(Some) mathematical measures

This page outlines, mainly for convenience, some useful measures/metrics utilised for several purposes.
The harmonic mean
It is the reciprocal of the mean of reciprocals:
h=n∑i=1i=n1xih = \frac{n}{\sum_{i=1}^{i=n} \frac{1}{x_i}}h=∑i=1i=n​xi​1​n​
Common measures of distance/similarity
Euclidean
The euclidean distance of two vectorsva=(xi)av_a = (x^i)_ava​=(xi)a​
andvb=(xi)bv_b = (x^i)_bvb​=(xi)b​
is the norml2l_2l2​
of the vector connecting them (it measures its length):
d=∑i(xai−xbi)2=∣∣vA−vB∣∣2d = \sqrt{\sum_i (x^i_a - x^i_b)^2} = ||v_A - v_B||_2d=i∑​(xai​−xbi​)2​=∣∣vA​−vB​∣∣2​
Hamming
The Hamming distance expresses the number of different elements in two lists/strings:
A=110101;B=111001;dAB=2A = 110101; B = 111001; d_{AB} = 2A=110101;B=111001;dAB​=2
Jaccard (index)
Given two finite sets A and B, the Jaccard index gives a measure of how much they overlap, as
JAB=∣A∩B∣∣A∪B∣J_{AB} = \frac{|A \cap B|}{|A \cup B|}JAB​=∣A∪B∣∣A∩B∣​
Manhattan
Also called cityblock, the Manhattan distance between two points is the norml1l_1l1​
of the shortest path a car would take between these two points in Manhattan (which has a grid layout):
d=∑i∣ui−vi∣d = \sum_i |u_i - v_i|d=i∑​∣ui​−vi​∣
Minkowski
The Minkowski distance is a generalisation of both the euclidean and the Manhattan to a generic p:
d=(∑i∣xi−yi∣p)1/pd = \left(\sum_i |x_i - y_i|^p\right)^{1/p}d=(i∑​∣xi​−yi​∣p)1/p
Cosine
The cosine similarity is given by the cosine of the angleθ\thetaθ
spanned by the two vectors
d=cos⁡θ=uˉ⋅vˉ∣uˉ∣∣vˉ∣d = \cos \theta = \frac{\bar u \cdot \bar v}{|\bar u| |\bar v|}d=cosθ=∣uˉ∣∣vˉ∣uˉ⋅vˉ​
So two perfectly overlapping vectors would have a cosine similarity of 1 and vectors at 90∘90^{\circ}90∘
would have a cosine similarity of 0.
Chebyshev
​
 
It is also called chessboard distance. In the game of chess, the Chebyshev distance between the centers of the squares is the minimum number of moves a king needs to go from a square to another one.
max⁡i∣ui−vi∣\max_i |u_i - v_i|imax​∣ui​−vi​∣
See the figure here, it reports in red all the Chebyshev distance value from where the king (well, there's a drawing for it ...) sits to cell; note that the king can move horizontally, vertically and diagonally.

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The harmonic mean

Common measures of distance/similarity