arXiv subjects

Nadja Klein

Publications and source records attributed to Nadja Klein.

At least 73 records · Page 4Linked to original sources

arXiv2022

Boosting Multivariate Structured Additive Distributional Regression Models

We develop a model-based boosting approach for multivariate distributional regression within the framework of generalized additive models for location, scale, and shape. Our approach enables the simultaneous modeling of all distribution parameters of an arbitrary parametric distribution of a multivariate response conditional on explanatory variables, while being applicable to potentially high-dimensional data. Moreover, the boosting algorithm incorporates data-driven variable selection, taking various different types of effects into account. As a special merit of our approach, it allows for modelling the association between multiple continuous or discrete outcomes through the relevant covariates. After a detailed simulation study investigating estimation and prediction performance, we demonstrate the full flexibility of our approach in three diverse biomedical applications. The first is based on high-dimensional genomic cohort data from the UK Biobank, considering a bivariate binary response (chronic ischemic heart disease and high cholesterol). Here, we are able to identify genetic variants that are informative for the association between cholesterol and heart disease. The second application considers the demand for health care in Australia with the number of consultations and the number of prescribed medications as a bivariate count response. The third application analyses two dimensions of childhood undernutrition in Nigeria as a bivariate response and we find that the correlation between the two undernutrition scores is considerably different depending on the child's age and the region the child lives in.

stat.ME↗

arXiv2022

Semi-Structured Distributional Regression -- Extending Structured Additive Models by Arbitrary Deep Neural Networks and Data Modalities

Combining additive models and neural networks allows to broaden the scope of statistical regression and extend deep learning-based approaches by interpretable structured additive predictors at the same time. Existing attempts uniting the two modeling approaches are, however, limited to very specific combinations and, more importantly, involve an identifiability issue. As a consequence, interpretability and stable estimation are typically lost. We propose a general framework to combine structured regression models and deep neural networks into a unifying network architecture. To overcome the inherent identifiability issues between different model parts, we construct an orthogonalization cell that projects the deep neural network into the orthogonal complement of the statistical model predictor. This enables proper estimation of structured model parts and thereby interpretability. We demonstrate the framework's efficacy in numerical experiments and illustrate its special merits in benchmarks and real-world applications.

stat.ML↗

arXiv2022

Bayesian Conditional Transformation Models

Recent developments in statistical regression methodology shift away from pure mean regression towards distributional regression models. One important strand thereof is that of conditional transformation models (CTMs). CTMs infer the entire conditional distribution directly by applying a transformation function to the response conditionally on a set of covariates towards a simple log-concave reference distribution. Thereby, CTMs allow not only variance, kurtosis or skewness but the complete conditional distribution to depend on the explanatory variables. We propose a Bayesian notion of conditional transformation models (BCTMs) focusing on exactly observed continuous responses, but also incorporating extensions to randomly censored and discrete responses. Rather than relying on Bernstein polynomials that have been considered in likelihood-based CTMs, we implement a spline-based parametrization for monotonic effects that are supplemented with smoothness priors. Furthermore, we are able to benefit from the Bayesian paradigm via easily obtainable credible intervals and other quantities without relying on large sample approximations. A simulation study demonstrates the competitiveness of our approach against its likelihood-based counterpart but also Bayesian additive models of location, scale and shape and Bayesian quantile regression. Two applications illustrate the versatility of BCTMs in problems involving real world data, again including the comparison with various types of competitors.

stat.ME↗

arXiv2022

Posterior Concentration Rates for Bayesian Penalized Splines

Despite their widespread use in practice, the asymptotic properties of Bayesian penalized splines have not been investigated so far. We close this gap and study posterior concentration rates for Bayesian penalized splines in a Gaussian nonparametric regression model. A key feature of the approach is the hyperprior on the smoothing variance, which allows for adaptive smoothing in practice but complicates the theoretical analysis considerably as it destroys conjugacy and precludes analytic expressions for the posterior moments. To derive our theoretical results, we rely on several new concepts including a carefully defined proper version of the partially improper penalized splines prior as well as an innovative spline estimator that projects the observations onto the first basis functions of a Demmler-Reinsch basis. Our results show that posterior concentration at near optimal rate can be achieved if the hyperprior on the smoothing variance strikes a fine balance between oversmoothing and undersmoothing, which can for instance be met by a Weibull hyperprior with shape parameter 1/2. We complement our theoretical results with empirical evidence demonstrating the adaptivity of the hyperprior in practice.

math.ST↗

arXiv2022

deepregression: a Flexible Neural Network Framework for Semi-Structured Deep Distributional Regression

In this paper we describe the implementation of semi-structured deep distributional regression, a flexible framework to learn conditional distributions based on the combination of additive regression models and deep networks. Our implementation encompasses (1) a modular neural network building system based on the deep learning library \pkg{TensorFlow} for the fusion of various statistical and deep learning approaches, (2) an orthogonalization cell to allow for an interpretable combination of different subnetworks, as well as (3) pre-processing steps necessary to set up such models. The software package allows to define models in a user-friendly manner via a formula interface that is inspired by classical statistical model frameworks such as \pkg{mgcv}. The packages' modular design and functionality provides a unique resource for both scalable estimation of complex statistical models and the combination of approaches from deep learning and statistics. This allows for state-of-the-art predictive performance while simultaneously retaining the indispensable interpretability of classical statistical models.

stat.ML↗

arXiv2022

Boosting Distributional Copula Regression

Capturing complex dependence structures between outcome variables (e.g., study endpoints) is of high relevance in contemporary biomedical data problems and medical research. Distributional copula regression provides a flexible tool to model the joint distribution of multiple outcome variables by disentangling the marginal response distributions and their dependence structure. In a regression setup each parameter of the copula model, i.e. the marginal distribution parameters and the copula dependence parameters, can be related to covariates via structured additive predictors. We propose a framework to fit distributional copula regression models via a model-based boosting algorithm. Model-based boosting is a modern estimation technique that incorporates useful features like an intrinsic variable selection mechanism, parameter shrinkage and the capability to fit regression models in high dimensional data setting, i.e. situations with more covariates than observations. Thus, model-based boosting does not only complement existing Bayesian and maximum-likelihood based estimation frameworks for this model class but rather enables unique intrinsic mechanisms that can be helpful in many applied problems. The performance of our boosting algorithm in the context of copula regression models with continuous margins is evaluated in simulation studies that cover low- and high-dimensional data settings and situations with and without dependence between the responses. Moreover, distributional copula boosting is used to jointly analyze and predict the length and the weight of newborns conditional on sonographic measurements of the fetus before delivery together with other clinical variables.

stat.ME↗

arXiv2022

Deselection of Base-Learners for Statistical Boosting -- with an Application to Distributional Regression

We present a new procedure for enhanced variable selection for component-wise gradient boosting. Statistical boosting is a computational approach that emerged from machine learning, which allows to fit regression models in the presence of high-dimensional data. Furthermore, the algorithm can lead to data-driven variable selection. In practice, however, the final models typically tend to include too many variables in some situations. This occurs particularly for low-dimensional data (p<n), where we observe a slow overfitting behavior of boosting. As a result, more variables get included into the final model without altering the prediction accuracy. Many of these false positives are incorporated with a small coefficient and therefore have a small impact, but lead to a larger model. We try to overcome this issue by giving the algorithm the chance to deselect base-learners with minor importance. We analyze the impact of the new approach on variable selection and prediction performance in comparison to alternative methods including boosting with earlier stopping as well as twin boosting. We illustrate our approach with data of an ongoing cohort study for chronic kidney disease patients, where the most influential predictors for the health-related quality of life measure are selected in a distributional regression approach based on beta regression.

stat.ME↗

arXiv2021

Flexible Specification Testing in Quantile Regression Models

We propose three novel consistent specification tests for quantile regression models which generalize former tests in three ways. First, we allow the covariate effects to be quantile-dependent and nonlinear. Second, we allow parameterizing the conditional quantile functions by appropriate basis functions, rather than parametrically. We are hence able to test for functional forms beyond linearity, while retaining the linear effects as special cases. In both cases, the induced class of conditional distribution functions is tested with a Cramér-von Mises type test statistic for which we derive the theoretical limit distribution and propose a bootstrap method. Third, to increase the power of the tests, we further suggest a modified test statistic. We highlight the merits of our tests in a detailed MC study and two real data examples. Our first application to conditional income distributions in Germany indicates that there are not only still significant differences between East and West but also across the quantiles of the conditional income distributions, when conditioning on age and year. The second application to data from the Australian national electricity market reveals the importance of using interaction effects for modelling the highly skewed and heavy-tailed distributions of energy prices conditional on day, time of day and demand.

stat.ME↗

arXiv2021

Bivariate Analysis of Birth Weight and Gestational Age Depending on Environmental Exposures: Bayesian Distributional Regression with Copulas

In this article, we analyze perinatal data with birth weight (BW) as primarily interesting response variable. Gestational age (GA) is usually an important covariate and included in polynomial form. However, in opposition to this univariate regression, bivariate modeling of BW and GA is recommended to distinguish effects on each, on both, and between them. Rather than a parametric bivariate distribution, we apply conditional copula regression, where marginal distributions of BW and GA (not necessarily of the same form) can be estimated independently, and where the dependence structure is modeled conditional on the covariates separately from these marginals. In the resulting distributional regression models, all parameters of the two marginals and the copula parameter are observation-specific. Besides biometric and obstetric information, data on drinking water contamination and maternal smoking are included as environmental covariates. While the Gaussian distribution is suitable for BW, the skewed GA data are better modeled by the three-parametric Dagum distribution. The Clayton copula performs better than the Gumbel and the symmetric Gaussian copula, indicating lower tail dependence (stronger dependence when both variables are low), although this non-linear dependence between BW and GA is surprisingly weak and only influenced by Cesarean section. A non-linear trend of BW on GA is detected by a classical univariate model that is polynomial with respect to the effect of GA. Linear effects on BW mean are similar in both models, while our distributional copula regression also reveals covariates' effects on all other parameters.

stat.ME↗

arXiv2021

Marginally calibrated response distributions for end-to-end learning in autonomous driving

End-to-end learners for autonomous driving are deep neural networks that predict the instantaneous steering angle directly from images of the ahead-lying street. These learners must provide reliable uncertainty estimates for their predictions in order to meet safety requirements and initiate a switch to manual control in areas of high uncertainty. Yet end-to-end learners typically only deliver point predictions, since distributional predictions are associated with large increases in training time or additional computational resources during prediction. To address this shortcoming we investigate efficient and scalable approximate inference for the implicit copula neural linear model of Klein, Nott and Smith (2021) in order to quantify uncertainty for the predictions of end-to-end learners. The result are densities for the steering angle that are marginally calibrated, i.e.~the average of the estimated densities equals the empirical distribution of steering angles. To ensure the scalability to large $n$ regimes, we develop efficient estimation based on variational inference as a fast alternative to computationally intensive, exact inference via Hamiltonian Monte Carlo. We demonstrate the accuracy and speed of the variational approach in comparison to Hamiltonian Monte Carlo on two end-to-end learners trained for highway driving using the comma2k19 data set. The implicit copula neural linear model delivers accurate calibration, high-quality prediction intervals and allows to identify overconfident learners. Our approach also contributes to the explainability of black-box end-to-end learners, since predictive densities can be used to understand which steering actions the end-to-end learner sees as valid.

stat.ML↗

arXiv2021

Variational Inference and Sparsity in High-Dimensional Deep Gaussian Mixture Models

Gaussian mixture models are a popular tool for model-based clustering, and mixtures of factor analyzers are Gaussian mixture models having parsimonious factor covariance structure for mixture components. There are several recent extensions of mixture of factor analyzers to deep mixtures, where the Gaussian model for the latent factors is replaced by a mixture of factor analyzers. This construction can be iterated to obtain a model with many layers. These deep models are challenging to fit, and we consider Bayesian inference using sparsity priors to further regularize the estimation. A scalable natural gradient variational inference algorithm is developed for fitting the model, and we suggest computationally efficient approaches to the architecture choice using overfitted mixtures where unnecessary components drop out in the estimation. In a number of simulated and two real examples, we demonstrate the versatility of our approach for high-dimensional problems, and demonstrate that the use of sparsity inducing priors can be helpful for obtaining improved clustering results.

stat.ME↗

arXiv2021

A multivariate Gaussian random field prior against spatial confounding

Spatial models are used in a variety research areas, such as environmental sciences, epidemiology, or physics. A common phenomenon in many spatial regression models is spatial confounding. This phenomenon takes place when spatially indexed covariates modeling the mean of the response are correlated with the spatial random effect. As a result, estimates for regression coefficients of the covariates can be severely biased and interpretation of these is no longer valid. Recent literature has shown that typical solutions for reducing spatial confounding can lead to misleading and counterintuitive results. In this paper, we develop a computationally efficient spatial model in a Bayesian framework integrating novel prior structure that reduces spatial confounding. Starting from the univariate case, we extend our prior structure to case of multiple spatially confounded covariates. In a simulation study, we show that our novel model flexibly detects and reduces spatial confounding in spatial datasets, and it performs better than typically used methods such as restricted spatial regression. These results are promising for any applied researcher who wishes to interpret covariate effects in spatial regression models. As a real data illustration, we study the effect of elevation and temperature on the mean of daily precipitation in Germany.

stat.ME↗

arXiv2021

Deep Distributional Time Series Models and the Probabilistic Forecasting of Intraday Electricity Prices

Recurrent neural networks (RNNs) with rich feature vectors of past values can provide accurate point forecasts for series that exhibit complex serial dependence. We propose two approaches to constructing deep time series probabilistic models based on a variant of RNN called an echo state network (ESN). The first is where the output layer of the ESN has stochastic disturbances and a shrinkage prior for additional regularization. The second approach employs the implicit copula of an ESN with Gaussian disturbances, which is a deep copula process on the feature space. Combining this copula with a non-parametrically estimated marginal distribution produces a deep distributional time series model. The resulting probabilistic forecasts are deep functions of the feature vector and also marginally calibrated. In both approaches, Bayesian Markov chain Monte Carlo methods are used to estimate the models and compute forecasts. The proposed models are suitable for the complex task of forecasting intraday electricity prices. Using data from the Australian National Electricity Market, we show that our deep time series models provide accurate short term probabilistic price forecasts, with the copula model dominating. Moreover, the models provide a flexible framework for incorporating probabilistic forecasts of electricity demand as additional features, which increases upper tail forecast accuracy from the copula model significantly.

stat.ME↗

arXiv2021

Bayesian Effect Selection for Additive Quantile Regression with an Analysis to Air Pollution Thresholds

Statistical techniques used in air pollution modelling usually lack the possibility to understand which predictors affect air pollution in which functional form; and are not able to regress on exceedances over certain thresholds imposed by authorities directly. The latter naturally induce conditional quantiles and reflect the seriousness of particular events. In the present paper we focus on this important aspect by developing quantile regression models further. We propose a general Bayesian effect selection approach for additive quantile regression within a highly interpretable framework. We place separate normal beta prime spike and slab priors on the scalar importance parameters of effect parts and implement a fast Gibbs sampling scheme. Specifically, it enables to study quantile-specific covariate effects, allows these covariates to be of general functional form using additive predictors, and facilitates the analysts' decision whether an effect should be included linearly, non-linearly or not at all in the quantiles of interest. In a detailed analysis on air pollution data in Madrid (Spain) we find the added value of modelling extreme nitrogen dioxide (NO2) concentrations and how thresholds are driven differently by several climatological variables and traffic as a spatial proxy. Our results underpin the need of enhanced statistical models to support short-term decisions and enable local authorities to mitigate or even prevent exceedances of NO2 concentration limits.

stat.ME↗

arXiv2020

Marginally-calibrated deep distributional regression

Deep neural network (DNN) regression models are widely used in applications requiring state-of-the-art predictive accuracy. However, until recently there has been little work on accurate uncertainty quantification for predictions from such models. We add to this literature by outlining an approach to constructing predictive distributions that are `marginally calibrated'. This is where the long run average of the predictive distributions of the response variable matches the observed empirical margin. Our approach considers a DNN regression with a conditionally Gaussian prior for the final layer weights, from which an implicit copula process on the feature space is extracted. This copula process is combined with a non-parametrically estimated marginal distribution for the response. The end result is a scalable distributional DNN regression method with marginally calibrated predictions, and our work complements existing methods for probability calibration. The approach is first illustrated using two applications of dense layer feed-forward neural networks. However, our main motivating applications are in likelihood-free inference, where distributional deep regression is used to estimate marginal posterior distributions. In two complex ecological time series examples we employ the implicit copulas of convolutional networks, and show that marginal calibration results in improved uncertainty quantification. Our approach also avoids the need for manual specification of summary statistics, a requirement that is burdensome for users and typical of competing likelihood-free inference methods.

stat.ME↗

arXiv2020

Bayesian Variable Selection for Non-Gaussian Responses: A Marginally Calibrated Copula Approach

We propose a new highly flexible and tractable Bayesian approach to undertake variable selection in non-Gaussian regression models. It uses a copula decomposition for the joint distribution of observations on the dependent variable. This allows the marginal distribution of the dependent variable to be calibrated accurately using a nonparametric or other estimator. The family of copulas employed are `implicit copulas' that are constructed from existing hierarchical Bayesian models widely used for variable selection, and we establish some of their properties. Even though the copulas are high-dimensional, they can be estimated efficiently and quickly using Markov chain Monte Carlo (MCMC). A simulation study shows that when the responses are non-Gaussian the approach selects variables more accurately than contemporary benchmarks. A real data example in the Web Appendix illustrates that accounting for even mild deviations from normality can lead to a substantial increase in accuracy. To illustrate the full potential of our approach we extend it to spatial variable selection for fMRI. Using real data, we show our method allows for voxel-specific marginal calibration of the magnetic resonance signal at over 6,000 voxels, leading to an increase in the quality of the activation maps.

stat.ME↗

arXiv2020

Multivariate Conditional Transformation Models

Regression models describing the joint distribution of multivariate response variables conditional on covariate information have become an important aspect of contemporary regression analysis. However, a limitation of such models is that they often rely on rather simplistic assumptions, e.g. a constant dependency structure that is not allowed to vary with the covariates or the restriction to linear dependence between the responses only. We propose a general framework for multivariate conditional transformation models that overcomes these limitations and describes the entire distribution in a tractable and interpretable yet flexible way conditional on nonlinear effects of covariates. The framework can be embedded into likelihood-based inference, including results on asymptotic normality, and allows the dependence structure to vary with covariates. In addition, the framework scales well beyond bivariate response situations, which were the main focus of most earlier investigations. We illustrate the application of multivariate conditional transformation models in a trivariate analysis of childhood undernutrition and demonstrate empirically that our approach can be beneficial compared to existing benchmarks such that complex truly multivariate data-generating processes can be inferred from observations.

stat.ME↗

arXiv2020

Bayesian Inference for Regression Copulas

We propose a new semi-parametric distributional regression smoother that is based on a copula decomposition of the joint distribution of the vector of response values. The copula is high-dimensional and constructed by inversion of a pseudo regression, where the conditional mean and variance are semi-parametric functions of covariates modeled using regularized basis functions. By integrating out the basis coefficients, an implicit copula process on the covariate space is obtained, which we call a `regression copula'. We combine this with a non-parametric margin to define a copula model, where the entire distribution - including the mean and variance - of the response is a smooth semi-parametric function of the covariates. The copula is estimated using both Hamiltonian Monte Carlo and variational Bayes; the latter of which is scalable to high dimensions. Using real data examples and a simulation study we illustrate the efficacy of these estimators and the copula model. In a substantive example, we estimate the distribution of half-hourly electricity spot prices as a function of demand and two time covariates using radial bases and horseshoe regularization. The copula model produces distributional estimates that are locally adaptive with respect to the covariates, and predictions that are more accurate than those from benchmark models.

stat.ME↗