Similarity analysis
In statistics , especially multivariate statistics , one is interested in the measurement of the similarity between different objects and defines similarity and distance measures for this purpose . This is not to measure the mathematical sense , the term applies only to the measurement of a certain size.
Typically, similarity measures are used for nominally or ordinally scaled variables and distance measures are used for metrically scaled variables (i.e. for interval and ratio scales ).
Similarity measure
definition
Let be a finite set. A function is called a similarity measure or similarity function if the following applies to all :
 and
 .
In addition, it is often required that the following applies to everyone :
 and .
The function values can be a symmetric  Matrix arrange. This matrix is called the similarity matrix . In this context it is also referred to as the similarity coefficient.
Application in bioinformatics
Similarity matrices such as PAM or BLOSUM play an important role in sequence alignment . Similar proteins , nucleotides or amino acids receive higher scores (i.e. similarity values) than dissimilar ones. The similarity is defined here by the chemical properties of the building blocks and their mutation rates .
Example (AGCT stands for the four nucleobases adenine , guanine , cytosine and thymine ):
A.  G  C.  T  

A.  10  −1  −3  −4 
G  −1  7th  −5  −3 
C.  −3  −5  9  0 
T  −4  −3  0  8th 
The molecules whose similarity is to be specified are sorted in the same order by columns and rows. The value at the position thus indicates how similar the molecule at column position i is to that at row position j.
According to the above similarity matrix, cytosine and tymine (similarity score 0) are more similar to one another than guanine to cytosine (similarity score 5).
Similarity measures for binary variables
For binary variables and two observations and be
 , ,
 , and
 .
Then you can define the following dimensions:
Similarity measure  

brown  
Dice  
Hamann  
Jaccard ( S coefficient )  
Kappa  
Kulczynski  
Ochiai  
Phi  
Russel Rao  
Simple matching ( Mcoefficient )  
Simpson  
Sneath  
Tanimoto ( Rogers )  
Yule 
For nonbinary nominal or ordinal variables, one defines a binary variable for each category of the variable and can then use the similarity measures for binary variables.
Choice of the degree of similarity
Which degree of similarity is chosen for the analysis depends on the problem. However, there are some indications as to when which measure is best depending on the properties of the binary variable:
 Is the variable symmetric, i. H. Both categories are equally important (e.g. gender), then the same presence ( ) or the same absence ( ) is often important for a similarity measure. Then Simple Matching, Hamman or Tanimoto can be used.
 Is the variable asymmetrical, i. H. only one category plays an essential role (e.g. disease occurred), then often only the same occurrence ( ) plays a role. Then Dice, Jaccard, Kulczynskl, Ochiai, Braun, Simpson or Sneath can be used.
 Kappa, Phi and Yule can be used in both the symmetrical and the asymmetrical case.
When choosing the similarity measure, connections between the measures should also be taken into account:
 Dice, Jaccard and Sneath are monotonous functions of each other:
 Looking at Simpson and Braun, the harmonic mean is Dice, the arithmetic mean is Kulczynski, and the geometric mean is Ochiai. From the inequality of the mean values it follows:
 Hamman, Rogers and Simple matching also show a connection.
Distance measure
definition
Let be a finite set. A function is called a distance measure or distance function if the following applies to all :
 such as
 and .
The function values can be a symmetrical  Matrix arrange. This matrix is called the distance matrix .
If the function also satisfies the triangle inequality , it is a metric . A metric is often referred to as a distance function .
Some distance measurements
For scale variables and two observations and one can define the following measures:
Distance measure  

Euclidean 

Pearson 
with the standard deviation of the variable 
City block Manhattan 

Gower 
with the range of the variable 
Mahalanobis 
with the sample covariance matrix of the variables 
Relationship between similarity and distance measures
In general, one can define a distance measure from a similarity measure by
 .
However, a distance measure obtained in this way generally does not satisfy the triangle inequality and is therefore not a metric.
See also
literature
 Joachim Hartung , Bärbel Elpelt: Multivariate Statistics. Teaching and handbook of applied statistics. Oldenbourg Verlag, Munich 1984, ISBN 3486284517
 Ludwig Fahrmeir, Alfred Hamerle: Multivariate statistical methods. de Gruyter, Berlin 1984, ISBN 3110085097
Individual evidence
 ^ PF Russel, TR Rao: On habitat and association of species of Anopheline larvae . In: Southeastern Madras, Journal of Malaria Institute India . 3, 1940, pp. 153178.
 ^ DJ Rogers and TT Tanimoto: A Computer Program for Classifying Plants . In: Science . 132, No. 3434, October 21, 1960, pp. 11151118. doi : 10.1126 / science.132.3434.1115 .
 ↑ ShengLi Tzeng, HanMing Wu, ChunHouh Chen: Selection of Proximity Measures for Matrix Visualization of Binary Data . In: Biomedical Engineering and Informatics, 2009. BMEI '09. 2nd International Conference on . October 30, 2009, p. 1–9 , doi : 10.1109 / BMEI.2009.5305137 .
 ↑ JC Gower: A General Coefficient of Similarity and Some of Its Properties . In: Biometrics . 27, No. 4, December 1971, pp. 857871.
 ^ Wolfgang Härdle , Léopold Simar: Applied Multivariate Statistical Analysis . 1st edition. Springer Verlag, Berlin 2003, ISBN 3540030794 , pp. 381 .