8.2 Background Preparation for Integrated Analysis

Similarity Measures

Given two vectors,  and  

 = (x1, x2, ..., xn)T

 = (y1, y2, ..., yn)T

where n is the number of elements in a vector, the similarity between  and  , g( ,  ) ³ 0 , can be defined by

1) dot product,

 =  

where M > 0 and M ³  ·  

2) angular cosine

 

3) correlation coefficient

 

where xm =  , ym =  

4) exponential similarity

 

where Sk is a positive number determined by the user.

5) Non-parametric method

Let x'k = xk - xm , y'k = yk - ym and xm and ym

are the same as in 3)

n+ = the number of elements which are greater than 0 is  

n- = the number of elements which are less than 0 in the same set, then g( ,  ) is  

6) minimum - maximum ratio

 

7) minimum arithmetic average

 

8) minimum geometric average

 

9) sum of absolute differences, or city-bock distances, Manhattan distances

(i) e 

(ii)  

where M is selected such that 0 ² g( ,  ) ² 1

10) Euclidian distances

 

where M is selected such that 0 ² g( ,  ) ² 1