# Fit the reduced-rank envelope model Fit the reduced-rank envelope model with rank d and fixed envelope dimension u. ```r rrenv(X, Y, u, d, asy = TRUE) ``` ## Arguments - `X`: Predictors. An n by p matrix, p is the number of predictors. The predictors can be univariate or multivariate, discrete or continuous. - `Y`: Multivariate responses. An n by r matrix, r is the number of responses and n is number of observations. The responses must be continuous variables. - `u`: Dimension of the envelope. An integer between 0 and r. - `d`: The rank of the coefficient matrix. An integer between 0 and u. - `asy`: Flag for computing the asymptotic variance of the reduced rank envelope estimator. The default is `TRUE`. When p and r are large, computing the asymptotic variance can take much time and memory. If only the envelope estimators are needed, the flag can be set to `asy = FALSE`. ## Details This function fits the reduced rank envelope model to the responses and predictors, $$ Y_{i} = \alpha + \Gamma\eta BX_{i}+\varepsilon_{i}, \Sigma=\Gamma\Omega\Gamma'+\Gamma_{0}\Omega_{0}\Gamma'_{0}, i=1, ..., n, $$ using the maximum likelihood estimation. The errors $\varepsilon_{i}$ follow a normal distribution. When 0 < d < u < r, the estimation procedure in Cook et al. (2015) is implemented. When d < u = r, then the model is equivalent to a reduced rank regression model. When d = u, or d = p < r, then B can be taken as the identity matrix and the model reduces to a response envelope model. When the dimension is d = u = r, then the envelope model degenerates to the standard multivariate linear regression. When the u = 0, it means that X and Y are uncorrelated, and the fitting is different. If the error covariance matrix is nonconstant, see the function `rrenv.apweights`. ## Returns The output is a list that contains the following components: - **Gamma**: An orthogonal basis of the envelope subspace. - **Gamma0**: An orthogonal basis of the complement of the envelope subspace. - **mu**: The estimated intercept. - **beta**: The envelope estimator of the regression coefficients. - **Sigma**: The envelope estimator of the error covariance matrix. - **eta**: The eta matrix in the coefficient matrix. - **B**: The B matrix in the coefficient matrix. - **Omega**: The coordinates of Sigma with respect to Gamma. - **Omega0**: The coordinates of Sigma with respect to Gamma0. - **loglik**: The maximized log likelihood function. - **covMatrix**: The asymptotic covariance of vec(beta). The covariance matrix returned are asymptotic. For the actual standard errors, multiply by 1 / n. - **asySE**: The asymptotic standard error for elements in beta under the reduced rank envelope model. The standard errors returned are asymptotic, for actual standard errors, multiply by 1 / sqrt(n). - **ratio**: The asymptotic standard error ratio of the standard multivariate linear regression estimator (with consideration of nonconstant variance) over the envelope estimator, for each element in beta. - **n**: The number of observations in the data. ## References Cook, R. D., Forzani, L. and Zhang, X. (2015). Envelopes and reduced-rank regression. Biometrika 102, 439-456. Forzani, L. and Su, Z. (2021). Envelopes for elliptical multivariate linear regression. Statist. Sinica 31, 301-332. ## Examples ```r data(vehicles) X <- vehicles[, 1:11] Y <- vehicles[, 12:15] X <- scale(X) Y <- scale(Y) # The scales of Y are vastly different, so scaling is reasonable here d <- d.select(X, Y, 0.01) d ## Not run: u <- u.rrenv(X, Y, 2) ## Not run: u m <- rrenv(X, Y, 4, 2) m m$beta ```

rrenv function