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I have two lists which are circular, in that given list

['A','B','C','D','E']

'A' and 'E' are the same distance apart as 'A' and 'B' and so on.

Given I have rankings

R1=['A','B','C','D','E']

R2=['E','A','B','C','D']

how can I compute their similarity or distance?

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  • $\begingroup$ What are 'A','B','C','D','E'? Real numbers? Integers? Matrices? $\endgroup$ – Vladislav Gladkikh Mar 30 '20 at 3:24
  • $\begingroup$ they are categorical variables representing colors for example. $\endgroup$ – ha554an Mar 30 '20 at 13:42
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The variables $A$, $B$, $C$, $D$, and $E$ can be encoded as points on a circle $S^1$ that is by angles between $0$ ans $2\pi$ if you measure the angles in radians. Vectors $R_1$ and $R_2$ are thus points on a five-dimensional torus $$T^5=\underbrace{S^1\times ...\times S^1}_\text{5}$$ The distance $d_{12}$ between $R_1$ and $R_2$ is the distance between these points on the torus $T^5$.

It's more convenient to measure the angles on your circles in $2\pi\cdot\mathrm{radians}$. Then $A$, $B$, $C$, $D$, and $E$ are real numbers $x_1, ..., x_5$ all between $0$ and $1$. Then $R_1=(x_{11},..., x_{15})$, $R_2=(x_{21},..., x_{25})$, and

$$d_{12} = \sum\limits_{i=1}^{5}s_i,$$ where \begin{equation} s_i = \begin{cases} |x_{2i}-x_{1i}|, & \text{if $|x_{2i}-x_{1i}|\leq 0.5$}\\ 1-|x_{2i}-x_{1i}|, & \text{if $|x_{2i}-x_{1i}| > 0.5$} \end{cases} \end{equation}

An explanation of this formula for a two-dimensional torus is given here: link.

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