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Here https://stats.stackexchange.com/a/313138 @whuber describes a beautiful solution to generating a correlated vector to an existing one. The thing i cant figure out is $SD()$ in following expression: $$X_{Y;\rho} = \rho\, \operatorname{SD}(Y^\perp)Y + \sqrt{1-\rho^2}\,\operatorname{SD}(Y)Y^\perp.$$ And also this sentence:

("$\operatorname{SD}$" stands for any calculation proportional to a standard deviation.)

The exact question is: what does standard deviation of two orthogonal vectors or something proportional to it have to do with finding suitable linear combination of them with intended $\theta$?

EDIT2: another way of asking, what is wrong with this equation?: $$X_{Y;\rho} = \rho\, Y \frac{1}{\begin{Vmatrix}Y\end{Vmatrix}} + \sqrt{1-\rho^2}\,Y^\perp \frac{1}{\begin{Vmatrix}Y^\perp\end{Vmatrix}}.$$

Or if it is supposed to be correct, how $\operatorname{SD}(Y^\perp)$ is related to $\frac{1}{\begin{Vmatrix}Y\end{Vmatrix}}$, and $\operatorname{SD}(Y)$ to $\frac{1}{\begin{Vmatrix}Y^\perp\end{Vmatrix}}$ ?

Hooman
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  • By multiplying each variate by the standard deviation of the other, the standard deviation of each term is the same and equal to the standard deviation of $X$. – Xi'an Mar 09 '21 at 13:54
  • Could you elaborate more, or give a reference or title to follow? Why equalizing $sd$ is important for two bases and how is it going to be the $sd$ of $X$? – Hooman Mar 09 '21 at 14:48
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    I think the original answer of @whuber is detailed enough. – Xi'an Mar 09 '21 at 14:49
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    Re the edit: just calculate the correlation coefficient for the wrong equation to see why you don't necessarily come up with the desired value of $\rho.$ – whuber Mar 15 '21 at 22:02
  • @whuber Thank you for your comment. I did a horrible mistake in the previous edit. I corrected it. i ran both codes. your solution generates the exact $\rho$ ($\rho = 0.2$), but mine(edit2) gives just an accurate approximation ($\rho = 0.1994477$). And still i cant figure out why. – Hooman Mar 18 '21 at 15:18
  • It's difficult to tell without having any information about your data. Are you using 181 observations by any chance? – whuber Mar 18 '21 at 15:51
  • i used `y – Hooman Mar 18 '21 at 16:05
  • Since, as you say, my code works and yours doesn't, your code must be different! – whuber Mar 18 '21 at 16:28
  • can you give a hint, reference, article or anything about bringing standard deviation into this quest for a proper linear combination? – Hooman Mar 18 '21 at 16:42
  • I posted a thorough explanation at https://stats.stackexchange.com/a/444058/919. – whuber Mar 18 '21 at 17:39

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