# In multivariable quantile regression, how can we determine which observation corresponds with a particular quantile?

I'm trying to use quantile regression to solve an areal interpolation problem as described in this paper: https://www.tandfonline.com/doi/abs/10.1080/00045608.2011.627054

Basically I have a set of polygons covering census output areas (source zones), and a population for each one. I want to estimate the population for an arbitrary different set of polygons (target zones).

A naive approach would be to just use the proportions of area intersections between the source and target zones to weight the population values, but this assumes the population density is the same everywhere within the source zones.

In reality, the population density is likely to vary depending on the land use within each source zone (e.g. urban land is likely to be more densely populated than rural land).

So I have some land cover data and can use this to calculate the area within each source zone for each type of land cover.

Here's where quantile regression comes in: by using the area of each land cover class within each source zone as independent variables, and the population as the dependent variable, I can obtain predictions for the population density of each land cover class in each source zone by running a quantile regression for each source zone at the quantile that corresponds with that source zone.

If there was only one land cover class, I can work out which quantile each source zone corresponds with by doing a percentile rank based on the dependent variable (the slope of the regression line at that quantile is then the population density for the single land cover class at that source zone).

However, this doesn't seem to be the case when there are multiple independent variables.

Is there a way to determine a quantile parameter that will give an exact fit for a particular observation?