Nonlinear functional models for functional responses in reproducing kernel hilbert spaces

Lian, Heng

doi:10.1002/cjs.5550350410

Cited by 56 publications

(43 citation statements)

References 10 publications

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“…There are relatively few published recommendations in the statistical literature on how to construct

k_{1} (\cdot, \cdot)

. For example, Lian [] writes “[...]the construction of k 1 in general is difficult and a search of the literature does not seem to provide us with any clues about how to construct a positive definite kernel in general.” Nonetheless, if we shift our attention to the machine learning literature, we see that

k_{1} (t, t_{i}) = G (t, t_{i})

, where

G (t, t_{i})

is a Green's function of the linear differential operator

L y (t)

[Fasshauer, ; Fasshauer and Ye, ; Poggio and Girosi, ; Rasmussen and Williams, ]. Note that the Green's function also depends on the boundary conditions.…”

Section: Methodsmentioning

confidence: 99%

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Uncovering Local Trends in Genetic Effects of Multiple Phenotypes via Functional Linear Models

et al. 2016

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Recent technological advances equipped researchers with capabilities that go beyond traditional genotyping of loci known to be polymorphic in a general population. Genetic sequences of study participants can now be assessed directly. This capability removed technology-driven bias toward scoring predominantly common polymorphisms and let researchers reveal a wealth of rare and sample-specific variants. While the relative contributions of rare and common polymorphisms to trait variation are being debated, researchers are faced with the need for new statistical tools for simultaneous evaluation of all variants within a region. Several research groups demonstrated flexibility and good statistical power of the functional linear model approach. In this work we extend previous developments to allow inclusion of multiple traits and adjustment for additional covariates. Our functional approach is unique in that it provides a nuanced depiction of effects and interactions for the variables in the model by representing them as curves varying over a genetic region. We demonstrate flexibility and competitive power of our approach by contrasting its performance with commonly used statistical tools and illustrate its potential for discovery and characterization of genetic architecture of complex traits using sequencing data from the Dallas Heart Study.

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“…There are relatively few published recommendations in the statistical literature on how to construct

k_{1} (\cdot, \cdot)

k_{1} (t, t_{i}) = G (t, t_{i})

, where

G (t, t_{i})

is a Green's function of the linear differential operator

L y (t)

[Fasshauer, ; Fasshauer and Ye, ; Poggio and Girosi, ; Rasmussen and Williams, ]. Note that the Green's function also depends on the boundary conditions.…”

Section: Methodsmentioning

confidence: 99%

“…Other choices of basis functions can also be used with corresponding changes to penalized terms. Possible choices include, but are not limited to, (a) truncated power basis

{(t - k_{i})}_{+}^{p}

, (b) O'Sullivan splines [Wand and Ormerod, ], (c) thin plate splines [Ivanescu et al., ], or (d) the Gaussian kernel [Lian, ].…”

Section: Methodsmentioning

confidence: 99%

Uncovering Local Trends in Genetic Effects of Multiple Phenotypes via Functional Linear Models

et al. 2016

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“…Let

F_{K}

be the reproducing kernel Hilbert space generated by the kernel

K false(\cdot, \cdot false)

. See Lian () for more about the RKHS

F_{K}

. It can be shown that the solution of with

scriptF = F_{K}

takes the form

f false(X false) = c_{0} + \sum_{j = 1}^{n} c_{j} K false(X false(\cdot false), {\hat{X}}_{j} false(\cdot false) false)

.…”

Section: Functional Nonparametric Regressionmentioning

confidence: 99%

“…See the discussion paper by Ramsay & Dalzell () for an introduction to functional linear regression. For methods extending multivariate nonparametric regression to functional data, see Ferraty & Vieu (), Horvath & Kokoszka (), Ferraty, Mas, & Vieu (), Ferraty & Vieu (), Lian (), Kadri et al (), and references therein. In particular, the book Ferraty & Vieu () introduces various methodologies for studying functional data in a nonparametric way based on kernel‐type estimators.…”

Section: Introductionmentioning

confidence: 99%

“…The main difference between our method and functional linear and quadratic regression is that we do not specify any parametric relationship between the predictor trajectory and the response. This work is motivated by Lian () and Kadri et al () who assume that the whole predictor trajectory is observable. In contrast, the method we propose can be applied to sparse and irregular functional data.…”

Section: Introductionmentioning

confidence: 99%

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RKHS‐based functional nonparametric regression for sparse and irregular longitudinal data

Avery

Zhang

et al. 2014

Can J Statistics

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This paper focuses on sparse and irregular longitudinal data with a scalar response. The predictor consists of sparse and irregular observations on predictor trajectories, potentially contaminated with measurement errors. For this type of data, Yao, Müller, & Wang (2005a) proposed a principal components analysis through conditional expectation (PACE) approach, which is capable of predicting each predictor trajectory based on sparse and irregular observations. Nonparametric functional data analysis provides an attractive alternative due to its high flexibility. Early work includes functional additive models as in Müller & Yao (2008) and Ferraty & Vieu (2006), which are mainly based on kernel smoothing methods. In this work, we propose a new functional nonparametric regression framework based on reproducing kernel Hilbert spaces (RKHS). The proposed method involves two steps. The first step is to estimate each predictor trajectory based on sparse and irregular observations using PACE. The second step is to conduct a RKHS‐based nonparametric regression using the estimated predictor trajectories. Our approach shows improvement over existing methods in simulation studies as well as in a real data example. The Canadian Journal of Statistics 42: 204–216; 2014 © 2014 Statistical Society of Canada

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Representer Theorem

Wahba¹,

Wang²

2019

Wiley StatsRef: Statistics Reference Online

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The representer theorem plays an outsized role in a large class of learning problems. It provides a means to reduce infinite dimensional optimization problems to tractable finite dimensional ones. This article reviews the representer theorem for various learning problems under the reproducing kernel Hilbert spaces framework. We present solutions to the penalized least squares and penalized likelihood for nonparametric regression and support vector machines for classification as a solution to the penalized hinge loss. We discuss extensions of the representer theorem for regression with functional data.

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Nonlinear functional models for functional responses in reproducing kernel hilbert spaces

Cited by 56 publications

References 10 publications

Uncovering Local Trends in Genetic Effects of Multiple Phenotypes via Functional Linear Models

Uncovering Local Trends in Genetic Effects of Multiple Phenotypes via Functional Linear Models

RKHS‐based functional nonparametric regression for sparse and irregular longitudinal data

Representer Theorem

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