Torsten Hothorn

Gabriele Campanella, Lucas Kook, Ida Häggström et al.

An every increasing number of clinical trials features a time-to-event outcome and records non-tabular patient data, such as magnetic resonance imaging or text data in the form of electronic health records. Recently, several neural-network based solutions have been proposed, some of which are binary classifiers. Parametric, distribution-free approaches which make full use of survival time and censoring status have not received much attention. We present deep conditional transformation models (DCTMs) for survival outcomes as a unifying approach to parametric and semiparametric survival analysis. DCTMs allow the specification of non-linear and non-proportional hazards for both tabular and non-tabular data and extend to all types of censoring and truncation. On real and semi-synthetic data, we show that DCTMs compete with state-of-the-art DL approaches to survival analysis.

Lucas Kook, Andrea Götschi, Philipp FM Baumann et al.

Ensembles improve prediction performance and allow uncertainty quantification by aggregating predictions from multiple models. In deep ensembling, the individual models are usually black box neural networks, or recently, partially interpretable semi-structured deep transformation models. However, interpretability of the ensemble members is generally lost upon aggregation. This is a crucial drawback of deep ensembles in high-stake decision fields, in which interpretable models are desired. We propose a novel transformation ensemble which aggregates probabilistic predictions with the guarantee to preserve interpretability and yield uniformly better predictions than the ensemble members on average. Transformation ensembles are tailored towards interpretable deep transformation models but are applicable to a wider range of probabilistic neural networks. In experiments on several publicly available data sets, we demonstrate that transformation ensembles perform on par with classical deep ensembles in terms of prediction performance, discrimination, and calibration. In addition, we demonstrate how transformation ensembles quantify both aleatoric and epistemic uncertainty, and produce minimax optimal predictions under certain conditions.

David Rügamer, Philipp F. M. Baumann, Thomas Kneib et al.

Probabilistic forecasting of time series is an important matter in many applications and research fields. In order to draw conclusions from a probabilistic forecast, we must ensure that the model class used to approximate the true forecasting distribution is expressive enough. Yet, characteristics of the model itself, such as its uncertainty or its feature-outcome relationship are not of lesser importance. This paper proposes Autoregressive Transformation Models (ATMs), a model class inspired by various research directions to unite expressive distributional forecasts using a semi-parametric distribution assumption with an interpretable model specification. We demonstrate the properties of ATMs both theoretically and through empirical evaluation on several simulated and real-world forecasting datasets.

Lucas Kook, Lisa Herzog, Torsten Hothorn et al.

Outcomes with a natural order commonly occur in prediction tasks and often the available input data are a mixture of complex data like images and tabular predictors. Deep Learning (DL) models are state-of-the-art for image classification tasks but frequently treat ordinal outcomes as unordered and lack interpretability. In contrast, classical ordinal regression models consider the outcome's order and yield interpretable predictor effects but are limited to tabular data. We present ordinal neural network transformation models (ONTRAMs), which unite DL with classical ordinal regression approaches. ONTRAMs are a special case of transformation models and trade off flexibility and interpretability by additively decomposing the transformation function into terms for image and tabular data using jointly trained neural networks. The performance of the most flexible ONTRAM is by definition equivalent to a standard multi-class DL model trained with cross-entropy while being faster in training when facing ordinal outcomes. Lastly, we discuss how to interpret model components for both tabular and image data on two publicly available datasets.

Philipp F. M. Baumann, Torsten Hothorn, David Rügamer

Learning the cumulative distribution function (CDF) of an outcome variable conditional on a set of features remains challenging, especially in high-dimensional settings. Conditional transformation models provide a semi-parametric approach that allows to model a large class of conditional CDFs without an explicit parametric distribution assumption and with only a few parameters. Existing estimation approaches within this class are, however, either limited in their complexity and applicability to unstructured data sources such as images or text, lack interpretability, or are restricted to certain types of outcomes. We close this gap by introducing the class of deep conditional transformation models which unifies existing approaches and allows to learn both interpretable (non-)linear model terms and more complex neural network predictors in one holistic framework. To this end we propose a novel network architecture, provide details on different model definitions and derive suitable constraints as well as network regularization terms. We demonstrate the efficacy of our approach through numerical experiments and applications.

Beate Sick, Torsten Hothorn, Oliver Dürr

We present a deep transformation model for probabilistic regression. Deep learning is known for outstandingly accurate predictions on complex data but in regression tasks, it is predominantly used to just predict a single number. This ignores the non-deterministic character of most tasks. Especially if crucial decisions are based on the predictions, like in medical applications, it is essential to quantify the prediction uncertainty. The presented deep learning transformation model estimates the whole conditional probability distribution, which is the most thorough way to capture uncertainty about the outcome. We combine ideas from a statistical transformation model (most likely transformation) with recent transformation models from deep learning (normalizing flows) to predict complex outcome distributions. The core of the method is a parameterized transformation function which can be trained with the usual maximum likelihood framework using gradient descent. The method can be combined with existing deep learning architectures. For small machine learning benchmark datasets, we report state of the art performance for most dataset and partly even outperform it. Our method works for complex input data, which we demonstrate by employing a CNN architecture on image data.

Torsten Hothorn, Achim Zeileis

Regression models for supervised learning problems with a continuous target are commonly understood as models for the conditional mean of the target given predictors. This notion is simple and therefore appealing for interpretation and visualisation. Information about the whole underlying conditional distribution is, however, not available from these models. A more general understanding of regression models as models for conditional distributions allows much broader inference from such models, for example the computation of prediction intervals. Several random forest-type algorithms aim at estimating conditional distributions, most prominently quantile regression forests (Meinshausen, 2006, JMLR). We propose a novel approach based on a parametric family of distributions characterised by their transformation function. A dedicated novel "transformation tree" algorithm able to detect distributional changes is developed. Based on these transformation trees, we introduce "transformation forests" as an adaptive local likelihood estimator of conditional distribution functions. The resulting models are fully parametric yet very general and allow broad inference procedures, such as the model-based bootstrap, to be applied in a straightforward way.