# Conditional Sample from Gaussian Copula

I'm trying to generate a set of conditional samples from a Gaussian copula. However, after implementing (what it seemed to me the right process), I'm observing a weird behavior of the conditioned variables.

Lets say, there are 3 variables: $X_1$, $X_2$ and $X_3$. Each of them is defined by its parametric cumulative density function: $F_{X_1}$, $F_{X_2}$ and $F_{X_3}$ and by a correlation matrix $\mathbf{ \rho } \in \mathbb{R}^{3x3}$.

Drawing samples of a random gaussian copula would involve:

1. Draw N samples from a $Z = MVN([0,0,0], \Sigma = \rho)$, where $Z = \{Z_1, Z_2, Z_3\}$
2. Convert those variables to a unit cube: as $U_i = F_{N(0,1)}(Z_i)$, being $F_{N(0,1)}$ the cdf of a standard normal distribution.
3. Back-transform them to each of the marginals as : $Sample_i = F_{X_i}^{-1}(U_i)$.

Now, I tried to extend it to the case in which one of the samples is conditioned, for example, $X_3 = X_3^{cond}$:

1. First I transformed the conditional value to a standard Gaussian: $Z_3^{cond} = F_{N(0,1)}^{-1}(F_{X_3}(X_3^{cond}))$
2. Generate a set of N samples from a conditional multivariate normal (initially $\mu = [0,0,0]$ and $\rho = \Sigma$) : $Z_i = MVN(\hat{\mu}, \hat{\Sigma})$ where $i=1,2$ and $\{\hat{\mu}, \hat{\Sigma}\}$ are the mean and covariance matrix of the conditional gaussian distribution. Which can be calculated as: $$\hat{\mu} = \Sigma_{12} \cdot \Sigma_{22}^{-1} \cdot Z_3^{cond}$$ $$\hat{\Sigma} = \Sigma_{12} - \Sigma_{12} \cdot \Sigma_{22}^{-1} \cdot \Sigma_{21}$$ when we split the correlation matrix in $\rho = \begin{bmatrix} \Sigma_{11} & \Sigma_{12} \\ \Sigma_{21} & \Sigma_{22} \end{bmatrix}$. This multivariate distribution will generate normally distributed samples with the provided correlation matrix and with the condition that $Z_3 = Z_3^{cond}$.
3. Then proceed as in the normal gaussian copula: Convert the 2 free variables to the unit cube, $U_i = F_{N(0,1)}(Z_i)$ and back transform them to their marginal distributions $Sample_i = F_{X_i}^{-1}(U_i)$.

An example of this process can be provided as: $$X_a \sim N(\mu = 100, \sigma = 30)$$ $$X_b \sim Lognormal(s = 0.5, loc = 1, scale = 100)^*$$ $$X_c \sim N(\mu = 30, \sigma = 10)$$

$$\rho = \begin{bmatrix} 1 & 0.2 & 0.5 \\ 0.2 & 1 & 0.9 \\ 0.5 & 0.9 & 1 \\ \end{bmatrix}$$ *Note: Notation as in python scipy https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.lognorm.html

By conditioning $X_c = 10$ and drawing 5000 samples as previously described. I observed that the correlation between $X_a^{sample}$ and $X_b^{sample}$ has turned negative, $\rho_{X_a^{sample}, X_b^{sample}}^{spearman} = - 0.662$

Here there are two images showing the joint distribution X_a and X_b by sampling the copula without conditioning and conditional to X_c = 10

Joint distribution X_a and X_b from an Un-Conditional Copula sample

Joint distribution X_a and X_b from a Conditional Copula sample X_c = 10

My question is: Is there something wrong in this procedure to sample a conditional gaussian copula?

This negative correlation is coming from the conditioning of the correlation matrix: $\hat{\Sigma} = \Sigma_{12} - \Sigma_{12} \cdot \Sigma_{22}^{-1} \cdot \Sigma_{21}$ which in the example provided would be $\hat{\Sigma} = \begin{bmatrix} 1 & 0.2 \\ 0.2 & 1 \end{bmatrix} - \begin{bmatrix} 0.5 \\ 0.9 \end{bmatrix} \cdot  \cdot \begin{bmatrix} 0.5 & 0.9 \end{bmatrix} = \begin{bmatrix} 0.75 & - 0.25 \\ - 0.25 & 0.19 \end{bmatrix}$

The conditioned correlation matrix is independent on the conditioned value, and only on the structure of the original correlation matrix $\rho$. However, this correlation is amplified when transforming the data, and it doesn't seem intuitive to me.

Samples from this MVN were later transformed assuming it was coming from a set of marginal standard normal $F_{N(0,1)}$ to keep the "drag" imposed by the condition $X_c = 10$. However, it is not clear to me wether this should be like this, perhaps the problem is there.