# Fitting distribution to spatial data

Cross posting my question from mathoverflow to find some stats specific help.

I am studying a physical process generating data which projects nicely into two dimensions with non-negative values. Each process has a (projected) track of $x$-$y$ points -- see the image below.

The sample tracks are blue, a troublesome type of track has been hand drawn in green, and a region of concern drawn in red:

Each track is the result of an independent experiment. Twenty million experiments have been conducted across several years, but from those only two thousand exhibit the feature which we plot as a track. We are only concerned with the experiments which generate a track, so our data set is the (approximately) two thousand tracks.

It is possible for a track to enter the region of concern, and we expect on the order of $1$ in $10^4$ tracks to do so. Estimating that number is the question at hand:

How can we calculate the likelihood of an arbitrary track entering the region of concern?

It is not possible to conduct experiments quickly enough to see how often tracks are generated which enter the region of concern, so we need to extrapolate from the available data.

We have fitted for example $x$ values given $y\ge200$, but this does not sufficiently handle data such as the green track -- it seems necessary to have a model encompassing both dimensions.

We have fitted the minimum distance from each track to the region of concern, but we are unconvinced this is producing a justifiable result.

1) Is there a known way to fit a distribution to this type of data for extrapolation?

-or-

2) Is there an obvious way to use this data to create a model for generating tracks? E.g., use principal component analysis on the tracks as points in a large space, then fit a distribution (Pearson?) to the tracks projected onto those components.

• I must not give more information about the particular process, but I will update the question to speak to how data are collected. – Jeff Snider Feb 4 '14 at 15:45
• I have updated the language to try and be more concrete. Imagine we're throwing pebbles at an open top floor window, and we only care how the pebbles which go through the window bounce on the floor inside. We've thrown millions of pebbles, and about 2000 have gone through the window. When a pebble goes through the window we draw a track of its progress across the floor. Given a pebble goes through the window, we want to estimate the likelihood it goes through the region of concern. – Jeff Snider Feb 4 '14 at 16:36