There are no Kalman filters for beginners because no one is out there trying to make maths more complicated than it needs to be.

To understand what’s happening you need work linear algebra l, probability theory and distribution theory. There’s no way to explain it without because it makes no sense without. Kalman filters are born of those things.

You could, but you should follow it up with "where one of those sources is your (incomplete, noisy) information about what the value _used_ to be and how it changes over time".

If you just have (say) three different ways of estimating the position of an aeroplane, then you don't need a Kalman filter. The place where the Kalman filter adds value is where you measure its position _repeatedly_ and make use of the fact that it's known to be moving approximately in a straight line.

How do EKF and ensemble Kalman Filters fit into this?

The Kalman filter assumes that all errors are gaussian, that all updates are linear (i.e., new state = linear function of old state), and that observations are linear (i.e., what you measure is a noisy version of a linear function of the state).

The extended Kalman filter allows for state updates and observations to be nonlinear, by the straightforward expedient of replacing them with linear approximations near to the current estimate.

The ensemble Kalman filter also allows for nonlinear updates and observations; instead of keeping track of the expectation and covariance of the state (i.e., everything you need to define a Gaussian model, things that behave nicely under linear transformations but not under nonlinear ones) it keeps track of an "ensemble" of sample values, applies the update and observation functions to those, and then estimates expectations and covariances from this ensemble. It's a sort of Monte Carlo Kalman filter.

A couple of other things worth knowing about:

Intermediate between the extended KF and the ensemble KF is the "unscented Kalman filter". Like the ensemble KF it estimates things using a number of samples; but instead of propagating those samples through repeated steps, it picks the sample points at each step on the basis of the estimated expectation and covariance, and uses them only to compute new expectations and covariances. More expensive than the EKF but copes better with substantial nonlinearities.

Extrapolating beyond the ensemble KF is the "particle filter", which uses the same "track an ensemble of samples" approach but gives up the assumption that all the errors are Gaussian. You need a larger ensemble to get good results, I think, but it can cope with a wider range of scenarios. (I find the name "particle filter" annoyingly distracting; the "particles" are the samples, which I guess you're supposed to think of as a cloud of points in possible-configuration-of-the-system space, and of course it's a "filter" in the same way as the Kalman filter is.)

You may have chanced upon this already but here is a wonderfully explained (IMHO) intro to Kalman filters: https://www.bzarg.com/p/how-a-kalman-filter-works-in-picture...

I don't know man, the link you clicked on says "Derive Yourself a Kalman Filter", so complaining about a formal treatment seems misguided rather than simply unfair.

you are assuming that the reader is familiar with this specific understanding of "derive". someone not familiar with formal methods wouldn't.

what other understanding of derive is there???

Derived data - e.g. indexes or summaries.

Derivative financial instruments - e.g. options, CDSs.

I'm sure there are many more examples.

You could, kind of. Basically, Kalman filter is this: you have 2 noisy data sources with a known real-life connection (like, vehicle position and velocity), so instead of an accurate measurement for each you have 2 "wide" gaussians. When you multiply them, you still get a gaussian, but a much narrower one, meaning: you still don't know the exact position for sure, but (kind of surprisingly) now you know it with a much greater precision than you had with the direct measurement.

In fact, I agree that this one is not really good explanation. I cannot remember, where I saw my favourite one, but there are really plenty of them in the internet, many are well illustrated and pretty clear overall, so if you are actually interested, I think you won't have much trouble understanding it after a few links from the first page of your favorite search-engine.

Hi,

Trying to simplify even further: you don't need 2 sources, just one, for which the distribution of noise is gaussian. And a model of how the data should evolve over time. Straight-line constant-speed for example, but also, static, turning, accelerating/decelerating... You need to be able to predict the next measurement from the previous one. The Kalman filter then uses the difference between the prediction from the model and the measurement, the incertitude on the measurement, and gives you: a smoothed estimation of the 'real' position of the object, an estimation of the 'noisyness' of your input data (i.e. How much does the data fit the model), and the ability to predict next positions accurately...

Well, not sure this word-soup simplifies anything...

Kalman filters are a bit counter intuitive if you don't have a strong background in probabilities... They clicked for me when 1- shown in comparison with other simpler filters 2- I played with each dimension of the formula on a toy example...

They're also very fun because they can be used in so many ways (sensor data fusion, non gaussian noise models, incredibly complex trajectory models...).

"it can be insulting"

Insulting, really? Somebody cares to try to explain you something and the word you can find is "insulting"?

It's like everyone owes it to you to chew knowledge in small enough bits that you can swallow them without any effort.

What a kindergarden.

Well, I dont agree with OP but i think you miss the mark. it's not deigning to provide an explanation that would be insulting, but the (apparent) promise of it being an accessible and intuitive explanation, and quite obviously not being one that OP considered insulting.

I understood OP exactly as you said. This doesn't justify OP's whining, not one bit.

A Kalman filter is a specific instantiation of the Bayes filter, which is very elegant and intuitive. Then once you plug in a bunch of Gaussians and linearity in the Bayes filter and crunch the math, a Kalman filter falls out.

This is how it's treated in the book Probabilistic Robotics by Thrun, Burgard, Fox and I found it to be one of the best treatments on Kalman filters. A really great book overall.

Yeah, this piece reads like the Wikipedia page for Kalman filters. It’s right, but it’s not very informative if you don’t already understand the topic. There’s definitely no attempt to simplify anything here.

Yes, when people try to explain things they will often say it’s easy in hopes that it will help the learner relax and not feel overwhelmed.

But that backfires the moment something is not easy for the learner. It’s completely demotivating.

It’s better in my opinion to say, “it’s okay if you don’t understand everything at first.” If the goal is to keep the learner from being immediately discouraged.

There's a place for both kinds of explanations. This is a good way of explaining what a Kalman filter is to someone who has a mathematical background, just like if you go look up a woodworking video, it might show an interesting way for someone who is familiar with woodworking to build a chair.

Neither is useful to the novice, but an article like this going out of its way to explain, say, matrix notation, or a video about building chairs that spends the first ten minutes explaining how to use a table saw, is less useful and interesting to the experienced reader or viewer.

I suppose I agree that articles like this shouldn't be presented as easy, or for the beginner, but there's a difference between "simple" and "easy".

When it comes to the woodworking: a japanese pull saw ( https://en.m.wikipedia.org/wiki/Japanese_saw ) is a great and fairly cheap way to get started. It enables high accuracy and is still quite fast.

Then you'll need a cheap but sturdy table, a hand plane, some vices, a tape measure, a carpenter's square, a cheap cordless drill. Screwing and glueing is the best way to get solid joints for a novice.

Thanks for your comment (and thanks to everyone who answered below). I have been way too many times in the same situation where someone says something is "trivial" or "easy" so I completely get your point and will try not to make the same mistake for the next posts.

Regarding your question, I think the answers you have had are on point!