Regression on a triangular shaped region of points representing a symmetric relation

Question

I plotted a set of about 200,000 points and got a triangular shaped region. The shape is roughly like the triangle made by the points $(1,0)$, $(0,1)$ and $(0,0)$. My points have the property that if $(a,b)$ is a member of the set, then $(b,a)$ is also a member. I wanted to do a regression on the variable associated with the second coordinate y on the variable associated with the first coordinate x. I did a linear regression but I'm bothered by the fact that the region is shaped nothing like a line. Is there a better way to do this?

Edit: The points represent a type of game. For each player i and j, i can attack j or j can attack i. The points are produced by looking at (number of times where i attacks j, of times j attacks i) for all pairs i and j. This is the reason that we get (a,b) being a member of my set if (b,a) is a member. I felt that the triangle region suggests that players who don't attack enough have a greater chance of being attacked. Just to emphasize, my points fill the whole region bounded by a triangle similar to (1,0), (0,1) and (0,0), not necessarily uniformly.

Second Edit:The region is shaped like the triangle I mentioned but bigger. Lets say for arguments sake, it roughly looks like the triangle (1000,0), (0,1000), (0,0). There are a few scattered points outside the triangle and the triangle itself is not completely filled out.

My goal is to characterize or relate the number of attacks that i makes on j to the number of attacks that j makes on i.

Answer 1

I may, after all this time, finally have understood the question. The data, if I'm correct, are a set of tuples $(i, j, y(i,j))$ where $i$ is one player, $j \ne i$ is another player, and $y(i,j)$ is the number of attacks of $i$ on $j$. In this notation the objective is to relate $y(i,j)$ to $y(j,i)$. There are some natural ways to do this, including:

1. Analyze the data set $\{(y(i,j), y(j,i))\}$ by means of a scatterplot or PCA (to find the principal eigenvalue). Note this is not a regression situation because both components of each ordered pair are observed: neither can be considered under the control of an experimenter nor observed without error. It is this scatterplot, I believe, that appears triangular. This already suggests that any attempt to describe it succinctly, such as by means of a principal direction, is doomed.

2. Model $y(i,j)$ in terms of characteristics of $i$ and $j$. This is a classic regression situation. The solution provides an indirect, but possibly powerful, way to relate $y(i,j)$ to $y(j,i)$.

   In this case, also consider re-expressing the data in terms of relative numbers of attacks. That is, instead of using $y(i,j)$ use $x(i,j) = y(i,j)/\sum_{j}{y(i,j)}$.