# Weighted metric (or semi-metric) with incorporated distance between Dimensions

Im trying to construct a distance Measure between two vectors, that takes into account also the distance between the Dimensions. I will illustrate my Problem with some examples:

$$x,y \in \mathbb{R}^n$$, $$d(x,y)$$ fullfills the metric-axioms (but im not shure if the triangle-equality must hold - so a semi-metric could also probably do the job).

1.)

$$x_1=1,x_2=1,x_3=1,x_4=1$$;

$$y_1=0,9;y_2=1,1;y_3=1;y_4=1$$;

$$d(x,y)=0,1$$

In this example 0.1 is moved from $$𝑥_1$$ to $$𝑦_2$$, so the distance between the coordinates is 1, times 0.1 gives $$d(x,y)=0.1$$.

2.) $$x_1=1;x_2=1;x_3=1;x_4=1$$;

$$y_1=0,9;y_2=1;y_3=1;y_4=1,1$$;

$$d(x,y)=0.3$$

In the second case 0.1 is moved from $$x_1$$ to $$𝑦_4$$, so the distance between the coordinates is 3 times 0.1 gives $$d(x,y)=0.3$$.

3.)

$$x_1=2;x_2=2;x_3=2;x_4=2$$;

$$y_1=1,9;y_2=2,1;y_3=1.9;y_4=2,1;$$

$$d(x,y)=0.2$$

In this example 0.1 is moved from 𝑥1 to 𝑦1 equates to 0.1. Also 0.1 is moved from $$𝑥_3$$ to $$𝑦_4$$. In total we moved 0.2 mass from $$x_i$$ to $$x_{i+1}$$. Equates to $$d(x,y)=0.2$$

4.)

$$x_1=2;x_2=2;x_3=2;x_4=2$$;

$$y_1=1,9;y_2=1,8;y_3=2,2;y_4=2,1$$;

$$d(x,y)=0.5$$

In this case either 0.1 is moved from $$𝑥_1$$ to $$𝑦_4$$ and 0.2 from $$𝑥_2$$ to $$𝑦_3$$. Equates to $$d(x,y)=0.3+0.2$$. Some more explanations are also possible,i.e: 0.1 is moved from $$x_1$$ to $$y_3$$ equals to $$d_1(x,y)=0.2$$. 0.1 are moved from $$x_2$$ to $$y_3$$, $$d_2(x,y)=0.1$$. 0.1 is moved from $$x_2$$ to $$y_4$$, $$d_3(x,y)=0.2$$. \$d(x,y)=d_1+d_2+d_3=0.5.

Im trying to define $$d(x,y)$$ very general. My first guess was to somehow combine the euclidean distance $$\sqrt{(x_i-y_j)^2}$$ and the euclidean distance $$\sqrt{(i-j)^2}$$ but it didnt worked out to well :) I would appreciate any help!

Let $$c\colon[n]\times[n]\to\mathbb R$$ be any (say transportation cost) function, which may or may not be a metric; here $$[n]:=\{1,\dots,n\}$$. Then what you apparently want is the Kantorovich--Rubinstein--Wasserstein distance corresponding to the transportation cost function $$c$$, defined by the formula $$d(x,y):=\min\Big\{\sum_{i,j=1}^n c(i,j)m_{i,j}\,\colon m\in M_{x,y}\Big\}$$ for all $$x=(x_1,\dots,x_n)$$ and $$y=(y_1,\dots,y_n)$$ in $$\mathbb{R}_+^n$$, where $$M_{x,y}$$ is the set of all $$n\times n$$ matrices $$m$$ with nonnegative real entries $$m_{i,j}$$ such that $$\sum_{j=1}^n m_{i,j}=x_i\quad\text{and}\quad \sum_{i=1}^n m_{i,j}=y_j$$ for all $$i$$ and $$j$$ in $$[n]$$.
For instance, here is Mathematica's calculation of the ($$\ell^1$$-)optimal transportation plan for your Example 4) -- which gives the same result, $$1/2$$, as yours: