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Mahalanobis' Distance

Introduction


Today, I will write about Mahalanobis’ Distance.
Mahalanobis’ Distance is used when each dimension has a relationship.
This distance is fulfilled definition of distance.
Mahalanobis’ Distance is important for Statics.
If you interested in Statics or Machine Learning, Please see my this blog.

Overview

  • definition of distance
  • deficition of Mahalanobis’ Distance
  • image of Mahalanobis’ Distance

definition of distance

if d is distance function, d if fulfilled following condtion.
\(d:X \times X -> R\)
  • \(d(x,y) \geq 0\)
  • \(d(x,y) = 0 \leftrightarrow x = y\)
  • \(d(x,y) = d(y,x)\)
  • \(d(x,z) \leq d(x,y) + d(y,z)\)

Mahalanobis’ Distance

Mahalanobis’ Distance is distance function.
Mahalanobis’ Distance is defined by following from
\[D_{M}(x) = \sqrt{(x-\mu)^T \Sigma^{-1} (x-\mu)}\]
here, \(\mu\) is mean vector
\[\mu = (\mu_1,\mu_2,....,\mu_n)\]
and, \(\Sigma\) is variance-convariance matrix.
Mahalanobis’ Distance between x and y is
\begin{eqnarray*} d(x,y) &=& \sqrt{(x-\mu-(y-\mu)^T \Sigma^{-1} (x-\mu-(y-\mu)}\\ &=& \sqrt{(x-y)^T \Sigma^{-1} (x-y)} \end{eqnarray*}

Image of Mahalanobis’ Distance

first, I think of Eculid distance. Eculid distance is following form.
\[d(x,y) = \sqrt{<x^T,y>}\]
The eculid distance regard distance between \(x\) and \(y\) as same if x and y exists over same circle.
enter image description here
This thinking has reason that data distributed like following.
enter image description here
but ,if data distributed like ellipse, it is not good at to use Euclid distance.
enter image description here
Because I want to regard distance between X and Y as same.
Mahalanobis’ Distance is regard distance between X and Y as same if X and Y have existed over the same ellipse.
enter image description here
Distance is always used Machine Learning. Machine Learning use Eculid distance, but We get interesting result by using Mahalanobis’ Distance.
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