The usual correlation coefficient (in 2d) measures how well a set of points can be described by a line, and if yes, its sign tells us whether we have a positive or negative correlation. But this assumes that coordinates of the points can actually interpreted quantitatively for instance as measurements.
If you cannot do that but you can still order the coordinates, there is the rank correlation coefficient: It measures how well the points can be described by a monotonic function.
Given a list of 2d points, determine their rank correlation coefficient.
- You can assume the input to be positive integers (but you don't have to), or any other "sortable" values.
- The points can be taken as a list of points, or two lists for the x- and y-coordinates or a matrix or 2d array etc.
- The output must be a floating point or rational type, as it should represent a real number between 0 and 1.
Rank: Given a list of numbers
X=[x(1),...,x(n)] we can assign a positive number
rx(i) called rank to each entry
x(i). We do so by sorting the list and assigning the index of
x(i) in the sorted list
rx(i). If two or more
x(i) have the same value, then we just use the arithmetic mean of all the corresponding indices as rank. Example:
List: [21, 10, 10, 25, 3] Indices sorted: [4, 2, 3, 5, 1]
10 appears twice here. In the sorted list it would occupy the indices
3. The arithmetic mean of those is
2.5 so the ranks are
Ranks: [4, 2.5, 2.5, 5, 1]
Rank Correlation Coefficient: Let
[(x(1),y(1)),(x(2),y(2)),...,(x(n),y(n))] be the given points where each
y(i) is a real number (wlog. you can assume it is an integer)
i=1,...,n we compute the rank
d(i) = rx(i)-ry(i) be the rank difference and let
S be the sum
S = d(1)^2 + d(2)^2 + ... + d(n)^2. Then the rank correlation coefficient
rho is given by
rho = 1 - 6 * S / (n * (n^2-1))
x y rx ry d d^2 21 15 4 5 -1 1 10 6 2&3 -> 2.5 2 0.5 0.25 10 7 2&3 -> 2.5 3 -0.5 0.25 25 11 5 4 1 1 3 5 1 1 0 0 rho = 1 - 6 * (1+0.25+0.25+1)/(5*(5^2-1)) = 0.875