数据挖掘：最全聚类分析 k-means+DBSCAN-CFANZ编程社区

文章目录

聚类算法

Applications of cluster analysis
- understanding: group related documents for browsing, group genes and proteins that have similar functionality, or group stocks
- 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
  - Clustering is a grouping of objects based on the data
- Graph partitioning
  - Some mutual relevance and synergy, but areas are not identical
- Association analysis
  - Local vs. global connections
Notion of a cluster can be ambiguous

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
- clusters of widely different sizes, shapes, and densities

Types of clusters

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.
center-based clusters
- 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
contiguous clusters (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
density-based clusters
- 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.
shared property or conceptual clusters
- find clusters that share some common property or represent a particular concept.
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.
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.
- what to minimize the edge weight between clusters and maximize the edge weight within clusters
type of proximity or density measure
- this is a derived measure, but central to clustering
sparseness
- dictates types of similarity
- adds to efficiency
attribute type
- dictates type pf 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
Number of clusters, K, must be specified
Each cluster is associated with a centroid (center point)
Each point is assigned to the cluster with the closest centroid
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.
  $\sum_{i=1}^K \sum_{x\in C_i} dist^2 (m_i,x)$
- 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 sets of clusters, we prefer 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

K-means as an Optimization Problem

Objective: Minimize the Sum of Squared Error (SSE)
$\sum_{i=1}^K \sum_{x\in C_i} dist^2 (m_i,x)$
We fix the center, if SSE is not optimal,

$c_j = \arg \min _{i \in \{1,2,…,k\}} dist(m_i,j)$

Then, we fix the cluster assignment, derive the new center
$m_i = \frac{1}{|C_i|}\sum_{x\in C_i}x$

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
$\frac{K!n^K}{Kn)^K} = \frac{K!}{K^K}$
for example, if K=10, then probability = 10!/10^10 = 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

Solutions to initial centroids problem

muptiple 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
postprecessing
bisecting K-means
- not as susceptible to initialization issues

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 peocess

Bisecting K-means 二分K均值算法

Bisecting K-means algorithm
- Variant of K-means that can produce a partitional or a
  hierarchical clustering

Limitation of K-means

K-means has problems when clusters are different in:
- Sizes
- Densities
- Non-globular shapes
K-means has problems when the data contains outliers.

Overcoming K-means limitations

One solution is to use many clusters.

Find parts of clusters, but need to put together.

handing empty clusters

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

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

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

Key operation is the computation of the proximity of two clusters

Different approaches to defining the distance between clusters distinguish the different algorithms

Starting situation

start with clusters of individual points and a proximity matrix

Intermediate Situation

after some merging steps, we have some clusters

we want to merge the two closest clusters and update the proximity matrix

After Merging

the question is “How do we update the proximity matrix?”

How to Define Inter-Cluster Similarity

Cluster Similarity: MIN or Single Link

Similarity of two clusters is based on the two most similar(closest) points in the different clusters
- determined by one pair of points, i.e., by one link in the proximity graph.

Strength of MIN

Cluster Similarity: MAX or Complete Linkage

similarity of two clusters is based on the two least similar (most distant) points in the different cluters

determined by all pairs of points in the two clusters

Cluster Similarity: Group Average

Proximity of two clusters is the average of pairwise proximity between points in the two clusters.
$proximity(cluster_i,cluster_j)= \frac{\sum_{p_i\in cluster_i,p_j\in cluster_j}proximity(p_i,p_j)}{|cluster_i|*|cluster_j|}$
Need to use average connectivity for scalability since total proximity favors large clusters

Hierarchical Clustering: Group Average

Compromise between single and complete link

strengths: less susceptible to noise and outliers

limitations: biased towards globular clusters

Cluster Similarity: Ward’s 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

$O(N^2)$ space since it uses the proximity matrix.

N is the number of points

$O(N^3)$ time in many cases

There are N steps and at each step the size, $N^2$ ,proximity matrix must be updated and searched
Complexity can be reduced to $O(N^2log(N))$ 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

MST: Divisive Hierarchical Clustering

Build MST (Minimum Spanning Tree)
- Start with a tree that consists of any point
- In successive steps, look for the closest pair of points (p, q) such
  that one point § is in the current tree but the other (q) is not
- Add q to the tree and put an edge between p and q

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
- 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.

When DBSCAN Works Well

Resistant to Noise
Can handle clusters of different shapes and sizes

When DBSCAN Does NOT Work Well

Varying densities
High-dimensional data

DBSCAN: Determining EPS and MinPts

Idea is that for points in a cluster, their $k^{th}$ nearest neighbors are at roughly the same distance
Nose points have the $k^{th}$ nearest neighbor at farther distance
So, plot sorted distance of every point to its $k^{th}$ nearest neighbor

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

Clusters found in Random Data

Different Aspects of Cluster Validation

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.

Measures of Cluster Validity

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 Via Correlation

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.

Using similarity matrix for cluster validation

Order the similarity matrix with respect to cluster labels and inspect visually.

Internal measures: SSE

Clusters in more complicated figures aren’t well separated
Internal Index: Used to measure the goodness of a clustering
structure without respect to external information - SSE
SSE is good for comparing two clusterings or two clusters (average SSE).
Can also be used to estimate the number of clusters

Framework for cluster validity

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

Statistical framework for SSE

Statistical framework for correlation

Internal measures: cohesion and separation

cluster cohesion: measures how closely related are objects in a cluster
cluster separation: measure how distinct or well-separated a cluster is from other clusters
- cohesion is measured by the within cluster sum of squares(SSE)
- separation is measured by the between cluster sum of squares
  - where $C_i|$ is the size of cluster i
$\sum_i \sum_{x\in C_i} (x-m_i)^2\\ BSS = \sum|C_i|(m-m_i)^2\\ BSS+WSS = constant$

A proximity graph based approach can also be used for cohesion and separation.

Cluster cohesion is the sum of the weight of all links within a cluster.
Cluster separation is the sum of the weights between nodes in the cluster and nodes outside the cluster.

Internal measures: silhouette coefficient

Silhouette Coefficient combine ideas of both cohesion and separation,
but for individual points, as well as clusters and clusterings
For an individual point, i

Calculate a = average distance of i to the points in its cluster
Calculate b = min (average distance of i to points in another cluster)
The silhouette coefficient for a point is then given by
$\quad if \quad a < b,$ (or s = b/a - 1 if a ≥ b, not the usual case)
- Typically between 0 and 1.
- The closer to 1 the better.

Can calculate the Average Silhouette width for a cluster or a clustering

Final comment for cluster validity

“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.”

数据挖掘：最全聚类分析 k-means+DBSCAN