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The computer science appendix

# (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:

$h = \frac{n}{\sum_{i=1}^{i=n} \frac{1}{x_i}}$

# Common measures of distance/similarity

## Euclidean

The euclidean distance of two vectors$v_a = (x^i)_a$and$v_b = (x^i)_b$is the norm$l_2$of the vector connecting them (it measures its length):

$d = \sqrt{\sum_i (x^i_a - x^i_b)^2} = ||v_A - v_B||_2$

## Hamming

The Hamming distance expresses the number of different elements in two lists/strings:

$A = 110101; B = 111001; d_{AB} = 2$

## Jaccard (index)

Given two finite sets A and B, the Jaccard index gives a measure of how much they overlap, as

$J_{AB} = \frac{|A \cap B|}{|A \cup B|}$

## Manhattan

Also called cityblock, the Manhattan distance between two points is the norm$l_1$of the shortest path a car would take between these two points in Manhattan (which has a grid layout):

$d = \sum_i |u_i - v_i|$

## Minkowski

The Minkowski distance is a generalisation of both the euclidean and the Manhattan to a generic p:

$d = \left(\sum_i |x_i - y_i|^p\right)^{1/p}$

## Cosine

The cosine similarity is given by the cosine of the angle$\theta$spanned by the two vectors

$d = \cos \theta = \frac{\bar u \cdot \bar v}{|\bar u| |\bar v|}$

So two perfectly overlapping vectors would have a cosine similarity of 1 and vectors at $90^{\circ}$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.

$\max_i |u_i - v_i|$

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.