Package 'MTE'

Huber-Lasso estimator

Description

This function is L1 penalized Huber estimator for linear regression under both fixed and high-dimensional settings. Currently, the function does not support automatic selection of huber tuning parameter.

Usage

huber.lasso(
  X,
  y,
  beta.ini,
  lambda,
  alpha = 2,
  adaptive = TRUE,
  intercept = TRUE,
  penalty.factor = rep(1, ncol(X))
)

Arguments

X	design matrix, standardization is recommended.
y	response vector.
beta.ini	initial estimates of beta. If not specified, LADLasso estimates from rq.lasso.fit() in rqPen is used. Otherwise, robust estimators are strongly recommended.
lambda	regularization parameter of Lasso or adaptive Lasso (if adaptive=TRUE).
alpha	1/alpha is the huber tuning parameter. Larger alpha results in smaller portion of squared loss.
adaptive	logical input that indicates if adaptive Lasso is used. Default is TRUE.
intercept	logical input that indicates if intercept needs to be estimated. Default is FALSE.
penalty.factor	can be used to force nonzero coefficients. Default is rep(1, ncol(X)) as in glmnet.

Value

beta	the regression coefficient estimates.
fitted	predicted response.
iter.steps	iteration steps.

Examples

set.seed(2017)
n=200; d=500
X=matrix(rnorm(n*d), nrow=n, ncol=d)
beta=c(rep(2,6), rep(0, d-6))
y=X%*%beta+c(rnorm(150), rnorm(30,10,10), rnorm(20,0,100))
output.HuberLasso=huber.lasso(X,y)
beta.est=output.HuberLasso$beta

---

Huber estimation for linear regression

Description

This function produces Huber estimates for linear regression. Initial estimates is required. Currently, the function does not support automatic selection of huber tuning parameter.

Usage

huber.reg(y, X, beta.ini, alpha, intercept = FALSE)

Arguments

y	the response vector
X	design matrix
beta.ini	initial value of estimates, could be from OLS.
alpha	1/alpha is the huber tuning parameter delta. Larger alpha results in smaller portion of squared loss.
intercept	logical input that indicates if intercept needs to be estimated. Default is FALSE.

Value

beta	the regression coefficient estimates
fitted.value	predicted response
iter.steps	iteration steps.

Examples

set.seed(2017)
n=200; d=4
X=matrix(rnorm(n*d), nrow=n, ncol=d)
beta=c(1, -1, 2, -2)
y=-2+X%*%beta+c(rnorm(150), rnorm(30,10,10), rnorm(20,0,100))
beta0=beta.ls=lm(y~X)$coeff
beta.huber=huber.reg(y, X, beta0, 2, intercept=TRUE)$beta
cbind(c(-2,beta), beta.ls, beta.huber)

---

Huber Loss

Description

Huber Loss

Usage

huberloss(r, alpha)

Arguments

r	residual, y-Xbeta
alpha	1/alpha is the huber tuning parameter delta. Larger alpha results in smaller portion of squared loss.

Value

it returns huber loss that will be called in Huber estimation.

---

Least Absolute Deviance Estimator for Linear Regression

Description

Least Absolute Deviance Estimator for Linear Regression

Usage

LAD(X, y, intercept = FALSE)

Arguments

X	design matrix
y	reponse vector
intercept	logical input that indicates if intercept needs to be estimated. Default is FALSE.

Value

coefficient estimates

Examples

set.seed(1989)
n=200; d=4
X=matrix(rnorm(n*d), nrow=n, ncol=d)
beta=c(1, -1, 2, -2)
y=-2+X%*%beta+c(rnorm(150), rnorm(30,10,10), rnorm(20,0,100))
beta.ls=lm(y~X)$coeff
beta.LAD=LAD(X,y,intercept=TRUE)
cbind(c(-2,beta), beta.ls, beta.LAD)

---

LAD-Lasso for Linear Regression

Description

LAD-Lasso for Linear Regression

Usage

LADlasso(
  X,
  y,
  beta.ini,
  lambda = NULL,
  adaptive = TRUE,
  intercept = FALSE,
  penalty.factor = rep(1, ncol(X))
)

Arguments

X	design matrix, standardization is recommended.
y	reponse vector
beta.ini	initial estimates of beta. Using unpenalized LAD is recommended under high-dimensional setting.
lambda	regularization parameter of Lasso or adaptive Lasso (if adaptive=TRUE).
adaptive	logical input that indicates if adaptive Lasso is used. Default is TRUE.
intercept	logical input that indicates if intercept needs to be estimated. Default is FALSE.
penalty.factor	can be used to force nonzero coefficients. Default is rep(1, ncol(X)) as in glmnet.

Value

beta	the regression coefficient estimates.
fitted	predicted response.
iter.steps	iteration steps.

Examples

set.seed(2017)
n=200; d=50
X=matrix(rnorm(n*d), nrow=n, ncol=d)
beta=c(rep(2,6), rep(0, 44))
y=X%*%beta+c(rnorm(150), rnorm(30,10,10), rnorm(20,0,100))
output.LADLasso=LADlasso(X, y, beta.ini=LAD(X, y))
beta.est=output.LADLasso$beta

---

Maximum Tangent-likelihood Estimation

Description

It estimates linear regression coefficient using MTE. The function produces robust estimates of linear regression. Outliers and contamination would be downweighted. It is robust to Gaussian assumption of the error term. Initial estimates need to be provided.

Usage

MTE(y, X, beta.ini, t, p, intercept = FALSE)

Arguments

y	the response vector
X	design matrix
beta.ini	initial value of estimates, could be from OLS.
t	the tangent point. You may specify a sequence of values, so that the function automatically select the optimal one.
p	Taylor expansion order, up to 3.
intercept	logical input that indicates if intercept needs to be estimated. Default is FALSE.

Value

Returns estimates from MTE method.

beta	the regression coefficient estimates
fitted.value	predicted response
t	the optimal tangent point through data-driven method

Examples

set.seed(2017)
n=200; d=4
X=matrix(rnorm(n*d), nrow=n, ncol=d)
beta=c(1, -1, 2, -2)
y=-2+X%*%beta+c(rnorm(150), rnorm(30,10,10), rnorm(20,0,100))
beta0=beta.ls=lm(y~X)$coeff
beta.MTE=MTE(y,X,beta0,0.1,2, intercept=TRUE)$beta
cbind(c(-2,beta), beta.ls, beta.MTE)

---

MTE-Lasso estimator

Description

MTELasso is the penalized MTE for robust estimation and variable selection for linear regression. It can deal with both fixed and high-dimensional settings.

Usage

MTElasso(
  X,
  y,
  beta.ini,
  p = 2,
  lambda = NULL,
  adaptive = TRUE,
  t = 0.01,
  intercept = TRUE,
  penalty.factor = rep(1, ncol(X)),
  ...
)

Arguments

X	design matrix, standardization is recommended.
y	response vector.
beta.ini	initial estimates of beta. If not specified, LADLasso estimates from rq.lasso.fit() in rqPen is used. Otherwise, robust estimators are strongly recommended.
p	Taylor expansion order.
lambda	regularization parameter for LASSO, but not necessary if "adaptive=TRUE".
adaptive	logic argument to indicate if Adaptive-Lasso is used. Default is TRUE.
t	the tuning parameter that controls for the tradeoff between robustness and efficiency. Default is t=0.01.
intercept	logical input that indicates if intercept needs to be estimated. Default is FALSE.
penalty.factor	can be used to force nonzero coefficients. Default is rep(1, ncol(X)) as in glmnet.
...	other arguments that are used in glmnet.

Value

It returns a sparse vector of estimates of linear regression. It has two types of penalty, LASSO and AdaLasso. Coordinate descent algorithm is used for iteratively updating coefficients.

beta	sparse regression coefficient
fitted	predicted response

Examples

set.seed(2017)
n=200; d=500
X=matrix(rnorm(n*d), nrow=n, ncol=d)
beta=c(rep(2,6), rep(0, d-6))
y=X%*%beta+c(rnorm(150), rnorm(30,10,10), rnorm(20,0,100))
output.MTELasso=MTElasso(X, y, p=2, t=0.01)
beta.est=output.MTELasso$beta

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