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shape-constrained Symbolic regression-Improving Extrapolation with Prior Knowledge

EVOLUTIONARY COMPUTATION 2022年第1期30卷 75-98页

作者： Kronberger, G. de Franca, F. O. Burlacu, B. Haider, C. Kommenda, M. Univ Appl Sci Upper Austria Josef Ressel Ctr Symbol Regress Softwarepk 11 A-4232 Hagenberg Austria Fed Univ ABC Ctr Math Computat & Cognit CMCC Heurist Anal & Learning Lab HAL Santo Andre SP Brazil

We investigate the addition of constraints on the function image and its derivatives for the incorporation of prior knowledge in symbolic regression. The approach is called shape-constrained symbolic regression and allows us to enforce, for example, monotonicity of the function over selected inputs. The aim is to find models which conform to expected behavior and which have improved extrapolation capabilities. We demonstrate the feasibility of the idea and propose and compare two evolutionary algorithms for shape-constrained symbolic regression: (i) an extension of tree-based genetic programming which discards infeasible solutions in the selection step, and (ii) a two-population evolutionary algorithm that separates the feasible from the infeasible solutions. In both algorithms we use interval arithmetic to approximate bounds for models and their partial derivatives. The algorithms are tested on a set of 19 synthetic and four real-world regression problems. Both algorithms are able to identify models which conform to shape constraints which is not the case for the unmodified symbolic regression algorithms. However, the predictive accuracy of models with constraints is worse on the training set and the test set. shape-constrained polynomial regression produces the best results for the test set but also significantly larger models.

关键词： Symbolic regression genetic programming shape-constrained regression

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shape-constrained multi-objective genetic programming for symbolic regression

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APPLIED SOFT COMPUTING 2023年 132卷

作者： Haider, C. de Franca, F. O. Burlacu, B. Kronberger, G. Univ Appl Sci Upper Austria Josef Ressel Ctr Symbol Regress Softwarepark 11 A-4232 Hagenberg Austria Fed Univ ABC Ctr Math Computat & Cognit CMCC Heurist Anal & Learning Lab HAL Santo Andre Brazil

We describe and analyze algorithms for shape-constrained symbolic regression, which allow the inclusion of prior knowledge about the shape of the regression function. This is relevant in many areas of engineering - in particular, when data-driven models, which are based on data of measurements must exhibit certain properties (e.g. positivity, monotonicity, or convexity/concavity). To satisfy these properties, we have extended multi-objective algorithms with shape constraints. A soft-penalty approach is used to minimize both the constraint violations and the prediction error. We use the non -dominated sorting genetic algorithm (NSGA-II) as well as the multi-objective evolutionary algorithm based on decomposition (MOEA/D). The algorithms are tested on a set of models from physics textbooks and compared against previous results achieved with single objective algorithms. Further, we generated out-of-domain samples to test the extrapolation behavior using shape constraints and added a different level of noise on the training data to verify if shape constraints can still help maintain the prediction errors to a minimum and generate valid models. The results showed that the multi-objective algorithms were capable of finding mostly valid models, also when using a soft-penalty approach. Further, we investigated that NSGA-II achieved the best overall ranks on high noise instances.(c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://***/licenses/by/4.0/).

关键词： Genetic Programming Multi-objective optimization shape-constrained regression Symbolic regression

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Distribution-free properties of isotonic regression

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ELECTRONIC JOURNAL OF STATISTICS 2019年第2期13卷 3243-3253页

作者： Soloff, Jake A. Guntuboyina, Adityanand Pitman, Jim Univ Calif Berkeley Dept Stat Berkeley CA 94720 USA

It is well known that the isotonic least squares estimator is characterized as the derivative of the greatest convex minorant of a random walk. Provided the walk has exchangeable increments, we prove that the slopes of the greatest convex minorant are distributed as order statistics of the running averages. This result implies an exact non-asymptotic formula for the squared error risk of least squares in homoscedastic isotonic regression when the true sequence is constant that holds for every exchangeable error distribution.

关键词： shape-constrained regression statistical dimension convex minorant fluctuation theory quantiles of stochastic processes

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Semi-nonparametric estimation of the call-option price surface under strike and time-to-expiry no-arbitrage constraints

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JOURNAL OF ECONOMETRICS 2015年第2期184卷 242-261页

作者： Fengler, Matthias R. Hin, Lin-Yee Univ St Gallen Dept Econ CH-9000 St Gallen Switzerland Curtin Univ Dept Math & Stat Bentley WA 6102 Australia

We suggest a semi-nonparametric estimator for the call-option price surface. The estimator is a bivariate tensor-product B-spline. To enforce no-arbitrage constraints across strikes and expiry dates, we establish sufficient no-arbitrage conditions on the control net of the B-spline surface. The conditions are linear and therefore allow for an implementation of the estimator by means of standard quadratic programming techniques. The consistency of the estimator is proved. By means of simulations, we explore the statistical efficiency benefits that are associated with estimating option price surfaces and state-price densities under the full set of no-arbitrage constraints. We estimate a call-option price surface, families of first-order strike derivatives, and state-price densities for S&P 500 option data. (C) 2014 Elsevier B.V. All rights reserved.

关键词： B-splines No-arbitrage constraints Option pricing function Semi-nonparametric estimation shape-constrained regression State-price density

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Nonparametric, tuning-free estimation of S-shaped functions

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JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY 2022年第4期84卷 1324-1352页

作者： Feng, Oliver Y. Chen, Yining Han, Qiyang Carroll, Raymond J. Samworth, Richard J. Univ Cambridge Stat Lab Cambridge England London Sch Econ & Polit Sci Dept Stat London England Rutgers State Univ Dept Stat Piscataway NJ USA Texas A&M Univ Dept Stat College Stn TX USA Univ Technol Sydney Sch Math & Phys Sci Sydney NSW Australia

We consider the nonparametric estimation of an S-shaped regression function. The least squares estimator provides a very natural, tuning-free approach, but results in a non-convex optimization problem, since the inflection point is unknown. We show that the estimator may nevertheless be regarded as a projection onto a finite union of convex cones, which allows us to propose a mixed primal-dual bases algorithm for its efficient, sequential computation. After developing a projection framework that demonstrates the consistency and robustness to misspecification of the estimator, our main theoretical results provide sharp oracle inequalities that yield worst-case and adaptive risk bounds for the estimation of the regression function, as well as a rate of convergence for the estimation of the inflection point. These results reveal not only that the estimator achieves the minimax optimal rate of convergence for both the estimation of the regression function and its inflection point (up to a logarithmic factor in the latter case), but also that it is able to achieve an almost-parametric rate when the true regression function is piecewise affine with not too many affine pieces. Simulations and a real data application to air pollution modelling also confirm the desirable finite-sample properties of the estimator, and our algorithm is implemented in the R package Sshaped.

关键词： sequential algorithm shape-constrained regression S-shaped functions

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Semiparametric Bayesian Inference for Local Extrema of Functions in the Presence of Noise

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JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION 2024年第548期119卷 3127-3140页

作者： Li, Meng Liu, Zejian Yu, Cheng-Han Vannucci, Marina Rice Univ Dept Stat Houston TX 77005 USA Marquette Univ Dept Math & Stat Sci Milwaukee WI USA

There is a wide range of applications where the local extrema of a function are the key quantity of interest. However, there is surprisingly little work on methods to infer local extrema with uncertainty quantification in the presence of noise. By viewing the function as an infinite-dimensional nuisance parameter, a semiparametric formulation of this problem poses daunting challenges, both methodologically and theoretically, as (i) the number of local extrema may be unknown, and (ii) the induced shape constraints associated with local extrema are highly irregular. In this article, we build upon a derivative-constrained Gaussian process prior recently proposed by Yu et al. to derive what we call an encompassing approach that indexes possibly multiple local extrema by a single parameter. We provide closed-form characterization of the posterior distribution and study its large sample behavior under this unconventional encompassing regime. We show that the posterior measure converges to a mixture of Gaussians with the number of components matching the underlying truth, leading to posterior exploration that accounts for multi-modality. Point and interval estimates of local extrema with frequentist properties are also provided. The encompassing approach leads to a remarkably simple, fast semiparametric approach for inference on local extrema. We illustrate the method through simulations and a real data application to event-related potential analysis. Supplementary materials for this article are available online.

关键词： Bernstein-von Mises theorem Gaussian process Local extrema Semiparametric shape-constrained regression

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A mathematical optimization approach to shape-constrained generalized additive models

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EXPERT SYSTEMS WITH APPLICATIONS 2024年第PartC期255卷

作者： Navarro-Garcia, Manuel Guerrero, Vanesa Durban, Maria Univ Carlos III Madrid Dept Stat Calle Madrid 126 Getafe 28901 Spain Komorebi AI Technol Ave Gen Peron 26Planta 4a Madrid 28020 Spain

The vast amount of data generated nowadays demands innovative and flexible techniques that allow to accommodate expert knowledge and help in decision-making. In this work, we address the problem of estimating a generalized additive regression model in which conditions about the sign, monotonicity, or curvature need to be satisfied by the functions involved in its terms. The univariate and multivariate functions, i.e., interaction terms, involved in the regression model, are defined through a B-splines basis and fitted using a penalized splines ( P-splines) approach. In the multivariate case the shape constraints are imposed into a finite set of curves belonging to the hypersurface defined by the function, thus defining a skeleton in which the required conditions have to be verified. To do so, new conic optimization models are proposed which can accommodate different conditions along each covariate involved in the regression model. Furthermore, our approach can be used for a continuous response variable, as well as for Poisson and logistic regression. Therefore, a new mathematical optimization framework for shape-constrained regression is stated which copes with different model specifications, involving main and/or interaction effects, and types of response variables. We prove that our methodology is competitive in terms of accuracy and computational times, which are improved in some cases by more than two orders of magnitude, with other state-of-the-art approaches in both simulated and real data sets with applications in economics, social sciences, and medicine.

关键词： shape-constrained regression Generalized additive models Penalized splines Conic optimization Smooth-ANOVA decomposition Data envelopment analysis

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Robust nonparametric frontier estimation in two steps

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ECONOMETRIC REVIEWS 2023年第7期42卷 612-634页

作者： Chen, Yining Torrent, Hudson S. Ziegelmann, Flavio A. London Sch Econ & Polit Sci Dept Stat London England Univ Fed Rio Grande do Sul Dept Stat Porto Alegre RS Brazil

We propose a robust methodology for estimating production frontiers with multi-dimensional input via a two-step nonparametric regression, in which we estimate the level and shape of the frontier before shifting it to an appropriate position. Our main contribution is to derive a novel frontier estimation method under a variety of flexible models which is robust to the presence of outliers and possesses some inherent advantages over traditional frontier estimators. Our approach may be viewed as a simplification, yet a generalization, of those proposed by Martins-Filho and coauthors, who estimate frontier surfaces in three steps. In particular, outliers, as well as commonly seen shape constraints of the frontier surfaces, such as concavity and monotonicity, can be straightforwardly handled by our estimation procedure. We show consistency and asymptotic distributional theory of our resulting estimators under standard assumptions in the multi-dimensional input setting. The competitive finite-sample performances of our estimators are highlighted in both simulation studies and empirical data analysis.

关键词： Concavity local polynomial smoothing monotonicity outlier detection shape-constrained regression >

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