Dai Feng

Lili Zhao, Dai Feng

There has been increasing interest in modelling survival data using deep learning methods in medical research. Current approaches have focused on designing special cost functions to handle censored survival data. We propose a very different method with two steps. In the first step, we transform each subject's survival time into a series of jackknife pseudo conditional survival probabilities and then use these pseudo probabilities as a quantitative response variable in the deep neural network model. By using the pseudo values, we reduce a complex survival analysis to a standard regression problem, which greatly simplifies the neural network construction. Our two-step approach is simple, yet very flexible in making risk predictions for survival data, which is very appealing from the practice point of view. The source code is freely available at http://github.com/lilizhaoUM/DNNSurv.

Dai Feng, Richard Baumgartner

Kernels ensuing from tree ensembles such as random forest (RF) or gradient boosted trees (GBT), when used for kernel learning, have been shown to be competitive to their respective tree ensembles (particularly in higher dimensional scenarios). On the other hand, it has been also shown that performance of the kernel algorithms depends on the degree of the kernel-target alignment. However, the kernel-target alignment for kernel learning based on the tree ensembles has not been investigated and filling this gap is the main goal of our work. Using the eigenanalysis of the kernel matrix, we demonstrate that for continuous targets good performance of the tree-based kernel learning is associated with strong kernel-target alignment. Moreover, we show that well performing tree ensemble based kernels are characterized by strong target aligned components that are expressed through scalar products between the eigenvectors of the kernel matrix and the target. This suggests that when tree ensemble based kernel learning is successful, relevant information for the supervised problem is concentrated near lower dimensional manifold spanned by the target aligned components. Persistence of the strong target aligned components in tree ensemble based kernels is further supported by sensitivity analysis via landmark learning. In addition to a comprehensive simulation study, we also provide experimental results from several real life data sets that are in line with the simulations.

Dai Feng, Lili Zhao

There has been increasing interest in modeling survival data using deep learning methods in medical research. In this paper, we proposed a Bayesian hierarchical deep neural networks model for modeling and prediction of survival data. Compared with previously studied methods, the new proposal can provide not only point estimate of survival probability but also quantification of the corresponding uncertainty, which can be of crucial importance in predictive modeling and subsequent decision making. The favorable statistical properties of point and uncertainty estimates were demonstrated by simulation studies and real data analysis. The Python code implementing the proposed approach was provided.

Dai Feng, Richard Baumgartner

Tree based ensembles such as Breiman's random forest (RF) and Gradient Boosted Trees (GBT) can be interpreted as implicit kernel generators, where the ensuing proximity matrix represents the data-driven tree ensemble kernel. Kernel perspective on the RF has been used to develop a principled framework for theoretical investigation of its statistical properties. Recently, it has been shown that the kernel interpretation is germane to other tree-based ensembles e.g. GBTs. However, practical utility of the links between kernels and the tree ensembles has not been widely explored and systematically evaluated. Focus of our work is investigation of the interplay between kernel methods and the tree based ensembles including the RF and GBT. We elucidate the performance and properties of the RF and GBT based kernels in a comprehensive simulation study comprising of continuous and binary targets. We show that for continuous targets, the RF/GBT kernels are competitive to their respective ensembles in higher dimensional scenarios, particularly in cases with larger number of noisy features. For the binary target, the RF/GBT kernels and their respective ensembles exhibit comparable performance. We provide the results from real life data sets for regression and classification to show how these insights may be leveraged in practice. Overall, our results support the tree ensemble based kernels as a valuable addition to the practitioner's toolbox. Finally, we discuss extensions of the tree ensemble based kernels for survival targets, interpretable prototype and landmarking classification and regression. We outline future line of research for kernels furnished by Bayesian counterparts of the frequentist tree ensembles.