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STRONG ORACLE OPTIMALITY OF FOLDED CONCAVE PENALIZED ESTIMATION
Folded concave penalization methods have been shown to enjoy the strong oracle property for high-dimensional sparse estimation. However, a folded concave penalization problem usually has multiple local solutions and the oracle property is established only for one of the unknown local solutions. A challenging fundamental issue still remains that it is not clear whether the local optimum computed by a given optimization algorithm possesses those nice theoretical properties. To close this important theoretical gap in over a decade, we provide a unified theory to show explicitly how to obtain the oracle solution via the local linear approximation algorithm. For a folded concave penalized estimation problem, we show that as long as the problem is localizable and the oracle estimator is well behaved, we can obtain the oracle estimator by using the one-step local linear approximation. In addition, once the oracle estimator is obtained, the local linear approximation algorithm converges, namely it produces the same estimator in the next iteration. The general theory is demonstrated by using four classical sparse estimation problems, that is, sparse linear regression, sparse logistic regression, sparse precision matrix estimation and sparse quantile regression.
Journal Article
The tiger prince
When a tigress whose cubs were killed by hunters ravages villages, the king gathers his army but Lao Lao, a seer, advises him to send his son, Wen, to the tiger, instead.
Book
The Adaptive Lasso and Its Oracle Properties
The lasso is a popular technique for simultaneous estimation and variable selection. Lasso variable selection has been shown to be consistent under certain conditions. In this work we derive a necessary condition for the lasso variable selection to be consistent. Consequently, there exist certain scenarios where the lasso is inconsistent for variable selection. We then propose a new version of the lasso, called the adaptive lasso, where adaptive weights are used for penalizing different coefficients in the ℓ
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penalty. We show that the adaptive lasso enjoys the oracle properties; namely, it performs as well as if the true underlying model were given in advance. Similar to the lasso, the adaptive lasso is shown to be near-minimax optimal. Furthermore, the adaptive lasso can be solved by the same efficient algorithm for solving the lasso. We also discuss the extension of the adaptive lasso in generalized linear models and show that the oracle properties still hold under mild regularity conditions. As a byproduct of our theory, the nonnegative garotte is shown to be consistent for variable selection.
Journal Article
The seer and the city : religion, politics, and colonial ideology in ancient Greece
\"Seers featured prominently in ancient Greek culture, but they rarely appear in colonial discourse from the archaic and classical periods. Margaret Foster exposes the ideological motivations behind this discrepancy and reveals how colonial discourse's privileging of the city's founder and his dependence on Delphi, the colonial oracle par excellence, entails a corresponding suppression of the seer. Foster explains why the seer's authority conflicts with that of the founder and investigates a sequence of literary works from a range of genres that showcase this dynamic. The first study to analyze the seer and the Delphi-sanctioned founder relationally, this volume illuminates the contests between religious and political powers in archaic and classical Greece.\"--Provided by publisher.
Book
Bio-ORACLE v2.0: Extending marine data layers for bioclimatic modelling
Motivation: The availability of user-friendly, high-resolution global environmental datasets is crucial for bioclimatic modelling. For terrestrial environments, WorldClim has served this purpose since 2005, but equivalent marine data only became available in 2012, with pioneer initiatives like Bio-ORACLE providing data layers for several ecologically relevant variables. Currently, the available marine data packages have not yet been updated to the most recent Intergovernmental Panel on Climate Change (IPCC) predictions nor to present times, and are mostly restricted to the top surface layer of the oceans, precluding the modelling of a large fraction of the benthic diversity that inhabits deeper habitats. To address this gap, we present a significant update of Bio-ORACLE for new future climate scenarios, present-day conditions and benthic layers (near sea bottom). The reliability of data layers was assessed using a cross-validation framework against in situ quality-controlled data. This test showed a generally good agreement between our data layers and the global climatic patterns. We also provide a package of functions in the R software environment (sdmpredictors) to facilitate listing, extraction and management of data layers and allow easy integration with the available pipelines for bioclimatic modelling. Main types of variable contained: Surface and benthic layers for water temperature, salinity, nutrients, chlorophyll, sea ice, current velocity, phytoplankton, primary productivity, iron and light at bottom. Spatial location and grain: Global at 5 arcmin (c. 0.08° or 9.2 km at the equator). Time period and grain: Present (2000–2014) and future (2040–2050 and 2090–2100) environmental conditions based on monthly averages. Major taxa and level of measurement: Marine biodiversity associated with sea surface and epibenthic habitats. Software format: ASCII and TIFF grid formats for geographical information systems and a package of functions developed for R software.
Journal Article
The shadow behind the stars
Chloe, Serena, and Xinot--the Fates--live on a secluded island spinning, measuring, and cutting the threads of human life, but when Aglaia, a mortal, finds them, Chloe must try to keep her sisters from getting attached to the girl and involved in a dark fate that could unravel the world.
Book
One-Step Sparse Estimates in Nonconcave Penalized Likelihood Models
Fan and Li propose a family of variable selection methods via penalized likelihood using concave penalty functions. The nonconcave penalized likelihood estimators enjoy the oracle properties, but maximizing the penalized likelihood function is computationally challenging, because the objective function is nondifferentiable and nonconcave. In this article, we propose a new unified algorithm based on the local linear approximation (LLA) for maximizing the penalized likelihood for a broad class of concave penalty functions. Convergence and other theoretical properties of the LLA algorithm are established. A distinguished feature of the LLA algorithm is that at each LLA step, the LLA estimator can naturally adopt a sparse representation. Thus, we suggest using the one-step LLA estimator from the LLA algorithm as the final estimates. Statistically, we show that if the regularization parameter is appropriately chosen, the one-step LLA estimates enjoy the oracle properties with good initial estimators. Computationally, the one-step LLA estimation methods dramatically reduce the computational cost in maximizing the nonconcave penalized likelihood. We conduct some Monte Carlo simulation to assess the finite sample performance of the one-step sparse estimation methods. The results are very encouraging.
Journal Article
The curse of Deadman's Forest
According to prophecy, a trip through the magical portal near the Dover, England, orphanage where Ian and Theo live will bring them to the third Oracle, a child with extraordinary healing powers to help defeat a great evil, but it will also lead to Ian's death.
Book
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 L₁-penalty, called weighted robust Lasso (WR-Lasso), in which weights are introduced to ameliorate the bias problem induced by the L₁-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 L₁-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.
Journal Article