21,693 research outputs found
Direction of arrival estimation using robust complex Lasso
The Lasso (Least Absolute Shrinkage and Selection Operator) has been a
popular technique for simultaneous linear regression estimation and variable
selection. In this paper, we propose a new novel approach for robust Lasso that
follows the spirit of M-estimation. We define -Lasso estimates of regression
and scale as solutions to generalized zero subgradient equations. Another
unique feature of this paper is that we consider complex-valued measurements
and regression parameters, which requires careful mathematical characterization
of the problem. An explicit and efficient algorithm for computing the -Lasso
solution is proposed that has comparable computational complexity as
state-of-the-art algorithm for computing the Lasso solution. Usefulness of the
-Lasso method is illustrated for direction-of-arrival (DoA) estimation with
sensor arrays in a single snapshot case.Comment: Paper has appeared in the Proceedings of the 10th European Conference
on Antennas and Propagation (EuCAP'2016), Davos, Switzerland, April 10-15,
201
Adaptive robust variable selection
Heavy-tailed high-dimensional data are commonly encountered in various
scientific fields and pose great challenges to modern statistical analysis. A
natural procedure to address this problem is to use penalized quantile
regression with weighted -penalty, called weighted robust Lasso
(WR-Lasso), in which weights are introduced to ameliorate the bias problem
induced by the -penalty. In the ultra-high dimensional setting, where the
dimensionality can grow exponentially with the sample size, we investigate the
model selection oracle property and establish the asymptotic normality of the
WR-Lasso. We show that only mild conditions on the model error distribution are
needed. Our theoretical results also reveal that adaptive choice of the weight
vector is essential for the WR-Lasso to enjoy these nice asymptotic properties.
To make the WR-Lasso practically feasible, we propose a two-step procedure,
called adaptive robust Lasso (AR-Lasso), in which the weight vector in the
second step is constructed based on the -penalized quantile regression
estimate from the first step. This two-step procedure is justified
theoretically to possess the oracle property and the asymptotic normality.
Numerical studies demonstrate the favorable finite-sample performance of the
AR-Lasso.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1191 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Robust Estimation of High-Dimensional Mean Regression
Data subject to heavy-tailed errors are commonly encountered in various
scientific fields, especially in the modern era with explosion of massive data.
To address this problem, procedures based on quantile regression and Least
Absolute Deviation (LAD) regression have been devel- oped in recent years.
These methods essentially estimate the conditional median (or quantile)
function. They can be very different from the conditional mean functions when
distributions are asymmetric and heteroscedastic. How can we efficiently
estimate the mean regression functions in ultra-high dimensional setting with
existence of only the second moment? To solve this problem, we propose a
penalized Huber loss with diverging parameter to reduce biases created by the
traditional Huber loss. Such a penalized robust approximate quadratic
(RA-quadratic) loss will be called RA-Lasso. In the ultra-high dimensional
setting, where the dimensionality can grow exponentially with the sample size,
our results reveal that the RA-lasso estimator produces a consistent estimator
at the same rate as the optimal rate under the light-tail situation. We further
study the computational convergence of RA-Lasso and show that the composite
gradient descent algorithm indeed produces a solution that admits the same
optimal rate after sufficient iterations. As a byproduct, we also establish the
concentration inequality for estimat- ing population mean when there exists
only the second moment. We compare RA-Lasso with other regularized robust
estimators based on quantile regression and LAD regression. Extensive
simulation studies demonstrate the satisfactory finite-sample performance of
RA-Lasso
Quantile regression in high-dimension with breaking
The paper considers a linear regression model in high-dimension for which the
predictive variables can change the influence on the response variable at
unknown times (called change-points). Moreover, the particular case of the
heavy-tailed errors is considered. In this case, least square method with LASSO
or adaptive LASSO penalty can not be used since the theoretical assumptions do
not occur or the estimators are not robust. Then, the quantile model with SCAD
penalty or median regression with LASSO-type penalty allows, in the same time,
to estimate the parameters on every segment and eliminate the irrelevant
variables. We show that, for the two penalized estimation methods, the oracle
properties is not affected by the change-point estimation. Convergence rates of
the estimators for the change-points and for the regression parameters, by the
two methods are found. Monte-Carlo simulations illustrate the performance of
the methods
The Influence Function of Penalized Regression Estimators
To perform regression analysis in high dimensions, lasso or ridge estimation
are a common choice. However, it has been shown that these methods are not
robust to outliers. Therefore, alternatives as penalized M-estimation or the
sparse least trimmed squares (LTS) estimator have been proposed. The robustness
of these regression methods can be measured with the influence function. It
quantifies the effect of infinitesimal perturbations in the data. Furthermore
it can be used to compute the asymptotic variance and the mean squared error.
In this paper we compute the influence function, the asymptotic variance and
the mean squared error for penalized M-estimators and the sparse LTS estimator.
The asymptotic biasedness of the estimators make the calculations nonstandard.
We show that only M-estimators with a loss function with a bounded derivative
are robust against regression outliers. In particular, the lasso has an
unbounded influence function.Comment: appears in Statistics: A Journal of Theoretical and Applied
Statistics, 201
LAD-LASSO: SIMULATION STUDY OF ROBUST REGRESSION IN HIGH DIMENSIONAL DATA
The common issues in regression, there are a lot of cases in the condition number of predictor variables more than number of observations ( ) called high dimensional data. The classical problem always lies in this case, that is multicolinearity. It would be worse when the datasets subject to heavy-tailed errors or outliers that may appear in the responses and/or the predictors. As this reason, Wang et al in 2007 developed combined methods from Least Absolute Deviation (LAD) regression that is useful for robust regression, and also LASSO that is popular choice for shrinkage estimation and variable selection, becoming LAD-LASSO. Extensive simulation studies demonstrate satisfactory using LAD-LASSO in high dimensional datasets that lies outliers better than using LASSO.Keywords: high dimensional data, LAD-LASSO, robust regressio
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