# Initial Process Covariance in 1-D Kalman Filter

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?

$$P$$ 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.
• 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. Commented Feb 20, 2023 at 17:02