.6033 0.5947 0.6633 0.5663 0.5577 0.6157 0.5332 0.5295 Ward 0.6832 0.4956 0.4422 0.6384 0.4124 0.3496 0.5895 0.3367 0.2664 Weighted 0.7367 0.5428 0.4692 0.7064 0.5024 0.4105 0.6749 0.4791 0.3941

Cosine Average 0.7590 0.6277 0.5839 0.6994 0.5143 0.4524 0.6441 0.4152 0.3413

Centroid 0.7518 0.6045 0.5478 0.6866 0.4711 0.3894 0.6097 0.3242 0.2312 Complete 0.7230 0.5782 0.5320 0.6501 0.4518 0.3929 0.5808 0.3428 0.2700 Median 0.7340 0.5530 0.5040 0.6681 0.4273 0.3546 0.5911 0.3004 0.2146 Single 0.6931 0.4695 0.3898 0.6034 0.3070 0.2202 0.5211 0.2017 0.1275 Ward 0.7381 0.6142 0.5730 0.6716 0.4979 0.4418 0.6083 0.3974 0.3280 Weighted 0.7433 0.5786 0.5336 0.6854 0.4711 0.4093 0.6311 0.3834 0.3069

Correlation Average 0.8217 0.7470 0.7226 0.7229 0.5581 0.5065 0.6507 0.4268 0.3542

Centroid 0.8169 0.7358 0.7037 0.7124 0.5245 0.4594 0.6212 0.3441 0.2518 Complete 0.7979 0.7083 0.6775 0.6791 0.4987 0.4514 0.5893 0.3545 0.2855 Median 0.7982 0.6886 0.6621 0.6952 0.4754 0.4128 0.6028 0.3177 0.2324 Single 0.7939 0.6715 0.6285 0.6389 0.3624 0.2781 0.5307 0.2124 0.1355 Ward 0.8069 0.7388 0.7155 0.6974 0.5429 0.4938 0.6170 0.4099 0.3426 Weighted 0.8052 0.7061 0.6816 0.7075 0.5105 0.4534 0.6379 0.3916 0.3185

Spearman Average 0.8207 0.7600 0.7441 0.7240 0.5636 0.5163 0.6487 0.4274 0.3567

Centroid 0.8116 0.7413 0.7199 0.7132 0.5336 0.4715 0.6184 0.3452 0.2511 Complete 0.8094 0.7788 0.7780 0.6815 0.5337 0.4966 0.5884 0.3572 0.2896 Median 0.7903 0.7162 0.7114 0.6958 0.4818 0.4235 0.5990 0.3175 0.2329 Single 0.6854 0.6737 0.6738 0.6397 0.4010 0.1830 0.5287 0.2198 0.1427 Ward 0.7892 0.7358 0.7275 0.6973 0.5488 0.5030 0.6145 0.4093 0.3434 Weighted 0.8118 0.7788 0.7780 0.7086 0.5154 0.4672 0.6357 0.3916 0.3194

Cophenetic correlation coefcientx = 3 x = 5 x = 10n = 10 n = 50 n = 100 n = 10 n = 50 n = 100 n = 10 n = 50 n = 100

Euclidean Average 0.7552 0.6358 0.6009 0.7255 0.6017 0.5753 0.6922 0.5728 0.5339

Centroid 0.6927 0.6393 0.6038 0.6463 0.6028 0.5557 0.5829 0.5759 0.5605 Complete 0.6858 0.4940 0.4313 0.6397 0.4135 0.3386 0.5920 0.3439 0.2640 Median 0.7309 0.5600 0.4797 0.6897 0.5412 0.4605 0.6220 0.5338 0.4989 Single 0.7034 0.6033 0.5947 0.6633 0.5663 0.5577 0.6157 0.5332 0.5295 Ward 0.6832 0.4956 0.4422 0.6384 0.4124 0.3496 0.5895 0.3367 0.2664 Weighted 0.7367 0.5428 0.4692 0.7064 0.5024 0.4105 0.6749 0.4791 0.3941

Squared

Euclidean

Clustering method

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Table 1 (Continued)

Distance measure

Table 2 The cophenetic correlation coefcient values for = 0, 2 = 1 (with outliers)

Distance measure

Average 0.8434 0.8088 0.7859 0.8239 0.7837 0.7636 0.8087 0.7663 0.7490 Centroid 0.8386 0.8107 0.7900 0.8123 0.7880 0.7730 0.7768 0.7695 0.7629 Complete 0.8022 0.7331 0.7027 0.7724 0.6937 0.6580 0.7505 0.6550 0.6166 Median 0.8289 0.7652 0.7275 0.8017 0.7525 0.7177 0.7637 0.7417 0.7313 Single 0.8142 0.7838 0.7774 0.7865 0.7584 0.7575 0.7633 0.7397 0.7424 Ward 0.7996 0.7337 0.7070 0.7725 0.6929 0.6645 0.7506 0.6532 0.6191 Weighted 0.8333 0.7592 0.7197 0.8141 0.7358 0.6956 0.7997 0.7192 0.6824

Mahalanobis Average 0.8103 0.8565 0.8315 0.6965 0.8239 0.8053 NaN 0.7705 0.7782

Centroid 0.7976 0.8570 0.8333 0.6701 0.8276 0.8113 NaN 0.7770 0.7882 Complete 0.6966 0.8051 0.7787 0.4529 0.7575 0.7380 NaN 0.6480 0.6848 Median 0.7895 0.8219 0.7843 0.6695 0.7875 0.7600 NaN 0.7453 0.7309 Single 0.7908 0.8220 0.8030 0.6633 0.7841 0.7680 NaN 0.7510 0.7515 Ward 0.7313 0.8018 0.7755 0.5497 0.7523 0.7350 NaN 0.6442 0.6839 Weighted 0.7980 0.8181 0.7848 0.6899 0.7824 0.7531 NaN 0.7230 0.7173

Cityblock Average 0.8404 0.7982 0.7767 0.8206 0.7707 0.7518 0.8027 0.7522 0.7352

Centroid 0.8367 0.7997 0.7805 0.8113 0.7741 0.7611 0.7739 0.7552 0.7482 Complete 0.7995 0.7226 0.6935 0.7727 0.6761 0.6484 0.7464 0.6418 0.6059 Median 0.8267 0.7566 0.7196 0.8018 0.7412 0.7074 0.7623 0.7306 0.7174 Single 0.8077 0.7737 0.7676 0.7804 0.7448 0.7447 0.7533 0.7248 0.7269 Ward 0.8004 0.7243 0.7003 0.7744 0.6817 0.6568 0.7493 0.6446 0.6131 Weighted 0.8305 0.7507 0.7114 0.8111 0.7241 0.6867 0.7936 0.7102 0.6718

Minkowski Average 0.8478 0.8065 0.7848 0.8280 0.7818 0.7629 0.8102 0.7647 0.7488

Centroid 0.8441 0.8088 0.7883 0.8179 0.7860 0.7721 0.7797 0.7695 0.7628 Complete 0.8095 0.7273 0.7006 0.7808 0.6865 0.6551 0.7535 0.6494 0.6136 Median 0.8342 0.7644 0.7262 0.8073 0.7509 0.7182 0.7661 0.7432 0.7311 Single 0.8168 0.7836 0.7774 0.7903 0.7582 0.7578 0.7653 0.7400 0.7426 Ward 0.8064 0.7278 0.7050 0.7801 0.6869 0.6606 0.7531 0.6464 0.6161 Weighted 0.8382 0.7551 0.7185 0.8182 0.7352 0.6918 0.8006 0.7197 0.6835

Cosine Average 0.7689 0.6484 0.6138 0.7107 0.5463 0.4946 0.6596 0.4549 0.3908

Centroid 0.7618 0.6285 0.5866 0.6982 0.5106 0.4462 0.6270 0.3717 0.2936 Complete 0.7320 0.5960 0.5568 0.6604 0.4773 0.4226 0.5977 0.3743 0.3064 Median 0.7438 0.5702 0.5265 0.6802 0.4521 0.3862 0.6058 0.3286 0.2517 Single 0.7082 0.4942 0.4287 0.6182 0.3484 0.2651 0.5429 0.2451 0.1693 Ward 0.7467 0.6325 0.5991 0.6808 0.5240 0.4733 0.6213 0.4232 0.3521 Weighted 0.7534 0.5961 0.5531 0.6961 0.4958 0.4296 0.6454 0.4104 0.3378

Cophenetic correlation coefcientx = 3 x = 5 x = 10n = 10 n = 50 n = 100 n = 10 n = 50 n = 100 n = 10 n = 50 n = 100

Chebychev Average 0.7375 0.6183 0.5804 0.6933 0.5595 0.5141 0.6448 0.4958 0.4523 Centroid 0.7317 0.6241 0.5870 0.6811 0.5693 0.5334 0.6100 0.5067 0.4805 Complete 0.6647 0.4780 0.4235 0.6035 0.3824 0.3164 0.5423 0.2962 0.2281 Median 0.7140 0.5431 0.4630 0.6625 0.5036 0.4287 0.5928 0.4628 0.4249 Single 0.6833 0.5792 0.5695 0.6223 0.5199 0.5084 0.5536 0.4468 0.4405 Ward 0.6680 0.4852 0.4317 0.6128 0.3949 0.3341 0.5595 0.3140 0.2423 Weighted 0.7189 0.5255 0.4494 0.6759 0.4734 0.3878 0.6294 0.4249 0.3525

Clustering method

Cophenetic correlation coefcientx = 3 x = 5 x = 10n = 10 n = 50 n = 100 n = 10 n = 50 n = 100 n = 10 n = 50 n = 100

Euclidean Average 0.8478 0.8065 0.7848 0.8280 0.7818 0.7629 0.8102 0.7647 0.7488

Centroid 0.8188 0.8061 0.7875 0.7872 0.7816 0.7704 0.7484 0.7638 0.7606 Complete 0.8095 0.7273 0.7006 0.7808 0.6865 0.6551 0.7535 0.6494 0.6136 Median 0.8342 0.7644 0.7262 0.8073 0.7509 0.7182 0.7661 0.7432 0.7311 Single 0.8168 0.7836 0.7774 0.7903 0.7582 0.7578 0.7653 0.7400 0.7426 Ward 0.8064 0.7278 0.7050 0.7801 0.6869 0.6606 0.7531 0.6464 0.6161 Weighted 0.8382 0.7551 0.7185 0.8182 0.7352 0.6918 0.8006 0.7197 0.6835

Squared

Euclidean

Clustering method

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Table 2 (Continued)

Distance measure

One may conclude that the results of this study, which is similar to ndings of Johnson and Wichern [], indicate the data set with outliers have higher cophenetic correlation values than the data set without outliers.

This study hopes to contribute to literature for making better decisions on selection of appropriate cluster methods by using subgroup sizes, variable numbers, subgroup means and variances.

Competing interests

The authors declare that they have no competing interests.

Authors contributions

SS has made intellectual contributions in order to carry out this study and also has carried out the simulation study. ND has determined the research design as well as has coordinated the whole process. ID has made theoretical contributions and has performed statistical analysis of the study. All authors read and approved the nal manuscript.

Author details

1Department of Statistics, Faculty of Arts and Sciences, Afyon Kocatepe University, Afyonkarahisar, 03200, Turkey.

2Department of Biostatistics, Faculty of Medicine, Afyon Kocatepe University, Afyonkarahisar, 03200, Turkey.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

The authors would like to thank Rdvan NAL for support of technical help. He is a lecturer at the Afyon Kocatepe University, Faculty of Science, Department of Physics, Afyonkarahisar/Turkey.

Received: 31 December 2012 Accepted: 10 April 2013 Published: 23 April 2013

References

1. Carmichael, JW, George, JA, Julius, RS: Finding natural clusters. Syst. Zool. 17(2), 144-150 (1968)2. Lapointe, FJ, Legendre, P: Comparison tests for dendrograms: a comparative evaluation. J. Classif. 12, 265-282 (1995)3. Kettenring, JR: The practice of cluster analysis. J. Classif. 23, 3-30 (2006)4. Gunnarsson, J: Portfolio-Based Segmentation and Consumer Behaviour: Empirical Evidence and Methodological Issues. Ph.D. Dissertation, Stockholm School of Economics, The Economic Research Institute, p. 274 (1999)

5. Steinbach, M, Karypis, G, Kumar, V: A comparison of document clustering techniques. Text mining workshop. In: Proc. of the Sixth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD 2000), Boston, MA, pp. 20-23 (2000)

6. Hill, T, Lewicki, P: STATISTICS: Methods and Applications. StatSoft, Tulsa (2007)

Cophenetic correlation coefcientx = 3 x = 5 x = 10n = 10 n = 50 n = 100 n = 10 n = 50 n = 100 n = 10 n = 50 n = 100

Correlation Average 0.8214 0.7466 0.7239 0.7213 0.5584 0.5060 0.6508 0.4269 0.3550

Centroid 0.8171 0.7330 0.7064 0.7117 0.5248 0.4587 0.6210 0.3434 0.2560 Complete 0.7978 0.7102 0.6794 0.6770 0.4992 0.4457 0.5887 0.3561 0.2874 Median 0.7984 0.6852 0.6596 0.6935 0.4756 0.4128 0.6027 0.3150 0.2328 Single 0.7931 0.6647 0.6319 0.6374 0.3624 0.2748 0.5293 0.2157 0.1354 Ward 0.8069 0.7378 0.7162 0.6955 0.5431 0.4940 0.6161 0.4096 0.3428 Weighted 0.8052 0.7017 0.6843 0.7059 0.5106 0.4557 0.6373 0.3914 0.3191

Spearman Average 0.8198 0.7583 0.7455 0.7233 0.5638 0.5159 0.6505 0.4267 0.3567

Centroid 0.8113 0.7396 0.7212 0.7133 0.5336 0.4729 0.6199 0.3429 0.2537 Complete 0.8090 0.7788 0.7762 0.6802 0.5341 0.4983 0.5909 0.3558 0.2887 Median 0.7887 0.7136 0.7140 0.6960 0.4819 0.4238 0.6021 0.3126 0.2304 Single 0.6861 0.6736 0.6742 0.6404 0.4008 0.1821 0.5302 0.2190 0.1404 Ward 0.7881 0.7364 0.7273 0.6963 0.5488 0.5037 0.6170 0.4081 0.3425 Weighted 0.8112 0.7788 0.7762 0.7076 0.5157 0.4687 0.6366 0.3926 0.3197

Chebychev Average 0.8373 0.7945 0.7740 0.8094 0.7588 0.7398 0.7824 0.7246 0.7059

Centroid 0.8338 0.7966 0.7774 0.8008 0.7627 0.7483 0.7601 0.7280 0.7181 Complete 0.7965 0.7182 0.6953 0.7581 0.6707 0.6424 0.7262 0.6226 0.5951 Median 0.8244 0.7546 0.7195 0.7913 0.7315 0.6996 0.7510 0.7061 0.6907 Single 0.8044 0.7700 0.7640 0.7663 0.7329 0.7324 0.7289 0.6940 0.6951 Ward 0.7978 0.7216 0.6996 0.7647 0.6761 0.6495 0.7351 0.6345 0.6038 Weighted 0.8279 0.7470 0.7135 0.8007 0.7171 0.6783 0.7745 0.6895 0.6557

Clustering method

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7. Lessig, VP: Comparing cluster analyses with cophenetic correlation. J. Mark. Res. 9(1), 82-84 (1972)8. Sneath, HA, Sokal, RR: Numerical Taxonomy: The Principles and Practice of Numerical Classication, p. 573. Freeman, San Francisco (1973)

9. Mathworks statistics toolbox: http://www.mathworks.com/help/stats/cophenet.html

Web End =http://www.mathworks.com/help/stats/cophenet.html (2012)10. Sokal, RR, Rohlf, FJ: The comparison of dendrograms by objective methods. Taxon 11, 33-40 (1962)11. Farris, JS: On the cophenetic correlation coecient. Syst. Zool. 18(3), 279-285 (1969)12. Rohlf, FJ, David, LF: Test for hierarchical structure in random data sets. Syst. Zool. 17, 407-412 (1968)13. Baker, FB: Stability of two hierarchical grouping techniques - case I: sensitivity to data errors. J. Am. Stat. Assoc. 69, 440-445 (1974)

14. Hubert, L: Approximate evaluation techniques for the single-link and complete-link hierarchical clustering procedures. J. Am. Stat. Assoc. 69, 698-704 (1974)

15. Kuiper, FK, Fisher, LA: A Monte Carlo comparison of six clustering procedures. Biometrics 31, 777-783 (1975)16. Blasheld, RK: Mixture model tests of cluster analysis: accuracy of four agglomerative hierarchical methods. Psychol. Bull. 83, 377-388 (1976)

17. Milligan, GW: An examination of the eect of six types of error perturbation on fteen clustering algorithms. Psychometrika 45, 325-342 (1980)

18. Hands, S, Everitt, B: A Monte Carlo study of the recovery of cluster structure in binary data by hierarchical clustering techniques. Multivar. Behav. Res. 22, 235-243 (1987)

19. Yao, KB: A comparison of clustering methods for unsupervised anomaly detection in network trac. Ph.D. Thesis, University of Copenhagen (2006)

20. Ferreira, L, Hitchcock, DB: A comparison of hierarchical methods for clustering functional data. Commun. Stat., Simul. Comput., 38, 1925-1949 (2009)

21. Milligan, GW, Cooper, MC: A study of standardization of variables in cluster analysis. J. Classif. 5, 181-204 (1988)22. Dixon, WJ: Analysis of extreme value. Ann. Math. Stat. 21, 488-506 (1950)23. Johnson, RA, Wichern, DW: Applied Multivariate Statistical Analysis, 5th edn. Prentice Hall, New York (2002)

doi:10.1186/1029-242X-2013-203Cite this article as: Sarali et al.: Comparison of hierarchical cluster analysis methods by cophenetic correlation. Journal of Inequalities and Applications 2013 2013:203.