Shape restricted nonparametric regression with Bernstein polynomials

Abstract

WANG, JIANGDIAN. Shape Restricted Nonparametric Regression with Bernstein Polynomials. (Under the direction of Sujit K. Ghosh.) There has been increasing interest in estimating a multivariate regression function subject to various shape restrictions, such as nonnegativity, isotonicity, convexity and concavity among many others. The estimation of such shape-restricted regression curves is more challenging for multivariate predictors, especially for functions with compact support. Most of the currently available statistical estimation methods for shape restricted regression functions are generally computationally very intensive. Some of the existing methods have perceptible boundary biases. This thesis considers a suitable class of univariate and multivariate Bernstein polynomials and proposes sieved estimators obtained from a nested sequence of shape-restricted multivariate Bernstein polynomials. The proposed nonparametric estimators are shown to be: (i) the regression function estimate is shown to be the solution of a quadratic programming problem; making it computationally attractive (ii) the nonparametric estimator is shown to be universally consistent under some mild regularity conditions and (iii) the estimation methodology is flexible in the sense that it can be easily adapted to accommodate many popular multivariate shape restrictions. Numerical results derived from simulated data sets and real data analysis are used to illustrate the superior performance of the proposed estimators compared to a few other existing estimators in terms of various goodness of fit metrics.

Shape-restricted nonparametric regression with overall noisy measurements

“…Several special cases have been investigated, such as unimodal regression (Frisen 1986), convex regression (Birke and Dette 2007), monotonic regression (Birke and Dette 2008), translation families of a given shape (Kneip and Engel 1995), smoothness (Mammen 1991) or constraints on partial derivatives (Beresteanu 2004). Other methods use general parametric classes of candidate regression functions, such as Bernstein polynomials (Wang and Gosh 2012), splines (Meyer 2008) or epi-splines (Ryan and Wets 2013). The literature dealing with the combined problem of shape-restricted estimation under errors in variables is quite rare.…”

Photometric Stereo Using Constrained Bivariate Regression for General Isotropic Surfaces

“…We need to choose g k carefully because estimating a multivariate regression function subject to shape restrictions with compact support is challenging and usually very time-consuming [18]. Here we adopt the bivariate Bernstein polynomials [13], where the shape-restricted regression function estimation is shown to be the solution of a quadratic programming problem [5,18], making it computationally attractive.…”

“…We need to choose g k carefully because estimating a multivariate regression function subject to shape restrictions with compact support is challenging and usually very time-consuming [18]. Here we adopt the bivariate Bernstein polynomials [13], where the shape-restricted regression function estimation is shown to be the solution of a quadratic programming problem [5,18], making it computationally attractive. Furthermore, the Bernstein polynomials approximation naturally selects smooth functions with little computational effort, unlike other nonparametric regression functions (e.g., smoothing spline [4]), which implicitly enforces the smoothness of BRDF as in [2].…”

“…(9) are easily encoded via linear constraints, that is Aβ ≥ 0 and Cβ = 0, where A, C are shape restriction matrices. Following [18], the shape restriction matrices required for our problem are defined as follows. From the definition in Eq.…”

Data Sharpening Guided by Global Constraint in Local Regression

“…However, imposing a global constraint on the resulting curve is not always straightforward. This may explain why most published research on nonparametric/semiparametric regression subject to a global (or shape) constraint is usually based on other approaches, such as smoothing splines (Ramsay and Silverman, 2005) or nonparametric regression with Bernstein polynomials (Wang and Ghosh, 2012). Notable exceptions include the weighted kernel estimator (otherwise, known as "tilting") proposed by Hall and Huang (2001) and its extension provided in Du et al (2013), where the weights are chosen according to the primary constraint and a few other requirements.…”