Intercepts
By Elijah Bernstein-Cooper, May 30, 2015, 0 comments.

Determine relevance of including an intercept in the model. This means we would solve for the DGR, $HI$ width and an intercept for each cloud. Previous attempts to include an intercept led to large intercepts, on order of negative several magnitudes in $A_V$. See this post for more details.

Plot $A_V$ vs. $HI$ with the fitted DGR with and without an intercept for each cloud.

Currently I am experiencing problems deriving a reasonable intercept. Below is a hess diagram of $A_V$ vs. $N(HI)$ for California of the masked pixels. The black line shows the derived relationship between $A_V$ and $N(HI)$ using the MLE method i.e. the best-fit DGR + $HI$ width. The red line is a least-squares fit to the given $N(HI)$, i.e., not fitting for $N(HI)$. It seems quite obvious that an intercept is needed to correctly describe the data.

Below is a likelihood space for California without fitting for an intercept.

The likelihood spaces for California while fitting for an intercept are distorted and do not have a clear MLE.