# Parameters of conditional multivariate normal distribution This function calculates mean (expectation) and covariance matrix of conditional multivariate normal distribution. ```r cmnorm( mean, sigma, given_ind, given_x, dependent_ind = numeric(), is_validation = TRUE, is_names = TRUE, control = NULL, n_cores = 1L ) ``` ## Arguments - `mean`: numeric vector representing expectation of multivariate normal vector (distribution). - `sigma`: positively defined numeric matrix representing covariance matrix of multivariate normal vector (distribution). - `given_ind`: numeric vector representing indexes of multivariate normal vector which are conditioned at values given by `given_x` argument. - `given_x`: numeric vector which `i`-th element corresponds to the given value of the `given_ind[i]`-th element (component) of multivariate normal vector. If `given_x` is numeric matrix then it's rows are such vectors of given values. - `dependent_ind`: numeric vector representing indexes of unconditional elements (components) of multivariate normal vector. - `is_validation`: logical value indicating whether input arguments should be validated. Set it to `FALSE` to get performance boost (default value is `TRUE`). - `is_names`: logical value indicating whether output values should have row and column names. Set it to `FALSE` to get performance boost (default value is `TRUE`). - `control`: a list of control parameters. See Details. - `n_cores`: positive integer representing the number of CPU cores used for parallel computing. Currently it is not recommended to set `n_cores > 1` if vectorized arguments include less then 100000 elements. ## Returns This function returns an object of class "mnorm_cmnorm". An object of class "mnorm_cmnorm" is a list containing the following components: * `mean` - conditional mean. * `sigma` - conditional covariance matrix. * `sigma_d` - covariance matrix of unconditioned elements. * `sigma_g` - covariance matrix of conditioned elements. * `sigma_dg` - matrix of covariances between unconditioned and conditioned elements. * `s12s22` - equals to the matrix product of `sigma_dg` and `solve(sigma_g)`. Note that `mean` corresponds to $\mu_{c}$ while `sigma` represents $\Sigma_{c}$. Moreover `sigma_d` is $\Sigma_{I_{d}, I_{d}}$, `sigma_g` is $\Sigma_{I_{g}, I_{g}}$ and `sigma_dg` is $\Sigma_{I_{d}, I_{g}}$. Since $\Sigma_{c}$ do not depend on $X^{(g)}$ the output `sigma` does not depend on `given_x`. In particular output `sigma` remains the same independent of whether `given_x` is a matrix or vector. Oppositely if `given_x` is a matrix then output `mean` is a matrix which rows correspond to conditional means associated with given values provided by corresponding rows of `given_x`. The order of elements of output `mean` and output `sigma` depends on the order of `dependet_ind` elements that is ascending by default. The order of `given_ind` elements does not matter. But, please, check that the order of `given_ind` match the order of given values i.e. the order of `given_x` columns. ## Details Consider $m$-dimensional multivariate normal vector $X=(X_{1},...,X_{m})^{T}~\sim N(\mu,\Sigma)$, where $E(X)=\mu$ and $Cov(X)=\Sigma$ are expectation (mean) and covariance matrix respectively. Let's denote vectors of indexes of conditioned and unconditioned elements of $X$ by $I_{g}$ and $I_{d}$ respectively. By $x^{(g)}$ denote deterministic (column) vector of given values of $X_{I_{g}}$. The function calculates expected value and covariance matrix of conditioned multivariate normal vector $X_{I_{d}} | X_{I_{g}} = x^{(g)}$. For example if $I_{g}=(1, 3)$ and $x^{(g)}=(-1, 1)$ then $I_{d}=(2, 4, 5)$ so the function calculates: $$ \mu_{c}=E\left(\left(X_{2}, X_{4}, X_{5}\right) | X_{1} = -1, X_{3} = 1\right) $$ $$ \Sigma_{c}=Cov\left(\left(X_{2}, X_{4}, X_{5}\right) |X_{1} = -1, X_{3} = 1\right) $$ In general case: $$ \mu_{c} = E\left(X_{I_{d}} | X_{I_{g}} = x^{(g)}\right) =\mu_{I_{d}} +\left(x^{(g)} - \mu_{I_{g}}\right)\left(\Sigma_{(I_{d}, I_{g})}\Sigma_{(I_{g}, I_{g})}^{-1}\right)^{T} $$ $$ \Sigma_{c} = Cov\left(X_{I_{d}} | X_{I_{g}} = x^{(g)}\right) =\Sigma_{(I_{d}, I_{d})} -\Sigma_{(I_{d}, I_{g})}\Sigma_{(I_{g}, I_{g})}^{-1}\Sigma_{(I_{g}, I_{d})} $$ Note that $\Sigma_{(A, B)}$, where $A,B\in\{d, g\}$, is a submatrix of $\Sigma$ generated by intersection of $I_{A}$ rows and $I_{B}$ columns of $\Sigma$. Below there is a correspondence between aforementioned theoretical (mathematical) notations and function arguments: * `mean` - $\mu$. * `sigma` - $\Sigma$. * `given_ind` - $I_{g}$. * `given_x` - $x^{(g)}$. * `dependent_ind` - $I_{d}$ Moreover $\Sigma_{(I_{g}, I_{d})}$ is a theoretical (mathematical) notation for `sigma[given_ind, dependent_ind]`. Similarly $\mu_{g}$ represents `mean[given_ind]`. By default `dependent_ind` contains all indexes that are not in `given_ind`. It is possible to omit and duplicate indexes of `dependent_ind`. But at least single index should be provided for `given_ind` without any duplicates. Also `dependent_ind` and `given_ind` should not have the same elements. Moreover `given_ind` should not be of the same length as `mean` so at least one component should be unconditioned. If `given_x` is a vector then (if possible) it will be treated as a matrix with the number of columns equal to the length of `mean`. Currently `control` has no input arguments intended for the users. This argument is used for some internal purposes of the package. ## Examples ```r # Consider multivariate normal vector: # X = (X1, X2, X3, X4, X5) ~ N(mean, sigma) # Prepare multivariate normal vector parameters # expected value mean <- c(-2, -1, 0, 1, 2) n_dim <- length(mean) # correlation matrix cor <- c( 1, 0.1, 0.2, 0.3, 0.4, 0.1, 1, -0.1, -0.2, -0.3, 0.2, -0.1, 1, 0.3, 0.2, 0.3, -0.2, 0.3, 1, -0.05, 0.4, -0.3, 0.2, -0.05, 1) cor <- matrix(cor, ncol = n_dim, nrow = n_dim, byrow = TRUE) # covariance matrix sd_mat <- diag(c(1, 1.5, 2, 2.5, 3)) sigma <- sd_mat %*% cor %*% t(sd_mat) # Estimate parameters of conditional distribution i.e. # when the first and the third components of X are conditioned: # (X2, X4, X5 | X1 = -1, X3 = 1) given_ind <- c(1, 3) given_x <- c(-1, 1) par <- cmnorm(mean = mean, sigma = sigma, given_ind = given_ind, given_x = given_x) # E(X2, X4, X5 | X1 = -1, X3 = 1) par$mean # Cov(X2, X4, X5 | X1 = -1, X3 = 1) par$sigma # Additionally calculate E(X2, X4, X5 | X1 = 2, X3 = 3) given_x_mat <- rbind(given_x, c(2, 3)) par1 <- cmnorm(mean = mean, sigma = sigma, given_ind = given_ind, given_x = given_x_mat) par1$mean # Duplicates and omitted indexes are allowed for dependent_ind # For given_ind duplicates are not allowed # Let's calculate conditional parameters for (X5, X2, X5 | X1 = -1, X3 = 1): dependent_ind <- c(5, 2, 5) par2 <- cmnorm(mean = mean, sigma = sigma, given_ind = given_ind, given_x = given_x, dependent_ind = dependent_ind) # E(X5, X2, X5 | X1 = -1, X3 = 1) par2$mean # Cov(X5, X2, X5 | X1 = -1, X3 = 1) par2$sigma ```

cmnorm function