eppls: Fit the Envelope-based Partial Partial Least Squares model

Description

Fit the Envelope-based Partial Partial Least Squares model for multivariate linear regression with dimension u.

Usage

eppls(X1, X2, Y, u, asy = TRUE, init = NULL)

Value

The output is a list that contains the following components:

muY: The estimator of mean of Y.
mu1: The estimator of mean of X1.
mu2: The estimator of mean of X2.
beta1: A $p1$ by $r$ matrix for the estimator of regression coefficients for continuous predictors.
beta2: A $p2$ by $r$ matrix for the estimator of regression coefficients for categorical predictors.
Gamma: An $p1$ by $d$ matrix for the orthogonal basis of the Envelope-based Partial Partial Least Squares.
Gamma0: An $p1$ by $(p1-d)$ matrix for the orthogonal basis of the complement of the Envelope-based Partial Partial Least Squares.
gamma: A $p2$ by $p1$ matrix for the estimator of regression coefficients based on the regression of X1 on X2.
eta: A $d$ by $p1$ matrix for the coordinates of beta1 with respect to Gamma.
Omega: A $d$ by $d$ matrix for the coordinates of SigmaX1 with respect to Gamma.
Omega0: A $(p1-d)$ by $(p1-d)$ matrix for the coordinates of SigmaX1 with respect to Gamma0.
SigmaX1: The estimator of error covariance matrix Sigma[1|2].
SigmaYcX: The estimator of error covariance matrix Sigma[Y|X].
loglik: The maximized log likelihood function.
n: The number of observations in the data.
covMatrix1: The asymptotic covariance of vec(beta1). The covariance matrix returned are asymptotic. For the actual standard errors, multiply by 1 / n.
covMatrix2: The asymptotic covariance of vec(beta2). The covariance matrix returned are asymptotic. For the actual standard errors, multiply by 1 / n.
asySE1: The asymptotic standard error matrix for elements in beta1. The multiplication by the reciprocal of square root of $n$ returns actual standard errors.
asySE2: The asymptotic standard error matrix for elements in beta2. The multiplication by the reciprocal of square root of $n$ returns actual standard errors.

Arguments

X1: An $n$ by $p1$ matrix of continuous predictors, where $p1$ is the number of continuous predictors with $p1 < n$.
X2: An $n$ by $p2$ matrix of categorical predictors, where $p2$ is the number of categorical predictors with $p2 < n$.
Y: An $n$ by $r$ matrix of multivariate responses, where $r$ is the number of responses.
u: A given dimension of the Envelope-based Partial Partial Least Squares. It should be an interger between $0$ and $p1$.
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. An $r$ by $u$ matrix. The default is the one generated by function envMU.

Details

This function the Envelope-based Partial Partial Least Squares model for multivariate linear regression with dimension $u$, $$ Y = \mu + \Gamma\eta X + \varepsilon, \Sigma=\Gamma\Omega\Gamma' + \Gamma_{0}\Omega_{0}\Gamma'_{0} $$ using the maximum likelihood estimation. When the dimension of the envelope is between 1 and $p1$-1, the starting value and blockwise coordinate descent algorithm in Cook et al. (2016) is implemented. When the dimension is $p1$, then the envelope model degenerates to the standard multivariate linear regression. When the dimension is 0, it means that X and Y are uncorrelated, and the fitting is different.

References

Park, Y., Su, Z. and Chung, D. (2022+) Envelope-based Partial Partial Least Squares with Application to Cytokine-based Biomarker Analysis for COVID-19.

Examples

Run this code

  data(amitriptyline)
  
  Y <- amitriptyline[ , 1:2]
  X1 <- amitriptyline[ , 4:7]
  X2 <- amitriptyline[ , 3]
  u <- u.eppls(X1, X2, Y)
  
  u
  
  m <- eppls(X1, X2, Y, 2)
  m

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