If function A is basically a flat plane, and function B is basically a possibly different flat plane, then attempting to find the solution B = \alpha A + \beta is effectively a degenerate system. As a result of this, if A does have a deviation from the flat plane model, then the minimization process fitting \alpha and \beta are very likely to set \alpha to a very small/large value to effectively smooth that deviation out, using \beta to compensate for the introduced DC offset.
I'm 90% sure that this is the cause of the bad fits I'm seeing, and why I can't find the deviations I expect to exist.
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