Kalman filters might be one of those weird cases in mathematics where the 'simple' version is simplified beyond all recognition.
I mean what you're really doing is take a measurement then simulate the possible future states and combine this information with the next measurement and repeat.
You can imagine e.g. taking multiple pictures of a tennis-ball, estimate its position and speed from the first picture, simulate where it's going to end up, and compare this with the next picture to see which estimate is closer to the truth. Or more old school, measure the inclination of the sun and compare the resulting line of possible locations on a map to the spot you thought you were.
Of course the exact calculations are beyond impractical. So you use sampling to simplify. However that still makes it difficult so you assume the distribution is somewhat close to a Gaussian distribution. And then you simplify even more by assuming the evolution of the system is just a linear transformation. And that's how you end up with the Kalman filter discussed here.
I'd be amazed if anyone could really understand what's going on just based on the linear algebra.
I mean what you're really doing is take a measurement then simulate the possible future states and combine this information with the next measurement and repeat.
You can imagine e.g. taking multiple pictures of a tennis-ball, estimate its position and speed from the first picture, simulate where it's going to end up, and compare this with the next picture to see which estimate is closer to the truth. Or more old school, measure the inclination of the sun and compare the resulting line of possible locations on a map to the spot you thought you were.
Of course the exact calculations are beyond impractical. So you use sampling to simplify. However that still makes it difficult so you assume the distribution is somewhat close to a Gaussian distribution. And then you simplify even more by assuming the evolution of the system is just a linear transformation. And that's how you end up with the Kalman filter discussed here.
I'd be amazed if anyone could really understand what's going on just based on the linear algebra.