Kalman forward filter causing jaggedness in example notebook

[pavelkomarov]

Something about #98 caused jaggedness in the Kalman results that I didn't notice until today. The unit test results aren't jagged, just the results on the simulation used for 1_basic_tutorial.ipynb.

I've restored the old code and pasted in my new filter function, and the jaggedness appears, so it's not the backward step, not the "correction", not the code restructuring. It's likely I introduced a bug, although it's possible the old code was actually not right. I'm doing ye olde ablation testing on the old code to see exactly where modifying it into the new causes this symptom. That will be a clue.

[pavelkomarov]

Okay, here's some weirdness for you:

If the Kalman filtering code has this line

xhat = xhat_ + K@(y[n] - C @ xhat_)

then happy, everything is fine. But if it has these lines

xhat = xhat_
xhat += K@(y[n] - C @ xhat_)

jagged! Very strange, because the value of xhat_ is unchanged by the assignment on the first line, and += should do what we expect.

[pavelkomarov]

Okay, the issue is that assignment of xhat to xhat_ aliases, so when xhat's value is changed, xhat_ and locations of the data array xhat_pre that contain xhat_s will get changed too, and this throws off the algorithm. Ugh. Fix incoming.

[pavelkomarov]

Oh yeah. Check out this printout from the unit tests:

tests/test_diff_methods.py ..........................Improvement detected for method constant_acceleration
Improvement detected for method constant_acceleration
.Improvement detected for method constant_acceleration
Improvement detected for method constant_acceleration
.Improvement detected for method constant_jerk
Improvement detected for method constant_jerk
.Improvement detected for method constant_jerk
Improvement detected for method constant_jerk
.....................................Improvement detected for method constant_velocity
Improvement detected for method constant_velocity
.Improvement detected for method constant_velocity
Improvement detected for method constant_velocity
.Improvement detected for method constant_acceleration
Improvement detected for method constant_acceleration
Improvement detected for method constant_acceleration
Improvement detected for method constant_acceleration
.Improvement detected for method constant_acceleration
Improvement detected for method constant_acceleration
Improvement detected for method constant_acceleration
Improvement detected for method constant_acceleration
.Improvement detected for method constant_jerk
Improvement detected for method constant_jerk
Improvement detected for method constant_jerk
.Improvement detected for method constant_jerk
Improvement detected for method constant_jerk
Improvement detected for method constant_jerk
.....................................Improvement detected for method constant_velocity
Improvement detected for method constant_velocity
.Improvement detected for method constant_velocity
Improvement detected for method constant_velocity
.Improvement detected for method constant_acceleration
Improvement detected for method constant_acceleration
Improvement detected for method constant_acceleration
Improvement detected for method constant_acceleration
.Improvement detected for method constant_acceleration
Improvement detected for method constant_acceleration
Improvement detected for method constant_acceleration
Improvement detected for method constant_acceleration
.Improvement detected for method constant_jerk
Improvement detected for method constant_jerk
Improvement detected for method constant_jerk
Improvement detected for method constant_jerk
.Improvement detected for method constant_jerk
Improvement detected for method constant_jerk
Improvement detected for method constant_jerk
Improvement detected for method constant_jerk
.....................................Improvement detected for method constant_velocity
Improvement detected for method constant_velocity
.Improvement detected for method constant_velocity
Improvement detected for method constant_velocity
.Improvement detected for method constant_acceleration
Improvement detected for method constant_acceleration
Improvement detected for method constant_acceleration
Improvement detected for method constant_acceleration
.Improvement detected for method constant_acceleration
Improvement detected for method constant_acceleration
Improvement detected for method constant_acceleration
Improvement detected for method constant_acceleration
.Improvement detected for method constant_jerk
Improvement detected for method constant_jerk
Improvement detected for method constant_jerk
.Improvement detected for method constant_jerk
Improvement detected for method constant_jerk
Improvement detected for method constant_jerk
.......................................Improvement detected for method constant_acceleration
Improvement detected for method constant_acceleration
.Improvement detected for method constant_acceleration
Improvement detected for method constant_acceleration
.Improvement detected for method constant_jerk
Improvement detected for method constant_jerk
.Improvement detected for method constant_jerk
Improvement detected for method constant_jerk
.......................................................

although it's unfortunately not quite as obvious to the eye on the test functions. Here is before the change:

and here is after:

But the notebook is happy again, with actually slightly lower RMSEs than in the past: 0.55, 0.50, 0.49 for vel, acc, jerk, vs. 0.55, 0.51, 0.53 for the same settings of r and q.