Stefan K.
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 Dec29 comment How to estimate variances for Kalman filter from real sensor measurements without underestimating process noise. @PseudoRandom: (1) What kind of "nontrivial problems" are you talking about?; (2) When finally filtering the data (not analyzing historical data to extract variances) I would agree. But for this proof it is needed to use continuous time; (3) The error residuals $e_{t+\delta}$ and $e_{t}$ are assumed to be independently Gaussian distributed. Thus, the limit of $e_{t+\delta}-e_t$ is just the term itself. But you are right if you mean that there could be some correlation (e.g. same fluctuation in power supply causing the error). Dec14 awarded Organizer Dec14 revised How to derive the process noise co-variance matrix Q in this Kalman Filter example? Reworked citation including it's correct formatting (now one can see, what is part of the question and what's actually text from the wikipedia article); especially some terms as "ak" were rewritten to use correct math subscript Dec14 revised How to estimate variances for Kalman filter from real sensor measurements without underestimating process noise. Corrected all variances: \sigma => \sigma^2 Dec14 suggested approved edit on How to derive the process noise co-variance matrix Q in this Kalman Filter example? Dec13 asked How to estimate variances for Kalman filter from real sensor measurements without underestimating process noise. Feb22 comment Minimize combined variance of multiple measurements with known (but varying) variance I read the wikipedia article and found that covariance measures the "coherence" or "correlation" between two random values. In fact I was regarding the "real value" of the measurement as a random variable (e.g. changing value when time passes, with more or less known distribution), too. But for multiple measurements of the same property - at the same time -, it's static / non random. Feb22 comment Minimize combined variance of multiple measurements with known (but varying) variance Ok, I checked myself that $cov(x_i, x) = - = <(y+e_i)*y> - = - * y = y² + y * - y² - y * = 0$ when $x = y$ (real value) and $x_i = y + e_i$ (measurement with gaussian noise). This means I have no covariance (and therefore correlation) between my measurements and my real value, correct? Is that, what you wanted to say. If so, I understood now. But I actually wonder what covariance/correlation is all about in that case... What does it actually mean? Feb22 comment Minimize combined variance of multiple measurements with known (but varying) variance Ok, that was nonsense. But actually $cov(x_i,x_j)$ would depend only on the error residuals, not the real value. Hm... need to think about that.. Feb22 comment Minimize combined variance of multiple measurements with known (but varying) variance Well, if you are really correct with that long explanation/formula above, that would mean, that $corr(x_i,y_i)=\frac{cov(x_i,y_i)}{(var(x_i) * var(x_j))^0.5}=0$ since $cov(x_i,x_j)=0$ Feb22 comment Minimize combined variance of multiple measurements with known (but varying) variance Ah!!! Ok, expected value E[x]. Now it's starts making sense... And you probably refer to the definition of the covariance (using the $x_i$ and $x_j$ instead of the usual X and Y): $cov(x_i,x_j) = E[( x_i - E[x_i]) * ( y_i - E[y_i] )] = E[ x_i * x_j ] - E[x_i] * E[x_j]$ Feb22 comment Minimize combined variance of multiple measurements with known (but varying) variance @joriki: Correct, I actually did not understand the fomula in your comment above. I therefore tried to repeat my thoughts to verify, if you still think it's wrong. -- My main problem with the formula is/was the notation, thought. $\langle x x_i\rangle$ means a vector of length 1, with it's value "$x$ multiplied by $x_i$". Right? -- I will now try to understand the longer form... I'll need a moment for that. Feb22 comment Minimize combined variance of multiple measurements with known (but varying) variance @joriki: Maybe I mixed things up... We have 1. a real value 'y'; 2. three measurements x_1, x_2, y_3, which consist of the real value plus some error / residual x_n = y + e_n . The measurement errors are assumed to be Gaussian distributed (for simplicity). The measurements are of course correlated to each other, as all of them contain "y". The errors e_n MAY be correlated, in this cases they most probably are. For example the ambient temperature affects the readings of all sensors and thus cause a correlation of their errors. Feb22 awarded Commentator Feb22 accepted Minimize combined variance of multiple measurements with known (but varying) variance Feb22 comment Minimize combined variance of multiple measurements with known (but varying) variance Thanks a lot for the variation with covariance matrix! Feb22 comment Minimize combined variance of multiple measurements with known (but varying) variance The errors are/can be correlated, too. As e.g. the same type sensor may be affected by the same external influence. Feb22 comment Minimize combined variance of multiple measurements with known (but varying) variance If all measurements correlate (coefficient >0) with the real value, don't they have to correlate with each other, too? Feb22 revised Minimize combined variance of multiple measurements with known (but varying) variance added 142 characters in body Feb22 comment Minimize combined variance of multiple measurements with known (but varying) variance Yes, I noticed that covariances are needed, as the measurements are in fact correlated. So I would have a covariance matrix... Do you also have a solution with known covariances?