Originally Posted by
Double Back
Kalman filter... Learned that during my (50 Years ago, state paid
) ab initio airline pilot training, 2,5 Yr course, 250 hrs of which close to 100 hrs on twin engine jets... )... Had trouble to understand it completely, it was at the time no one owned a PC, programming was for aliens only.
The idea behind Kalman filter is very simple and easy to understand, at least if you limit it to a single variable.
- The first step is about computing a new value from the previous state of the system, using physics equations.
This step is equivalent to dead reckoning; the current position is equal to the position of the previous state, plus the speed vector of the previous state multiplied by the elapsed time.
- Then this value is fitted with a gaussian probability distribution, centered on the computed value. This distribution represents the probability that the system departed from a steady evolution.
The width or variance of the gaussian may be dependent on other variables of the state (eg: dead reckoning with a slow speed is more precise than with a high speed).
- Then the measured value of the same variable is also prepared by applying a gaussian distribution on it, to model the measurement errors or noise.
For dead reckoning a horizontal position, it means taking the current GPS position and applying an error distribution using, for instance, the current GPS Horizontal Figure of Merit (HFOM).
- Then the 2 distribution are combined (in this case, multiplied) and the maximum of the resulting curve is the most probable value for the variable and it is chosen as representing the variable value for the current state.
Multiplying the 2 distributions means computing a function that describes the probability that any given "actual" value of the variable could be represented by both the computed value (dead reckoning) and the measured value.
The maximum of that function is the most probable "actual" value for the current situation.
The calculation is relatively quick (hence, its implementation in the Apollo hardware) because the product of the Probability Density Functions (PDF) of 2 Gaussian variables is another PDF function, with the mean and variance equal to:
μ = (μ1 σ2² + μ2 σ1²) / (σ2² + σ1²)
σ² = σ1² σ2² / (σ2² + σ1²)
where μ is the mean value and σ² is the variance (or σ is the standard deviation).
So there is no need to make complex calculations on the plots of these functions.
It is to be noted that the theoretical ground for the Kalman filter reasoning is the Bayes theorem, published in
1763.
Kalman filter is a special and simple case of the more general Bayes filter and is an application of a branch of statistics named after Bayes; Bayesian statistics.
Another of the Bayesian statistics applications is computing the evolution of infected populations in the context of an infectious disease outbreak and its use for monitoring and predicting COVID evolution made it known to the general public.