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Given two $n-$dimensional vector, namely $U = [u_1,u_2 \cdots u_n]$ and $V=[v_1,v_2 \cdots v_n]$ such that $\sum_i u_i = \sum_i v_i = 1$ and $u_i \in [0,1]$ and $v_i \in [0,1]$, i would like to derive a $\text{Share_coefficient}$ and $\text{NoShare_coefficient}$ measure which satisfies the below constraint.

  1. $\text{Share_coefficient} + \text{NoShare_coefficient} = 1$
  2. $\text{Share_coefficient}\geq 0$ and $\text{NoShare_coefficient}\geq 0$
  3. When the vector $U$ and $V$ are identical then $\text{Share_coefficient}=1$
  4. When the vector $U$ and $V$ are disjoint (e.g., $U=[1\;\; 0\;\; 0]$ and $V = [ 0\;\; 0.5 \;\; 0.5]$) then $\text{NoShare_coefficient}=1$
  5. For other combination of vector $U$ and $V$, the coefficients are assigned accordingly; i.e. when the overlap between the vectors is more, then $\text{Share_coefficient}$ should be greater than $\text{NoShare_coefficient}$ and vice versa.

I derived the following measure.
$\text{Share_coefficient} = 1- \frac{1}{2}\sum_i|u_i-v_i|\quad\text{ and }\quad \text{NoShare_coefficient} = \frac{1}{2}\sum_i|u_i-v_i|$

But it is unable to distinguish the below two example

  1. $U=[1\;\; 0\;\; 0]$ and $V=[0.5\;\; 0.5\;\; 0]$ (only 2-dimension are involved out of 3-dimension)
  2. $U=[0.5\;\; 0.5\;\; 0]$ and $V=[0\;\; 0.5\;\; 0.5]$ (all the dimension are involved)

Both these example provides $\text{Share_coefficient} = \text{NoShare_coefficient} = 0.5$. However I would like to distinguish these two example by assigning different values.

Can anyone help me out in deriving a measure which distinguish the above example and satisfies the mentioned constraint?

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    @Learner: Great, then Thomas' suggestion is perfect. (Dilip's, which is the square root of Thomas', also works, but Thomas' avoids the square roots and also has a nice symmetry between share and no-share.)2011-12-20

1 Answers 1

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If two vectors are "disjoint" in your sense, then they are perpendicular - the angle between them is $\frac{\pi}2$. On the other hand, the angle between a vector and itself is $0$. Since you want the share value in these cases to be $0$ and $1$, respectively, this seems like the cosine function.

Since you want the values for share and no-share to add to $1$, I'd choose the squares of the cosine and sine of the angle between the two vectors to be your two functions.

Since the cosine of the angle between $U$ and $V$ can be written as $\frac{U\cdot V}{|U||V|}$, you can write:

$\text{Share_coefficient}(U,V)=\frac{(U\cdot V)^2}{(U\cdot U)(V\cdot V)}$