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statisticswolfram-mathematicaformulaalgebraeuclidean-distance
Luca Zenesini
Luca Zenesini

Reputation: 21

Understand normalized squared euclidean distance?

I'm trying to understand the normalized squared euclidean distance formula from the Wolfram documentation:

1/2*Norm[(u-Mean[u])-(v-Mean[v])]^2/(Norm[u-Mean[u]]^2+Norm[v-Mean[v]]^2)

I searched around for this formula on the web but couldn't find it. Can someone explain how this formula is derived?

Upvotes: 2

Views: 3833

Answers (2)

Chris Degnen
Chris Degnen

Reputation: 8655

Further to Luca's comment, here is an example showing the "distance between two vectors where their lengths have been scaled to have unit norm". It doesn't equal the normalised square Euclidean distance. The former is coloured blue in the graphic below. The standard Euclidean distance is coloured red.

(* Leave this unevaluated to see symbolic expressions *)
{{a, b, c}, {d, e, f}} = {{1, 2, 3}, {3, 5, 10}};

N[EuclideanDistance[{a, b, c}, {d, e, f}]]

7.87401

Norm[{a, b, c} - {d, e, f}]

SquaredEuclideanDistance[{a, b, c}, {d, e, f}]

Norm[{a, b, c} - {d, e, f}]^2

N[NormalizedSquaredEuclideanDistance[{a, b, c}, {d, e, f}]]

0.25

(1/2 Norm[({a, b, c} - Mean[{a, b, c}]) - ({d, e, f} - Mean[{d, e, f}])]^2)/
 (Norm[{a, b, c} - Mean[{a, b, c}]]^2 + Norm[{d, e, f} - Mean[{d, e, f}]]^2)

1/2 Variance[{a, b, c} - {d, e, f}]/(Variance[{a, b, c}] + Variance[{d, e, f}])

{a2, b2, c2} = Normalize[{a, b, c}];
{d2, e2, f2} = Normalize[{d, e, f}];

N[EuclideanDistance[{a2, b2, c2}, {d2, e2, f2}]]

0.120185

Graphics3D[{Line[{{0, 0, 0}, {1, 2, 3}}], 
  Line[{{0, 0, 0}, {3, 5, 10}}],
  Red, Thick, Line[{{1, 2, 3}, {3, 5, 10}}],
  Blue, Line[{{a2, b2, c2}, {d2, e2, f2}}]},
 Axes -> True, AspectRatio -> 1, 
 PlotRange -> {{0, 10}, {0, 10}, {0, 10}},
 AxesLabel -> Map[Style[#, Bold, 16] &, {"x", "y", "z"}],
 AxesEdge -> {{-1, -1}, {-1, -1}, {-1, -1}},
 ViewPoint -> {1.275, -2.433, -1.975}, 
 ViewVertical -> {0.551, -0.778, 0.302}]

enter image description here

Upvotes: 1

Daniel
Daniel

Reputation: 8677

Meaning of this formula is the following:

Distance between two vectors where there lengths have been scaled to have unit norm. This is helpful when the direction of the vector is meaningful but the magnitude is not.

https://stats.stackexchange.com/questions/136232/definition-of-normalized-euclidean-distance

Upvotes: 2

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