How do I interpret the uncertainty of each of your parameters?
Here's how I think about it.
Maximum likelihood curve fitting seeks a set of parameters that
minimizes the improbability of the data, according to how our
chosen parameterized model judges the improbability.
In our case, we are searching for the lowest point on a taco-shaped
landscape. Two issues:
-- The uncertainty is wildly different in different directions.
-- The principal axes of the taco are not aligned with the
directions defined by wiggling parameters one at a time. In
other words, there are terrible, horrible, no good, very bad
correlations.
Let's take it row by row. The last 5 numbers are the components of a
unit vector in a particular direction. In the errorbar column is the
error bar associated with that direction, i.e. the halfwidth of the
distribution in that direction.
The top row has only 0.5% uncertainty. This is mostly in the "slow amplitude"
direction, weakly correlated with the slow decay constant and the baseline
height.
The next row involves some weird mixture of slow amplitude, slow decay constant,
and baseline. The mixture has 1.8% uncertainty.
The bottom row has 20.5% uncertainty. This is more than 40× more uncertainty
than the top row. This involves a mixture of baseline, slow decay constant,
and fast amplitude.
At the next level of detail, here is how this is calculated.
Start with the Mahalanobis metric Mij.
This measures the cost squared of wiggling each parameter and
each combination of two parameters:
limp = ½ Xi Mij Xj
where limp means log improbability, i.e. log(1/P).
This makes sense provided the limp contours are quadratic (or
nearly so) in the neighborhood of the minimum in parameter space.
We calculate Mij as the second derivative (i.e. Hessian) of
the limp. That is easy to do, just by displacing the parameters
slightly from their best-fit values.
In one dimension M11 is just 1/σ² i.e. the inverse error bar
squared.
In higher dimensionality, if the Mahalanobis metric happens to
be diagonal, then all the parameters are uncorrelated and the
diagonal elements are just the inverse error bars squared.
In general, the matrix inverse of Mij is the covariance matrix.
It gives sorta the worst-case error bars whereas Mij gives
sorta the best-case error bars. Neither of those is nearly as
informative as the SVD, as we now discuss.
In our case Mij is nowhere near diagonal, so AFAICT the
only sensible way to proceed is to perform a Singular Value
Decomposition (SVD) to find the eigenvalues and associated
eivenvectors of Mij. In the table given above, the cost²
column is the eivenvalue and the last 5 numbers are the
components of the associated unit eigenvector.
The error bar is the reciprocal square root of the eivenvalue.
You can verify that each row of the 5×5 matrix is a unit vector,
and that each row is orthogonal to all of the others.
Don't ask me why it's called SVD. To me it looks like "eigen
decomposition". Even so, you need to know the name SVD if you
want to try googling for more information.