Comment fonctionne la formule de génération de variables aléatoires corrélées?

Si nous avons 2 variables aléatoires normales non corrélées $X_1, X_2$ nous pouvons créer 2 variables aléatoires corrélées avec la formule

$Y=\rho X_1+ \sqrt{1-\rho^2} X_2$

puis aura une corrélation ρ avec X 1 .$Y$$\rho$$X_1$

Quelqu'un peut-il expliquer d'où vient cette formule?

Supposons que vous vouliez trouver une combinaison linéaire de et X 2 telle que$X_1$$X_2$

$$\text{corr}(\alpha X_1 + \beta X_2, X_1) = \rho$$

Notez que si vous multipliez à la fois et β par la même constante (non nulle), la corrélation ne changera pas. Ainsi, nous allons ajouter une condition pour conserver la variance: var ( α X 1 + β X 2 ) = $\alpha$$\beta$$\text{var}(\alpha X_1 + \beta X_2) = \text{var}(X_1)$

Cela équivaut à

$$\rho = \frac{\text{cov}(\alpha X_1 + \beta X_2, X_1)}{\sqrt{\text{var}(\alpha X_1 + \beta X_2) \text{var}(X_1)}} = \frac{\alpha \overbrace{\text{cov}(X_1, X_1)}^{=\text{var}(X_1)} + \overbrace{\beta \text{cov}(X_2, X_1)}^{=0}}{\sqrt{\text{var}(\alpha X_1 + \beta X_2) \text{var}(X_1)}} = \alpha \sqrt{\frac{\text{var}(X_1)}{\alpha^2 \text{var}(X_1) + \beta^2 \text{var}(X_2)}}$$

Assuming both random variables have the same variance (this is a crucial assumption!) ($\text{var}(X_1) = \text{var}(X_2)$), we get

$$\rho \sqrt{\alpha^2 + \beta^2} = \alpha$$

There are many solutions to this equation, so it's time to recall variance-preserving condition:

$$\text{var}(X_1) = \text{var}(\alpha X_1 + \beta X_2) = \alpha^2 \text{var}(X_1) + \beta^2 \text{var}(X_2) \Rightarrow \alpha^2 + \beta^2 = 1$$

And this leads us to

$$\alpha = \rho \\ \beta = \pm \sqrt{1-\rho^2}$$

UPD. Regarding the second question: yes, this is known as whitening.

The equation is a simplified bivariate form of Cholesky decomposition. This simplified equation is sometimes called the Kaiser-Dickman algorithm (Kaiser & Dickman, 1962).

Note that $X_1$ and $X_2$ must have the same variance for this algorithm to work properly. Also, the algorithm is typically used with normal variables. If $X_1$ or $X_2$ are not normal, $Y$ might not have the same distributional form as $X_2$.

References:

Kaiser, H. F., & Dickman, K. (1962). Sample and population score matrices and sample correlation matrices from an arbitrary population correlation matrix. Psychometrika, 27(2), 179-182.

Correlation coefficient is the $\cos$ between two series if they are treated as vectors (with $n^{th}$ data point being $n^{th}$ dimension of a vector). The above formula simply creates a decomposition of a vector into its $\cos\theta$, $sin\theta$ components (with respect to $X_1,X_2$).
if $\rho = cos \theta$ , then $\sqrt{1-{\rho}^2}=\pm sin \theta$.

Because if $X_1, X_2$ are uncorrelated, the angle between them is a right angle (ie, they can be considered as orthogonal, albeit non-normalized, basis vectors ).