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Proximity matrix in data mining

29/11/2021 Client: muhammad11 Deadline: 2 Day

Data Mining
Cluster Analysis: Basic Concepts
and Algorithms

Lecture Notes for Chapter 8

Introduction to Data Mining

Tan, Steinbach, Kumar

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
Clustering precipitation in Australia

Discovered Clusters
Industry Group

Applied-Matl-DOWN,Bay-Network-Down,3-COM-DOWN,

Cabletron-Sys-DOWN,CISCO-DOWN,HP-DOWN,

DSC-Comm-DOWN,INTEL-DOWN,LSI-Logic-DOWN,

Micron-Tech-DOWN,Texas-Inst-Down,Tellabs-Inc-Down,

Natl-Semiconduct-DOWN,Oracl-DOWN,SGI-DOWN,

Sun-DOWN

Technology1-DOWN
2

Apple-Comp-DOWN,Autodesk-DOWN,DEC-DOWN,

ADV-Micro-Device-DOWN,Andrew-Corp-DOWN,

Computer-Assoc-DOWN,Circuit-City-DOWN,

Compaq-DOWN, EMC-Corp-DOWN, Gen-Inst-DOWN,

Motorola-DOWN,Microsoft-DOWN,Scientific-Atl-DOWN

Technology2-DOWN

Fannie-Mae-DOWN,Fed-Home-Loan-DOWN,

MBNA-Corp-DOWN,Morgan-Stanley-DOWN

Financial-DOWN

Baker-Hughes-UP,Dresser-Inds-UP,Halliburton-HLD-UP,

Louisiana-Land-UP,Phillips-Petro-UP,Unocal-UP,

Schlumberger-UP

Oil-UP

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
(C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002

Notion of a Cluster can be Ambiguous

How many clusters?

Four Clusters

Two Clusters

Six Clusters

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
(C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002

Partitional Clustering

Original Points

A Partitional Clustering

Hierarchical Clustering

Traditional Hierarchical Clustering

Non-traditional Hierarchical Clustering

Non-traditional Dendrogram

Traditional Dendrogram

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
(C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002

Types of Clusters

Well-separated clusters
Center-based clusters
Contiguous clusters
Density-based clusters
Property or Conceptual
Described by an Objective Function
(C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002

Types of Clusters: Well-Separated

Well-Separated Clusters:
A cluster is a set of points such that any point in a cluster is closer (or more similar) to every other point in the cluster than to any point not in the cluster.
3 well-separated clusters

Types of Clusters: Center-Based

Center-based
A cluster is a set of objects such that an object in a cluster is closer (more similar) to the “center” of a cluster, than to the center of any other cluster
The center of a cluster is often a centroid, the average of all the points in the cluster, or a medoid, the most “representative” point of a cluster
4 center-based clusters

Types of Clusters: Contiguity-Based

Contiguous Cluster (Nearest neighbor or Transitive)
A cluster is a set of points such that a point in a cluster is closer (or more similar) to one or more other points in the cluster than to any point not in the cluster.
8 contiguous clusters

Types of Clusters: Density-Based

Density-based
A cluster is a dense region of points, which is separated by low-density regions, from other regions of high density.
Used when the clusters are irregular or intertwined, and when noise and outliers are present.
6 density-based clusters

Types of Clusters: Conceptual Clusters

Shared Property or Conceptual Clusters
Finds clusters that share some common property or represent a particular concept.
.

2 Overlapping Circles

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.
(C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002

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
(C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002

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
(C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002

Clustering Algorithms

K-means and its variants
Hierarchical clustering
Density-based clustering
(C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002

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
(C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002

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
(C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002

Two different K-means Clusterings

Original Points

Sub-optimal Clustering

Optimal Clustering

Importance of Choosing Initial Centroids

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
(C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002

Importance of Choosing Initial Centroids …

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 don’t
Consider an example of five pairs of clusters
(C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002

10 Clusters Example

Starting with two initial centroids in one cluster of each pair of clusters

10 Clusters Example

Starting with two initial centroids in one cluster of each pair of clusters

10 Clusters Example

Starting with some pairs of clusters having three initial centroids, while other have only one.

10 Clusters Example

Starting with some pairs of clusters having three initial centroids, while other have only one.

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
(C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002

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.
(C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002

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
(C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002

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
(C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002

Bisecting K-means

Bisecting K-means algorithm
Variant of K-means that can produce a partitional or a hierarchical clustering
(C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002

Bisecting K-means Example

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.
(C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002

Limitations of K-means: Differing Sizes

Original Points

K-means (3 Clusters)

Limitations of K-means: Differing Density

Original Points

K-means (3 Clusters)

Limitations of K-means: Non-globular Shapes

Original Points

K-means (2 Clusters)

Overcoming K-means Limitations

Original Points K-means Clusters

One solution is to use many clusters.

Find parts of clusters, but need to put together.

Overcoming K-means Limitations

Original Points K-means Clusters

Overcoming K-means Limitations

Original Points K-means Clusters

Hierarchical Clustering

Produces a set of nested clusters organized as a hierarchical tree
Can be visualized as a dendrogram
A tree like diagram that records the sequences of merges or splits
(C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002

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, …)
(C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002

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
(C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002

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
(C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002

Starting Situation

Start with clusters of individual points and a proximity matrix
Proximity Matrix

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