Regression Trees for Longitudinal Data

While studying response trajectory, often the population of interest may be diverse enough to exist distinct subgroups within it and the longitudinal change in response may not be uniform in these subgroups. That is, the timeslope and/or influence of covariates in longitudinal profile may vary among these different subgroups. For example, Raudenbush (2001) used depression as an example to argue that it is incorrect to assume that all the people in a given population would be experiencing either increasing or decreasing levels of depression. In such cases, traditional linear mixed effects model (assuming common parametric form for covariates and time) is not directly applicable for the entire population as a group-averaged trajectory can mask important subgroup differences. Our aim is to identify and characterize longitudinally homogeneous subgroups based on the combination of baseline covariates in the most parsimonious way. This goal can be achieved via constructing regression tree for longitudinal data using baseline covariates as partitioning variables. We have proposed LongCART algorithm to construct regression tree for the longitudinal data. In each node, the proposed LongCART algorithm determines the need for further splitting (i.e. whether parameter(s) of longitudinal profile is influenced by any baseline attributes) via parameter instability tests and thus the decision of further splitting is type-I error controlled. We have obtained the asymptotic results for the proposed instability test and examined finite sample behavior of the whole algorithm through simulation studies. Finally, we have applied the LongCART algorithm to study the longitudinal changes in choline level among HIV patients

On Theory for BART

Ensemble learning is a statistical paradigm built on the premise that many weak learners can perform exceptionally well when deployed collectively. The BART method of Chipman et al. (2010) is a prominent example of Bayesian ensemble learning, where each learner is a tree. Due to its impressive performance, BART has received a lot of attention from practitioners. Despite its wide popularity, however, theoretical studies of BART have begun emerging only very recently. Laying the foundations for the theoretical analysis of Bayesian forests, Rockova and van der Pas (2017) showed optimal posterior concentration under conditionally uniform tree priors. These priors deviate from the actual priors implemented in BART. Here, we study the exact BART prior and propose a simple modification so that it also enjoys optimality properties. To this end, we dive into branching process theory. We obtain tail bounds for the distribution of total progeny under heterogeneous Galton-Watson (GW) processes exploiting their connection to random walks. We conclude with a result stating the optimal rate of posterior convergence for BART.Comment: 2

BET: Bayesian Ensemble Trees for Clustering and Prediction in Heterogeneous Data

We propose a novel "tree-averaging" model that utilizes the ensemble of classification and regression trees (CART). Each constituent tree is estimated with a subset of similar data. We treat this grouping of subsets as Bayesian ensemble trees (BET) and model them as an infinite mixture Dirichlet process. We show that BET adapts to data heterogeneity and accurately estimates each component. Compared with the bootstrap-aggregating approach, BET shows improved prediction performance with fewer trees. We develop an efficient estimating procedure with improved sampling strategies in both CART and mixture models. We demonstrate these advantages of BET with simulations, classification of breast cancer and regression of lung function measurement of cystic fibrosis patients. Keywords: Bayesian CART; Dirichlet Process; Ensemble Approach; Heterogeneity; Mixture of Trees

Causal Tree Estimation of Heterogeneous Household Response to Time-Of-Use Electricity Pricing Schemes

We examine the household-specific effects of the introduction of Time-of-Use (TOU) electricity pricing schemes. Using a causal forest (Athey and Imbens, 2016; Wager and Athey, 2018; Athey et al., 2019), we consider the association between past consumption and survey variables, and the effect of TOU pricing on household electricity demand. We describe the heterogeneity in household variables across quartiles of estimated demand response and utilise variable importance measures. Household-specific estimates produced by a causal forest exhibit reasonable associations with covariates. For example, households that are younger, more educated, and that consume more electricity, are predicted to respond more to a new pricing scheme. In addition, variable importance measures suggest that some aspects of past consumption information may be more useful than survey information in producing these estimates.Comment: 34 pages, 6 figure

Interpretable Clustering via Optimal Trees

State-of-the-art clustering algorithms use heuristics to partition the feature space and provide little insight into the rationale for cluster membership, limiting their interpretability. In healthcare applications, the latter poses a barrier to the adoption of these methods since medical researchers are required to provide detailed explanations of their decisions in order to gain patient trust and limit liability. We present a new unsupervised learning algorithm that leverages Mixed Integer Optimization techniques to generate interpretable tree-based clustering models. Utilizing the flexible framework of Optimal Trees, our method approximates the globally optimal solution leading to high quality partitions of the feature space. Our algorithm, can incorporate various internal validation metrics, naturally determines the optimal number of clusters, and is able to account for mixed numeric and categorical data. It achieves comparable or superior performance on both synthetic and real world datasets when compared to K-Means while offering significantly higher interpretability.Comment: Machine Learning for Health (ML4H) Workshop at NeurIPS 2018 arXiv:1811.0721

Integrating Economic Knowledge in Data Mining Algorithms

The assessment of knowledge derived from databases depends on many factors. Decision makers often need to convince others about the correctness and effectiveness of knowledge induced from data.The current data mining techniques do not contribute much to this process of persuasion.Part of this limitation can be removed by integrating knowledge from experts in the field, encoded in some accessible way, with knowledge derived form patterns stored in the database.In this paper we will in particular discuss methods for implementing monotonicity constraints in economic decision problems.This prior knowledge is combined with data mining algorithms based on decision trees and neural networks.The method is illustrated in a hedonic price model.knowledge;neural network;data mining;decision trees

Confidence Decision Trees via Online and Active Learning for Streaming (BIG) Data

Decision tree classifiers are a widely used tool in data stream mining. The use of confidence intervals to estimate the gain associated with each split leads to very effective methods, like the popular Hoeffding tree algorithm. From a statistical viewpoint, the analysis of decision tree classifiers in a streaming setting requires knowing when enough new information has been collected to justify splitting a leaf. Although some of the issues in the statistical analysis of Hoeffding trees have been already clarified, a general and rigorous study of confidence intervals for splitting criteria is missing. We fill this gap by deriving accurate confidence intervals to estimate the splitting gain in decision tree learning with respect to three criteria: entropy, Gini index, and a third index proposed by Kearns and Mansour. Our confidence intervals depend in a more detailed way on the tree parameters. We also extend our confidence analysis to a selective sampling setting, in which the decision tree learner adaptively decides which labels to query in the stream. We furnish theoretical guarantee bounding the probability that the classification is non-optimal learning the decision tree via our selective sampling strategy. Experiments on real and synthetic data in a streaming setting show that our trees are indeed more accurate than trees with the same number of leaves generated by other techniques and our active learning module permits to save labeling cost. In addition, comparing our labeling strategy with recent methods, we show that our approach is more robust and consistent respect all the other techniques applied to incremental decision trees

Deep Embedding Forest: Forest-based Serving with Deep Embedding Features

Deep Neural Networks (DNN) have demonstrated superior ability to extract high level embedding vectors from low level features. Despite the success, the serving time is still the bottleneck due to expensive run-time computation of multiple layers of dense matrices. GPGPU, FPGA, or ASIC-based serving systems require additional hardware that are not in the mainstream design of most commercial applications. In contrast, tree or forest-based models are widely adopted because of low serving cost, but heavily depend on carefully engineered features. This work proposes a Deep Embedding Forest model that benefits from the best of both worlds. The model consists of a number of embedding layers and a forest/tree layer. The former maps high dimensional (hundreds of thousands to millions) and heterogeneous low-level features to the lower dimensional (thousands) vectors, and the latter ensures fast serving. Built on top of a representative DNN model called Deep Crossing, and two forest/tree-based models including XGBoost and LightGBM, a two-step Deep Embedding Forest algorithm is demonstrated to achieve on-par or slightly better performance as compared with the DNN counterpart, with only a fraction of serving time on conventional hardware. After comparing with a joint optimization algorithm called partial fuzzification, also proposed in this paper, it is concluded that the two-step Deep Embedding Forest has achieved near optimal performance. Experiments based on large scale data sets (up to 1 billion samples) from a major sponsored search engine proves the efficacy of the proposed model.Comment: 14 pages, 3 figures, 5 table

Some methods for heterogeneous treatment effect estimation in high-dimensions

When devising a course of treatment for a patient, doctors often have little quantitative evidence on which to base their decisions, beyond their medical education and published clinical trials. Stanford Health Care alone has millions of electronic medical records (EMRs) that are only just recently being leveraged to inform better treatment recommendations. These data present a unique challenge because they are high-dimensional and observational. Our goal is to make personalized treatment recommendations based on the outcomes for past patients similar to a new patient. We propose and analyze three methods for estimating heterogeneous treatment effects using observational data. Our methods perform well in simulations using a wide variety of treatment effect functions, and we present results of applying the two most promising methods to data from The SPRINT Data Analysis Challenge, from a large randomized trial of a treatment for high blood pressure

Algorithm Runtime Prediction: Methods & Evaluation

Perhaps surprisingly, it is possible to predict how long an algorithm will take to run on a previously unseen input, using machine learning techniques to build a model of the algorithm's runtime as a function of problem-specific instance features. Such models have important applications to algorithm analysis, portfolio-based algorithm selection, and the automatic configuration of parameterized algorithms. Over the past decade, a wide variety of techniques have been studied for building such models. Here, we describe extensions and improvements of existing models, new families of models, and -- perhaps most importantly -- a much more thorough treatment of algorithm parameters as model inputs. We also comprehensively describe new and existing features for predicting algorithm runtime for propositional satisfiability (SAT), travelling salesperson (TSP) and mixed integer programming (MIP) problems. We evaluate these innovations through the largest empirical analysis of its kind, comparing to a wide range of runtime modelling techniques from the literature. Our experiments consider 11 algorithms and 35 instance distributions; they also span a very wide range of SAT, MIP, and TSP instances, with the least structured having been generated uniformly at random and the most structured having emerged from real industrial applications. Overall, we demonstrate that our new models yield substantially better runtime predictions than previous approaches in terms of their generalization to new problem instances, to new algorithms from a parameterized space, and to both simultaneously.Comment: 51 pages, 13 figures, 8 tables. Added references, feature cost, and experiments with subsets of features; reworded Sections 1&