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Mahalanobis distance
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==Relationship to normal random variables== In general, given a normal ([[Gaussian distribution|Gaussian]]) random variable <math>X</math> with variance <math>S=1</math> and mean <math>\mu = 0</math>, any other normal random variable <math>R</math> (with mean <math>\mu_1</math> and variance <math>S_1</math>) can be defined in terms of <math>X</math> by the equation <math>R = \mu_1 + \sqrt{S_1}X.</math> Conversely, to recover a normalized random variable from any normal random variable, one can typically solve for <math>X = (R - \mu_1)/\sqrt{S_1} </math>. If we square both sides, and take the square-root, we will get an equation for a metric that looks a lot like the Mahalanobis distance: <math display="block">D = \sqrt{X^2} = \sqrt{(R - \mu_1)^2/S_1} = \sqrt{(R - \mu_1) S_1^{-1} (R - \mu_1) }.</math> The resulting magnitude is always non-negative and varies with the distance of the data from the mean, attributes that are convenient when trying to define a model for the data.
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