(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=ni=1i=n1xih = \frac{n}{\sum_{i=1}^{i=n} \frac{1}{x_i}}

Common measures of distance/similarity


The euclidean distance of two vectors
va=(xi)av_a = (x^i)_a
vb=(xi)bv_b = (x^i)_b
is the norm
of the vector connecting them (it measures its length):
d=i(xaixbi)2=vAvB2d = \sqrt{\sum_i (x^i_a - x^i_b)^2} = ||v_A - v_B||_2


The Hamming distance expresses the number of different elements in two lists/strings:
A=110101;B=111001;dAB=2A = 110101; B = 111001; d_{AB} = 2

Jaccard (index)

Given two finite sets A and B, the Jaccard index gives a measure of how much they overlap, as
JAB=ABABJ_{AB} = \frac{|A \cap B|}{|A \cup B|}


Also called cityblock, the Manhattan distance between two points is the norm
of the shortest path a car would take between these two points in Manhattan (which has a grid layout):
d=iuivid = \sum_i |u_i - v_i|


The Minkowski distance is a generalisation of both the euclidean and the Manhattan to a generic p:
d=(ixiyip)1/pd = \left(\sum_i |x_i - y_i|^p\right)^{1/p}


The cosine similarity is given by the cosine of the angle
spanned by the two vectors
d=cosθ=uˉvˉuˉvˉd = \cos \theta = \frac{\bar u \cdot \bar v}{|\bar u| |\bar v|}
So two perfectly overlapping vectors would have a cosine similarity of 1 and vectors at
would have a cosine similarity of 0.


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.
maxiuivi\max_i |u_i - v_i|
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.