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Trevor Hastie

Trevor Hastie appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

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Machine LearningMethodologyComputationApplicationsmath.STStatistics Theoryq-fin.MFq-fin.RMq-fin.STQuantitative Methods

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Published work

25 published item(s)

preprint2026arXiv

Enhancing a Risk Model by Adding Transient Statistical Factors

Estimating the covariance of asset returns, i.e., the risk model, is a key component of financial portfolio construction and evaluation. Most risk modeling approaches produce a factor model that decomposes the asset variability into two components: the first attributed to a small number of factors that are common among the assets and the second attributed to the idiosyncratic behavior of each asset. Third-party providers typically provide risk models to investors, and while these models are typically of high quality, they may fail to capture important information, e.g., changing market regimes and transient factors. To overcome these limitations, we propose a systematic method based on maximum likelihood estimation to enhance an existing factor model by both refining the given model and adding new statistical factors. Our approach relies only on the observed sequence of realized returns and on the choice of two hyperparameters: the number of additional factors and the half-life parameter that determines the weights assigned to returns in the log-likelihood objective. Importantly, our methodology applies to the situation where asset returns may be missing, making it suitable for typical equity datasets. We demonstrate our approach on the Barra short-term US risk model, a high-quality risk model used in practice, for a universe of US high-capitalization equities. We show that the proposed extension captures structure in the returns that is missed by the original model.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Confidence Intervals for the Generalisation Error of Random Forests

Out-of-bag error is commonly used as an estimate of generalisation error in ensemble-based learning models such as random forests. We present confidence intervals for this quantity using the delta-method-after-bootstrap and the jackknife-after-bootstrap techniques. These methods do not require growing any additional trees. We show that these new confidence intervals have improved coverage properties over the naive confidence interval, in real and simulated examples.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Estimating Heterogeneous Treatment Effects for General Responses

Heterogeneous treatment effect models allow us to compare treatments at subgroup and individual levels, and are of increasing popularity in applications like personalized medicine, advertising, and education. In this talk, we first survey different causal estimands used in practice, which focus on estimating the difference in conditional means. We then propose DINA, the difference in natural parameters, to quantify heterogeneous treatment effect in exponential families and the Cox model. For binary outcomes and survival times, DINA is both convenient and more practical for modeling the influence of covariates on the treatment effect. Second, we introduce a meta-algorithm for DINA, which allows practitioners to use powerful off-the-shelf machine learning tools for the estimation of nuisance functions, and which is also statistically robust to errors in inaccurate nuisance function estimation. We demonstrate the efficacy of our method combined with various machine learning base-learners on simulated and real datasets.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Generalized Matrix Factorization: efficient algorithms for fitting generalized linear latent variable models to large data arrays

Unmeasured or latent variables are often the cause of correlations between multivariate measurements, which are studied in a variety of fields such as psychology, ecology, and medicine. For Gaussian measurements, there are classical tools such as factor analysis or principal component analysis with a well-established theory and fast algorithms. Generalized Linear Latent Variable models (GLLVMs) generalize such factor models to non-Gaussian responses. However, current algorithms for estimating model parameters in GLLVMs require intensive computation and do not scale to large datasets with thousands of observational units or responses. In this article, we propose a new approach for fitting GLLVMs to high-dimensional datasets, based on approximating the model using penalized quasi-likelihood and then using a Newton method and Fisher scoring to learn the model parameters. Computationally, our method is noticeably faster and more stable, enabling GLLVM fits to much larger matrices than previously possible. We apply our method on a dataset of 48,000 observational units with over 2,000 observed species in each unit and find that most of the variability can be explained with a handful of factors. We publish an easy-to-use implementation of our proposed fitting algorithm.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

Elastic Net Regularization Paths for All Generalized Linear Models

The lasso and elastic net are popular regularized regression models for supervised learning. Friedman, Hastie, and Tibshirani (2010) introduced a computationally efficient algorithm for computing the elastic net regularization path for ordinary least squares regression, logistic regression and multinomial logistic regression, while Simon, Friedman, Hastie, and Tibshirani (2011) extended this work to Cox models for right-censored data. We further extend the reach of the elastic net-regularized regression to all generalized linear model families, Cox models with (start, stop] data and strata, and a simplified version of the relaxed lasso. We also discuss convenient utility functions for measuring the performance of these fitted models.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

LinCDE: Conditional Density Estimation via Lindsey's Method

Conditional density estimation is a fundamental problem in statistics, with scientific and practical applications in biology, economics, finance and environmental studies, to name a few. In this paper, we propose a conditional density estimator based on gradient boosting and Lindsey's method (LinCDE). LinCDE admits flexible modeling of the density family and can capture distributional characteristics like modality and shape. In particular, when suitably parametrized, LinCDE will produce smooth and non-negative density estimates. Furthermore, like boosted regression trees, LinCDE does automatic feature selection. We demonstrate LinCDE's efficacy through extensive simulations and three real data examples.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Assessment of Heterogeneous Treatment Effect Estimation Accuracy via Matching

We study the assessment of the accuracy of heterogeneous treatment effect (HTE) estimation, where the HTE is not directly observable so standard computation of prediction errors is not applicable. To tackle the difficulty, we propose an assessment approach by constructing pseudo-observations of the HTE based on matching. Our contributions are three-fold: first, we introduce a novel matching distance derived from proximity scores in random forests; second, we formulate the matching problem as an average minimum-cost flow problem and provide an efficient algorithm; third, we propose a match-then-split principle for the assessment with cross-validation. We demonstrate the efficacy of the assessment approach on synthetic data and data generated from a real dataset.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Feature-weighted elastic net: using "features of features" for better prediction

In some supervised learning settings, the practitioner might have additional information on the features used for prediction. We propose a new method which leverages this additional information for better prediction. The method, which we call the feature-weighted elastic net ("fwelnet"), uses these "features of features" to adapt the relative penalties on the feature coefficients in the elastic net penalty. In our simulations, fwelnet outperforms the lasso in terms of test mean squared error and usually gives an improvement in true positive rate or false positive rate for feature selection. We also apply this method to early prediction of preeclampsia, where fwelnet outperforms the lasso in terms of 10-fold cross-validated area under the curve (0.86 vs. 0.80). We also provide a connection between fwelnet and the group lasso and suggest how fwelnet might be used for multi-task learning.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Surprises in High-Dimensional Ridgeless Least Squares Interpolation

Interpolators -- estimators that achieve zero training error -- have attracted growing attention in machine learning, mainly because state-of-the art neural networks appear to be models of this type. In this paper, we study minimum $\ell_2$ norm ("ridgeless") interpolation in high-dimensional least squares regression. We consider two different models for the feature distribution: a linear model, where the feature vectors $x_i \in {\mathbb R}^p$ are obtained by applying a linear transform to a vector of i.i.d. entries, $x_i = Σ^{1/2} z_i$ (with $z_i \in {\mathbb R}^p$); and a nonlinear model, where the feature vectors are obtained by passing the input through a random one-layer neural network, $x_i = φ(W z_i)$ (with $z_i \in {\mathbb R}^d$, $W \in {\mathbb R}^{p \times d}$ a matrix of i.i.d. entries, and $φ$ an activation function acting componentwise on $W z_i$). We recover -- in a precise quantitative way -- several phenomena that have been observed in large-scale neural networks and kernel machines, including the "double descent" behavior of the prediction risk, and the potential benefits of overparametrization.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Confounder Adjustment in Multiple Hypothesis Testing

We consider large-scale studies in which thousands of significance tests are performed simultaneously. In some of these studies, the multiple testing procedure can be severely biased by latent confounding factors such as batch effects and unmeasured covariates that correlate with both primary variable(s) of interest (e.g. treatment variable, phenotype) and the outcome. Over the past decade, many statistical methods have been proposed to adjust for the confounders in hypothesis testing. We unify these methods in the same framework, generalize them to include multiple primary variables and multiple nuisance variables, and analyze their statistical properties. In particular, we provide theoretical guarantees for RUV-4 and LEAPP, which correspond to two different identification conditions in the framework: the first requires a set of "negative controls" that are known a priori to follow the null distribution; the second requires the true non-nulls to be sparse. Two different estimators which are based on RUV-4 and LEAPP are then applied to these two scenarios. We show that if the confounding factors are strong, the resulting estimators can be asymptotically as powerful as the oracle estimator which observes the latent confounding factors. For hypothesis testing, we show the asymptotic z-tests based on the estimators can control the type I error. Numerical experiments show that the false discovery rate is also controlled by the Benjamini-Hochberg procedure when the sample size is reasonably large.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Customized training with an application to mass spectrometric imaging of cancer tissue

We introduce a simple, interpretable strategy for making predictions on test data when the features of the test data are available at the time of model fitting. Our proposal - customized training - clusters the data to find training points close to each test point and then fits an $\ell_1$-regularized model (lasso) separately in each training cluster. This approach combines the local adaptivity of $k$-nearest neighbors with the interpretability of the lasso. Although we use the lasso for the model fitting, any supervised learning method can be applied to the customized training sets. We apply the method to a mass-spectrometric imaging data set from an ongoing collaboration in gastric cancer detection which demonstrates the power and interpretability of the technique. Our idea is simple but potentially useful in situations where the data have some underlying structure.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Sparse Quadratic Discriminant Analysis and Community Bayes

We develop a class of rules spanning the range between quadratic discriminant analysis and naive Bayes, through a path of sparse graphical models. A group lasso penalty is used to introduce shrinkage and encourage a similar pattern of sparsity across precision matrices. It gives sparse estimates of interactions and produces interpretable models. Inspired by the connected-components structure of the estimated precision matrices, we propose the community Bayes model, which partitions features into several conditional independent communities and splits the classification problem into separate smaller ones. The community Bayes idea is quite general and can be applied to non-Gaussian data and likelihood-based classifiers.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Generalized Additive Model Selection

We introduce GAMSEL (Generalized Additive Model Selection), a penalized likelihood approach for fitting sparse generalized additive models in high dimension. Our method interpolates between null, linear and additive models by allowing the effect of each variable to be estimated as being either zero, linear, or a low-complexity curve, as determined by the data. We present a blockwise coordinate descent procedure for efficiently optimizing the penalized likelihood objective over a dense grid of the tuning parameter, producing a regularization path of additive models. We demonstrate the performance of our method on both real and simulated data examples, and compare it with existing techniques for additive model selection.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

Bias Correction in Species Distribution Models: Pooling Survey and Collection Data for Multiple Species

Presence-only records may provide data on the distributions of rare species, but commonly suffer from large, unknown biases due to their typically haphazard collection schemes. Presence-absence or count data collected in systematic, planned surveys are more reliable but typically less abundant. We proposed a probabilistic model to allow for joint analysis of presence-only and survey data to exploit their complementary strengths. Our method pools presence-only and presence-absence data for many species and maximizes a joint likelihood, simultaneously estimating and adjusting for the sampling bias affecting the presence-only data. By assuming that the sampling bias is the same for all species, we can borrow strength across species to efficiently estimate the bias and improve our inference from presence-only data. We evaluate our model's performance on data for 36 eucalypt species in southeastern Australia. We find that presence-only records exhibit a strong sampling bias toward the coast and toward Sydney, the largest city. Our data-pooling technique substantially improves the out-of-sample predictive performance of our model when the amount of available presence-absence data for a given species is scarce. If we have only presence-only data and no presence-absence data for a given species, but both types of data for several other species that suffer from the same spatial sampling bias, then our method can obtain an unbiased estimate of the first species' geographic range.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

Confidence Intervals for Random Forests: The Jackknife and the Infinitesimal Jackknife

We study the variability of predictions made by bagged learners and random forests, and show how to estimate standard errors for these methods. Our work builds on variance estimates for bagging proposed by Efron (1992, 2012) that are based on the jackknife and the infinitesimal jackknife (IJ). In practice, bagged predictors are computed using a finite number B of bootstrap replicates, and working with a large B can be computationally expensive. Direct applications of jackknife and IJ estimators to bagging require B on the order of n^{1.5} bootstrap replicates to converge, where n is the size of the training set. We propose improved versions that only require B on the order of n replicates. Moreover, we show that the IJ estimator requires 1.7 times less bootstrap replicates than the jackknife to achieve a given accuracy. Finally, we study the sampling distributions of the jackknife and IJ variance estimates themselves. We illustrate our findings with multiple experiments and simulation studies.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

Finite-sample equivalence in statistical models for presence-only data

Statistical modeling of presence-only data has attracted much recent attention in the ecological literature, leading to a proliferation of methods, including the inhomogeneous Poisson process (IPP) model, maximum entropy (Maxent) modeling of species distributions and logistic regression models. Several recent articles have shown the close relationships between these methods. We explain why the IPP intensity function is a more natural object of inference in presence-only studies than occurrence probability (which is only defined with reference to quadrat size), and why presence-only data only allows estimation of relative, and not absolute intensity of species occurrence. All three of the above techniques amount to parametric density estimation under the same exponential family model (in the case of the IPP, the fitted density is multiplied by the number of presence records to obtain a fitted intensity). We show that IPP and Maxent give the exact same estimate for this density, but logistic regression in general yields a different estimate in finite samples. When the model is misspecified - as it practically always is - logistic regression and the IPP may have substantially different asymptotic limits with large data sets. We propose ``infinitely weighted logistic regression,'' which is exactly equivalent to the IPP in finite samples. Consequently, many already-implemented methods extending logistic regression can also extend the Maxent and IPP models in directly analogous ways using this technique.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

Local case-control sampling: Efficient subsampling in imbalanced data sets

For classification problems with significant class imbalance, subsampling can reduce computational costs at the price of inflated variance in estimating model parameters. We propose a method for subsampling efficiently for logistic regression by adjusting the class balance locally in feature space via an accept-reject scheme. Our method generalizes standard case-control sampling, using a pilot estimate to preferentially select examples whose responses are conditionally rare given their features. The biased subsampling is corrected by a post-hoc analytic adjustment to the parameters. The method is simple and requires one parallelizable scan over the full data set. Standard case-control sampling is inconsistent under model misspecification for the population risk-minimizing coefficients $θ^*$. By contrast, our estimator is consistent for $θ^*$ provided that the pilot estimate is. Moreover, under correct specification and with a consistent, independent pilot estimate, our estimator has exactly twice the asymptotic variance of the full-sample MLE - even if the selected subsample comprises a miniscule fraction of the full data set, as happens when the original data are severely imbalanced. The factor of two improves to $1+\frac{1}{c}$ if we multiply the baseline acceptance probabilities by $c>1$ (and weight points with acceptance probability greater than 1), taking roughly $\frac{1+c}{2}$ times as many data points into the subsample. Experiments on simulated and real data show that our method can substantially outperform standard case-control subsampling.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

Matrix Completion and Low-Rank SVD via Fast Alternating Least Squares

The matrix-completion problem has attracted a lot of attention, largely as a result of the celebrated Netflix competition. Two popular approaches for solving the problem are nuclear-norm-regularized matrix approximation (Candes and Tao, 2009, Mazumder, Hastie and Tibshirani, 2010), and maximum-margin matrix factorization (Srebro, Rennie and Jaakkola, 2005). These two procedures are in some cases solving equivalent problems, but with quite different algorithms. In this article we bring the two approaches together, leading to an efficient algorithm for large matrix factorization and completion that outperforms both of these. We develop a software package "softImpute" in R for implementing our approaches, and a distributed version for very large matrices using the "Spark" cluster programming environment.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

A Blockwise Descent Algorithm for Group-penalized Multiresponse and Multinomial Regression

In this paper we purpose a blockwise descent algorithm for group-penalized multiresponse regression. Using a quasi-newton framework we extend this to group-penalized multinomial regression. We give a publicly available implementation for these in R, and compare the speed of this algorithm to a competing algorithm --- we show that our implementation is an order of magnitude faster than its competitor, and can solve gene-expression-sized problems in real time.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

False Variable Selection Rates in Regression

There has been recent interest in extending the ideas of False Discovery Rates (FDR) to variable selection in regression settings. Traditionally the FDR in these settings has been defined in terms of the coefficients of the full regression model. Recent papers have struggled with controlling this quantity when the predictors are correlated. This paper shows that this full model definition of FDR suffers from unintuitive and potentially undesirable behavior in the presence of correlated predictors. We propose a new false selection error criterion, the False Variable Rate (FVR), that avoids these problems and behaves in a more intuitive manner. We discuss the behavior of this criterion and how it compares with the traditional FDR, as well as presenting guidelines for determining which is appropriate in a particular setting. Finally, we present a simple estimation procedure for FVR in stepwise variable selection. We analyze the performance of this estimator and draw connections to recent estimators in the literature.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

Learning interactions through hierarchical group-lasso regularization

We introduce a method for learning pairwise interactions in a manner that satisfies strong hierarchy: whenever an interaction is estimated to be nonzero, both its associated main effects are also included in the model. We motivate our approach by modeling pairwise interactions for categorical variables with arbitrary numbers of levels, and then show how we can accommodate continuous variables and mixtures thereof. Our approach allows us to dispense with explicitly applying constraints on the main effects and interactions for identifiability, which results in interpretable interaction models. We compare our method with existing approaches on both simulated and real data, including a genome wide association study, all using our R package glinternet.

0 citations0 reviewsNo code linkedOpen access
preprint2012arXiv

The Graphical Lasso: New Insights and Alternatives

The graphical lasso \citep{FHT2007a} is an algorithm for learning the structure in an undirected Gaussian graphical model, using $\ell_1$ regularization to control the number of zeros in the precision matrix ${\BΘ}={\BΣ}^{-1}$ \citep{BGA2008,yuan_lin_07}. The {\texttt R} package \GL\ \citep{FHT2007a} is popular, fast, and allows one to efficiently build a path of models for different values of the tuning parameter. Convergence of \GL\ can be tricky; the converged precision matrix might not be the inverse of the estimated covariance, and occasionally it fails to converge with warm starts. In this paper we explain this behavior, and propose new algorithms that appear to outperform \GL. By studying the "normal equations" we see that, \GL\ is solving the {\em dual} of the graphical lasso penalized likelihood, by block coordinate ascent; a result which can also be found in \cite{BGA2008}. In this dual, the target of estimation is $\BΣ$, the covariance matrix, rather than the precision matrix $\BΘ$. We propose similar primal algorithms \PGL\ and \DPGL, that also operate by block-coordinate descent, where $\BΘ$ is the optimization target. We study all of these algorithms, and in particular different approaches to solving their coordinate sub-problems. We conclude that \DPGL\ is superior from several points of view.

0 citations0 reviewsNo code linkedOpen access
preprint2011arXiv

Exact covariance thresholding into connected components for large-scale Graphical Lasso

We consider the sparse inverse covariance regularization problem or graphical lasso with regularization parameter $ρ$. Suppose the co- variance graph formed by thresholding the entries of the sample covariance matrix at $ρ$ is decomposed into connected components. We show that the vertex-partition induced by the thresholded covariance graph is exactly equal to that induced by the estimated concentration graph. This simple rule, when used as a wrapper around existing algorithms, leads to enormous performance gains. For large values of $ρ$, our proposal splits a large graphical lasso problem into smaller tractable problems, making it possible to solve an otherwise infeasible large scale graphical lasso problem.

0 citations0 reviewsNo code linkedOpen access
preprint2010arXiv

Strong rules for discarding predictors in lasso-type problems

We consider rules for discarding predictors in lasso regression and related problems, for computational efficiency. El Ghaoui et al (2010) propose "SAFE" rules that guarantee that a coefficient will be zero in the solution, based on the inner products of each predictor with the outcome. In this paper we propose strong rules that are not foolproof but rarely fail in practice. These can be complemented with simple checks of the Karush- Kuhn-Tucker (KKT) conditions to provide safe rules that offer substantial speed and space savings in a variety of statistical convex optimization problems.

0 citations0 reviewsNo code linkedOpen access
preprint2007arXiv

"Pre-conditioning" for feature selection and regression in high-dimensional problems

We consider regression problems where the number of predictors greatly exceeds the number of observations. We propose a method for variable selection that first estimates the regression function, yielding a "pre-conditioned" response variable. The primary method used for this initial regression is supervised principal components. Then we apply a standard procedure such as forward stepwise selection or the LASSO to the pre-conditioned response variable. In a number of simulated and real data examples, this two-step procedure outperforms forward stepwise selection or the usual LASSO (applied directly to the raw outcome). We also show that under a certain Gaussian latent variable model, application of the LASSO to the pre-conditioned response variable is consistent as the number of predictors and observations increases. Moreover, when the observational noise is rather large, the suggested procedure can give a more accurate estimate than LASSO. We illustrate our method on some real problems, including survival analysis with microarray data.

0 citations0 reviewsNo code linkedOpen access