第十九章 聚类分析 (Clustering Analysis) - 中南大学

rij
( X X )( X X ) (X X ) (X X
i i j j 2 i i j
j
)
2
(19-1)
The two variables tend to be more similar when the absolute value increases. Similarly, Spearman rank correlation coefficient can be used to define the similarity coefficient of non-normal variables. But when the variables are all qualitative variables, it’s best to use contingency coefficient.
For example, m refers to the number of variables(i.e. indexes)
while n refers to that of cases(i.e. samples) ,you can do as follows: (1) R-type clustering: also called index clustering. The method to sort the m kinds of indexes, aiming at lowering the
individuals to the correct population.
Clustering Analysis: a statistic method for grouping objects of random kind into respective categories. It’s used when there’s no priori hypotheses, but trying to find the most appropriate sorting method resorting to mathematical statistics and some collected information. It has become the first selected means to uncover great capacity of genetic messages.
The calculation of similarity coefficient between clusters
Each step of hierarchical clustering has to calculate the similarity coefficient among clusters. When there is only one sample or variable in each of the two clusters, the similarity coefficient between them equals to that of the two samples or the two variables, or compute according to section one. When there are more than one sample or variable in each cluster, many kinds of methods can be used to compute similarity coefficient. Just list 5 kinds of methods as follows.G p and Gq refer to the two clusters, which respectively has np or nq kinds of samples or variables.
Discriminant Analysis : having known with certainty to
come from two or more populations, it’s a method to acquire the discriminate model that will allocate further
distance or maximum correlation coefficient) are merged into a new
cluster. Compute the similarity coefficient between the new cluster with other clusters. Repeat step two until all of the samples (or variables) are merged into one cluster.
2． Similarity coefficient commonly used in Q-type clustering : Suppose there are n cases regard as n spots in a m dimensions space, distance between two spots can be used to define similarity coefficient, the two samples tend to be more similar when the distance declines.
（1）Euclidean distance
dij
(X
i
X j )2
(19-3)
（2）Manhattan distance
dij
|X
i
Xj |
(19-4)
（3）Minkowski distance：
dij
q

| XiAbsolute distance refers to Minkowski distance when q=1；Euclidean distance is direct-viewing and simple to compute, but having not regarded the correlated relations among variables. That’s why Manhattan distance was introduced.
The matrix is made up of similarity coefficients between samples (or
variables). Similarity coefficient matrix is a symmetrical matrix. 2）The two clusters with the maximum similarity coefficient( minimum
§ 2 Hierarchical Clustering Analysis
Hierarchical clustering analysis is a most commonly used method to sort out similar samples or variables. The process is as follows: 1）At the beginning, samples(or variables) are regarded respectively as one single cluster, that is, each cluster contains only one sample(or variable). Then work out similarity coefficient matrix among clusters.
(19-6)
When it’s a unit matrix, Mahalanobis distance equals to the square of Euclidean distance.
All of the four distances refer to quantitative variables, for the qualitative variables and ordinal variables, quantization is needed before using.
commonness among them.
The most important thing for both R-type
clustering and Q-type clustering is the
definition of similarity, that is how to
quantify similarity. The first step of
clustering is to define the metric similarity
between two indexes or two samplessimilarity coefficient
§ 1 similarity coefficient
1 similarity coefficient of R-type clustering Suppose there are m kinds of variables: X1，X2，…，Xm. Rtype clustering usually use the absolute value of simple correlation coefficient to define the similarity coefficient among variables:
1．The
maximum similarity coefficient method If there’re respectively np , nq samples(or variables) in cluster Gp and Gq , here’re altogether n p and nq 2 similarity coefficients between the two clusters, but only the maximum is considered as the similarity coefficient of the two clusters.