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Having a bit of confusion about what the initial process covariance (P) should be. Assume a 1-D tracking problem where I am measuring the distance/position of a static object. Would P not just be process error (R) in this case?

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Based on Tim's provided info, my thinking is that the accuracy of my measurement system would basically be this initial uncertainty. For example, if I know the object should be a distance 1 meter away but my measuring system, say a tape measure, is good to only 1cm, then the values within the P matrix should be the square of this uncertainty, or in this case just 1. But, to keep units and magnitude consistent throughout the entire Kalman filter setup using meters (1cm -> 0.01m), this would be a P of 0.0001 then?

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$P$ is an expression of your initial uncertainty in the position. If you start out knowing $x$ exactly, then $P = 0$. If you start out with no clue of where $x$ is then, in theory, $P = \infty$, although the math gets harder.

The process error expresses how much you expect $x$ to change at each timestep because of stuff happening.

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  • $\begingroup$ Thanks for your response! That helps a lot. Would the accuracy of my measurement system basically be the initial uncertainty? For example, if I know the object should be a distance 1 meter away but my measuring system, say a tape measure, is good to only ± 1cm. Looking at some examples online, the values within the P matrix should be the square of this uncertainty, or in this just 1. But, to keep units and magnitude consistent throughout the entire Kalman filter setup using meters, this would be a P of 0.0001 then? $\endgroup$
    – 6900HS
    Nov 19, 2022 at 15:59
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    $\begingroup$ OK, Stackexchange is not your usual forum. They want a closed-form question, with closed-form answers, and they don't want the real question and answer to be buried in the comments. They do want you to edit your question or start a new one. In this case, I think this further query is closely related to the rest of your question, so please edit your question -- I or someone else will see when you've done that, and you'll get a more complete answer. $\endgroup$
    – TimWescott
    Nov 19, 2022 at 16:05
  • $\begingroup$ Sorry about that, question has been updated. $\endgroup$
    – 6900HS
    Nov 19, 2022 at 16:26
  • $\begingroup$ This is not accurate. The estimation noise is a function of the measurement noise and the process noise. The Kalman filter, in contrast to the LS, assumes the model is stochastic. Hence, even if you have a perfect measurement at a single point of time, it doesn't meant your estimation covariance is 0. $\endgroup$
    – Royi
    Feb 20, 2023 at 8:34
  • $\begingroup$ I didn't say that $x$ was measured perfectly, I said it was known perfectly, by however means. This fits just fine within the Kalman framework: it just expresses a system where you start out with perfect knowledge of $x$ and then see your knowledge of $x$ degrade over time as it is moved by unknown forces. At any rate, I also wasn't saying that one could start with a perfectly-known $x$ -- I was trying to explain the meaning of a covariance matrix, in simple, if over-the-top, terms. $\endgroup$
    – TimWescott
    Feb 20, 2023 at 17:02

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