In Table , only dichotomous data are reported since we see only the an- swers "yes" or "no". Subjects nos. Thus, they can be considered as being mutually more similar than subjects nos. Similarity now can be measured by the number attributes which are equally answered by the subjects. After having defined the term "similarity", how will we perform a classification of the data?

The answer is rather easily given if the data are normally distributed with different means for the different groups, and if the number of different groups is known in advance. Figure gives an example of a sample composed of objects from two different groups with normally distributed measurements. The task of a cluster-detecting algorithm is to split the sample into two disjoint clusters.

This task is optimally solved and the within-groups similarities are high if the sum of the within- groups variances is low. And this is the case when we draw a straight line from top to bottom right across Figure , separating the two ellipsoidal collections of points. Such classification procedures, which are based on the minimization of the variability of the sum of within-groups variances, for example , are called "variance-based procedures".

The situation changes abruptly if the assumption of normally distributed data is wrong. Everybody sees immediately how to split the objects of Figure into two disjoint clusters: A curve should be drawn separating the clusters where the point density is low. Variance-based procedures, however, are linear in the sense that they ag!

In two-dimensional data, this misbehaviour can be easily checked. Cluster analyses, however, are performed very often with high- dimensional data. Bad or wrong classification then cannot be seen by pure visual inspection. Another reason not to choose procedures which are based on variances is that these procedures need not work correctly when the number of real groups is unknown.

Therefore, if natural groups within a data set are to be uncovered, it is better to rely on classification methods which do not depend on the distribution of the variables or assume any knowledge of the true number of the groups. One such method is the single-Ihlkage or nearest-neighbour procedure. Let us again consider the data of Table We define a threshold d and call two patients similar if the distance between their measurements does not exceed this value of d. The points of each pair of mutually similar subjects are joined by line segments.

### Definitions

No line is drawn between points associated with dissimilar persons. Those points which are connected either directly by a line or by an alternating sequence of lines and other points, belong to the same cluster. This procedure defines a graph, see Section 1. We therefore speak of graph- theoretic cluster analysis.

We speak of "single-linkage clusters", since clusters are joined by the single shortest link that is, by the mutually closest elements. A single object is attached to an already existing group if its distance to only one element of that group is not larger than a threshold d.

Two clusters are joined if their mutually closest elements have a distance at most d. For this procedure, no knowledge about the number of groups is required. It also can outline twisted or sickle-shaped groups like those of Figure The classification of the data of Table is given in Figure for a special thresh- old d.

### Background

As opposed to methods which are based on variance criteria, single-linkage procedures can also be applied to qualitative data, since they use similarities for the classification process instead of the geometric distribution of the points. Single-linkage procedures, however, also have some disadvantages. The main disappointment is that the threshold d must be chosen "appropriately", but how can we do that without proper information? If d is too small then we get too many clusters with few elements only. This is indicated at the upper left corner of in Figure Only one line is pictured, connecting the objects no.

If d is too large, however, the whole sample is joined to one single cluster see the upper right part of Figure This problem can be by-passed if we don't specify a single threshold d. We let d increase from 0 to 00 and get dendrogram. The lower part of Figure shows the dendrogram of the single-linkage procedure applied to the four points above. If d is increased to 0. Obviously, dendrograms give a deeper insight into the structure of a data set than classifications at a particular level d.

Another displeasing property of single-linkage procedures is an effect called "bridg- ing" or "chaining": Otherwise well-separated clusters may be linked together by a single path as is shown in Figure the dashed lines indicate the bridge. This disappoint- ing property lead to different modifications of the graph- theoretic definition of a cluster. Each of the two objects of the bridge in Figure is similar to only two other objects. A side effect of this modification of the term "cluster" is that the shape of the clusters to be uncovered can be controlled by k.

For small values of k, the clusters can be stretched and twisted. For large values of k, however, they must be compact and ellipsoidal. Up to this point attention has been focussed on samples of data units with at- tributes of the same type. In our first example, the data all are continuous. The data of the hypertension study are dichotomous. Many data sets from the life sciences or from medicine, however, involve a mixture of data from different scale levels.

If, in the hypertension study, the original values of blood pressure at rest and during bicycle exercise had been listed, we would have had continuous data together with qualitative data. We speak of mixed data. Mixed data as well as missing values and outliers cause problems in defining similarities or distances between objects. Like all methods of exploratory data analysis, cluster analysis helps to "explore" and uncover structures within a data set.

Methods of cluster analysis should primarily be used to formulate hypotheses mathematical models which are well-fitted to the data. Every researcher, however, must note that cluster analyses are very subjective even if we use "objective" mathematical methods to outline the different groups.

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This holds since the resulting clusters depend not only on-the computational procedure, but also on the choice of attributes to be measured. And since the researcher alone or together with a biometrician decides on the-basis of his or her personal knowledge which attributes and objects should be drawn for a sample, this choice may be biased. Therefore, the results of a cluster analysis--oare chiefly valid for the specific sample only, and we cannot generalize them to a larger population without careful inspection.

Moreover, every application of a clustering algorithm to a set of data results in a classification of objects, whether or not the data exhibit a true or "natural" grouping structure. This can happen if in single-linkage classification the threshold d has not been chosen appropriately.

This is irrelevant if clustering is done to obtain a practical stratification of the given set of objects for organisational purposes. In exploratory data analysis however, the interest is in detecting an unknown clustering structure from the data. Here, the result of a clustering procedure should reflect the real structure real or natural clusters. From the group structure of the objects of a sample we usually make inference about the whole population. Thus, an artificial clustering is not acceptable. The classes resulting from the algorithm therefore must be investigated for their relevance and their validity.

While the researcher may judge the relevance from a possibly more qualitative point of view, the statistician has to rely on appropriate significance tests or other methods of data analysis.

Using graph-theoretic models, we can develop test procedures for testing the validity of the clusters outlined by a classification procedure. One model is as follows.

We interpret homogeneity within a data set as random attachment of the distances to the pairs of objects. If now PnN Y ; k ; Q holds for a given error probability Q of the first kind, then the null hypothesis of a random attachment of distances to pairs of objects can be rejected at this level Q. On the other hand, we do not expect that the five smallest distances mutually connect four objects within homogeneous data, that is, under random conditions.

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## Adjacency matrix - Wikipedia

The probability for this eVf:'nt is smaller than see the upper part of Figure The four points connected by the five smallest distances can be considered as the kernel of a real group. These developments show that, today, we can use methods of cluster analysis not only for pure exploration of samples but also to test statistical hypotheses. This presupposes a well-defined mathematical classification model like that of single-linkage clusters.

The development of test procedures to test the homogeneity of a sample and thus to infer from a sample to a larger population as sketched above, is a first step from pure exploration to confirmatory statistics. It illustrates the strong intercorrelations between exploratory and confirmatory statistics.

## Graph and Network Algorithms

The objects upon which the measurements were taken were assumed to be homogeneous. Rarely was interest in the possibility that a given set of objects could be grouped into subsets that displayed systematic differences.

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In many applications, however, there is reason to believe that a set of objects can be clustered into subgroups that differ in meaningful ways. In large data sets like those stored in clinical cancer registers, nobody would expect the objects to be homogeneous. The most commonly used term for the class of procedures that seek to separate the component data into groups is cluster analysis. Other names which are used synonymously are automatic classification, numerical classification, numerical taxonomy, and sometimes pattern recognition. Cluster analytical methods gained distinctive importance while being used as tools to derive dendrograms of animals or plants in biology.