tauRRR: Tau estimate for Reduced Rank Regression
In meszre/tauPFC: Computes Robust Estimators for the PFC Model

Description Usage Arguments Details Value References Examples

Fits a reduced-rank regression (RRR) model (for a description, see Izenman(2008)) with a robust procedure that can resist the presence of outliers. It computes tau type estimates, see Bergesio et al. (2020). It fits also a multivariate linear model (MLM) without rank restriction if the rank is chosen properly.

1	tauRRR(yy, XX, d, efficiency = 0.90)

`yy`	vector of response variables, n x p matrix, each row is a response vector
`XX`	vector of covariates in the multivariate regression problem. Is a n x r matrix, each row is the corresponding covariate vector
`d`	rank of coefficient matrix. If `d` = min(p,r), this routine produces robust multivariate linear regression estimators. If `d` < min(p,r), this routine produces a reduced rank regression model, with arbitrary covariance of errors
`efficiency`	efficiency of the robust estimators, 0.90 by default

We consider the multivariate linear reduced rank regression (RRR) model given by

Y = μ + ABX + ε,

where the variables are

Y is a p x 1 observed response vector
X is a r x 1 observed covariates vector
ε is unobserved p dimensional vector of errors

and the unknown parameters (to be estimated) are

μ a p x 1 vector of intercepts
A is a full-rank p x d matrix
B is a full-rank d x r matrix
cov(ε), the p x p covariance matrix of errors ε,

Both coefficient matrices A and B are not unique, but their product p x r matrix C = AB is unique. The rank of C is d ≤ min(p,r). The notation refers to Izenman (2008).

List with the following components, of the RRR model described above

`mu`	tau-estimator for intercept vector
`RRRcoef`	tau-estimator for the coefficient p x r matrix C of rank `d`
`AA`	tau-estimator for A, a full-rank p x d matrix
`BB`	tau-estimator for B, a full-rank d x r matrix
`cov.error`	tau-estimator for the covariance matrix of errors, cov(ε)

Izenman, A. J. (2008). Modern multivariate statistical techniques. Regression, classification and manifold learning, New York: Springer.

Bergesio, A., Szretter Noste, M. E. and Yohai, V. J. (2020). A robust proposal of estimation for the sufficient dimension reduction problem

# We work with Example 6.3.3, Chemical Composition of Tobacco, from
# Izenman (2008). Dataset available in \link[rrr]{tobacco} or in
# \url{https://astro.temple.edu/~alan/tobacco.txt}

library(rrr)
data(tobacco)

XX = as.matrix(tobacco[,4:9])  # covariates
yy = as.matrix(tobacco[,1:3])  # response vector

###############################
# RRR model with d=2
###############################
# robust MLM fit
robustRRR2 = tauRRR(yy, XX, d=2)

# maximum likelihood MLM, with arbitrary covariance of errors matrix
maxliRRR2 = MLE(yy, XX, d=2)


###############################
# robust MLM fit
###############################
robustMLM = tauRRR(yy, XX, d=3)

# classical MLM, with covariance of errors a multiple of identity
classicalMLM = lm(yy ~ XX)

# maximum likelihood MLM, with arbitrary covariance of errors matrix
maxlikMLM = MLE(yy, XX, d=3)

# to show that the three of them agree, we can do any of the following

# 1. Verify they span the same column space
library(pracma)
angle_AB(orth(robustMLM$RRRcoef),orth(t(classicalMLM$coefficients[2:7,])))
# equivalently
angle_AB(robustMLM$AA ,orth(t(classicalMLM$coefficients[2:7,])))
angle_AB(robustMLM$AA,maxlikMLM$gamma)

# 2. Plot coefficients estimated by every method
plot(robustMLM$RRRcoef,t(classicalMLM$coefficients[2:7,]))
abline(0,1)

points(t(classicalMLM$coefficients[2:7,]),maxlikMLM$gamma%*%maxlikMLM$beta,col="red")

# 3. Compute maximum absolute difference in coefficients estimated by every method
max(abs(t(classicalMLM$coefficients[2:7,])-maxlikMLM$gamma%*%maxlikMLM$beta))
max(abs(t(classicalMLM$coefficients[2:7,])-robustMLM$RRRcoef))