Given that the scrambled data and actual data PDCFs are basically consistent with each other but both inconsistent with zero, seems like there should be some sort of direct (i.e. not pairwise-differenced) \({E(B-V)}\)-residual-\({R(V)}\)-residual correlation. We can try to extract this by, e.g., extending the model we’ve used to remove large-scale \({R(V)}\) structure to include the near and far \({E(B-V)}\) residuals. Something where
\[\label{eqn:extended_geometric_RV} x_n = x_{n,0} + x_{n,1} \left( {E(B-V)}_{n} - g_n({\ell},\, {b}) \right)\\ x_f = x_{f,0} + x_{f,1} \left( {E(B-V)}_{f} - g_f({\ell},\, {b}) \right)\\\]
and where \(R(V)\) is computed from these \(x_n\) and \(x_f\) using equation \ref{eqn:geometric_RV}. The \(g_n({\ell},\, {b})\) and \(g_f({\ell},\, {b})\) terms represent the large-scale structure of \({E(B-V)}\); the \({\ell}\) dependence comes from fitting different \(g_n({b})\) and \(g_f({b})\) to each field.
I’ve done this, and \(x_{n,1}\) and \(x_{f,1}\) are both positive and produce noticeable (\(\sim 0.5\)) \({R(V)}\) changes over the \({E(B-V)}\) residual range (\(-0.5\) to \(+1.5\)). The usual triangle plot is shown in figure \ref{fig:RE_corr_triangle}.
While this all seems to be going in the right direction, this only produces a small reduction in the standard deviation of the standardized \(R(V)\) residuals, from 1.4 (the original geometric model) to 1.36 (extended). Also, throwing in these extra coefficients doesn’t help with getting rid of the obvious large-scale spatial structure in the \({R(V)}\) residuals in figure \ref{fig:deLSSd}.
Given that the large-scale structure seems to depend mostly on how we handle the distance-dependence of \({R(V)}\), perhaps the more productive thing to do is to more carefully consider how to include information from the \({E(B-V)}\) map in the large-scale structure part of the \({R(V)}\) model. For example, I could perhaps take Eddie’s distance-dependent \({R(V)}\) map and throw in a constant or heavily regularized \({R(V)}\)-increase per \({E(B-V)}\)-residual term.