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Comment by piracykills | Hacker News Reader

Interesting - I don't have the strongest math background so I'm usually somewhat intimidated by topics like this, but this makes it seem like a fairly simple weighted average based on squared error values like in standard deviation? I assume this would also only work on a normal distribution too?

Makes it seem a lot more approachable than many people have made it sound to me, but I may be horribly misunderstanding still.

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I struggled a long time trying to get past all the matrix math that the usual Kalman filter tutorial starts out with. KF's make a lot more sense if you start from an example. By the end of the example, you realize that matrix math simplifies the notation hugely. But without the intuition about what it is doing for you, it doesn't help much -- at least it doesn't help me much.

So... my over-simplified touch-stones for KF's:

1. Everything is a Baysian quantity -- a mean (best guess) and a std. dev. (confidence).

2. You have a model of the system state.

3. You have some sensors, and you have some model for the accuracy & precision (noise) in the measurements.

Now, two rules:

1. The model runs open loop and at a regular cadence: "Tick tock, tick tock, the model updates on the clock." Of course, since you are running open loop, your confidence in every value of the model gets worse with each iteration (std. dev. grows). How much you adjust (reduce) the confidence is based on the precision of the system and the control inputs.

2. Sensors readings are applied as they come in: "Use'em if you got'em."

So.... now the moving average part... if you are twice as confident in your current estimate as you are in the reliability of the new measurement, do a weighted average of 2/3 of the current estimate and 1/3 of the sensor. This is why the confidence is carried along for all state values and sensor readings. Of course compute the weights of the moving average according to the current confidence values with every update.

And that is about 103% of what I know about Kalman filters :)

That’s exactly right - new measurements (observations) or state predictions (estimates based on the assumed system model) are weighted based on the ratio of the standard deviation (uncertainty) of that observation/measurement to the historical standard deviation of the model to that point in time. For the closed form equations of the KF to work, you have to assume a) all measurements and parameters in the state model can be approximated by normal distributions with known standard deviations and means b) the underlying system model is linear. If you can’t assume b), then as the author mentions you can approximate the non-linear model with a linear one by taking the Jacobian and evaluating it at each step, and use the slightly modified EKF algorithm to bring everything together into an estimate of the system’s underlying state.

As an aside a nice property of particle filtering (a different approach for localization and SLAM) is that there’s no linearity assumption. A nice property of both particle filtering and the KF/EKF is that the Markov assumption holds. This simplifies the underlying mathematics, as well as benefits the implementation on a computer by reducing time and space complexity required to evaluate the algorithms at each step.

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