Kalman Filter Tutorial
kalmanfilter.net
kalmanfilter.net
If you're learning the Kalman Filter in isolation, you're kind of learning it backwards and missing out on huge "aha" moments that the surrounding theory can unlock.
To truly understand the Kalman Filter, you need to study Least Squares (aka linear regression), then recursive Least Squares, then the Information Filter (which is a different formulation of the KF). Then you'll realize the KF is just recursive Least Squares reformulated in a way to prioritize efficiency in the update step.
This PDF gives a concise overview:
[1] http://ais.informatik.uni-freiburg.de/teaching/ws13/mapping/...
For (1D) we get the prior from the linear prediction X'1 = X0*a + b, for which mean(X'1) = mean(X0)*a + b and var(X'1) = var(X0)*a^2, where a and b give the assumed dynamics.
The posterior for Gaussians is the precision weighted mean of the prior and the observation: X1 = (1 - K)*X'1 + Y*K, where the weighting K = (1/var(X'1))/(1/var(X'1) + 1/var(Y)), with Y being the Gaussian observation.
Iterating this gives the Kalman filter. Generalizing this to multiple dimensions is straightforward given the linearity of multidimensional Gaussians.
This is how (after I understood it) it makes it really simple to me, but things like linearity of (multidimensional) Gaussians and the posterior of Gaussians as such probably are not.
For UKF the Y is still a multidimensional Gaussian and computing K is the same. The mean and covariance of Y is computed from Z and the nonlinear measurement function using the unscented transform.
Why would anyone have people implementing Kalman filters who found the math behind them "esoteric"?
Back in the day, in my wet behind the ears phase, my first time implementing a Kalman Filter from scratch, the application was to perform magnetic heading normalisation for on mag data from an airborne geophysical survey - 3 axis nanotesla sensor inputs on each wing and tail boom requiring a per survey calibration pattern to normalise the readings over a fixed location regardless of heading.
This was buried as part of a suite requiring calculation of the geomagnetic reference field (a big paramaterised spherical harmonic equation), upward, downward and reduce to pole continuations of magnetic field equations, raw GPS post processing corrections, etc.
where "etc" goes on for a shelf full of books with a dense chunk of applied mathematics
I think the lesson there is that the Kalman filter is simpler in the "information form" where the Gaussian distribution is parameterized using the inverse of the covariance matrix.
If you don't already know what that means, you likely don't get much out of that. I think the more intuitive way is to first understand the 1D case where the filter result is weighted average of the prediction and the observation where the weights are the multiplicative inverses of the respective variances (the less uncertainty/"inprecision", the more you give weight).
In the multidimensional case the inverse is the matrix inverse but the logic is the same.
More generally the idea is to statistically predict the next step from the previous and then balance out the prediction and the noisy observation based on the confidence you have in each. This intuition covers all Bayesian filters. The Kalman filter is a special case of the Bayesian filter where the prediction is linear and all uncertainties are Gaussian, although it was understood this way only well after Kalman invented the eponymous filter.
Not sure how intuitive that's either, but don't be too worried if these things aren't obvious, because they aren't until you know all the previous steps. To implement or use a Kalman filter you don't really need this statistical understanding.
If you prefer to understand things more "procedually", check out the particle filter. It's conceptually the Bayesian filter but doesn't require the mathematical analysis. That's the way I really understood the underlying logic.
From a different perspective... I have no traditional background in mathematics or physics. I do not understand the first line of the pdf you posted nor do I understand the process for obtaining the context to understand it.
But I have intellectual curiosity. So the best path forward for me understanding is a path that can maintain that curiosity while making progress on understanding. I can reread the The Six (Not So ) Easy Pieces and not understand any of it and still find value in it. I can play with Arnold's cat and, slowly, through no scientific rigor other than the curiosity of the naked ape, I can experience these concepts that have traditionally been behind gates of context I do not possess keys to.
Truly appreciate the power of linear approximations by going through algebra, appreciate the tricks of calculus, marvel at the inherent tradeoffs of knowledge with estimator theory, and see the joy of the central limit theorem being true. All of this knowledge is free, and much more interesting than a formal restatement of "it was not supposed to rain, but I see clouds outside, I guess I'll expect light rain instead of a big thunderstorm".
I will think more about this, but I'm not sure I agree. I have enjoyed reading Feynman talk about twins and one going on a supersonic vacation without understanding the math. Verisimilitude allows a modeling of understanding with a scalar representation of scientific knowledge, so why not?
Of course I would like to understand the math in its purest forms–just the same as I wanted to read 1Q84 in Japanese to be able to fully experience it in its purest form, but my life isn't structured in a way were that is realistic even if the knowledge of the Japanese language is free.
> Truly appreciate the power of linear approximations by going through algebra, appreciate the tricks of calculus, marvel at the inherent tradeoffs of knowledge with estimator theory, and see the joy of the central limit theorem being true.
I can't even foil so the journey toward understanding can feel unattainable in the time resources I have. This absolutely may be a limiting belief, but the concept of knowledge being free ignores the time cost for some exploring these outside of academia or professional setting.
Since you mention Feynman, I would like to observe that many expositors who target the lay audience have the skill of making the audience believe that they have comprehended(1) something of an intellectual world that they have no technical grounding to truly comprehend(2). In my view these are two distinct types of comprehension/understanding. So long as the audience is clear on which type of understanding they are getting, and is not wasting time unwittingly pursuing one type at the expense of the other then I see no harm.
There is a risk however, that the pop expositors will put you in a headspace where even if you are faced with accessible, but type 2, material you will not be familiar with what really constitutes understanding. As a mature age student it took me quite a few years of maths exams to switch from 1 to 2. Nowadays I am more comfortable with admitting that I don't understand some piece of math (for that is the first step on the path to learning) than being satisfied with a pop-expository gist.
You need handwavy and vague versions of things to understand the shape of them and to build intuition.
Then you need to test the intuition and build up levels of rigor.
Especially in the context of the Kalman Filter. I just helped a bunch of middle school students build a system for field localization and position tracking. They don't have all kinds of knowledge. They don't have linear algebra or a real understanding of something being gaussian and have to have a bazillion variables. They understand that their estimates and the quality of stuff coming off their sensors have different qualities based on circumstances, and that gain needs to vary. They'll never hit the optimum parameters.
But: their system works. They understand how it works (even if they don't know how to quantify how well it works). They understand how changing parameters changes its behavior. When they learn tracking filters and control by root locus and all kinds of things later, they'll have an edge in understanding what things mean and how it actually works. I expect their intuition will give them an easier time in tackling harder problems.
Conversely, I've encountered a bunch of students who know what "multimodal" means but couldn't name a single example in the real world of such a thing. I would argue that they don't even know what they're talking about, even if they can calculate a mixture coefficient under ideal conditions.
Linear algebra is not something that takes years of patient study to gain basic competency. It had almost no prerequisites and can be understood enough to understand least squares in a focused weekend or two.
None of these are needed, or even useful, for understanding the Kalman filter.
While I appreciate rigor to really know deep details, is not only not a requirement for understanding, but a hurdle. A terrible insurmountable hurdle.
To first have understanding, I need some kind intuition. Some explanation that makes sense easily. That explanation is btw, what typically the inventor or discoverer had to begin with, before nailing it down with rigor.
You don't need to know what Gravity is to calculate the time it takes for an apple to fall from a tree. You just need to accept that g=9.8m/s2.
You also don't need to understand the chemistry of flour, salt, sugar, sodium, milk and eggs to bake a cake.
Meinhold, Richard J., and Nozer D. Singpurwalla. 1983. "Understanding the Kalman Filter." American Statistician 37 (May): 123–27.
However, to understand recursive least squares, in particular the covariance matrix update you're going to need a firm grounding in probability and statistics. Simon makes the case that probability theory is a less strict pre-requisite than multiple-input-multiple-output (state space) linear systems theory (for which I can recommend Chen's "Linear System Theory and Design").
So I would argue that to understand Kalman filters you need to know state space systems modelling, both continuous time and discrete time discretisation methods (this provides the dynamics that describe the time-update step), plus you need to know enough multivariate statistics to understand how the Kalman filter propagates the gaussian random variables (i.e. the Kalman state) through the dynamics and back and forth through the measurement matrices.
Using Jupyter notebooks is really great too.
Though I'm not sure Sympy can handle the conditional (Bayesian posterior) distribution needed for the Kalman filter.
In any case, you are better off working direcly with the mean and variance (or covariance matrix) if you want to play around with the Kalman filter with Sympy.
See:
https://reference.wolfram.com/language/howto/WorkWithStatist...
and:
https://reference.wolfram.com/language/ref/MultinormalDistri...
For example, if I'm tracking birds from video footage, I might choose a certain Q, but depending on the time of day the noise statistics might change. What do you do then?
Kalman filter from the ground up - https://news.ycombinator.com/item?id=37879715 - Oct 2023 (150 comments)
(also what's the best year to put in the title above?)
David G.\ Luenberger, {\it Optimization by Vector Space Methods,\/} John Wiley and Sons, Inc., New York, 1969.\ \
This would treat both lying and inaccuracy as "error"
I'm thinking of things like: reports of Phoenix lights or UFOs in general, ghosts, NDEs, and more prosaically, claims of rape
Edit: in fact, I see part three of the book in tfa is devoted to nonlinear Kalman filters. I suspect some of the crowd (myself included) just assumed we were talking about linear Kalman filters
Even if you aren't a python person, it's fantastic and really goes through everything.
- https://www.youtube.com/watch?v=CaCcOwJPytQ&list=PLX2gX-ftPV...
"The filter is named after Rudolf E. Kálmán (May 19, 1930 – July 2, 2016). In 1960, Kálmán published his famous paper describing a recursive solution to the discrete-data linear filtering problem."
"Past evidence that indicates" is deliberate phrasing, in the majority of these examples we are looking at acquired data with noise; errors, instrument noise, missing returns, etc.
"Tracking" is multi-stage, there's a desired target to be found (or to be declared absent) in noisy data .. that's pattern search and locking, the trajectory (the track) of that target must be best guessed, and the best guess forward prediction can be used to assist the search for the target in a new position.
This is not all that can be done with a Kalman filter but it's typical of a class of common applications.
"Tracking", here, means providing some kind of `f(time) -> space` API.
Dead reckoning is a mechanism for incorporating velocity and whatnot into a previously estimated position to estimate a new position (and is also one possible way to implement tracking, usually with compounding errors).
The Kalman filter example is better than just dead reckoning. For a simple example, imagine you're standing still but don't know exactly where. You have an API (like GPS) that can estimate your current position within some tolerance. If you're able to query that API repeatedly and the errors aren't correlated, you can pinpoint your location much more precisely.
Back to tracking with non-zero velocity, every new position estimate (e.g., from GPS) can be incorporated with all the information you've seen so far, adjusting your estimates of velocity, acceleration, and position and giving you a much more accurate current estimate but also better data for dead-reckoning estimates while you wait for your next external signal.
The technique (Kalman Filter) is pretty general. It's just merging all your noisy sources of information according to some ruleset (real-world physics being a common ruleset). You can tack on all sorts of other interesting information, like nearby wifi signals or whatever, and even very noisy signals can aggregate to give precise results.
Another application I threw it at once was estimating my true weight, glycogen reserves, ..., from a variety of noisy measurements. The sky's the limit. You just need multiple measurements and a rule for how they interact.
What you're describing is in general the Bayesian filter (or Bayesian smoothing if you don't have to give the result immediately).
I'm implementing real-time dual state/parameter estimators at the limits of computational tractability. Please tell me how "machine learning" can solve my problem better.
Easier to code, easier to tune parameters without a noise model
But I agree, if a one-pole filter will solve your problem. Kalman filter is an optimal state estimator, useful when you have:
- multiple input and/or multiple output system
- non-trivial dynamics (i.e. the system of interest is in motion and you have a state space description of the dynamics)
- noisy measurements
- statistical estimates of process and measurement noise are available
- ideally, iid gaussian noises
What have I missed?