Adaptive Multivariate Ridge Regression

Brown, Philip; Zidek, James V.

doi:10.1214/aos/1176344891

Cited by 89 publications

(71 citation statements)

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“…By assuming that variables are normally distributed, we have shown that BNs are in fact equivalent to multivariate GBLUP and, by extension, to single-trait GBLUP. Furthermore, the separation between structure and parameter learning makes it possible to accommodate different parametric assumptions with relatively few changes and subsume models such as univariate and multivariate ridge-regression (Hoerl and Kennard 1970;Brown and Zidek 1980). As far as inference is concerned, several established methods from the literature can be used to predict traits from SNPs and vice versa; two examples are logic sampling and likelihood weighting (Koller and Friedman 2009).…”

Section: Discussionmentioning

confidence: 99%

“…Otherwise, penalized estimators such as ridge regression (RR; Hoerl and Kennard 1970) can be used when G is dense. The resulting BN can then be considered a flexible implementation of multivariate ridge regression, which has a number of of desirable properties over OLS (Brown and Zidek 1980). Equivalently, we can describe a BN using its global distribution, denoted with P(X) in (1).…”

Section: Methodsmentioning

confidence: 99%

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Multiple Quantitative Trait Analysis Using Bayesian Networks

et al. 2014

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Models for genome-wide prediction and association studies usually target a single phenotypic trait. However, in animal and plant genetics it is common to record information on multiple phenotypes for each individual that will be genotyped. Modeling traits individually disregards the fact that they are most likely associated due to pleiotropy and shared biological basis, thus providing only a partial, confounded view of genetic effects and phenotypic interactions. In this article we use data from a Multiparent Advanced Generation Inter-Cross (MAGIC) winter wheat population to explore Bayesian networks as a convenient and interpretable framework for the simultaneous modeling of multiple quantitative traits. We show that they are equivalent to multivariate genetic best linear unbiased prediction (GBLUP) and that they are competitive with single-trait elastic net and single-trait GBLUP in predictive performance. Finally, we discuss their relationship with other additive-effects models and their advantages in inference and interpretation. MAGIC populations provide an ideal setting for this kind of investigation because the very low population structure and large sample size result in predictive models with good power and limited confounding due to relatedness.

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Section: Discussionmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Multiple Quantitative Trait Analysis Using Bayesian Networks

et al. 2014

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“…Brown and Zidek (1980) consider multivariate regression, although with 7 known, so that ; | ;tN(;, I) instead of (2.8). Assuming ;tN(0, I q 1 p_p ) for 1 partially known, they derive minimaxity results generalizing those of Thisted (1976) when the unknown hyperparameters in 1 are appropriately estimated from the data.…”

Section: Discussionmentioning

confidence: 99%

Minimax Hierarchical Empirical Bayes Estimation in Multivariate Regression

Oman¹

2002

Journal of Multivariate Analysis

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The multivariate normal regression model, in which a vector y of responses is to be predicted by a vector x of explanatory variables, is considered. A hierarchical framework is used to express prior information on both x and y. An empirical Bayes estimator is developed which shrinks the maximum likelihood estimator of the matrix of regression coefficients across rows and columns to nontrivial subspaces which reflect both types of prior information. The estimator is shown to be minimax and is applied to a set of chemometrics data for which it reduces the crossvalidated predicted mean squared error of the maximum likelihood estimator by 380. Elsevier ScienceAMS 1991 subject classifications: 62H12; 62J07.

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“…It is worth noting that if we replace Θ with I n , which may be thought of as a degenerate case of restriction to the span of a basis, then (9) becomes a special case of the (n-dimensional) multivariate ridge regression estimate of Brown and Zidek (1980).…”

Section: Responses In Raw Form: the General Casementioning

confidence: 99%

“…However, whereas the latter submatrices have n rows, the former have K-expressing the assumption that, even if the functional response data consist of a large number n of points, these contain no more than K "pieces of information" given by K basis coefficients. When J θθ = I K , (13) reduces again to the multivariate ridge regression estimate of Brown and Zidek (1980), but in this case the multivariate response is the basis coefficient vector.…”

mentioning

confidence: 99%

Fast Function-on-Scalar Regression with Penalized Basis Expansions

Reiss

Huang²,

Mennes³

2010

The International Journal of Biostatistics

117

118

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Regression models for functional responses and scalar predictors are often fitted by means of basis functions, with quadratic roughness penalties applied to avoid overfitting. The fitting approach described by Ramsay and Silverman in the 1990s amounts to a penalized ordinary least squares (P-OLS) estimator of the coefficient functions. We recast this estimator as a generalized ridge regression estimator, and present a penalized generalized least squares (P-GLS) alternative. We describe algorithms by which both estimators can be implemented, with automatic selection of optimal smoothing parameters, in a more computationally efficient manner than has heretofore been available. We discuss pointwise confidence intervals for the coefficient functions, simultaneous inference by permutation tests, and model selection, including a novel notion of pointwise model selection. P-OLS and P-GLS are compared in a simulation study. Our methods are illustrated with an analysis of age effects in a functional magnetic resonance imaging data set, as well as a reanalysis of a now-classic Canadian weather data set. An R package implementing the methods is publicly available.

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Adaptive Multivariate Ridge Regression

Cited by 89 publications

References 0 publications

Multiple Quantitative Trait Analysis Using Bayesian Networks

Multiple Quantitative Trait Analysis Using Bayesian Networks

Minimax Hierarchical Empirical Bayes Estimation in Multivariate Regression

Fast Function-on-Scalar Regression with Penalized Basis Expansions

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