The autocorrelation of sin(t) is defined as $$\displaystyle \int_{-\infty}^{\infty} \sin(t+\tau)\sin(t)d\tau$$ I’ve tried using the Wiener-Khinchin theorem which says that $$Corr(g,g)\Longleftrightarrow|G(f)|^2$$ I’ve tried reverting the FT squared of the sine wave, and don’t see any solution. I can’t find a derivation online for this. So I came here wondering if anyone of you could offer a […]

I have three vectors of numbers with the same dimensionality, $A$,$B$ and $C$. What is the most suitable number $x$, which maximizes the correlation of $A$ and $B+xC$ . To what extend can I increase the correlation. Thanks

So I was day dreaming about linear algebra today (in a class which had nothing to do with linear algebra), when I stumbled across an interesting relationship. I was thinking about how determinants are really the area spanned by column vectors, and I had the thought that one could measure linear independence (in $R^2$ in […]

I am reverse engineering custom software for a stepper motor. The original software eases in and out of any motion, and the duration of the ramping up to speed is directly related to the speed that the motor is ramping up to. In other words, when the motor is ramping up to a high speed, […]

Assume that all the entries of an $n \times n$ correlation matrix which are not on the main diagonal are equal to $q$. Find upper and lower bounds on the possible values of $q$. I know that the matrix should be positive semidefinite but how to proceed to get the upper and lower bounds? Thanks!

I understand how to create random variables with a prespecified correlational structure using a Cholsesky decomposition. But I would like to be able to solve the inverse problem: Given random variables $X_1, X_3, \dots X_n$ ,and two different linear sums of those variables $V_1=a_{11}X_1+a_{12}X_2+\dots a_{1n}X_n$, and $V_2=a_{21}X_1 + a_{22}X_2 +\dots+a_{2n}X_n$, I wish to calculate the […]

To show: If |Cor(X,Y)| = 1, then there exists a, b ∈ R s.t Y = bX + a. Any ideas or hints to proceed? Basically, I’ve to prove that if the absolute value of correlation b/w two random variables is 1, then they should be linearly related. So far, $$ |cor(X, Y)| = 1 […]

Assuming that the data set was $z$-standardized to zero mean and unit variance (also assuming that it does not contain constant vectors). Then Pearson’s r reduces to Covariance: $$\rho(X,Y) := \frac{Cov(X,Y)}{\sigma(X)\sigma(Y)} = Cov(X,Y)$$ Now I’m investigating the dissimilarity function $$d(X,Y):=\sqrt{1 – \rho(X,Y)}$$ which is the square root of a common transformation of $\rho$ for use […]

If we know that A has some correlation with B ($\rho_{AB}$), and that B has some with C ($\rho_{BC}$), is there something we know to say about the correlation between A and C ($\rho_{AC}$)? Thanks.

I understand that the variance of the sum of two independent normally distributed random variables is the sum of the variances, but how does this change when the two random variables are correlated?

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