TESI DOCTORAL - La Salle

1.2. Clustering in knowledge discovery and data mining
– Euclidean distance: possibly the most widely used metric, it is obtained as the
particularization of the Minkowski metric for n =2:
D(xi, xj) =
d
l=1
|xil
− xjl | 1
2
2
(1.2)
This is the distance measure used in the most classic implementation of the kmeans
clustering algorithm, and it tends to form hyperspherical clusters (Xu and
Wunsch II, 2005).
– Manhattan distance: also known as city block distance, it is defined as a particular
case of the Minkowski metric for n = 1 (see equation (1.3)), and it tends to
create hyperrectangular clusters (Xu and Wunsch II, 2005).
d
D(xi, xj) = |xil − xjl |
l=1
(1.3)
– Mahalanobis distance: it can be regarded as a modified version of the Euclidean
distance, that takes into account the covariance among the attributes. It is
defined as follows:
D(xi, xj) =(xi − xj) T S −1 (xi − xj) (1.4)
where S is the sample covariance matrix computed over all the data set (Jain,
Murty, and Flynn, 1999). Algorithms using this distance tend to create hyperellipsoidal
clusters (Xu and Wunsch II, 2005).
2. Similarity measures
– Cosine similarity: it consists of measuring the angle comprised between the vectors
representing the objects, and, as such, it does not depend on their length.
It is defined as follows:
S(xi, xj) = xT i xj
||xi|| ||xj||
(1.5)
where ||·||denotes vector norm.
– Pearson correlation coefficient: a classic concept in the probability theory and
statistics fields, correlation measures the strength and direction of the linear
relationship between vectors xi and xj. The most widely used correlation index
is the Pearson correlation coefficient, which is defined in equation (1.6):
d
(xil − ¯xi)(xjl − ¯xj)
l=1
S(xi, xj) =
d
d 2 2
(xil − ¯xi) (xjl − ¯xj)
l=1
l=1
where xil is the lth component of vector xi, and¯xi denotes its sample mean.
12
(1.6)