P-values for high-dimensional regression

Assigning significance in high-dimensional regression is challenging. Most computationally efficient selection algorithms cannot guard against inclusion of noise variables. Asymptotically valid p-values are not available. An exception is a recent proposal by Wasserman and Roeder (2008) which splits the data into two parts. The number of variables is then reduced to a manageable size using the first split, while classical variable selection techniques can be applied to the remaining variables, using the data from the second split. This yields asymptotic error control under minimal conditions. It involves, however, a one-time random split of the data. Results are sensitive to this arbitrary choice: it amounts to a `p-value lottery' and makes it difficult to reproduce results. Here, we show that inference across multiple random splits can be aggregated, while keeping asymptotic control over the inclusion of noise variables. We show that the resulting p-values can be used for control of both family-wise error (FWER) and false discovery rate (FDR). In addition, the proposed aggregation is shown to improve power while reducing the number of falsely selected variables substantially.Comment: 25 pages, 4 figure

Toward a unified theory of sparse dimensionality reduction in Euclidean space

Let $\Phi\in\mathbb{R}^{m\times n}$ be a sparse Johnson-Lindenstrauss transform [KN14] with $s$ non-zeroes per column. For a subset $T$ of the unit sphere, $\varepsilon\in(0,1/2)$ given, we study settings for $m,s$ required to ensure $$\mathop{\mathbb{E}}_\Phi \sup_{x\in T} \left|\|\Phi x\|_2^2 - 1 \right| < \varepsilon ,$$ i.e. so that $\Phi$ preserves the norm of every $x\in T$ simultaneously and multiplicatively up to $1+\varepsilon$. We introduce a new complexity parameter, which depends on the geometry of $T$, and show that it suffices to choose $s$ and $m$ such that this parameter is small. Our result is a sparse analog of Gordon's theorem, which was concerned with a dense $\Phi$ having i.i.d. Gaussian entries. We qualitatively unify several results related to the Johnson-Lindenstrauss lemma, subspace embeddings, and Fourier-based restricted isometries. Our work also implies new results in using the sparse Johnson-Lindenstrauss transform in numerical linear algebra, classical and model-based compressed sensing, manifold learning, and constrained least squares problems such as the Lasso

ACS Applied Materials & Interfaces

Key parameters that influence the specific energy of electrochemical double-layer capacitors (EDLCs) are the double-layer capacitance and the operating potential of the cell. The operating potential of the cell is generally limited by the electrochemical window of the electrolyte solution, that is, the range of applied voltages within which the electrolyte or solvent is not reduced or oxidized. Ionic liquids are of interest as electrolytes for EDLCs because they offer relatively wide potential windows. Here, we provide a systematic study of the influence of the physical properties of ionic liquid electrolytes on the electrochemical stability and electrochemical performance (double-layer capacitance, specific energy) of EDLCs that employ a mesoporous carbon model electrode with uniform, highly interconnected mesopores (3DOm carbon). Several ionic liquids with structurally diverse anions (tetrafluoroborate, trifluoromethanesulfonate, trifluoromethanesulfonimide) and cations (imidazolium, ammonium, pyridinium, piperidinium, and pyrrolidinium) were investigated. We show that the cation size has a significant effect on the electrolyte viscosity and conductivity, as well as the capacitance of EDLCs. Imidazolium- and pyridinium-based ionic liquids provide the highest cell capacitance, and ammonium-based ionic liquids offer potential windows much larger than imidazolium and pyridinium ionic liquids. Increasing the chain length of the alkyl substituents in 1-alkyl-3-methylimidazolium trifluoromethanesulfonimide does not widen the potential window of the ionic liquid. We identified the ionic liquids that maximize the specific energies of EDLCs through the combined effects of their potential windows and the double-layer capacitance. The highest specific energies are obtained with ionic liquid electrolytes that possess moderate electrochemical stability, small ionic volumes, low viscosity, and hence high conductivity, the best performing ionic liquid tested being 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide

Patient well‐being after general anaesthesia: a prospective, randomized, controlled multi‐centre trial comparing intravenous and inhalation anaesthesia

Background. The aim of this study was to assess postoperative patient well‐being after total i.v. anaesthesia compared with inhalation anaesthesia by means of validated psychometric tests. Methods. With ethics committee approval, 305 patients undergoing minor elective gynaecologic or orthopaedic interventions were assigned randomly to total i.v. anaesthesia using propofol or inhalation anaesthesia using sevoflurane. The primary outcome measurement was the actual mental state 90 min and 24 h after anaesthesia assessed by a blinded observer using the Adjective Mood Scale (AMS) and the State‐Trait‐Anxiety Inventory (STAI). Incidence of postoperative nausea and vomiting (PONV) and postoperative pain level were determined by Visual Analogue Scale (VAS) 90 min and 24 h after anaesthesia (secondary outcome measurements). Patient satisfaction was evaluated using a VAS 24 h after anaesthesia. Results. The AMS and STAI scores were significantly better 90 min after total i.v. anaesthesia compared with inhalation anaesthesia (P=0.02, P=0.05, respectively), but equal 24 h after both anaesthetic techniques (P=0.90, P=0.78, respectively); patient satisfaction was comparable (P=0.26). Postoperative pain was comparable in both groups 90 min and 24 h after anaesthesia (P=0.11, P=0.12, respectively). The incidence of postoperative nausea was reduced after total i.v. compared with inhalation anaesthesia at 90 min (7 vs 35%, P<0.001), and 24 h (33 vs 52%, P=0.001). Conclusion. Total i.v. anaesthesia improves early postoperative patient well‐being and reduces the incidence of PONV. Br J Anaesth 2003; 91: 631-

Handwritten digit recognition by bio-inspired hierarchical networks

The human brain processes information showing learning and prediction abilities but the underlying neuronal mechanisms still remain unknown. Recently, many studies prove that neuronal networks are able of both generalizations and associations of sensory inputs. In this paper, following a set of neurophysiological evidences, we propose a learning framework with a strong biological plausibility that mimics prominent functions of cortical circuitries. We developed the Inductive Conceptual Network (ICN), that is a hierarchical bio-inspired network, able to learn invariant patterns by Variable-order Markov Models implemented in its nodes. The outputs of the top-most node of ICN hierarchy, representing the highest input generalization, allow for automatic classification of inputs. We found that the ICN clusterized MNIST images with an error of 5.73% and USPS images with an error of 12.56%

Robust high-dimensional precision matrix estimation

The dependency structure of multivariate data can be analyzed using the covariance matrix $\Sigma$. In many fields the precision matrix $\Sigma^{-1}$ is even more informative. As the sample covariance estimator is singular in high-dimensions, it cannot be used to obtain a precision matrix estimator. A popular high-dimensional estimator is the graphical lasso, but it lacks robustness. We consider the high-dimensional independent contamination model. Here, even a small percentage of contaminated cells in the data matrix may lead to a high percentage of contaminated rows. Downweighting entire observations, which is done by traditional robust procedures, would then results in a loss of information. In this paper, we formally prove that replacing the sample covariance matrix in the graphical lasso with an elementwise robust covariance matrix leads to an elementwise robust, sparse precision matrix estimator computable in high-dimensions. Examples of such elementwise robust covariance estimators are given. The final precision matrix estimator is positive definite, has a high breakdown point under elementwise contamination and can be computed fast

Noisy Monte Carlo: Convergence of Markov chains with approximate transition kernels

Monte Carlo algorithms often aim to draw from a distribution $\pi$ by simulating a Markov chain with transition kernel $P$ such that $\pi$ is invariant under $P$. However, there are many situations for which it is impractical or impossible to draw from the transition kernel $P$. For instance, this is the case with massive datasets, where is it prohibitively expensive to calculate the likelihood and is also the case for intractable likelihood models arising from, for example, Gibbs random fields, such as those found in spatial statistics and network analysis. A natural approach in these cases is to replace $P$ by an approximation $\hat{P}$. Using theory from the stability of Markov chains we explore a variety of situations where it is possible to quantify how 'close' the chain given by the transition kernel $\hat{P}$ is to the chain given by $P$. We apply these results to several examples from spatial statistics and network analysis.Comment: This version: results extended to non-uniformly ergodic Markov chain

On the combination of omics data for prediction of binary outcomes

Enrichment of predictive models with new biomolecular markers is an important task in high-dimensional omic applications. Increasingly, clinical studies include several sets of such omics markers available for each patient, measuring different levels of biological variation. As a result, one of the main challenges in predictive research is the integration of different sources of omic biomarkers for the prediction of health traits. We review several approaches for the combination of omic markers in the context of binary outcome prediction, all based on double cross-validation and regularized regression models. We evaluate their performance in terms of calibration and discrimination and we compare their performance with respect to single-omic source predictions. We illustrate the methods through the analysis of two real datasets. On the one hand, we consider the combination of two fractions of proteomic mass spectrometry for the calibration of a diagnostic rule for the detection of early-stage breast cancer. On the other hand, we consider transcriptomics and metabolomics as predictors of obesity using data from the Dietary, Lifestyle, and Genetic determinants of Obesity and Metabolic syndrome (DILGOM) study, a population-based cohort, from Finland