I have worked with Kalman Filters for years, and gave this quick read. I saw the comments on Process Noise, so I focus there for now. I might get back to other sections tomorrow.
My simple head space (as I was taught and re-learned thru experience, and have passed on)
1. Kalman Gain close to 1 or 0 is a warning sign that careful consideration is needed.
This fact can be brought up immediately in example #5 and continued
2a. K close to 1.0 can be bad because..., however for some applications (dynamic models) it can be acceptable since...
2b. K close to 0.0 can be bad because... however for some applications (dynamic models) it can be acceptable since...
3. To solve the problem from step 2, As a first step, for those applications where K close to zero or one is bad... a fudge factor term (called Q for reasons discussed later) can be added to the Kalman Gain computation
3a. Choosing the correct fudge factor for the application is often very difficult and may require lots of simulation runs (a parameter study) with different measurement sequences (including some expected off-nominals) and various values for the process noise.
Remember we are designing a filter, likely for a new application (or a non-trivial extension of an existing application)... so all the elements of an engineering design are needed. Make solution hypothesis, test them, refine them, test them some more with greater realism and eventually real-world data, continue to refine the solution.
4. For easy case of a simple application and only a few unknown states, the process noise can be guesstimated from experience. For more complex applications (perhaps there are dozens of unknown states to estimate) a more rigorous approach to select the correct mathematical description of Process Noise is needed.
-- End of Fudge Factor discussion --
{I think you covered this section well} Then you can introduce the notion that the state dynamics cannot model everything and that unmodeled part can be approximated by Process Noise. For example an unmodeled constant acceleration, gives a process noise of ....
Here are some sentences I think are wrong or misleading
"As you can see, the Kalman Gain gradually decreases; therefore, the KF converges." However, the Kalman Filter may converge to garbage. This garbage could be a "lag", or just plain wrong.
"The process noise produces estimation errors."
A well chosen process noise is important to reduce estimation errors over an ensemble of conditions, by accommodating a range of unmodeled state dynamics. A poorly chosen process may not improve anything.
I have worked with Kalman Filters for years, and gave this quick read. I saw the comments on Process Noise, so I focus there for now. I might get back to other sections tomorrow.
My simple head space (as I was taught and re-learned thru experience, and have passed on)
1. Kalman Gain close to 1 or 0 is a warning sign that careful consideration is needed.
This fact can be brought up immediately in example #5 and continued
2a. K close to 1.0 can be bad because..., however for some applications (dynamic models) it can be acceptable since...
2b. K close to 0.0 can be bad because... however for some applications (dynamic models) it can be acceptable since...
3. To solve the problem from step 2, As a first step, for those applications where K close to zero or one is bad... a fudge factor term (called Q for reasons discussed later) can be added to the Kalman Gain computation
3a. Choosing the correct fudge factor for the application is often very difficult and may require lots of simulation runs (a parameter study) with different measurement sequences (including some expected off-nominals) and various values for the process noise.
Remember we are designing a filter, likely for a new application (or a non-trivial extension of an existing application)... so all the elements of an engineering design are needed. Make solution hypothesis, test them, refine them, test them some more with greater realism and eventually real-world data, continue to refine the solution.
4. For easy case of a simple application and only a few unknown states, the process noise can be guesstimated from experience. For more complex applications (perhaps there are dozens of unknown states to estimate) a more rigorous approach to select the correct mathematical description of Process Noise is needed.
-- End of Fudge Factor discussion --
5. Here you can introduce the notion that the state dynamics cannot model everything and that unmodeled part can be approximated by Process Noise. For example an unmodeled constant acceleration, gives dt^4
Here are some sentences I think are wrong or misleading
"As you can see, the Kalman Gain gradually decreases; therefore, the KF converges." However, the Kalman Filter may converge to garbage. This garbage could be a "lag", or just plain wrong.
"The process noise produces estimation errors."
A well chosen process noise is important to reduce estimation errors over an ensemble of conditions, by accommodating a range of unmodeled state dynamics. A poorly chosen process may not improve anything.
Secondly, I remember watching a few months ago a video from Michael Penn, about something called Padé Approximations: Pade Approximation – unfortunately missed in most Caclulus courses. It was a subject worth exploring.
Another shout out to Emmy Noether's (First) theorem.
Informally stated, if a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved.
As an illustration, if a physical system behaves the same regardless of how it is oriented in space, angular momentum of the system must be conserved, as a consequence of its laws of motion.
Another illustration, if a physical process exhibits the same outcomes regardless of place or time, then linear momentum and energy must be conserved.
It is regarded as the foundation of particle physics
> The question isn’t whether there are still architectures where bytes aren’t 8-bits (there are!) but whether these care about modern C++... and whether modern C++ cares about them.
they did acknowledge some alternatives, but I agree more discussion would have been nice.
FTA - Option 3: Using other existing REPL implementations: The authors looked at several alternatives like IPython, bpython, ptpython, and xonsh. While all the above are impressive projects, in the end PyREPL was chosen for its combination of maturity, feature set, and lack of additional dependencies. Another key factor was the alignment with PyPy’s implementation.
My simple head space (as I was taught and re-learned thru experience, and have passed on)
1. Kalman Gain close to 1 or 0 is a warning sign that careful consideration is needed.
This fact can be brought up immediately in example #5 and continued
2a. K close to 1.0 can be bad because..., however for some applications (dynamic models) it can be acceptable since...
2b. K close to 0.0 can be bad because... however for some applications (dynamic models) it can be acceptable since...
3. To solve the problem from step 2, As a first step, for those applications where K close to zero or one is bad... a fudge factor term (called Q for reasons discussed later) can be added to the Kalman Gain computation
3a. Choosing the correct fudge factor for the application is often very difficult and may require lots of simulation runs (a parameter study) with different measurement sequences (including some expected off-nominals) and various values for the process noise.
Remember we are designing a filter, likely for a new application (or a non-trivial extension of an existing application)... so all the elements of an engineering design are needed. Make solution hypothesis, test them, refine them, test them some more with greater realism and eventually real-world data, continue to refine the solution.
4. For easy case of a simple application and only a few unknown states, the process noise can be guesstimated from experience. For more complex applications (perhaps there are dozens of unknown states to estimate) a more rigorous approach to select the correct mathematical description of Process Noise is needed.
-- End of Fudge Factor discussion --
{I think you covered this section well} Then you can introduce the notion that the state dynamics cannot model everything and that unmodeled part can be approximated by Process Noise. For example an unmodeled constant acceleration, gives a process noise of ....
Here are some sentences I think are wrong or misleading
"As you can see, the Kalman Gain gradually decreases; therefore, the KF converges." However, the Kalman Filter may converge to garbage. This garbage could be a "lag", or just plain wrong.
"The process noise produces estimation errors." A well chosen process noise is important to reduce estimation errors over an ensemble of conditions, by accommodating a range of unmodeled state dynamics. A poorly chosen process may not improve anything.