How to write sum notation for an array of 2D points

Question

What is a correct way to write using sigma notation for a problem involving an array of 2-dimensional points. Say I have 2 arrays, $P_{e}$ and $P_{a}$, both containing $N$ elements. $P_{e}$ represents the expected values of each of the $N$ points, and $P_{a}$ represents the actual value of these $N$ points on a Euclidean plane.

I would like to know how to write the summation notation for the average distance between each of the pairs of points. I don't know if it's right to express it as simply the distance between $P_{e}$ and $P_{a}$, like $\frac{\sum_{n=0}^{N}|P_{e}(n) - P_{a}(n)|}{N}$, or if I have to write it out in terms of the $x$ and $y$ values of each pair, Pythagoras-style.

Could someone give me some pointers or help writing it in a mathematically sound way? This has to go on a thesis, so I'd like to make sure I'm doing it right.

Answer 1

You don't have to write it out with $x$ and $y$ coordinates-what you have done is just fine. You are really looking at $P_a$ and $P_e$ as one dimensional lists and averaging the difference between corresponding elements.

What you have done loses the information of the geometric relationship between the points. What you can do is define your points on a $2D$ grid. Now $P_a$ becomes $P_a(i,j)$ with $1 \le i \le m, 1 \le j \le n$ and the total number of points is $N=mn$. You even know where the point came from. If the grid spacing is $k$ and the $(1,1)$ point is the origin, the location of point $(i,j)$ is $((i-1)k,(j-1)k)$ Now your average becomes $$\frac 1N\sum_{i=1}^m\sum_{j=1}^n|P_e(i,j)-P_a(i,j)|$$ So far we haven't learned anything new, we just named the points differently. What this does allow you to do is to ask whether the errors grow along $x$, for example. This is a difficult question to ask when you just have the points in a list, but you can now weight the errors by $x$ coordinate by writing $$\frac 1N\sum_{i=1}^m\sum_{j=1}^n|P_e(i,j)-P_a(i,j)|(i-1)$$ and comparing this to the average error multiplied by the average $x$ location $$\frac 1N\frac{m+1}2\sum_{i=1}^m\sum_{j=1}^n|P_e(i,j)-P_a(i,j)|$$ If it is larger, your errors grow with $x$. You can also see if large errors are near each other, which is difficult if you just have the points in a list.