\[ \newcommand{\vbeta}{\boldsymbol{\beta}} \]


The L2E package implements the computational framework in “A User-Friendly Computational Framework for Robust Structured Regression with the L\(_2\) Criterion” by Jocelyn T. Chi and Eric C. Chi. This vignette provides code to replicate some examples illustrating the framework in that paper.


Multivariate Regression

The first example provides code to perform multivariate L\(_{2}\)E regression with the Italian bank data from the paper. We begin by loading the bank data from the L2E package.

y <- bank$y
X <- as.matrix(bank[,1:13])
X0 <- as.matrix(cbind(rep(1,length(y)), X))
tau_initial <- 1/mad(y)
beta_initial <- matrix(0, 14, 1)

The l2e_regression function performs multivariate regression via the L\(_{2}\) criterion, also called LTE multivariate regression, as described in the paper. It simultaneously obtains an estimate for the coefficient vector \(\vbeta\) and the precision \(\tau\).

sol <- l2e_regression(y, X0, tau_initial, beta_initial)

We can use the estimates for \(\vbeta\) and \(\tau\) in sol$beta and sol$tau to identify outlying observations, depicted in blue in the figure below.

betaEstimate <- sol$beta
tauEstimate <- sol$tau

r <- y - X0 %*% betaEstimate
outliers <- which(abs(r) > 3/tauEstimate)
l2e_fit <- X0 %*% betaEstimate
plot(y, l2e_fit, ylab='Predicted values', pch=16, cex=0.8)
points(y[outliers], l2e_fit[outliers], pch=16, col='blue', cex=0.8)