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seen Feb 22 '13 at 18:49

Feb
22
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.
Feb
22
comment Minimize combined variance of multiple measurements with known (but varying) variance
Ok, I checked myself that $cov(x_i, x) = <x_i x> - <x_i><x> = <(y+e_i)*y> - <y+e_i><y> = <y^2 + y*e_i> - <y+e_i> * y = y² + y * <e_i> - y² - y * <e_i> = 0$ 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?
Feb
22
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..
Feb
22
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 $cov(x_i,x_j)=0$
Feb
22
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): $cov(x_i,x_j) = E[( x_i - E[x_i]) * ( y_i - E[y_i] )] = E[ x_i * x_j ] - E[x_i] * E[x_j]$
Feb
22
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.
Feb
22
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.
Feb
22
awarded  Commentator
Feb
22
accepted Minimize combined variance of multiple measurements with known (but varying) variance
Feb
22
comment Minimize combined variance of multiple measurements with known (but varying) variance
Thanks a lot for the variation with covariance matrix!
Feb
22
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.
Feb
22
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?
Feb
22
revised Minimize combined variance of multiple measurements with known (but varying) variance
added 142 characters in body
Feb
22
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?
Feb
22
asked Minimize combined variance of multiple measurements with known (but varying) variance
Aug
21
comment Statistical Significance Dice Probability
@Sasha: I think I'll actually use a Clopper-Pearson-Interval as my number of samples tend to be quite small. And therefore the computational costs are acceptable too. The approximations (as normal distribution) will be probably to inaccurate for e.g. only 49 samples.
Aug
21
comment Statistical Significance Dice Probability
Because of one-sided test, right?
Aug
21
awarded  Student
Aug
21
comment Statistical Significance Dice Probability
What the... Ok, thank you very much for that posting. Must have taken you a while to produce it.
Aug
21
awarded  Scholar