Models which estimate main effects of individual variables alongside interaction effects have an identifiability challenge: effects can be freely moved between main effects and interaction effects without changing the model prediction. This is a critical problem for interpretability because it permits "contradictory" models to represent the same function. To solve this problem, we propose pure interaction effects: variance in the outcome which cannot be represented by any smaller subset of features. This definition has an equivalence with the Functional ANOVA decomposition. To compute this decomposition, we present a fast, exact algorithm that transforms any piecewise-constant function (such as a tree-based model) into a purified, canonical representation. We apply this algorithm to Generalized Additive Models with interactions trained on several datasets and show large disparity, including contradictions, between the effects before and after purification. These results underscore the need to specify data distributions and ensure identifiability before interpreting model parameters.
Over the last decades, the challenges in applied regression and in predictive modeling have been changing considerably: (1) More flexible model specifications are needed as big(ger) data become available, facilitated by more powerful computing infrastructure. (2) Full probabilistic modeling rather than predicting just means or expectations is crucial in many applications. (3) Interest in Bayesian inference has been increasing both as an appealing framework for regularizing or penalizing model estimation as well as a natural alternative to classical frequentist inference. However, while there has been a lot of research in all three areas, also leading to associated software packages, a modular software implementation that allows to easily combine all three aspects has not yet been available. For filling this gap, the R package bamlss is introduced for Bayesian additive models for location, scale, and shape (and beyond). At the core of the package are algorithms for highly-efficient Bayesian estimation and inference that can be applied to generalized additive models (GAMs) or generalized additive models for location, scale, and shape (GAMLSS), also known as distributional regression. However, its building blocks are designed as "Lego bricks" encompassing various distributions (exponential family, Cox, joint models, ...), regression terms (linear, splines, random effects, tensor products, spatial fields, ...), and estimators (MCMC, backfitting, gradient boosting, lasso, ...). It is demonstrated how these can be easily recombined to make classical models more flexible or create new custom models for specific modeling challenges.
InterpretML is an open-source Python package which exposes machine learning interpretability algorithms to practitioners and researchers. InterpretML exposes two types of interpretability - glassbox models, which are machine learning models designed for interpretability (ex: linear models, rule lists, generalized additive models), and blackbox explainability techniques for explaining existing systems (ex: Partial Dependence, LIME). The package enables practitioners to easily compare interpretability algorithms by exposing multiple methods under a unified API, and by having a built-in, extensible visualization platform. InterpretML also includes the first implementation of the Explainable Boosting Machine, a powerful, interpretable, glassbox model that can be as accurate as many blackbox models. The MIT licensed source code can be downloaded from github.com/microsoft/interpret.
Grounding is crucial for natural language understanding. An important subtask is to understand modified color expressions, such as 'dirty blue'. We present a model of color modifiers that, compared with previous additive models in RGB space, learns more complex transformations. In addition, we present a model that operates in the HSV color space. We show that certain adjectives are better modeled in that space. To account for all modifiers, we train a hard ensemble model that selects a color space depending on the modifier color pair. Experimental results show significant and consistent improvements compared to the state-of-the-art baseline model.
Among interpretable machine learning methods, the class of Generalised Additive Neural Networks (GANNs) is referred to as Self-Explaining Neural Networks (SENN) because of the linear dependence on explicit functions of the inputs. In binary classification this shows the precise weight that each input contributes towards the logit. The nomogram is a graphical representation of these weights. We show that functions of individual and pairs of variables can be derived from a functional Analysis of Variance (ANOVA) representation, enabling an efficient feature selection to be carried by application of the logistic Lasso. This process infers the structure of GANNs which otherwise needs to be predefined. As this method is particularly suited for tabular data, it starts by fitting a generic flexible model, in this case a Multi-layer Perceptron (MLP) to which the ANOVA decomposition is applied. This has the further advantage that the resulting GANN can be replicated as a SENN, enabling further refinement of the univariate and bivariate component functions to take place. The component functions are partial responses hence the SENN is a partial response network. The Partial Response Network (PRN) is equally as transparent as a traditional logistic regression model, but capable of non-linear classification with comparable or superior performance to the original MLP. In other words, the PRN is a fully interpretable representation of the MLP, at the level of univariate and bivariate effects. The performance of the PRN is shown to be competitive for benchmark data, against state-of-the-art machine learning methods including GBM, SVM and Random Forests. It is also compared with spline-based Sparse Additive Models (SAM) showing that a semi-parametric representation of the GAM as a neural network can be as effective as the SAM though less constrained by the need to set spline nodes.
Logistic regression (LR) is widely used in clinical prediction because it is simple to deploy and easy to interpret. Nevertheless, being a linear model, LR has limited expressive capability and often has unsatisfactory performance. Generalized additive models (GAMs) extend the linear model with transformations of input features, though feature interaction is not allowed for all GAM variants. In this paper, we propose a factored generalized additive model (F-GAM) to preserve the model interpretability for targeted features while allowing a rich model for interaction with features fixed within the individual. We evaluate F-GAM on prediction of two targets, postoperative acute kidney injury and acute respiratory failure, from a single-center database. We find superior model performance of F-GAM in terms of AUPRC and AUROC compared to several other GAM implementations, random forests, support vector machine, and a deep neural network. We find that the model interpretability is good with results with high face validity.
Analyzing mobility behavior of users is extremely useful to create or improve existing services. Several research works have been done in order to study mobility behavior of users that mainly use users' significant locations. However, these existing analysis are extremely intrusive because they require the knowledge of the frequently visited places of users, which thus makes it fairly easy to identify them. Consequently, in this paper, we present a privacy-aware methodology to analyze mobility behavior of users. We firstly propose a new metric based on the well-known Shannon entropy, called spatio-temporal entropy, to quantify the mobility level of a user during a time window. Then, we compute a sequence of spatio-temporal entropy from the location history of the user that expresses user's movements as rhythms. We secondly present how to study the effects of several groups of additional variables on the evolution of the spatio-temporal entropy of a user, such as spatio-temporal, demographic and mean of transportation variables. For this, we use Generalized Additive Models (GAMs). The results firstly show that the spatio-temporal entropy and GAMs are an ideal combination to understand mobility behavior of an individual user or a group of users. We also evaluate the prediction accuracy of a global GAM compared to individual GAMs and individual AutoRegressive Integrated Moving Average (ARIMA) models. These last results highlighted that the global GAM gives more accurate predictions of spatio-temporal entropy by checking the Mean Absolute Error (MAE). In addition, this research work opens various threads, such as the prediction of demographic data of users or the creation of personalized mobility prediction models by using movement rhythm characteristics of a user.
Explainable Artificial Intelligence (XAI)has received a great deal of attention recently. Explainability is being presented as a remedy for the distrust of complex and opaque models. Model agnostic methods such as LIME, SHAP, or Break Down promise instance-level interpretability for any complex machine learning model. But how faithful are these additive explanations? Can we rely on additive explanations for non-additive models? In this paper, we (1) examine the behavior of the most popular instance-level explanations under the presence of interactions, (2) introduce a new method that detects interactions for instance-level explanations, (3) perform a large scale benchmark to see how frequently additive explanations may be misleading.
We present a unified framework for estimation and analysis of generalized additive models in high dimensions. The framework defines a large class of penalized regression estimators, encompassing many existing methods. An efficient computational algorithm for this class is presented that easily scales to thousands of observations and features. We prove minimax optimal convergence bounds for this class under a weak compatibility condition. In addition, we characterize the rate of convergence when this compatibility condition is not met. Finally, we also show that the optimal penalty parameters for structure and sparsity penalties in our framework are linked, allowing cross-validation to be conducted over only a single tuning parameter. We complement our theoretical results with empirical studies comparing some existing methods within this framework.
We present a novel approach for nonparametric regression using wavelet basis functions. Our proposal, $\texttt{waveMesh}$, can be applied to non-equispaced data with sample size not necessarily a power of 2. We develop an efficient proximal gradient descent algorithm for computing the estimator and establish adaptive minimax convergence rates. The main appeal of our approach is that it naturally extends to additive and sparse additive models for a potentially large number of covariates. We prove minimax optimal convergence rates under a weak compatibility condition for sparse additive models. The compatibility condition holds when we have a small number of covariates. Additionally, we establish convergence rates for when the condition is not met. We complement our theoretical results with empirical studies comparing $\texttt{waveMesh}$ to existing methods.
Generalized additive models (GAMs) are a widely used class of models of interest to statisticians as they provide a flexible way to design interpretable models of data beyond linear models. We here propose a scalable and well-calibrated Bayesian treatment of GAMs using Gaussian processes (GPs) and leveraging recent advances in variational inference. We use sparse GPs to represent each component and exploit the additive structure of the model to efficiently represent a Gaussian a posteriori coupling between the components.
Generalized additive models (GAMs) are favored in many regression and binary classification problems because they are able to fit complex, nonlinear functions while still remaining interpretable. In the first part of this paper, we generalize a state-of-the-art GAM learning algorithm based on boosted trees to the multiclass setting, and show that this multiclass algorithm outperforms existing GAM learning algorithms and sometimes matches the performance of full complexity models such as gradient boosted trees. In the second part, we turn our attention to the interpretability of GAMs in the multiclass setting. Surprisingly, the natural interpretability of GAMs breaks down when there are more than two classes. Naive interpretation of multiclass GAMs can lead to false conclusions. Inspired by binary GAMs, we identify two axioms that any additive model must satisfy in order to not be visually misleading. We then develop a technique called Additive Post-Processing for Interpretability (API), that provably transforms a pre-trained additive model to satisfy the interpretability axioms without sacrificing accuracy. The technique works not just on models trained with our learning algorithm, but on any multiclass additive model, including multiclass linear and logistic regression. We demonstrate the effectiveness of API on a 12-class infant mortality dataset.
Multiple generalized additive models (GAMs) are a type of distributional regression wherein parameters of probability distributions depend on predictors through smooth functions, with selection of the degree of smoothness via $L_2$ regularization. Multiple GAMs allow finer statistical inference by incorporating explanatory information in any or all of the parameters of the distribution. Owing to their nonlinearity, flexibility and interpretability, GAMs are widely used, but reliable and fast methods for automatic smoothing in large datasets are still lacking, despite recent advances. We develop a general methodology for automatically learning the optimal degree of $L_2$ regularization for multiple GAMs using an empirical Bayes approach. The smooth functions are penalized by different amounts, which are learned simultaneously by maximization of a marginal likelihood through an approximate expectation-maximization algorithm that involves a double Laplace approximation at the E-step, and leads to an efficient M-step. Empirical analysis shows that the resulting algorithm is numerically stable, faster than all existing methods and achieves state-of-the-art accuracy. For illustration, we apply it to an important and challenging problem in the analysis of extremal data.
Additive models play an important role in semiparametric statistics. This paper gives learning rates for regularized kernel based methods for additive models. These learning rates compare favourably in particular in high dimensions to recent results on optimal learning rates for purely nonparametric regularized kernel based quantile regression using the Gaussian radial basis function kernel, provided the assumption of an additive model is valid. Additionally, a concrete example is presented to show that a Gaussian function depending only on one variable lies in a reproducing kernel Hilbert space generated by an additive Gaussian kernel, but does not belong to the reproducing kernel Hilbert space generated by the multivariate Gaussian kernel of the same variance.
Despite the major advances taken in causal modeling, causality is still an unfamiliar topic for many statisticians. In this paper, it is demonstrated from the beginning to the end how causal effects can be estimated from observational data assuming that the causal structure is known. To make the problem more challenging, the causal effects are highly nonlinear and the data are missing at random. The tools used in the estimation include causal models with design, causal calculus, multiple imputation and generalized additive models. The main message is that a trained statistician can estimate causal effects by judiciously combining existing tools.
We study the problem of variable selection in convex nonparametric regression. Under the assumption that the true regression function is convex and sparse, we develop a screening procedure to select a subset of variables that contains the relevant variables. Our approach is a two-stage quadratic programming method that estimates a sum of one-dimensional convex functions, followed by one-dimensional concave regression fits on the residuals. In contrast to previous methods for sparse additive models, the optimization is finite dimensional and requires no tuning parameters for smoothness. Under appropriate assumptions, we prove that the procedure is faithful in the population setting, yielding no false negatives. We give a finite sample statistical analysis, and introduce algorithms for efficiently carrying out the required quadratic programs. The approach leads to computational and statistical advantages over fitting a full model, and provides an effective, practical approach to variable screening in convex regression.
Many factors can affect the predictability of public bus services such as traffic, weather and local events. Other aspects, such as day of week or hour of day, may influence bus travel times as well, either directly or in conjunction with other variables. However, the exact nature of such relationships between travel times and predictor variables is, in most situations, not known. In this paper we develop a framework that allows for flexible modeling of bus travel times through the use of Additive Models. In particular, we model travel times as a sum of linear as well as nonlinear terms that are modeled as smooth functions of predictor variables. The proposed class of models provides a principled statistical framework that is highly flexible in terms of model building. The experimental results demonstrate uniformly superior performance of our best model as compared to previous prediction methods when applied to a very large GPS data set obtained from buses operating in the city of Rio de Janeiro.
We develop estimation for potentially high-dimensional additive structural equation models. A key component of our approach is to decouple order search among the variables from feature or edge selection in a directed acyclic graph encoding the causal structure. We show that the former can be done with nonregularized (restricted) maximum likelihood estimation while the latter can be efficiently addressed using sparse regression techniques. Thus, we substantially simplify the problem of structure search and estimation for an important class of causal models. We establish consistency of the (restricted) maximum likelihood estimator for low- and high-dimensional scenarios, and we also allow for misspecification of the error distribution. Furthermore, we develop an efficient computational algorithm which can deal with many variables, and the new method's accuracy and performance is illustrated on simulated and real data.
We propose a high dimensional classification method that involves nonparametric feature augmentation. Knowing that marginal density ratios are the most powerful univariate classifiers, we use the ratio estimates to transform the original feature measurements. Subsequently, penalized logistic regression is invoked, taking as input the newly transformed or augmented features. This procedure trains models equipped with local complexity and global simplicity, thereby avoiding the curse of dimensionality while creating a flexible nonlinear decision boundary. The resulting method is called Feature Augmentation via Nonparametrics and Selection (FANS). We motivate FANS by generalizing the Naive Bayes model, writing the log ratio of joint densities as a linear combination of those of marginal densities. It is related to generalized additive models, but has better interpretability and computability. Risk bounds are developed for FANS. In numerical analysis, FANS is compared with competing methods, so as to provide a guideline on its best application domain. Real data analysis demonstrates that FANS performs very competitively on benchmark email spam and gene expression data sets. Moreover, FANS is implemented by an extremely fast algorithm through parallel computing.
We establish minimax optimal rates of convergence for estimation in a high dimensional additive model assuming that it is approximately sparse. Our results reveal an interesting phase transition behavior universal to this class of high dimensional problems. In the {\it sparse regime} when the components are sufficiently smooth or the dimensionality is sufficiently large, the optimal rates are identical to those for high dimensional linear regression, and therefore there is no additional cost to entertain a nonparametric model. Otherwise, in the so-called {\it smooth regime}, the rates coincide with the optimal rates for estimating a univariate function, and therefore they are immune to the "curse of dimensionality".