Data Mining
Cluster Analysis: Basic Concepts
and Algorithms
Lecture Notes for Chapter 7
Introduction to Data Mining
by
Tan, Steinbach, Kumar
© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 *
What is Cluster Analysis?
Finding groups of objects such that the objects in a group will be similar (or related) to one another and different from (or unrelated to) the objects in other groups
Inter-cluster distances are maximized
Intra-cluster distances are minimized
Applications of Cluster Analysis
Understanding
Group related documents for browsing, group genes and proteins that have similar functionality, or group stocks with similar price fluctuations
Summarization
Reduce the size of large data sets
What is not Cluster Analysis?
Supervised classification
Have class label information
Simple segmentation
Dividing students into different registration groups alphabetically, by last name
Results of a query
Groupings are a result of an external specification
Graph partitioning
Some mutual relevance and synergy, but areas are not identical
Types of Clusterings
A clustering is a set of clusters
Important distinction between hierarchical and partitional sets of clusters
Partitional Clustering
A division data objects into non-overlapping subsets (clusters) such that each data object is in exactly one subset
Hierarchical clustering
A set of nested clusters organized as a hierarchical tree
Other Distinctions Between Sets of Clusters
Exclusive versus non-exclusive
In non-exclusive clusterings, points may belong to multiple clusters.
Can represent multiple classes or border points
Fuzzy versus non-fuzzy
In fuzzy clustering, a point belongs to every cluster with some weight between 0 and 1
Weights must sum to 1
Probabilistic clustering has similar characteristics
Partial versus complete
In some cases, we only want to cluster some of the data
Heterogeneous versus homogeneous
Cluster of widely different sizes, shapes, and densities
Types of Clusters
Well-separated clusters
Center-based clusters
Contiguous clusters
Density-based clusters
Property or Conceptual
Described by an Objective Function
Types of Clusters: Objective Function
Clusters Defined by an Objective Function
Finds clusters that minimize or maximize an objective function.
Enumerate all possible ways of dividing the points into clusters and evaluate the `goodness' of each potential set of clusters by using the given objective function. (NP Hard)
Can have global or local objectives.
Hierarchical clustering algorithms typically have local objectives
Partitional algorithms typically have global objectives
A variation of the global objective function approach is to fit the data to a parameterized model.
Parameters for the model are determined from the data.
Mixture models assume that the data is a mixture' of a number of statistical distributions.
Types of Clusters: Objective Function
Map the clustering problem to a different domain and solve a related problem in that domain
Proximity matrix defines a weighted graph, where the nodes are the points being clustered, and the weighted edges represent the proximities between points
Clustering is equivalent to breaking the graph into connected components, one for each cluster.
Want to minimize the edge weight between clusters and maximize the edge weight within clusters
Characteristics of the Input Data Are Important
Type of proximity or density measure
This is a derived measure, but central to clustering
Sparseness
Dictates type of similarity
Adds to efficiency
Attribute type
Dictates type of similarity
Type of Data
Dictates type of similarity
Other characteristics, e.g., autocorrelation
Dimensionality
Noise and Outliers
Type of Distribution
Clustering Algorithms
K-means and its variants
Hierarchical clustering
Density-based clustering
K-means Clustering
Partitional clustering approach
Each cluster is associated with a centroid (center point)
Each point is assigned to the cluster with the closest centroid
Number of clusters, K, must be specified
The basic algorithm is very simple
K-means Clustering Details
Initial centroids are often chosen randomly.
Clusters produced vary from one run to another.
The centroid is (typically) the mean of the points in the cluster.
Closeness is measured by Euclidean distance, cosine similarity, correlation, etc.
K-means will converge for common similarity measures mentioned above.
Most of the convergence happens in the first few iterations.
Often the stopping condition is changed to Until relatively few points change clusters
Complexity is O( n * K * I * d )
n = number of points, K = number of clusters,
I = number of iterations, d = number of attributes
Evaluating K-means Clusters
Most common measure is Sum of Squared Error (SSE)
For each point, the error is the distance to the nearest cluster
To get SSE, we square these errors and sum them.
x is a data point in cluster Ci and mi is the representative point for cluster Ci
can show that mi corresponds to the center (mean) of the cluster
Given two clusters, we can choose the one with the smallest error
One easy way to reduce SSE is to increase K, the number of clusters
A good clustering with smaller K can have a lower SSE than a poor clustering with higher K
Problems with Selecting Initial Points
If there are K real clusters then the chance of selecting one centroid from each cluster is small.
Chance is relatively small when K is large
If clusters are the same size, n, then
For example, if K = 10, then probability = 10!/1010 = 0.00036
Sometimes the initial centroids will readjust themselves in right way, and sometimes they dont
Consider an example of five pairs of clusters
Solutions to Initial Centroids Problem
Multiple runs
Helps, but probability is not on your side
Sample and use hierarchical clustering to determine initial centroids
Select more than k initial centroids and then select among these initial centroids
Select most widely separated
Postprocessing
Bisecting K-means
Not as susceptible to initialization issues
Handling Empty Clusters
Basic K-means algorithm can yield empty clusters
Several strategies
Choose the point that contributes most to SSE
Choose a point from the cluster with the highest SSE
If there are several empty clusters, the above can be repeated several times.
Updating Centers Incrementally
In the basic K-means algorithm, centroids are updated after all points are assigned to a centroid
An alternative is to update the centroids after each assignment (incremental approach)
Each assignment updates zero or two centroids
More expensive
Introduces an order dependency
Never get an empty cluster
Can use “weights” to change the impact
Pre-processing and Post-processing
Pre-processing
Normalize the data
Eliminate outliers
Post-processing
Eliminate small clusters that may represent outliers
Split loose clusters, i.e., clusters with relatively high SSE
Merge clusters that are close and that have relatively low SSE
Can use these steps during the clustering process
ISODATA
Bisecting K-means
Bisecting K-means algorithm
Variant of K-means that can produce a partitional or a hierarchical clustering
Limitations of K-means
K-means has problems when clusters are of differing
Sizes
Densities
Non-globular shapes
K-means has problems when the data contains outliers.
Strengths of Hierarchical Clustering
Do not have to assume any particular number of clusters
Any desired number of clusters can be obtained by cutting the dendogram at the proper level
They may correspond to meaningful taxonomies
Example in biological sciences (e.g., animal kingdom, phylogeny reconstruction, )
Hierarchical Clustering
Two main types of hierarchical clustering
Agglomerative:
Start with the points as individual clusters
At each step, merge the closest pair of clusters until only one cluster (or k clusters) left
Divisive:
Start with one, all-inclusive cluster
At each step, split a cluster until each cluster contains a point (or there are k clusters)
Traditional hierarchical algorithms use a similarity or distance matrix
Merge or split one cluster at a time
Agglomerative Clustering Algorithm
More popular hierarchical clustering technique
Basic algorithm is straightforward
Compute the proximity matrix
Let each data point be a cluster
Repeat
Merge the two closest clusters
Update the proximity matrix
Until only a single cluster remains
Key operation is the computation of the proximity of two clusters
Different approaches to defining the distance between clusters distinguish the different algorithms
Hierarchical Clustering: Group Average
Compromise between Single and Complete Link
Strengths
Less susceptible to noise and outliers
Limitations
Biased towards globular clusters
Cluster Similarity: Wards Method
Similarity of two clusters is based on the increase in squared error when two clusters are merged
Similar to group average if distance between points is distance squared
Less susceptible to noise and outliers
Biased towards globular clusters
Hierarchical analogue of K-means
Can be used to initialize K-means
Hierarchical Clustering: Time and Space requirements
O(N2) space since it uses the proximity matrix.
N is the number of points.
O(N3) time in many cases
There are N steps and at each step the size, N2, proximity matrix must be updated and searched
Complexity can be reduced to O(N2 log(N) ) time for some approaches
Hierarchical Clustering: Problems and Limitations
Once a decision is made to combine two clusters, it cannot be undone
No objective function is directly minimized
Different schemes have problems with one or more of the following:
Sensitivity to noise and outliers
Difficulty handling different sized clusters and convex shapes
Breaking large clusters
DBSCAN
DBSCAN is a density-based algorithm.
Density = number of points within a specified radius (Eps)
A point is a core point if it has more than a specified number of points (MinPts) within Eps
These are points that are at the interior of a cluster
A border point has fewer than MinPts within Eps, but is in the neighborhood of a core point
A noise point is any point that is not a core point or a border point.
Cluster Validity
For supervised classification we have a variety of measures to evaluate how good our model is
Accuracy, precision, recall
For cluster analysis, the analogous question is how to evaluate the “goodness” of the resulting clusters?
But “clusters are in the eye of the beholder”!
Then why do we want to evaluate them?
To avoid finding patterns in noise
To compare clustering algorithms
To compare two sets of clusters
To compare two clusters
Determining the clustering tendency of a set of data, i.e., distinguishing whether non-random structure actually exists in the data.
Comparing the results of a cluster analysis to externally known results, e.g., to externally given class labels.
Evaluating how well the results of a cluster analysis fit the data without reference to external information.
– Use only the data
Comparing the results of two different sets of cluster analyses to determine which is better.
Determining the correct number of clusters.
For 2, 3, and 4, we can further distinguish whether we want to evaluate the entire clustering or just individual clusters.
Different Aspects of Cluster Validation
Numerical measures that are applied to judge various aspects of cluster validity, are classified into the following three types.
External Index: Used to measure the extent to which cluster labels match externally supplied class labels.
Entropy
Internal Index: Used to measure the goodness of a clustering structure without respect to external information.
Sum of Squared Error (SSE)
Relative Index: Used to compare two different clusterings or clusters.
Often an external or internal index is used for this function, e.g., SSE or entropy
Sometimes these are referred to as criteria instead of indices
However, sometimes criterion is the general strategy and index is the numerical measure that implements the criterion.
Measures of Cluster Validity
Two matrices
Proximity Matrix
“Incidence” Matrix
One row and one column for each data point
An entry is 1 if the associated pair of points belong to the same cluster
An entry is 0 if the associated pair of points belongs to different clusters
Compute the correlation between the two matrices
Since the matrices are symmetric, only the correlation between
n(n-1) / 2 entries needs to be calculated.
High correlation indicates that points that belong to the same cluster are close to each other.
Not a good measure for some density or contiguity based clusters.
Measuring Cluster Validity Via Correlation
Need a framework to interpret any measure.
For example, if our measure of evaluation has the value, 10, is that good, fair, or poor?
Statistics provide a framework for cluster validity
The more “atypical” a clustering result is, the more likely it represents valid structure in the data
Can compare the values of an index that result from random data or clusterings to those of a clustering result.
If the value of the index is unlikely, then the cluster results are valid
These approaches are more complicated and harder to understand.
For comparing the results of two different sets of cluster analyses, a framework is less necessary.
However, there is the question of whether the difference between two index values is significant
Framework for Cluster Validity
Cluster Cohesion: Measures how closely related are objects in a cluster
Example: SSE
Cluster Separation: Measure how distinct or well-separated a cluster is from other clusters
Example: Squared Error
Cohesion is measured by the within cluster sum of squares (SSE)
Separation is measured by the between cluster sum of squares
Where |Ci| is the size of cluster i
Internal Measures: Cohesion and Separation
“The validation of clustering structures is the most difficult and frustrating part of cluster analysis.
Without a strong effort in this direction, cluster analysis will remain a black art accessible only to those true believers who have experience and great courage.”
Algorithms for Clustering Data, Jain and Dubes
Final Comment on Cluster Validity
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