Q in Kalman module is not valid discrete-time process noise covariance

[pavelkomarov]

I'm trying to extend the Kalman module to handle irregular time steps, and I realized that, per all the math this relies on, technically if you have process noise entering in the whichever-derivative-dimension, then when you integrate the continuous time system you should have entries all over the matrix in the discrete-time Q. Discrete-time Q with only entry q in its bottom-right corner is just adding more sampling noise at measurement time. It's approximately right, because all the other entries of Q get higher powers of dt and so shrink rapidly, and the parameter space is still only the q:r ratio (#139), but the frequency plot does change. Here is with old Q:

And here is with what Q should be:

I'm going to double check the math and make Q the correct thing in the discrete case.

[pavelkomarov]

Addressed in the code and in the paper.

[pavelkomarov]

Some follow-up results. Here is a performance plot of a dozen chosen methods across 6 simulations, averaged and sample-stddeved over 52 random seeds, before the change to $Q$:

And here it is after the changes associated with this issue, as well as #80, in #151:

The RTS results get a tiny bit better. That's encouraging, means I did something right.

Here is another such plot, before:

and after:

There the effect may be running a tiny bit the other way, but the difference is miniscule.