How does Kalman Filter provide information regarding the accuracy of the current estimate? I am a beginner in Kalman Filter and have been reading quite a lot on the Internet and books. I am stuck on how can the P matrix provide the accuracy information regarding the current estimate. Below is the quote from wiki. Note that k is subscript.
Pk,k, the a posteriori error covariance matrix (a measure of the estimated accuracy of the state estimate).
If I use an example of a car traveling in a straight line constant velocity throughout, my estimate position should be about a straight line with the slope being the speed. If I were to note down all the estimate points for example, 1m, 3m, 5m, 7m, etc (car with speed 2m/s, readings at 1sec interval), wouldn't the variance of this estimate position be a number which is constantly growing?
But base on the wiki description, the variance of the predicted position should be small when it can predict accurately. Are you guys able to tell me which portion am I interpreting wrongly? Is my way of calculating the variance of predicted position wrong?
 A: Let $x$ be the unknown state, and let $\hat{x}$ be the estimate. Assume that $x$ and $\hat{x}$ are both column vectors. Define the error covariance matrix $P$ as
$$
P = E\{(x-\hat{x})(x-\hat{x})^{\rm T}\} - E\{x-\hat{x}\} E\{x-\hat{x}\}^{\rm T}.
$$
Under certain assumptions, the Kalman filter is an unbiased estimator. Which means that 
$$
P = E\{(x-\hat{x})(x-\hat{x})^{\rm T}\}.
$$
So if your position grows, the estimated position should grow as well, and error covariance matrix $P$ gives you an idea of how close your estimate is to the truth.
A: Thank you Mr Fegur for the above formula.
After more readings, I realize that the P matrix is actually the theoretical error in estimate. If the process noise Q is zero, the variance will converge close to 0. If there is some process noise, the variance will just converge to some stable value. The measurements will not affect the P matrix as it is just the theoretical variances.
Initially I thought the P matrix is referring the the estimates only which is why I got confuse.
