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.
- $\text{Share_coefficient} + \text{NoShare_coefficient} = 1$
- $\text{Share_coefficient}\geq 0$ and $\text{NoShare_coefficient}\geq 0$
- When the vector $U$ and $V$ are identical then $\text{Share_coefficient}=1$
- When the vector $U$ and $V$ are disjoint (e.g., $U=[1\;\; 0\;\; 0]$ and $V = [ 0\;\; 0.5 \;\; 0.5]$) then $\text{NoShare_coefficient}=1$
- 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
- $U=[1\;\; 0\;\; 0]$ and $V=[0.5\;\; 0.5\;\; 0]$ (only 2-dimension are involved out of 3-dimension)
- $U=[0.5\;\; 0.5\;\; 0]$ 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?