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Peter Bühlmann

Publications and source records attributed to Peter Bühlmann.

124 records · Page 7Linked to original sources
arXiv2010

Estimation for High-Dimensional Linear Mixed-Effects Models Using $\ell_1$-Penalization

We propose an $\ell_1$-penalized estimation procedure for high-dimensional linear mixed-effects models. The models are useful whenever there is a grouping structure among high-dimensional observations, i.e. for clustered data. We prove a consistency and an oracle optimality result and we develop an algorithm with provable numerical convergence. Furthermore, we demonstrate the performance of the method on simulated and a real high-dimensional data set.

stat.ME↗
arXiv2009

High-dimensional additive modeling

We propose a new sparsity-smoothness penalty for high-dimensional generalized additive models. The combination of sparsity and smoothness is crucial for mathematical theory as well as performance for finite-sample data. We present a computationally efficient algorithm, with provable numerical convergence properties, for optimizing the penalized likelihood. Furthermore, we provide oracle results which yield asymptotic optimality of our estimator for high dimensional but sparse additive models. Finally, an adaptive version of our sparsity-smoothness penalized approach yields large additional performance gains.

stat.ML↗
arXiv2009

Decomposition and Model Selection for Large Contingency Tables

Large contingency tables summarizing categorical variables arise in many areas. For example in biology when a large number of biomarkers are cross-tabulated according to their discrete expression level. Interactions of the variables are generally studied with log-linear models and the structure of a log-linear model can be visually represented by a graph from which the conditional independence structure can then be read off. However, since the number of parameters in a saturated model grows exponentially in the number of variables, this generally comes with a heavy burden as far as computational power is concerned. If we restrict ourselves to models of lower order interactions or other sparse structures we face similar problems as the number of cells remains unchanged. We therefore present a divide-and-conquer approach, where we first divide the problem into several lower-dimensional problems and then combine these to form a global solution. Our methodology is computationally feasible for log-linear interaction modeling with many categorical variables each or some of them having many categories. We demonstrate the proposed method on simulated data and apply it to a bio-medical problem in cancer research.

stat.ME↗
arXiv2009

High dimensional sparse covariance estimation via directed acyclic graphs

We present a graph-based technique for estimating sparse covariance matrices and their inverses from high-dimensional data. The method is based on learning a directed acyclic graph (DAG) and estimating parameters of a multivariate Gaussian distribution based on a DAG. For inferring the underlying DAG we use the PC-algorithm and for estimating the DAG-based covariance matrix and its inverse, we use a Cholesky decomposition approach which provides a positive (semi-)definite sparse estimate. We present a consistency result in the high-dimensional framework and we compare our method with the Glasso for simulated and real data.

stat.ME↗
arXiv2009

Variable selection in high-dimensional linear models: partially faithful distributions and the PC-simple algorithm

We consider variable selection in high-dimensional linear models where the number of covariates greatly exceeds the sample size. We introduce the new concept of partial faithfulness and use it to infer associations between the covariates and the response. Under partial faithfulness, we develop a simplified version of the PC algorithm (Spirtes et al., 2000), the PC-simple algorithm, which is computationally feasible even with thousands of covariates and provides consistent variable selection under conditions on the random design matrix that are of a different nature than coherence conditions for penalty-based approaches like the Lasso. Simulations and application to real data show that our method is competitive compared to penalty-based approaches. We provide an efficient implementation of the algorithm in the R-package pcalg.

stat.ME↗
arXiv2009

On the conditions used to prove oracle results for the Lasso

Oracle inequalities and variable selection properties for the Lasso in linear models have been established under a variety of different assumptions on the design matrix. We show in this paper how the different conditions and concepts relate to each other. The restricted eigenvalue condition (Bickel et al., 2009) or the slightly weaker compatibility condition (van de Geer, 2007) are sufficient for oracle results. We argue that both these conditions allow for a fairly general class of design matrices. Hence, optimality of the Lasso for prediction and estimation holds for more general situations than what it appears from coherence (Bunea et al, 2007b,c) or restricted isometry (Candes and Tao, 2005) assumptions.

math.ST↗
arXiv2009

Estimating high-dimensional intervention effects from observational data

We assume that we have observational data generated from an unknown underlying directed acyclic graph (DAG) model. A DAG is typically not identifiable from observational data, but it is possible to consistently estimate the equivalence class of a DAG. Moreover, for any given DAG, causal effects can be estimated using intervention calculus. In this paper, we combine these two parts. For each DAG in the estimated equivalence class, we use intervention calculus to estimate the causal effects of the covariates on the response. This yields a collection of estimated causal effects for each covariate. We show that the distinct values in this set can be consistently estimated by an algorithm that uses only local information of the graph. This local approach is computationally fast and feasible in high-dimensional problems. We propose to use summary measures of the set of possible causal effects to determine variable importance. In particular, we use the minimum absolute value of this set, since that is a lower bound on the size of the causal effect. We demonstrate the merits of our methods in a simulation study and on a data set about riboflavin production.

stat.ME↗
arXiv2009

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.

stat.ME↗
arXiv2009

Adaptive Lasso for High Dimensional Regression and Gaussian Graphical Modeling

We show that the two-stage adaptive Lasso procedure (Zou, 2006) is consistent for high-dimensional model selection in linear and Gaussian graphical models. Our conditions for consistency cover more general situations than those accomplished in previous work: we prove that restricted eigenvalue conditions (Bickel et al., 2008) are also sufficient for sparse structure estimation.

math.ST↗
arXiv2008

Discussion of: Treelets--An adaptive multi-scale basis for sparse unordered data

We congratulate Lee, Nadler and Wasserman (henceforth LNW) on a very interesting paper on new methodology and supporting theory [arXiv:0707.0481]. Treelets seem to tackle two important problems of modern data analysis at once. For datasets with many variables, treelets give powerful predictions even if variables are highly correlated and redundant. Maybe more importantly, interpretation of the results is intuitive. Useful insights about relevant groups of variables can be gained. Our comments and questions include: (i) Could the success of treelets be replicated by a combination of hierarchical clustering and PCA? (ii) When choosing a suitable basis, treelets seem to be largely an unsupervised method. Could the results be even more interpretable and powerful if treelets would take into account some supervised response variable? (iii) Interpretability of the result hinges on the sparsity of the final basis. Do we expect that the selected groups of variables will always be sufficiently small to be amenable for interpretation?

stat.AP↗
arXiv2008

Boosting Algorithms: Regularization, Prediction and Model Fitting

We present a statistical perspective on boosting. Special emphasis is given to estimating potentially complex parametric or nonparametric models, including generalized linear and additive models as well as regression models for survival analysis. Concepts of degrees of freedom and corresponding Akaike or Bayesian information criteria, particularly useful for regularization and variable selection in high-dimensional covariate spaces, are discussed as well. The practical aspects of boosting procedures for fitting statistical models are illustrated by means of the dedicated open-source software package mboost. This package implements functions which can be used for model fitting, prediction and variable selection. It is flexible, allowing for the implementation of new boosting algorithms optimizing user-specified loss functions.

stat.ME↗
arXiv2007

Smoothing $\ell_1$-penalized estimators for high-dimensional time-course data

When a series of (related) linear models has to be estimated it is often appropriate to combine the different data-sets to construct more efficient estimators. We use $\ell_1$-penalized estimators like the Lasso or the Adaptive Lasso which can simultaneously do parameter estimation and model selection. We show that for a time-course of high-dimensional linear models the convergence rates of the Lasso and of the Adaptive Lasso can be improved by combining the different time-points in a suitable way. Moreover, the Adaptive Lasso still enjoys oracle properties and consistent variable selection. The finite sample properties of the proposed methods are illustrated on simulated data and on a real problem of motif finding in DNA sequences.

math.ST↗
arXiv2006

High-dimensional graphs and variable selection with the Lasso

The pattern of zero entries in the inverse covariance matrix of a multivariate normal distribution corresponds to conditional independence restrictions between variables. Covariance selection aims at estimating those structural zeros from data. We show that neighborhood selection with the Lasso is a computationally attractive alternative to standard covariance selection for sparse high-dimensional graphs. Neighborhood selection estimates the conditional independence restrictions separately for each node in the graph and is hence equivalent to variable selection for Gaussian linear models. We show that the proposed neighborhood selection scheme is consistent for sparse high-dimensional graphs. Consistency hinges on the choice of the penalty parameter. The oracle value for optimal prediction does not lead to a consistent neighborhood estimate. Controlling instead the probability of falsely joining some distinct connectivity components of the graph, consistent estimation for sparse graphs is achieved (with exponential rates), even when the number of variables grows as the number of observations raised to an arbitrary power.

math.ST↗
arXiv2006

Boosting for high-dimensional linear models

We prove that boosting with the squared error loss, $L_2$Boosting, is consistent for very high-dimensional linear models, where the number of predictor variables is allowed to grow essentially as fast as $O$(exp(sample size)), assuming that the true underlying regression function is sparse in terms of the $\ell_1$-norm of the regression coefficients. In the language of signal processing, this means consistency for de-noising using a strongly overcomplete dictionary if the underlying signal is sparse in terms of the $\ell_1$-norm. We also propose here an $\mathit{AIC}$-based method for tuning, namely for choosing the number of boosting iterations. This makes $L_2$Boosting computationally attractive since it is not required to run the algorithm multiple times for cross-validation as commonly used so far. We demonstrate $L_2$Boosting for simulated data, in particular where the predictor dimension is large in comparison to sample size, and for a difficult tumor-classification problem with gene expression microarray data.

math.ST↗