# Fit the envelope model in poisson regression Fit the envelope model in poisson regression with dimension u. ```r pois.env(X, Y, u, asy = TRUE, init = NULL) ``` ## Arguments - `X`: Predictors. An n by p matrix, p is the number of predictors and n is number of observations. The predictors must be continuous variables. - `Y`: Response. An n by 1 matrix. The univariate response must be counts. - `u`: Dimension of the envelope. An integer between 0 and p. - `asy`: Flag for computing the asymptotic variance of the 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`. - `init`: The user-specified value of Gamma for the envelope subspace in poisson regression. An p by u matrix. The default is the one generated by function pois.envMU. ## Details This function fits the envelope model in poisson regression, $$ Y = exp(\mu + \beta' X), \Sigma_{X}=\Gamma\Omega\Gamma'+\Gamma_{0}\Omega_{0}\Gamma'_{0} $$ using the maximum likelihood estimation. When the dimension of the envelope is between 1 and p-1, the starting value and blockwise coordinate descent algorithm in Cook et al. (2016) is implemented. This model works the best when X is multivariate normal. ## Returns The output is a list that contains the following components: - **beta**: The envelope estimator of the canonical parameter. - **SigmaX**: The envelope estimator of the covariance matrix of X. - **Gamma**: An orthonormal basis of the envelope subspace. - **Gamma0**: An orthonormal basis of the complement of the envelope subspace. - **eta**: The estimated beta of the canonical parameter with respect to Gamma. - **Omega**: The coordinates of SigmaX with respect to Gamma. - **Omega0**: The coordinates of SigmaX with respect to Gamma0. - **mu**: The estimated intercept of the canonical parameter. - **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 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 over the envelope estimator, for each element in beta. - **n**: The number of observations in the data. ## References Cook, R. D., Zhang, X. (2015). Foundations for Envelope Models and Methods. Journal of the American Statistical Association 110, 599 - 611. Cook, R. D., Forzani, L. and Su, Z. (2016) A Note on Fast Envelope Estimation. Journal of Multivariate Analysis. 150, 42-54. ## Examples ```r data(horseshoecrab) X1 <- as.numeric(horseshoecrab[ , 1] == 2) X2 <- as.numeric(horseshoecrab[ , 1] == 3) X3 <- as.numeric(horseshoecrab[ , 1] == 4) X4 <- as.numeric(horseshoecrab[ , 2] == 2) X5 <- as.numeric(horseshoecrab[ , 2] == 3) X6 <- horseshoecrab[ , 3] X7 <- horseshoecrab[ , 5] X <- cbind(X1, X2, X3, X4, X5, X6, X7) Y <- horseshoecrab[ , 4] ## Not run: u <- u.pois.env(X, Y) ## Not run: u m <- pois.env(X, Y, 1) m$beta ```

pois.env function