DISTANCES

PROPERTIES OF DISTANCES

• symmetry $Dist(p,q)=Dist(q,p)$
• positive definiteness $Dist(p,q)>=0\forall p,q$
• Triangle inequality $Dist(p,q)<=Dist(p,r)+Dist(r,q)\forall p,q,r$

EUCLIDEAN DISTANCE

$$\sqrt{\sum _{d=1}^D(p_d-q_d{)}^2}$$

Where $D$ is the number of dimensions (attributes) and $p_d$ and $q_d$ are, respectively, the d-th attributes (components) of data objects p and q. Standardization/Rescaling is necessary if scales differ

MINKOWSKI DISTANCE $L_r$

generalization of euclidean distance

$$(\sum _{d=1}^D|p_d-q_d{|}^r{)}^{\frac{1}{r}}$$

Where $D$ is the number of dimensions (attributes) and $p_d$ and $q_d$ are, respectively, the d-th attributes (components) of data objects p and q. Standardization/Rescaling is necessary if scales differ. $r$ is a parameter which is chosen depending on the data set and the application

cases

• $r=1$ Manhattan distance best at discriminate between 0 distance and near 0 distance

• $r=2$ Euclidean distance

• $r=\infty$ Chebyshev, supremum, $L_{max}$ norm, $L_{\infty }$ norm considers only the feature with the maximum difference

  $$\underset{d}{\max }|p_d-q_d|$$

MAHALANOBIS DISTANCE

The Mahlanobis distance between two points p and q decreases if, keeping the same euclidean distance, the segment connecting the points is stretched along a direction of greater variation of data. The distribution is described by the covariance matrix of the data set

$$\sqrt{(p-q)\sum ^{-1}(p-q{)}^T}$$

PREVIOUS NEXT