Many water-quality monitoring programs aim to measure turbidity to help guide effective management of waterways and catchments, yet distributing turbidity sensors throughout networks is typically cost prohibitive. To this end, we built and compared the ability of dynamic regression (ARIMA), long short-term memory neural nets (LSTM), and generalized additive models (GAM) to forecast stream turbidity one step ahead, using surrogate data from relatively low-cost in-situ sensors and publicly available databases. We iteratively trialled combinations of four surrogate covariates (rainfall, water level, air temperature and total global solar exposure) selecting a final model for each type that minimised the corrected Akaike Information Criterion. Cross-validation using a rolling time-window indicated that ARIMA, which included the rainfall and water-level covariates only, produced the most accurate predictions, followed closely by GAM, which included all four covariates. We constructed a meta-model, trained on time-series features of turbidity, to take advantage of the strengths of each model over different time points and predict the best model (that with the lowest forecast error one-step prior) for each time step. The meta-model outperformed all other models, indicating that this methodology can yield high accuracy and may be a viable alternative to using measurements sourced directly from turbidity-sensors where costs prohibit their deployment and maintenance, and when predicting turbidity across the short term. Our findings also indicated that temperature and light-associated variables, for example underwater illuminance, may hold promise as cost-effective, high-frequency surrogates of turbidity, especially when combined with other covariates, like rainfall, that are typically measured at coarse levels of spatial resolution.
Black-box models, such as deep neural networks, exhibit superior predictive performances, but understanding their behavior is notoriously difficult. Many explainable artificial intelligence methods have been proposed to reveal the decision-making processes of black box models. However, their applications in high-stakes domains remain limited. Recently proposed neural additive models (NAM) have achieved state-of-the-art interpretable machine learning. NAM can provide straightforward interpretations with slight performance sacrifices compared with multi-layer perceptron. However, NAM can only model 1$^{\text{st}}$-order feature interactions; thus, it cannot capture the co-relationships between input features. To overcome this problem, we propose a novel interpretable machine learning method called higher-order neural additive models (HONAM) and a feature interaction method for high interpretability. HONAM can model arbitrary orders of feature interactions. Therefore, it can provide the high predictive performance and interpretability that high-stakes domains need. In addition, we propose a novel hidden unit to effectively learn sharp-shape functions. We conducted experiments using various real-world datasets to examine the effectiveness of HONAM. Furthermore, we demonstrate that HONAM can achieve fair AI with a slight performance sacrifice. The source code for HONAM is publicly available.
Recent large language models (LLMs) have demonstrated remarkable prediction performance for a growing array of tasks. However, their proliferation into high-stakes domains (e.g. medicine) and compute-limited settings has created a burgeoning need for interpretability and efficiency. We address this need by proposing Augmented Interpretable Models (Aug-imodels), a framework for leveraging the knowledge learned by LLMs to build extremely efficient and interpretable models. Aug-imodels use LLMs during fitting but not during inference, allowing complete transparency and often a speed/memory improvement of greater than 1,000x for inference compared to LLMs. We explore two instantiations of Aug-imodels in natural-language processing: (i) Aug-GAM, which augments a generalized additive model with decoupled embeddings from an LLM and (ii) Aug-Tree, which augments a decision tree with LLM feature expansions. Across a variety of text-classification datasets, both outperform their non-augmented counterparts. Aug-GAM can even outperform much larger models (e.g. a 6-billion parameter GPT-J model), despite having 10,000x fewer parameters and being fully transparent. We further explore Aug-imodels in a natural-language fMRI study, where they generate interesting interpretations from scientific data. All code for using Aug-imodels and reproducing results is made available on Github.
The forecasting of credit default risk has been an active research field for several decades. Historically, logistic regression has been used as a major tool due to its compliance with regulatory requirements: transparency, explainability, and fairness. In recent years, researchers have increasingly used complex and advanced machine learning methods to improve prediction accuracy. Even though a machine learning method could potentially improve the model accuracy, it complicates simple logistic regression, deteriorates explainability, and often violates fairness. In the absence of compliance with regulatory requirements, even highly accurate machine learning methods are unlikely to be accepted by companies for credit scoring. In this paper, we introduce a novel class of monotonic neural additive models, which meet regulatory requirements by simplifying neural network architecture and enforcing monotonicity. By utilizing the special architectural features of the neural additive model, the monotonic neural additive model penalizes monotonicity violations effectively. Consequently, the computational cost of training a monotonic neural additive model is similar to that of training a neural additive model, as a free lunch. We demonstrate through empirical results that our new model is as accurate as black-box fully-connected neural networks, providing a highly accurate and regulated machine learning method.
While machine learning methods have significantly improved model performance over traditional methods, their black-box structure makes it difficult for researchers to interpret results. For highly regulated financial industries, model transparency is equally important to accuracy. Without understanding how models work, even highly accurate machine learning methods are unlikely to be accepted. We address this issue by introducing a novel class of transparent machine learning models known as generalized groves of neural additive models. The generalized groves of neural additive models separate features into three categories: linear features, individual nonlinear features, and interacted nonlinear features. Additionally, interactions in the last category are only local. A stepwise selection algorithm distinguishes the linear and nonlinear components, and interacted groups are carefully verified by applying additive separation criteria. Through some empirical examples in finance, we demonstrate that generalized grove of neural additive models exhibit high accuracy and transparency with predominantly linear terms and only sparse nonlinear ones.
Predictions and forecasts of machine learning models should take the form of probability distributions, aiming to increase the quantity of information communicated to end users. Although applications of probabilistic prediction and forecasting with machine learning models in academia and industry are becoming more frequent, related concepts and methods have not been formalized and structured under a holistic view of the entire field. Here, we review the topic of predictive uncertainty estimation with machine learning algorithms, as well as the related metrics (consistent scoring functions and proper scoring rules) for assessing probabilistic predictions. The review covers a time period spanning from the introduction of early statistical (linear regression and time series models, based on Bayesian statistics or quantile regression) to recent machine learning algorithms (including generalized additive models for location, scale and shape, random forests, boosting and deep learning algorithms) that are more flexible by nature. The review of the progress in the field, expedites our understanding on how to develop new algorithms tailored to users' needs, since the latest advancements are based on some fundamental concepts applied to more complex algorithms. We conclude by classifying the material and discussing challenges that are becoming a hot topic of research.
In explainable machine learning, local post-hoc explanation algorithms and inherently interpretable models are often seen as competing approaches. This work offers a partial reconciliation between the two by establishing a correspondence between Shapley Values and Generalized Additive Models (GAMs). We introduce $n$-Shapley Values, a parametric family of local post-hoc explanation algorithms that explain individual predictions with interaction terms up to order $n$. By varying the parameter $n$, we obtain a sequence of explanations that covers the entire range from Shapley Values up to a uniquely determined decomposition of the function we want to explain. The relationship between $n$-Shapley Values and this decomposition offers a functionally-grounded characterization of Shapley Values, which highlights their limitations. We then show that $n$-Shapley Values, as well as the Shapley Taylor- and Faith-Shap interaction indices, recover GAMs with interaction terms up to order $n$. This implies that the original Shapely Values recover GAMs without variable interactions. Taken together, our results provide a precise characterization of Shapley Values as they are being used in explainable machine learning. They also offer a principled interpretation of partial dependence plots of Shapley Values in terms of the underlying functional decomposition. A package for the estimation of different interaction indices is available at \url{https://github.com/tml-tuebingen/nshap}.
Risk management in many environmental settings requires an understanding of the mechanisms that drive extreme events. Useful metrics for quantifying such risk are extreme quantiles of response variables conditioned on predictor variables that describe, e.g., climate, biosphere and environmental states. Typically these quantiles lie outside the range of observable data and so, for estimation, require specification of parametric extreme value models within a regression framework. Classical approaches in this context utilise linear or additive relationships between predictor and response variables and suffer in either their predictive capabilities or computational efficiency; moreover, their simplicity is unlikely to capture the truly complex structures that lead to the creation of extreme wildfires. In this paper, we propose a new methodological framework for performing extreme quantile regression using artificial neutral networks, which are able to capture complex non-linear relationships and scale well to high-dimensional data. The "black box" nature of neural networks means that they lack the desirable trait of interpretability often favoured by practitioners; thus, we unify linear, and additive, regression methodology with deep learning to create partially-interpretable neural networks that can be used for statistical inference but retain high prediction accuracy. To complement this methodology, we further propose a novel point process model for extreme values which overcomes the finite lower-endpoint problem associated with the generalised extreme value class of distributions. Efficacy of our unified framework is illustrated on U.S. wildfire data with a high-dimensional predictor set and we illustrate vast improvements in predictive performance over linear and spline-based regression techniques.
The authors propose new additive models for binary outcomes, where the components are copula-based regression models (Noh et al, 2013), and designed such that the model may capture potentially complex interaction effects. The models do not require discretisation of continuous covariates, and are therefore suitable for problems with many such covariates. A fitting algorithm, and efficient procedures for model selection and evaluation of the components are described. Software is provided in the R-package copulaboost. Simulations and illustrations on data sets indicate that the method's predictive performance is either better than or comparable to the other methods.
ControlBurn is a Python package to construct feature-sparse tree ensembles that support nonlinear feature selection and interpretable machine learning. The algorithms in this package first build large tree ensembles that prioritize basis functions with few features and then select a feature-sparse subset of these basis functions using a weighted lasso optimization criterion. The package includes visualizations to analyze the features selected by the ensemble and their impact on predictions. Hence ControlBurn offers the accuracy and flexibility of tree-ensemble models and the interpretability of sparse generalized additive models. ControlBurn is scalable and flexible: for example, it can use warm-start continuation to compute the regularization path (prediction error for any number of selected features) for a dataset with tens of thousands of samples and hundreds of features in seconds. For larger datasets, the runtime scales linearly in the number of samples and features (up to a log factor), and the package support acceleration using sketching. Moreover, the ControlBurn framework accommodates feature costs, feature groupings, and $\ell_0$-based regularizers. The package is user-friendly and open-source: its documentation and source code appear on https://pypi.org/project/ControlBurn/ and https://github.com/udellgroup/controlburn/.
Interpretable machine learning offers insights into what factors drive a certain prediction of a black-box system. A large number of interpreting methods focus on identifying explanatory input features, which generally fall into two main categories: attribution and selection. A popular attribution-based approach is to exploit local neighborhoods for learning instance-specific explainers in an additive manner. The process is thus inefficient and susceptible to poorly-conditioned samples. Meanwhile, many selection-based methods directly optimize local feature distributions in an instance-wise training framework, thereby being capable of leveraging global information from other inputs. However, they can only interpret single-class predictions and many suffer from inconsistency across different settings, due to a strict reliance on a pre-defined number of features selected. This work exploits the strengths of both methods and proposes a framework for learning local explanations simultaneously for multiple target classes. Our model explainer significantly outperforms additive and instance-wise counterparts on faithfulness with more compact and comprehensible explanations. We also demonstrate the capacity to select stable and important features through extensive experiments on various data sets and black-box model architectures.
In recent years, a great emphasis has been put on engineering the acoustic signature of vehicles that represents the overall comfort level for passengers. Due to highly uncertain behavior of production cars, probabilistic metamodels or surrogates can be useful to estimate the NVH dispersion and assess different NVH risks. These metamodels follow physical behaviors and shall aid as a design space exploration tool during the early stage design process to support the NVH optimization. The measurement databases constitute different noise paths such as aerodynamic noise (wind-tunnel test), tire-pavement interaction noise (rolling noise), and noise due to electric motors (whining noise). This research work proposes a global NVH metamodeling technique for broadband noises such as aerodynamic and rolling noises exploiting the Bayesian framework that takes into account the prior (domain-expert) knowledge about complex physical mechanisms. Generalized additive models (GAMs) with polynomials and Gaussian basis functions are used to model the dependency of sound pressure level (SPL) on predictor variables. Moreover, parametric bootstrap algorithm based on data-generating mechanism using the point estimates is used to estimate the dispersion in unknown parameters. Probabilistic modelling is carried out using an open-source library PyMC3 that utilizes No-U-Turn sampler (NUTS) and the developed models are validated using Cross-Validation technique.
Trial-to-trial effects have been found in a number of studies, indicating that processing a stimulus influences responses in subsequent trials. A special case are priming effects which have been modelled successfully with error-driven learning (Marsolek, 2008), implying that participants are continuously learning during experiments. This study investigates whether trial-to-trial learning can be detected in an unprimed lexical decision experiment. We used the Discriminative Lexicon Model (DLM; Baayen et al., 2019), a model of the mental lexicon with meaning representations from distributional semantics, which models error-driven incremental learning with the Widrow-Hoff rule. We used data from the British Lexicon Project (BLP; Keuleers et al., 2012) and simulated the lexical decision experiment with the DLM on a trial-by-trial basis for each subject individually. Then, reaction times were predicted with Generalised Additive Models (GAMs), using measures derived from the DLM simulations as predictors. We extracted measures from two simulations per subject (one with learning updates between trials and one without), and used them as input to two GAMs. Learning-based models showed better model fit than the non-learning ones for the majority of subjects. Our measures also provide insights into lexical processing and individual differences. This demonstrates the potential of the DLM to model behavioural data and leads to the conclusion that trial-to-trial learning can indeed be detected in unprimed lexical decision. Our results support the possibility that our lexical knowledge is subject to continuous changes.
Machine learning (ML) interpretability techniques can reveal undesirable patterns in data that models exploit to make predictions--potentially causing harms once deployed. However, how to take action to address these patterns is not always clear. In a collaboration between ML and human-computer interaction researchers, physicians, and data scientists, we develop GAM Changer, the first interactive system to help domain experts and data scientists easily and responsibly edit Generalized Additive Models (GAMs) and fix problematic patterns. With novel interaction techniques, our tool puts interpretability into action--empowering users to analyze, validate, and align model behaviors with their knowledge and values. Physicians have started to use our tool to investigate and fix pneumonia and sepsis risk prediction models, and an evaluation with 7 data scientists working in diverse domains highlights that our tool is easy to use, meets their model editing needs, and fits into their current workflows. Built with modern web technologies, our tool runs locally in users' web browsers or computational notebooks, lowering the barrier to use. GAM Changer is available at the following public demo link: https://interpret.ml/gam-changer.
Due to the widespread use of complex machine learning models in real-world applications, it is becoming critical to explain model predictions. However, these models are typically black-box deep neural networks, explained post-hoc via methods with known faithfulness limitations. Generalized Additive Models (GAMs) are an inherently interpretable class of models that address this limitation by learning a non-linear shape function for each feature separately, followed by a linear model on top. However, these models are typically difficult to train, require numerous parameters, and are difficult to scale. We propose an entirely new subfamily of GAMs that utilizes basis decomposition of shape functions. A small number of basis functions are shared among all features, and are learned jointly for a given task, thus making our model scale much better to large-scale data with high-dimensional features, especially when features are sparse. We propose an architecture denoted as the Neural Basis Model (NBM) which uses a single neural network to learn these bases. On a variety of tabular and image datasets, we demonstrate that for interpretable machine learning, NBMs are the state-of-the-art in accuracy, model size, and, throughput and can easily model all higher-order feature interactions. Source code is available at https://github.com/facebookresearch/nbm-spam.
Generalized Additive Models (GAMs) have quickly become the leading choice for inherently-interpretable machine learning. However, unlike uninterpretable methods such as DNNs, they lack expressive power and easy scalability, and are hence not a feasible alternative for real-world tasks. We present a new class of GAMs that use tensor rank decompositions of polynomials to learn powerful, {\em inherently-interpretable} models. Our approach, titled Scalable Polynomial Additive Models (SPAM) is effortlessly scalable and models {\em all} higher-order feature interactions without a combinatorial parameter explosion. SPAM outperforms all current interpretable approaches, and matches DNN/XGBoost performance on a series of real-world benchmarks with up to hundreds of thousands of features. We demonstrate by human subject evaluations that SPAMs are demonstrably more interpretable in practice, and are hence an effortless replacement for DNNs for creating interpretable and high-performance systems suitable for large-scale machine learning. Source code is available at https://github.com/facebookresearch/nbm-spam.
A basic task in explainable AI (XAI) is to identify the most important features behind a prediction made by a black box function $f$. The insertion and deletion tests of Petsiuk et al. (2018) can be used to judge the quality of algorithms that rank pixels from most to least important for a classification. Motivated by regression problems we establish a formula for their area under the curve (AUC) criteria in terms of certain main effects and interactions in an anchored decomposition of $f$. We find an expression for the expected value of the AUC under a random ordering of inputs to $f$ and propose an alternative area above a straight line for the regression setting. We use this criterion to compare feature importances computed by integrated gradients (IG) to those computed by Kernel SHAP (KS) as well as LIME, DeepLIFT, vanilla gradient and input$\times$gradient methods. KS has the best overall performance in two datasets we consider but it is very expensive to compute. We find that IG is nearly as good as KS while being much faster. Our comparison problems include some binary inputs that pose a challenge to IG because it must use values between the possible variable levels and so we consider ways to handle binary variables in IG. We show that sorting variables by their Shapley value does not necessarily give the optimal ordering for an insertion-deletion test. It will however do that for monotone functions of additive models, such as logistic regression.
Deep neural networks (DNNs) are one of the most highlighted methods in machine learning. However, as DNNs are black-box models, they lack explanatory power for their predictions. Recently, neural additive models (NAMs) have been proposed to provide this power while maintaining high prediction performance. In this paper, we propose a novel NAM approach for multivariate nowcasting (NC) problems, which comprise an important focus area of machine learning. For the multivariate time-series data used in NC problems, explanations should be considered for every input value to the variables at distinguishable time steps. By employing generalized additive models, the proposed NAM-NC successfully explains each input value's importance for multiple variables and time steps. Experimental results involving a toy example and two real-world datasets show that the NAM-NC predicts multivariate time-series data as accurately as state-of-the-art neural networks, while also providing the explanatory importance of each input value. We also examine parameter-sharing networks using NAM-NC to decrease their complexity, and NAM-MC's hard-tied feature net extracted explanations with good performance.
Many high-dimensional statistical inference problems are believed to possess inherent computational hardness. Various frameworks have been proposed to give rigorous evidence for such hardness, including lower bounds against restricted models of computation (such as low-degree functions), as well as methods rooted in statistical physics that are based on free energy landscapes. This paper aims to make a rigorous connection between the seemingly different low-degree and free-energy based approaches. We define a free-energy based criterion for hardness and formally connect it to the well-established notion of low-degree hardness for a broad class of statistical problems, namely all Gaussian additive models and certain models with a sparse planted signal. By leveraging these rigorous connections we are able to: establish that for Gaussian additive models the "algebraic" notion of low-degree hardness implies failure of "geometric" local MCMC algorithms, and provide new low-degree lower bounds for sparse linear regression which seem difficult to prove directly. These results provide both conceptual insights into the connections between different notions of hardness, as well as concrete technical tools such as new methods for proving low-degree lower bounds.
Distributed statistical learning has become a popular technique for large-scale data analysis. Most existing work in this area focuses on dividing the observations, but we propose a new algorithm, DDAC-SpAM, which divides the features under a high-dimensional sparse additive model. Our approach involves three steps: divide, decorrelate, and conquer. The decorrelation operation enables each local estimator to recover the sparsity pattern for each additive component without imposing strict constraints on the correlation structure among variables. The effectiveness and efficiency of the proposed algorithm are demonstrated through theoretical analysis and empirical results on both synthetic and real data. The theoretical results include both the consistent sparsity pattern recovery as well as statistical inference for each additive functional component. Our approach provides a practical solution for fitting sparse additive models, with promising applications in a wide range of domains.