Bayesian Semiparametric Multivariate Density Deconvolution via Stochastic Rotation of Replicates

We consider the problem of multivariate density deconvolution where the distribution of a random vector needs to be estimated from replicates contaminated with conditionally heteroscedastic measurement errors. We propose a conceptually straightforward yet fundamentally novel and highly robust approach to multivariate density deconvolution by stochastically rotating the replicates toward the corresponding true latent values. We also address the additionally significantly challenging problem of accommodating conditionally heteroscedastic measurement errors in this newly introduced framework. We take a Bayesian route to estimation and inference, implemented via an efficient Markov chain Monte Carlo algorithm, appropriately accommodating uncertainty in all aspects of our analysis. Asymptotic convergence guarantees for the method are also established. We illustrate the method's empirical efficacy through simulation experiments and its practical utility in estimating the long-term joint average intakes of different dietary components from their measurement error contaminated 24-hour dietary recalls.Comment: arXiv admin note: text overlap with arXiv:1912.0508

Model-Robust Designs for Quantile Regression

We give methods for the construction of designs for linear models, when the purpose of the investigation is the estimation of the conditional quantile function and the estimation method is quantile regression. The designs are robust against misspecified response functions, and against unanticipated heteroscedasticity. The methods are illustrated by example, and in a case study in which they are applied to growth charts

Fast L1L_1-CkC^k polynomial spline interpolation algorithm with shape-preserving properties

International audienceIn this article, we address the interpolation problem of data points per regular $L_1$-spline polynomial curve that is invariant under a rotation of the data. We iteratively apply a minimization method on ¯ve data, belonging to a sliding window, in order to obtain this interpolating curve. We even show in the $C^k$-continuous interpolation case that this local minimization method preserves well the linear parts of the data, while a global $L_p$ (p >=1) minimization method does not in general satisfy this property. In addition, the complexity of the calculations of the unknown derivatives is a linear function of the length of the data whatever the order of smoothness of the curve

Normal-Mixture-of-Inverse-Gamma Priors for Bayesian Regularization and Model Selection in Structured Additive Regression Models

In regression models with many potential predictors, choosing an appropriate subset of covariates and their interactions at the same time as determining whether linear or more flexible functional forms are required is a challenging and important task. We propose a spike-and-slab prior structure in order to include or exclude single coefficients as well as blocks of coefficients associated with factor variables, random effects or basis expansions of smooth functions. Structured additive models with this prior structure are estimated with Markov Chain Monte Carlo using a redundant multiplicative parameter expansion. We discuss shrinkage properties of the novel prior induced by the redundant parameterization, investigate its sensitivity to hyperparameter settings and compare performance of the proposed method in terms of model selection, sparsity recovery, and estimation error for Gaussian, binomial and Poisson responses on real and simulated data sets with that of component-wise boosting and other approaches

Bayesian Nonparametric Methods For Causal Inference And Prediction

In this thesis we present novel approaches to regression and causal inference using popular Bayesian nonparametric methods. Bayesian Additive Regression Trees (BART) is a Bayesian machine learning algorithm in which the conditional distribution is modeled as a sum of regression trees. We extend BART into a semiparametric generalized linear model framework so that a portion of the covariates are modeled nonparametrically using BART and a subset of the covariates have parametric form. This presents an attractive option for research in which only a few covariates are of scientific interest but there are other covariates must be controlled for. Under certain causal assumptions, this model can be used as a structural mean model. We demonstrate this method by examining the effect of initiating certain antiretroviral medications has on mortality among HIV/HCV coinfected subjects. In later chapters, we propose a joint model for a continuous longitudinal outcome and baseline covariates using penalized splines and an enriched Dirichlet process (EDP) prior. This joint model decomposes into local linear mixed models for the outcome given the covariates and marginals for the covariates. The EDP prior that is placed on the regression parameters and the parameters on the covariates induces clustering among subjects determined by similarity in their regression parameters and nested within those clusters, sub-clusters based on similarity in the covariate space. When there are a large number of covariates, we find improved prediction over the same model with Dirichlet process (DP) priors. Since the model clusters based on regression parameters, this model also serves as a functional clustering algorithm where one does not have to choose the number of clusters beforehand. We use the method to estimate incidence rates of diabetes when longitudinal laboratory values from electronic health records are used to augment diagnostic codes for outcome identification. We later extend this work by using our EDP model in a causal inference setting using the parametric g-formula. We demonstrate this using electronic health record data consisting of subjects initiating second generation antipsychotics

Subdivision and manifold techniques for isogeometric design and analysis of surfaces

Design of surfaces and analysis of partial differential equations defined on them are of great importance in engineering applications, e.g., structural engineering, automotive and aerospace. This thesis focuses on isogeometric design and analysis of surfaces, which aims to integrate engineering design and analysis by using the same representation for both. The unresolved challenge is to develop a desirable surface representation that simultaneously satisfies certain favourable properties on meshes of arbitrary topology around the extraordinary vertices (EVs), i.e., vertices not shared by four quadrilaterals or three triangles. These properties include high continuity (geometric or parametric), optimal convergence in finite element analysis as well as simplicity in terms of implementation. To overcome the challenge, we further develop subdivision and manifold surface modelling techniques, and explore a possible scheme to combine the distinct appealing properties of the two. The unique advantages of the developed techniques have been confirmed with numerical experiments. Subdivision surfaces generate smooth surfaces from coarse control meshes of arbitrary topology by recursive refinement. Around EVs the optimal refinement weights are application-dependent. We first review subdivision-based finite elements. We then proceed to derive the optimal subdivision weights that minimise finite element errors and can be easily incorporated into existing implementations of subdivision schemes to achieve the same accuracy with much coarser meshes in engineering computations. To this end, the eigenstructure of the subdivision matrix is extensively used and a novel local shape decomposition approach is proposed to choose the optimal weights for each EV independently. Manifold-based basis functions are derived by combining differential-geometric manifold techniques with conformal parametrisations and the partition of unity method. This thesis derives novel manifold-based basis functions with arbitrary prescribed smoothness using quasi-conformal maps, enabling us to model and analyse surfaces with sharp features, such as creases and corners. Their practical utility in finite element simulation of hinged or rigidly joined structures is demonstrated with Kirchhoff-Love thin shell examples. We also propose a particular manifold basis reproducing subdivision surfaces away from EVs, i.e., B-splines, providing a way to combine the appealing properties of subdivision (available in industrial software) for design and manifold basis (relatively new) for analysis.Cambridge International Scholarship Scheme (CISS) by Cambridge Trus

Inverse passive learning of an input-output-map through update-spline-smoothing

Caption title.Includes bibliographical references (p. 26-31).Research supported by the Austrian "Fonds zur Förderung der wissenschaftlichen Forschung". J0514TECM. Heiss

Monotone approximation of aggregation operators using least squares splines

The need for monotone approximation of scattered data often arises in many problems of regression, when the monotonicity is semantically important. One such domain is fuzzy set theory, where membership functions and aggregation operators are order preserving. Least squares polynomial splines provide great flexbility when modeling non-linear functions, but may fail to be monotone. Linear restrictions on spline coefficients provide necessary and sufficient conditions for spline monotonicity. The basis for splines is selected in such a way that these restrictions take an especially simple form. The resulting non-negative least squares problem can be solved by a variety of standard proven techniques. Additional interpolation requirements can also be imposed in the same framework. The method is applied to fuzzy systems, where membership functions and aggregation operators are constructed from empirical data.<br /