Bayesian nonparametric modeling for functional analysis of variance

Nguyen, XuanLong; Gelfand, Alan E.

doi:10.1007/s10463-013-0436-7

Cited by 6 publications

(4 citation statements)

References 23 publications

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“…In this section, we discuss definitions of the biomarker functional profile,

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Measures of distributional change based on Bayesian nonparametric priors recently have been considered by Nguyen and Gelfand (), who describe

L^{r}

‐norm, variational and symmetrized Kullback–Leibler distances between realizations of a random probability measure in a functional ANOVA setting. Here, our objective is different, as our goals are to estimate how a biomarker changes due to treatment and relate this to the outcome

T .

For example, if on average a given treatment results in a decrease of a biomarker's levels, this information usually is hidden by area‐based measures such as those noted above.…”

Section: Biomarker Distributional Change Functionalsmentioning

confidence: 99%

Bayesian Nonparametric Estimation of Targeted Agent Effects on Biomarker Change to Predict Clinical Outcome

2014

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Summary The effect of a targeted agent on a cancer patient's clinical outcome putatively is mediated through the agent's effect on one or more early biological events. This is motivated by pre-clinical experiments with cells or animals that identify such events, represented by binary or quantitative biomarkers. When evaluating targeted agents in humans, central questions are whether the distribution of a targeted biomarker changes following treatment, the nature and magnitude of this change, and whether it is associated with clinical outcome. Major difficulties in estimating these effects are that a biomarker's distribution may be complex, vary substantially between patients, and have complicated relationships with clinical outcomes. We present a probabilistically coherent framework for modeling and estimation in this setting, including a hierarchical Bayesian nonparametric mixture model for biomarkers that we use to define a functional profile of pre-versus-post treatment biomarker distribution change. The functional is similar to the receiver operating characteristic used in diagnostic testing. The hierarchical model yields clusters of individual patient biomarker profile functionals, and we use the profile as a covariate in a regression model for clinical outcome. The methodology is illustrated by analysis of a dataset from a clinical trial in prostate cancer using imatinib to target platelet-derived growth factor, with the clinical aim to improve progression-free survival time.

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“…In this section, we discuss definitions of the biomarker functional profile,

Δ .

Measures of distributional change based on Bayesian nonparametric priors recently have been considered by Nguyen and Gelfand (), who describe

L^{r}

T .

For example, if on average a given treatment results in a decrease of a biomarker's levels, this information usually is hidden by area‐based measures such as those noted above.…”

Section: Biomarker Distributional Change Functionalsmentioning

confidence: 99%

Bayesian Nonparametric Estimation of Targeted Agent Effects on Biomarker Change to Predict Clinical Outcome

2014

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“…It is possible to model additive response surfaces with a GP by choosing a suitable covariance function. The first such effort can be traced to [60] and has been recently revisited by [61][62][63][64][65]. By exploiting the additive structure of response surfaces one can potentially deal with a few hundred to a few thousand input dimensions.…”

Section: Introductionmentioning

confidence: 99%

“…Another example of an exploitable response surface feature is active subspaces (AS) [66]. An AS is a low-dimensional linear manifold of the input space characterized by maximal response variation.…”

Section: Introductionmentioning

confidence: 99%

Gaussian processes with built-in dimensionality reduction: Applications to high-dimensional uncertainty propagation

Tripathy

Bilionis

Gonzalez

2016

Journal of Computational Physics

164

126

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Uncertainty quantification (UQ) tasks, such as model calibration, uncertainty propagation, and optimization under uncertainty, typically require several thousand evaluations of the underlying computer codes. To cope with the cost of simulations, one replaces the real response surface with a cheap surrogate based, e.g., on polynomial chaos expansions, neural networks, support vector machines, or Gaussian processes (GP). However, the number of simulations required to learn a generic multivariate response grows exponentially as the input dimension increases. This curse of dimensionality can only be addressed, if the response exhibits some special structure that can be discovered and exploited. A wide range of physical responses exhibit a special structure known as an active subspace (AS). An AS is a linear manifold of the stochastic space characterized by maximal response variation. The idea is that one should first identify this low dimensional manifold, project the high-dimensional input onto it, and then link the projection to the output. If the dimensionality of the AS is low enough, then learning the link function is a much easier problem than the original problem of learning a high-dimensional function. The classic approach to discovering the AS requires gradient information, a fact that severely limits its applicability. Furthermore, and partly because of its reliance to gradients, it is not able to handle noisy observations. The latter is an essential trait if one wants to be able to propagate uncertainty through stochastic simulators, e.g., through molecular dynamics codes. In this work, we develop a probabilistic version of AS which is gradient-free and robust to observational noise. Our approach relies on a novel Gaussian process regression with built-in dimensionality reduction. In particular, the AS is represented as an orthogonal projection matrix that serves as yet another covariance function hyper-parameter to be estimated from the data. To train the model, we design a two-step maximum likelihood optimization procedure that ensures the orthogonality of the projection matrix by exploiting recent results on the Stiefel manifold, i.e., the manifold of matrices with orthogonal columns. The additional benefit of our probabilistic formulation, is that it allows us to select the dimensionality of the AS via the Bayesian information criterion. We validate our approach by showing that it can discover the right AS in synthetic examples without gradient information using both noiseless and noisy observations. We demonstrate that our method is able to discover the same AS as the classical approach in a challenging one-hundred-dimensional problem involving an elliptic stochastic partial differential equation with random conductivity. Finally, we use our approach to study the effect of geometric and material uncertainties in the propagation of solitary waves in a one dimensional granular system.

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“…Recently, GP prior has also been increasingly employed in Bayesian FDA. For instance, GP has been used as the prior for functional batch effects (Kaufman et al 2010) and as the base measure for a Dirichlet process prior (Nguyen and Gelfand 2014) in functional ANOVA. GP regression has also been used in FDA (see, e.g., Shi and Choi (2011), Shi et al (2012), and Wang and Shi (2014)), providing a nonparametric linkage between a functional predictor and a functional response.…”

Section: Introductionmentioning

confidence: 99%

Smoothing and Mean–Covariance Estimation of Functional Data with a Bayesian Hierarchical Model

et al. 2016

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Functional data, with basic observational units being functions (e.g., curves, surfaces) varying over a continuum, are frequently encountered in various applications. While many statistical tools have been developed for functional data analysis, the issue of smoothing all functional observations simultaneously is less studied. Existing methods often focus on smoothing each individual function separately, at the risk of removing important systematic patterns common across functions. We propose a nonparametric Bayesian approach to smooth all functional observations simultaneously and nonparametrically. In the proposed approach, we assume that the functional observations are independent Gaussian processes subject to a common level of measurement errors, enabling the borrowing of strength across all observations. Unlike most Gaussian process regression models that rely on pre-specified structures for the covariance kernel, we adopt a hierarchical framework by assuming a Gaussian process prior for the mean function and an Inverse-Wishart process prior for the covariance function. These prior assumptions induce an automatic mean-covariance estimation in the posterior inference in addition to the simultaneous smoothing of all observations. Such a hierarchical framework is flexible enough to incorporate functional data with different characteristics, including data measured on either common or uncommon grids, and data with either stationary or nonstationary covariance structures. Simulations and real data analysis demonstrate that, in comparison with alternative methods, the proposed Bayesian approach achieves better smoothing accuracy and comparable mean-covariance estimation results. Furthermore, it can successfully retain the systematic patterns in the functional observations that are usually neglected by the existing functional data analyses based on individual-curve smoothing.

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Bayesian nonparametric modeling for functional analysis of variance

Cited by 6 publications

References 23 publications

Bayesian Nonparametric Estimation of Targeted Agent Effects on Biomarker Change to Predict Clinical Outcome

Bayesian Nonparametric Estimation of Targeted Agent Effects on Biomarker Change to Predict Clinical Outcome

Gaussian processes with built-in dimensionality reduction: Applications to high-dimensional uncertainty propagation

Smoothing and Mean–Covariance Estimation of Functional Data with a Bayesian Hierarchical Model

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