Meta & resources
Machine Learning: concepts & procedures
Machine Learning: fundamental algorithms
Machine Learning: model assessment
Artificial neural networks
Natural language processing
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.