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Linear optimization is a powerful modeling method for discovering the best solution to a problem among a set of available alternatives. It is one of today's most important branches of mathematics and computer science-and also a surprisingly rich medium for creating breathtaking works of art. Opt Art takes readers on an entertaining tour of linear optimization and its applications, showing along the way how it can be used to design visual art. Robert Bosch provides a lively and accessible introduction to the geometric, algebraic, and algorithmic foundations of optimization. He presents classical applications, such as the legendary Traveling Salesman Problem, and shows how to adapt them to make optimization art-opt art. Each chapter in this marvelously illustrated book begins with a problem or puzzle and demonstrates how the solution can be derived using a host of artistic methods and media, including 3D printing, laser cutting, and computer-controlled machining. Bosch focuses on mathematical modeling throughout-converting a problem into a workable mathematical form, solving it using optimization techniques, and examining the results, which can take the form of mosaics, line drawings, and even sculpture. All you need is some high-school algebra, geometry, and calculus to follow along. Featuring more than a hundred illustrations and photos of Bosch's own art, Opt Art demonstrates how mathematics and computing can be used to create beauty and express emotion through amazing works of art.

Many contemporary large-scale applications involve building interpretable models linking a large set of potential covariates to a response in a non-linear fashion, such as when the response is binary. Although this modelling problem has been extensively studied, it remains unclear how to control the fraction of false discoveries effectively even in high dimensional logistic regression, not to mention general high dimensional non-linear models. To address such a practical problem, we propose a new framework of ‘model-X’ knockoffs, which reads from a different perspective the knockoff procedure that was originally designed for controlling the false discovery rate in linear models. Whereas the knockoffs procedure is constrained to homoscedastic linear models with n ⩾ p, the key innovation here is that model-X knockoffs provide valid inference from finite samples in settings in which the conditional distribution of the response is arbitrary and completely unknown. Furthermore, this holds no matter the number of covariates. Correct inference in such a broad setting is achieved by constructing knockoff variables probabilistically instead of geometrically. To do this, our approach requires that the covariates are random (independent and identically distributed rows) with a distribution that is known, although we provide preliminary experimental evidence that our procedure is robust to unknown or estimated distributions. To our knowledge, no other procedure solves the controlled variable selection problem in such generality but, in the restricted settings where competitors exist, we demonstrate the superior power of knockoffs through simulations. Finally, we apply our procedure to data from a case–control study of Crohn’s disease in the UK, making twice as many discoveries as the original analysis of the same data.

\"David A. Freedman presents here a definitive synthesis of his approach to causal inference in the social sciences. He explores the foundations and limitations of statistical modeling, illustrating basic arguments with examples from political science, public policy, law, and epidemiology. Freedman maintains that many new technical approaches to statistical modeling constitute not progress, but regress. Instead, he advocates a 'shoe leather' methodology, which exploits natural variation to mitigate confounding and relies on intimate knowledge of the subject matter to develop meticulous research designs and eliminate rival explanations. When Freedman first enunciated this position, he was met with scepticism, in part because it was hard to believe that a mathematical statistician of his stature would favor 'low-tech' approaches. But the tide is turning. Many social scientists now agree that statistical technique cannot substitute for good research design and subject matter knowledge. This book offers an integrated presentation of Freedman's views\"--Provided by publisher.

We propose a general method for constructing confidence intervals and statistical tests for single or low-dimensional components of a large parameter vector in a high-dimensional model. It can be easily adjusted for multiplicity taking dependence among tests into account. For linear models, our method is essentially the same as in Zhang and Zhang [J. R. Stat. Soc. Ser. B Stat. Methodol. 76 (2014) 217-242]: we analyze its asymptotic properties and establish its asymptotic optimality in terms of semiparametric efficiency. Our method naturally extends to generalized linear models with convex loss functions. We develop the corresponding theory which includes a careful analysis for Gaussian, sub-Gaussian and bounded correlated designs.

Recent work by Reiss and Ogden provides a theoretical basis for sometimes preferring restricted maximum likelihood (REML) to generalized cross-validation (GCV) for smoothing parameter selection in semiparametric regression. However, existing REML or marginal likelihood (ML) based methods for semiparametric generalized linear models (GLMs) use iterative REML or ML estimation of the smoothing parameters of working linear approximations to the GLM. Such indirect schemes need not converge and fail to do so in a non-negligible proportion of practical analyses. By contrast, very reliable prediction error criteria smoothing parameter selection methods are available, based on direct optimization of GCV, or related criteria, for the GLM itself. Since such methods directly optimize properly defined functions of the smoothing parameters, they have much more reliable convergence properties. The paper develops the first such method for REML or ML estimation of smoothing parameters. A Laplace approximation is used to obtain an approximate REML or ML for any GLM, which is suitable for efficient direct optimization. This REML or ML criterion requires that Newton-Raphson iteration, rather than Fisher scoring, be used for GLM fitting, and a computationally stable approach to this is proposed. The REML or ML criterion itself is optimized by a Newton method, with the derivatives required obtained by a mixture of implicit differentiation and direct methods. The method will cope with numerical rank deficiency in the fitted model and in fact provides a slight improvement in numerical robustness on the earlier method of Wood for prediction error criteria based smoothness selection. Simulation results suggest that the new REML and ML methods offer some improvement in mean-square error performance relative to GCV or Akaike's information criterion in most cases, without the small number of severe undersmoothing failures to which Akaike's information criterion and GCV are prone. This is achieved at the same computational cost as GCV or Akaike's information criterion. The new approach also eliminates the convergence failures of previous REML- or ML-based approaches for penalized GLMs and usually has lower computational cost than these alternatives. Example applications are presented in adaptive smoothing, scalar on function regression and generalized additive model selection.

Background Quantitative trait locus (QTL) mapping in genetic data often involves analysis of correlated observations, which need to be accounted for to avoid false association signals. This is commonly performed by modeling such correlations as random effects in linear mixed models (LMMs). The R package lme4 is a well-established tool that implements major LMM features using sparse matrix methods; however, it is not fully adapted for QTL mapping association and linkage studies. In particular, two LMM features are lacking in the base version of lme4 : the definition of random effects by custom covariance matrices; and parameter constraints, which are essential in advanced QTL models. Apart from applications in linkage studies of related individuals, such functionalities are of high interest for association studies in situations where multiple covariance matrices need to be modeled, a scenario not covered by many genome-wide association study (GWAS) software. Results To address the aforementioned limitations, we developed a new R package lme4qtl as an extension of lme4 . First, lme4qtl contributes new models for genetic studies within a single tool integrated with lme4 and its companion packages. Second, lme4qtl offers a flexible framework for scenarios with multiple levels of relatedness and becomes efficient when covariance matrices are sparse. We showed the value of our package using real family-based data in the Genetic Analysis of Idiopathic Thrombophilia 2 (GAIT2) project. Conclusions Our software lme4qtl enables QTL mapping models with a versatile structure of random effects and efficient computation for sparse covariances. lme4qtl is available at https://github.com/variani/lme4qtl .

\"Mechanism design is an analytical framework for thinking clearly and carefully about what exactly a given institution can achieve when the information necessary to make decisions is dispersed and privately held. This analysis provides an account of the underlying mathematics of mechanism design based on linear programming. Three advantages characterize the approach. The first is simplicity: arguments based on linear programming are both elementary and transparent. The second is unity: the machinery of linear programming provides a way to unify results from disparate areas of mechanism design. The third is reach: the technique offers the ability to solve problems that appear to be beyond solutions offered by traditional methods. No claim is made that the approach advocated should supplant traditional mathematical machinery. Rather, the approach represents an addition to the tools of the economic theorist who proposes to understand economic phenomena through the lens of mechanism design\"-- Provided by publisher.

In this wIn this work, we explore finite-dimensional linear representations of nonlinear dynamical systems by restricting the Koopman operator to an invariant subspace spanned by specially chosen observable functions. The Koopman operator is an infinite-dimensional linear operator that evolves functions of the state of a dynamical system. Dominant terms in the Koopman expansion are typically computed using dynamic mode decomposition (DMD). DMD uses linear measurements of the state variables, and it has recently been shown that this may be too restrictive for nonlinear systems. Choosing the right nonlinear observable functions to form an invariant subspace where it is possible to obtain linear reduced-order models, especially those that are useful for control, is an open challenge. Here, we investigate the choice of observable functions for Koopman analysis that enable the use of optimal linear control techniques on nonlinear problems. First, to include a cost on the state of the system, as in linear quadratic regulator (LQR) control, it is helpful to include these states in the observable subspace, as in DMD. However, we find that this is only possible when there is a single isolated fixed point, as systems with multiple fixed points or more complicated attractors are not globally topologically conjugate to a finite-dimensional linear system, and cannot be represented by a finite-dimensional linear Koopman subspace that includes the state. We then present a data-driven strategy to identify relevant observable functions for Koopman analysis by leveraging a new algorithm to determine relevant terms in a dynamical system by ℓ1-regularized regression of the data in a nonlinear function space; we also show how this algorithm is related to DMD. Finally, we demonstrate the usefulness of nonlinear observable subspaces in the design of Koopman operator optimal control laws for fully nonlinear systems using techniques from linear optimal control.ork, we explore finite-dimensional linear representations of nonlinear dynamical systems by restricting the Koopman operator to an invariant subspace spanned by specially chosen observable functions. The Koopman operator is an infinite-dimensional linear operator that evolves functions of the state of a dynamical system. Dominant terms in the Koopman expansion are typically computed using dynamic mode decomposition (DMD). DMD uses linear measurements of the state variables, and it has recently been shown that this may be too restrictive for nonlinear systems. Choosing the right nonlinear observable functions to form an invariant subspace where it is possible to obtain linear reduced-order models, especially those that are useful for control, is an open challenge. Here, we investigate the choice of observable functions for Koopman analysis that enable the use of optimal linear control techniques on nonlinear problems. First, to include a cost on the state of the system, as in linear quadratic regulator (LQR) control, it is helpful to include these states in the observable subspace, as in DMD. However, we find that this is only possible when there is a single isolated fixed point, as systems with multiple fixed points or more complicated attractors are not globally topologically conjugate to a finite-dimensional linear system, and cannot be represented by a finite-dimensional linear Koopman subspace that includes the state. We then present a data-driven strategy to identify relevant observable functions for Koopman analysis by leveraging a new algorithm to determine relevant terms in a dynamical system by ℓ1-regularized regression of the data in a nonlinear function space; we also show how this algorithm is related to DMD. Finally, we demonstrate the usefulness of nonlinear observable subspaces in the design of Koopman operator optimal control laws for fully nonlinear systems using techniques from linear optimal control.