Clustering: An Introduction

What is Clustering?

Clustering can be considered the most important unsupervised learning problem; so, as every other problem of this kind, it deals with finding a structure in a collection of unlabeled data.

A loose definition of clustering could be “the process of organizing objects into groups whose members are similar in some way”.

A cluster is therefore a collection of objects which are “similar” between them and are “dissimilar” to the objects belonging to other clusters.

We can show this with a simple graphical example:

In this case we easily identify the 4 clusters into which the data can be divided; the similarity criterion is distance: two or more objects belong to the same cluster if they are “close” according to a given distance (in this case geometrical distance). This is calleddistance-based clustering.

Another kind of clustering is conceptual clustering: two or more objects belong to the same cluster if this one defines a concept common to all that objects. In other words, objects are grouped according to their fit to descriptive concepts, not according to simple similarity measures.

 

The Goals of Clustering

So, the goal of clustering is to determine the intrinsic grouping in a set of unlabeled data. But how to decide what constitutes a good clustering? It can be shown that there is no absolute “best” criterion which would be independent of the final aim of the clustering. Consequently, it is the user which must supply this criterion, in such a way that the result of the clustering will suit their needs.

For instance, we could be interested in finding representatives for homogeneous groups (data reduction), in finding “natural clusters” and describe their unknown properties (“natural” data types), in finding useful and suitable groupings (“useful” data classes) or in finding unusual data objects (outlier detection).

 

Possible Applications

Clustering algorithms can be applied in many fields, for instance:

• Marketing: finding groups of customers with similar behavior given a large database of customer data containing their properties and past buying records;
• Biology: classification of plants and animals given their features;
• Libraries: book ordering;
• Insurance: identifying groups of motor insurance policy holders with a high average claim cost; identifying frauds;
• City-planning: identifying groups of houses according to their house type, value and geographical location;
• Earthquake studies: clustering observed earthquake epicenters to identify dangerous zones;
• WWW: document classification; clustering weblog data to discover groups of similar access patterns.

 

Requirements

The main requirements that a clustering algorithm should satisfy are:

• scalability;
• dealing with different types of attributes;
• discovering clusters with arbitrary shape;
• minimal requirements for domain knowledge to determine input parameters;
• ability to deal with noise and outliers;
• insensitivity to order of input records;
• high dimensionality;
• interpretability and usability.

 

Problems

There are a number of problems with clustering. Among them:

• current clustering techniques do not address all the requirements adequately (and concurrently);
• dealing with large number of dimensions and large number of data items can be problematic because of time complexity;
• the effectiveness of the method depends on the definition of “distance” (for distance-based clustering);
• if an obvious distance measure doesn’t exist we must “define” it, which is not always easy, especially in multi-dimensional spaces;
• the result of the clustering algorithm (that in many cases can be arbitrary itself) can be interpreted in different ways.

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Clustering Algorithms

Classification

Clustering algorithms may be classified as listed below:

• Exclusive Clustering
• Overlapping Clustering
• Hierarchical Clustering
• Probabilistic Clustering

In the first case data are grouped in an exclusive way, so that if a certain datum belongs to a definite cluster then it could not be included in another cluster. A simple example of that is shown in the figure below, where the separation of points is achieved by a straight line on a bi-dimensional plane.

On the contrary the second type, the overlapping clustering, uses fuzzy sets to cluster data, so that each point may belong to two or more clusters with different degrees of membership. In this case, data will be associated to an appropriate membership value.

Instead, a hierarchical clustering algorithm is based on the union between the two nearest clusters. The beginning condition is realized by setting every datum as a cluster. After a few iterations it reaches the final clusters wanted.

Finally, the last kind of clustering use a completely probabilistic approach.

In this tutorial we propose four of the most used clustering algorithms:

• K-means
• Fuzzy C-means
• Hierarchical clustering
• Mixture of Gaussians

Each of these algorithms belongs to one of the clustering types listed above. So that,  is an exclusive clustering algorithm,  is an overlapping clustering algorithm,  is obvious and lastly  is aprobabilistic clustering algorithm. We will discuss about each clustering method in the following paragraphs.

 

Distance Measure

An important component of a clustering algorithm is the distance measure between data points. If the components of the data instance vectors are all in the same physical units then it is possible that the simple Euclidean distance metric is sufficient to successfully group similar data instances. However, even in this case the Euclidean distance can sometimes be misleading. Figure shown below illustrates this with an example of the width and height measurements of an object. Despite both measurements being taken in the same physical units, an informed decision has to be made as to the relative scaling. As the figure shows, different scalings can lead to different clusterings.

Notice however that this is not only a graphic issue: the problem arises from the mathematical formula used to combine the distances between the single components of the data feature vectors into a unique distance measure that can be used for clustering purposes: different formulas leads to different clusterings.

Again, domain knowledge must be used to guide the formulation of a suitable distance measure for each particular application.

Minkowski Metric

For higher dimensional data, a popular measure is the Minkowski metric,

where d is the dimensionality of the data. The Euclidean distance is a special case where p=2, while Manhattan metric has p=1. However, there are no general theoretical guidelines for selecting a measure for any given application.

It is often the case that the components of the data feature vectors are not immediately comparable. It can be that the components are not continuous variables, like length, but nominal categories, such as the days of the week. In these cases again, domain knowledge must be used to formulate an appropriate measure.

 

Bibliography

• • Tariq Rashid: “Clustering”
  • Osmar R. Zaïane: “Principles of Knowledge Discovery in Databases - Chapter 8: Data Clustering”
  • Pier Luca Lanzi: “Ingegneria della Conoscenza e Sistemi Esperti – Lezione 2: Apprendimento non supervisionato”

 

K-Means Clustering

 

The Algorithm

K-means () is one of the simplest unsupervised learning algorithms that solve the well known clustering problem. The procedure follows a simple and easy way to classify a given data set through a certain number of clusters (assume k clusters) fixed a priori. The main idea is to define k centroids, one for each cluster. These centroids shoud be placed in a cunning way because of different location causes different result. So, the better choice is to place them as much as possible far away from each other. The next step is to take each point belonging to a given data set and associate it to the nearest centroid. When no point is pending, the first step is completed and an early groupage is done. At this point we need to re-calculate k new centroids as barycenters of the clusters resulting from the previous step. After we have these k new centroids, a new binding has to be done between the same data set points and the nearest new centroid. A loop has been generated. As a result of this loop we may notice that the k centroids change their location step by step until no more changes are done. In other words centroids do not move any more.

Finally, this algorithm aims at minimizing an objective function, in this case a squared error function. The objective function

 

 ,

 

where  is a chosen distance measure between a data point  and the cluster centre , is an indicator of the distance of the n data points from their respective cluster centres.

 

The algorithm is composed of the following steps:

 

 

 

1. Place K points into the space represented by the objects that are being clustered. These points represent initial group centroids.
2. Assign each object to the group that has the closest centroid.
3. When all objects have been assigned, recalculate the positions of the K centroids.
4. Repeat Steps 2 and 3 until the centroids no longer move. This produces a separation of the objects into groups from which the metric to be minimized can be calculated.

 

Although it can be proved that the procedure will always terminate, the k-means algorithm does not necessarily find the most optimal configuration, corresponding to the global objective function minimum. The algorithm is also significantly sensitive to the initial randomly selected cluster centres. The k-means algorithm can be run multiple times to reduce this effect.

 

K-means is a simple algorithm that has been adapted to many problem domains. As we are going to see, it is a good candidate for extension to work with fuzzy feature vectors.

 

 

 

An example

Suppose that we have n sample feature vectors x₁, x₂, ..., x_n all from the same class, and we know that they fall into k compact clusters, k < n. Let m_i be the mean of the vectors in cluster i. If the clusters are well separated, we can use a minimum-distance classifier to separate them. That is, we can say that x is in cluster i if || x - m_i || is the minimum of all the k distances. This suggests the following procedure for finding the k means:

 

• Make initial guesses for the means m₁, m₂, ..., m_k
• Until there are no changes in any mean
  • Use the estimated means to classify the samples into clusters
  • For i from 1 to k
    • Replace m_i with the mean of all of the samples for cluster i
  • end_for
• end_until

 

Here is an example showing how the means m₁ and m₂ move into the centers of two clusters.

 

 

Remarks

This is a simple version of the k-means procedure. It can be viewed as a greedy algorithm for partitioning the n samples into k clusters so as to minimize the sum of the squared distances to the cluster centers. It does have some weaknesses:

 

• The way to initialize the means was not specified. One popular way to start is to randomly choose k of the samples.
• The results produced depend on the initial values for the means, and it frequently happens that suboptimal partitions are found. The standard solution is to try a number of different starting points.
• It can happen that the set of samples closest to m_i is empty, so that m_i cannot be updated. This is an annoyance that must be handled in an implementation, but that we shall ignore.
• The results depend on the metric used to measure || x - m_i ||. A popular solution is to normalize each variable by its standard deviation, though this is not always desirable.
• The results depend on the value of k.

 

This last problem is particularly troublesome, since we often have no way of knowing how many clusters exist. In the example shown above, the same algorithm applied to the same data produces the following 3-means clustering. Is it better or worse than the 2-means clustering?

 

 

Unfortunately there is no general theoretical solution to find the optimal number of clusters for any given data set. A simple approach is to compare the results of multiple runs with different k classes and choose the best one according to a given criterion (for instance the Schwarz Criterion - see ), but we need to be careful because increasing k results in smaller error function values by definition, but also an increasing risk of overfitting.

 

 

 

Bibliography

 

• J. B. MacQueen (1967): "Some Methods for classification and Analysis of Multivariate Observations, Proceedings of 5-th Berkeley Symposium on Mathematical Statistics and Probability", Berkeley, University of California Press, 1:281-297
• Andrew Moore: “K-means and Hierarchical Clustering - Tutorial Slides”
• Brian T. Luke: “K-Means Clustering”
• Tariq Rashid: “Clustering”
• Hans-Joachim Mucha and Hizir Sofyan: “Nonhierarchical Clustering”

 

Fuzzy C-Means Clustering

The Algorithm

Fuzzy c-means (FCM) is a method of clustering which allows one piece of data to belong to two or more clusters. This method (developed by  and improved by ) is frequently used in pattern recognition. It is based on minimization of the following objective function:

     ,     

where m is any real number greater than 1, u_ij is the degree of membership of x_i in the cluster j, x_i is the ith of d-dimensional measured data, c_j is the d-dimension center of the cluster, and ||*|| is any norm expressing the similarity between any measured data and the center.

Fuzzy partitioning is carried out through an iterative optimization of the objective function shown above, with the update of membership u_ij and the cluster centers c_j by:

     ,     

This iteration will stop when , where  is a termination criterion between 0 and 1, whereas k are the iteration steps. This procedure converges to a local minimum or a saddle point of J_m.

The algorithm is composed of the following steps:

1. Initialize U=[u_ij] matrix, U(0)
2. At k-step: calculate the centers vectors C^(k)=[c_j] with U(k)
   
3. Update U^(k) , U(k+1)
   
4. If || U^(k+1) - U^(k)||< then STOP; otherwise return to step 2.

Remarks

As already told, data are bound to each cluster by means of a Membership Function, which represents the fuzzy behaviour of this algorithm. To do that, we simply have to build an appropriate matrix named U whose factors are numbers between 0 and 1, and represent the degree of membership between data and centers of clusters.

For a better understanding, we may consider this simple mono-dimensional example. Given a certain data set, suppose to represent it as distributed on an axis. The figure below shows this:

Looking at the picture, we may identify two clusters in proximity of the two data concentrations. We will refer to them using ‘A’ and ‘B’. In the first approach shown in this tutorial - the k-means algorithm - we associated each datum to a specific centroid; therefore, this membership function looked like this:

In the FCM approach, instead, the same given datum does not belong exclusively to a well defined cluster, but it can be placed in a middle way. In this case, the membership function follows a smoother line to indicate that every datum may belong to several clusters with different values of the membership coefficient.

In the figure above, the datum shown as a red marked spot belongs more to the B cluster rather than the A cluster. The value 0.2 of ‘m’ indicates the degree of membership to A for such datum. Now, instead of using a graphical representation, we introduce a matrix U whose factors are the ones taken from the membership functions:

         

(a)                                  (b)

The number of rows and columns depends on how many data and clusters we are considering. More exactly we have C = 2 columns (C = 2 clusters) and N rows, where C is the total number of clusters and N is the total number of data. The generic element is so indicated: u_ij.

In the examples above we have considered the k-means (a) and FCM (b) cases. We can notice that in the first case (a) the coefficients are always unitary. It is so to indicate the fact that each datum can belong only to one cluster. Other properties are shown below:

• 
• 
• 

 

An Example

Here, we consider the simple case of a mono-dimensional application of the FCM. Twenty data and three clusters are used to initialize the algorithm and to compute the U matrix. Figures below (taken from our ) show the membership value for each datum and for each cluster. The color of the data is that of the nearest cluster according to the membership function.

In the simulation shown in the figure above we have used a fuzzyness coefficient m = 2 and we have also imposed to terminate the algorithm when . The picture shows the initial condition where the fuzzy distribution depends on the particular position of the clusters. No step is performed yet so that clusters are not identified very well. Now we can run the algorithm until the stop condition is verified. The figure below shows the final condition reached at the 8th step with m=2 and =0.3:

Is it possible to do better? Certainly, we could use an higher accuracy but we would have also to pay for a bigger computational effort. In the next figure we can see a better result having used the same initial conditions and =0.01, but we needed 37 steps!

It is also important to notice that different initializations cause different evolutions of the algorithm. In fact it could converge to the same result but probably with a different number of iteration steps.

Bibliography

• • J. C. Dunn (1973): "A Fuzzy Relative of the ISODATA Process and Its Use in Detecting Compact Well-Separated Clusters", Journal of Cybernetics 3: 32-57
  • J. C. Bezdek (1981): "Pattern Recognition with Fuzzy Objective Function Algoritms", Plenum Press, New York
  • Tariq Rashid: “Clustering”
  • Hans-Joachim Mucha and Hizir Sofyan: “Nonhierarchical Clustering”
     

Hierarchical Clustering Algorithms

How They Work

Given a set of N items to be clustered, and an N*N distance (or similarity) matrix, the basic process of hierarchical clustering (defined by ) is this:

1. Start by assigning each item to a cluster, so that if you have N items, you now have N clusters, each containing just one item. Let the distances (similarities) between the clusters the same as the distances (similarities) between the items they contain.
2. Find the closest (most similar) pair of clusters and merge them into a single cluster, so that now you have one cluster less.
3. Compute distances (similarities) between the new cluster and each of the old clusters.
4. Repeat steps 2 and 3 until all items are clustered into a single cluster of size N. (*)

Step 3 can be done in different ways, which is what distinguishes single-linkage from complete-linkage and average-linkage clustering.

In single-linkage clustering (also called the connectedness or minimum method), we consider the distance between one cluster and another cluster to be equal to the shortest distance from any member of one cluster to any member of the other cluster. If the data consist of similarities, we consider the similarity between one cluster and another cluster to be equal to the greatest similarity from any member of one cluster to any member of the other cluster.

In complete-linkage clustering (also called the diameter or maximum method), we consider the distance between one cluster and another cluster to be equal to the greatest distance from any member of one cluster to any member of the other cluster.

In average-linkage clustering, we consider the distance between one cluster and another cluster to be equal to the average distance from any member of one cluster to any member of the other cluster.

A variation on average-link clustering is the UCLUS method of  which uses the median distance, which is much more outlier-proof than the average distance.

This kind of hierarchical clustering is called agglomerative because it merges clusters iteratively. There is also a divisive hierarchical clustering which does the reverse by starting with all objects in one cluster and subdividing them into smaller pieces. Divisive methods are not generally available, and rarely have been applied.

(*) Of course there is no point in having all the N items grouped in a single cluster but, once you have got the complete hierarchical tree, if you want k clusters you just have to cut the k-1 longest links.

 

Single-Linkage Clustering: The Algorithm

Let’s now take a deeper look at how Johnson’s algorithm works in the case of single-linkage clustering.

The algorithm is an agglomerative scheme that erases rows and columns in the proximity matrix as old clusters are merged into new ones.

The N*N proximity matrix is D = [d(i,j)]. The clusterings are assigned sequence numbers 0,1,......, (n-1) and L(k) is the level of the kth clustering. A cluster with sequence number m is denoted (m) and the proximity between clusters (r) and (s) is denoted d [(r),(s)].

The algorithm is composed of the following steps:

1. Begin with the disjoint clustering having level L(0) = 0 and sequence number m = 0.
2. Find the least dissimilar pair of clusters in the current clustering, say pair (r), (s), according to
   d[(r),(s)] = min d[(i),(j)]
   where the minimum is over all pairs of clusters in the current clustering.
3. Increment the sequence number : m = m +1. Merge clusters (r) and (s) into a single cluster to form the next clustering m. Set the level of this clustering to
   L(m) = d[(r),(s)]
4. Update the proximity matrix, D, by deleting the rows and columns corresponding to clusters (r) and (s) and adding a row and column corresponding to the newly formed cluster. The proximity between the new cluster, denoted (r,s) and old cluster (k) is defined in this way:
   d[(k), (r,s)] = min d[(k),(r)], d[(k),(s)]
5. If all objects are in one cluster, stop. Else, go to step 2.

 

An Example

Let’s now see a simple example: a hierarchical clustering of distances in kilometers between some Italian cities. The method used is single-linkage.

Input distance matrix (L = 0 for all the clusters):

	BA	FI	MI	NA	RM	TO
BA	0	662	877	255	412	996
FI	662	0	295	468	268	400
MI	877	295	0	754	564	138
NA	255	468	754	0	219	869
RM	412	268	564	219	0	669
TO	996	400	138	869	669	0

The nearest pair of cities is MI and TO, at distance 138. These are merged into a single cluster called "MI/TO". The level of the new cluster is L(MI/TO) = 138 and the new sequence number is m = 1.

Then we compute the distance from this new compound object to all other objects. In single link clustering the rule is that the distance from the compound object to another object is equal to the shortest distance from any member of the cluster to the outside object. So the distance from "MI/TO" to RM is chosen to be 564, which is the distance from MI to RM, and so on.

After merging MI with TO we obtain the following matrix:

 

	BA	FI	MI/TO	NA	RM
BA	0	662	877	255	412
FI	662	0	295	468	268
MI/TO	877	295	0	754	564
NA	255	468	754	0	219
RM	412	268	564	219	0

min d(i,j) = d(NA,RM) = 219 => merge NA and RM into a new cluster called NA/RM

L(NA/RM) = 219

m = 2

 

	BA	FI	MI/TO	NA/RM
BA	0	662	877	255
FI	662	0	295	268
MI/TO	877	295	0	564
NA/RM	255	268	564	0

min d(i,j) = d(BA,NA/RM) = 255 => merge BA and NA/RM into a new cluster called BA/NA/RM

L(BA/NA/RM) = 255

m = 3

 

	BA/NA/RM	FI	MI/TO
BA/NA/RM	0	268	564
FI	268	0	295
MI/TO	564	295	0

min d(i,j) = d(BA/NA/RM,FI) = 268 => merge BA/NA/RM and FI into a new cluster called BA/FI/NA/RM

L(BA/FI/NA/RM) = 268

m = 4

 

	BA/FI/NA/RM	MI/TO
BA/FI/NA/RM	0	295
MI/TO	295	0

Finally, we merge the last two clusters at level 295.

The process is summarized by the following hierarchical tree:

 

Problems

The main weaknesses of agglomerative clustering methods are:

• they do not scale well: time complexity of at least O(n²), where n is the number of total objects;
• they can never undo what was done previously.

 

Bibliography

• S. C. Johnson (1967): "Hierarchical Clustering Schemes" Psychometrika, 2:241-254
• R. D'andrade (1978): "U-Statistic Hierarchical Clustering" Psychometrika, 4:58-67
• Andrew Moore: “K-means and Hierarchical Clustering - Tutorial Slides”
• Osmar R. Zaïane: “Principles of Knowledge Discovery in Databases - Chapter 8: Data Clustering”
• Stephen P. Borgatti: “How to explain hierarchical clustering”
• Maria Irene Miranda: “Clustering methods and algorithms”

 

 

Clustering as a Mixture of Gaussians

Introduction to Model-Based Clustering

There’s another way to deal with clustering problems: a model-based approach, which consists in using certain models for clusters and attempting to optimize the fit between the data and the model.

In practice, each cluster can be mathematically represented by a parametric distribution, like a Gaussian (continuous) or a Poisson (discrete). The entire data set is therefore modelled by a mixture of these distributions. An individual distribution used to model a specific cluster is often referred to as a component distribution.

A mixture model with high likelihood tends to have the following traits:

• component distributions have high “peaks” (data in one cluster are tight);
• the mixture model “covers” the data well (dominant patterns in the data are captured by component distributions).

Main advantages of model-based clustering:

• well-studied statistical inference techniques available;
• flexibility in choosing the component distribution;
• obtain a density estimation for each cluster;
• a “soft” classification is available.

 

Mixture of Gaussians

The most widely used clustering method of this kind is the one based on learning a mixture of Gaussians: we can actually consider clusters as Gaussian distributions centred on their barycentres, as we can see in this picture, where the grey circle represents the first variance of the distribution:

The algorithm works in this way:

• it chooses the component (the Gaussian) at random with probability ;
• it samples a point .

Let’s suppose to have:

• x₁, x₂,..., x_N
• 

We can obtain the likelihood of the sample: .

What we really want to maximise is  (probability of a datum given the centres of the Gaussians).

is the base to write the likelihood function:

Now we should maximise the likelihood function by calculating , but it would be too difficult. That’s why we use a simplified algorithm called EM (Expectation-Maximization).

 

The EM Algorithm

The algorithm which is used in practice to find the mixture of Gaussians that can model the data set is called EM (Expectation-Maximization) (). Let’s see how it works with an example.

Suppose x_k are the marks got by the students of a class, with these probabilities:

x₁ = 30             

x₂ = 18             

x₃ = 0               

x₄ = 23             

First case: we observe that the marks are so distributed among students:

x₁ : a students

x₂ : b students

x₃ : c students

x₄ : d students

We should maximise this function by calculating . Let’s instead calculate the logarithm of the function and maximise it:

Supposing a = 14, b = 6, c = 9 and d = 10 we can calculate that .

Second case: we observe that marks are so distributed among students:

x₁ + x₂ : h students

x₃ : c students

x₄ : d students

We have so obtained a circularity which is divided into two steps:

• expectation: 
• maximization: 

This circularity can be solved in an iterative way.

Let’s now see how the EM algorithm works for a mixture of Gaussians (parameters estimated at the pth iteration are marked by a superscript (p):

1. Initialize parameters:
   
2. E-step:
   
3. M-step:
   
   where R is the number of records.

 

Bibliography

• A.P. Dempster, N.M. Laird, and D.B. Rubin (1977): "Maximum Likelihood from Incomplete Data via theEM algorithm", Journal of the Royal Statistical Society, Series B, vol. 39, 1:1-38
• Osmar R. Zaïane: “Principles of Knowledge Discovery in Databases - Chapter 8: Data Clustering”
• Jia Li: “Data Mining - Clustering by Mixture Models”

 

 

Links

 

Andrew Moore: “K-means and Hierarchical Clustering - Tutorial Slides”

Brian T. Luke: “K-Means Clustering”

Tariq Rashid: “Clustering”

Hans-Joachim Mucha and Hizir Sofyan: “Cluster Analysis”

Frigui Hichem: “Similarity Measures and Criterion Functions for clustering”

Osmar R. Zaïane: “Principles of Knowledge Discovery in Databases - Chapter 8: Data Clustering”

Pier Luca Lanzi: “Ingegneria della Conoscenza e Sistemi Esperti – Lezione 2: Apprendimento non supervisionato”

Stephen P. Borgatti: “How to explain hierarchical clustering”

Maria Irene Miranda: “Clustering methods and algorithms”

Jia Li: “Data Mining - Clustering by Mixture Models”

 

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