SIMILARITY AND DISSIMILARITY

SIMILARITY

numerical measure of how alike two data objects are, higher when objects are more alike ( range of $[0-1]$ )

DISSIMILARITY

numerical measure of how different two data objects are, lower when objects are more alike ( minimum value $0$ upper bound varies )

ATTRIBUTE TYPE	DISSIMILARITY	SIMILARITY
NOMINAL	$0$ if $p=q$ and viceversa	$1$ if $p=q$ and viceversa
ORDINAL	$\frac{\Vert p-q\Vert }{V-1}$	$1-\frac{\Vert p-q\Vert }{V-1}$
INTERVAL	$\Vert p-q\Vert$	$\frac{1}{1+d}$

PROPERTIES OF SIMILARITY

• $Sim(p,q)=1⟺p=q$
• $Sim(p,q)=Sim(q,p)$

SIMILARITY BETWEEN VECTORS

SIMPLE MATCHING COEFFICIENT

the ratio between the number of matches and the number of attributes

$$\frac{M_{00}+M_{11}}{M_{00}+M_{10}+M_{01}+M_{11}}$$

JACCARD COEFFICIENT

the ratio between the number of $11$ matches and the number of non $00$ attributes

$$\frac{M_{11}}{M_{10}+M_{01}+M_{11}}$$

COSINE COEFFICIENT

the cosine between the vectors

$$\frac{p\ast q}{|p|\ast |q|}$$

EXTENDED JACCARD COEFFICIENT TANIMOTO

the jaccard coefficient for continuous attributes

$$\frac{p\ast q}{|p{|}^2\ast |q{|}^2-p\ast q}$$

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