## Initial Analysis:

While the plot in the introductory post made it seem like the data might allow for a mapping from the distance fields to an x, y coordinate system, I wanted to double check the idea before proceeding. To do so I looked at the dimensions of the inputs and outputs for the proposed mapping. If the dimension of the information being input is smaller than the dimension I would like the inverse process to output, that would suggest the idea is untenable.

Id | Elevation | Vertical Distance To Hydrology | Horizontal Distance To Hydrology | Horizontal Distance To Road | Horizontal Distance To Fire Points |
---|---|---|---|---|---|

2 | 2590 | -6 | 212 | 390 | 6225 |

5 | 2595 | -1 | 153 | 391 | 6172 |

Each sample point contributes three pieces of individual data directly related to re-creating x, y coordinates for the points: the three horizontal distance fields. If I have a group of **n** sample points with the same 3 reference points, then there will be **3n** pieces of information. For the output I will need **2n** pieces of information for the sample points (their x, y coordinates), plus **6** total for the x, y coordinates of the 3 reference points. So if **3n** is at least as big as **2n + 6** it seems plausible that we might have enough information to perform the inversion - this should be the case for **n** at least **6**. In practice it is a bit more complicated than that since these are not linear systems, but for me it was enough justification to give it a try.

While proceeding, I will make the assumption that the reference points can be treated as discrete points. For example, if a set of points have a pond as their closest water source my assumption is that the distances were all measured to the same location, rather than each measuring to whichever point on the pond edge is closest to the sample location. I am not certain if this assumption is warranted and if not whether I will be able to adjust my algorithm to compensate.

My next post should be a description of my experimental setup to test my process in a controlled system.

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