arXiv subjects

Andreas Buja

Publications and source records attributed to Andreas Buja.

At least 19 recordsLinked to original sources

arXiv2022

Hole or grain? A Section Pursuit Index for Finding Hidden Structure in Multiple Dimensions

Multivariate data is often visualized using linear projections, produced by techniques such as principal component analysis, linear discriminant analysis, and projection pursuit. A problem with projections is that they obscure low and high density regions near the center of the distribution. Sections, or slices, can help to reveal them. This paper develops a section pursuit method, building on the extensive work in projection pursuit, to search for interesting slices of the data. Linear projections are used to define sections of the parameter space, and to calculate interestingness by comparing the distribution of observations, inside and outside a section. By optimizing this index, it is possible to reveal features such as holes (low density) or grains (high density). The optimization is incorporated into a guided tour so that the search for structure can be dynamic. The approach can be useful for problems when data distributions depart from uniform or normal, as in visually exploring nonlinear manifolds, and functions in multivariate space. Two applications of section pursuit are shown: exploring decision boundaries from classification models, and exploring subspaces induced by complex inequality conditions from multiple parameter model. The new methods are available in R, in the tourr package.

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arXiv2021

Uniform-in-Submodel Bounds for Linear Regression in a Model Free Framework

For the last two decades, high-dimensional data and methods have proliferated throughout the literature. Yet, the classical technique of linear regression has not lost its usefulness in applications. In fact, many high-dimensional estimation techniques can be seen as variable selection that leads to a smaller set of variables (a ``sub-model'') where classical linear regression applies. We analyze linear regression estimators resulting from model-selection by proving estimation error and linear representation bounds uniformly over sets of submodels. Based on deterministic inequalities, our results provide ``good'' rates when applied to both independent and dependent data. These results are useful in meaningfully interpreting the linear regression estimator obtained after exploring and reducing the variables and also in justifying post model-selection inference. All results are derived under no model assumptions and are non-asymptotic in nature.

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arXiv2021

Semi-Supervised linear regression

We study a regression problem where for some part of the data we observe both the label variable ($Y$) and the predictors (${\bf X}$), while for other part of the data only the predictors are given. Such a problem arises, for example, when observations of the label variable are costly and may require a skilled human agent. When the conditional expectation $E[Y | {\bf X}]$ is not exactly linear, one can consider the best linear approximation to the conditional expectation, which can be estimated consistently by the least squares estimates (LSE). The latter depends only on the labeled data. We suggest improved alternative estimates to the LSE that use also the unlabeled data. Our estimation method can be easily implemented and has simply described asymptotic properties.The new estimates asymptotically dominate the usual standard procedures under certain non-linearity condition of $E[Y | {\bf X}]$; otherwise, they are asymptotically equivalent.The performance of the new estimator for small sample size is investigated in an extensive simulation study. A real data example of inferring homeless population is used to illustrate the new methodology.

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arXiv2019

All of Linear Regression

Least squares linear regression is one of the oldest and widely used data analysis tools. Although the theoretical analysis of the ordinary least squares (OLS) estimator is as old, several fundamental questions are yet to be answered. Suppose regression observations $(X_1,Y_1),\ldots,(X_n,Y_n)\in\mathbb{R}^d\times\mathbb{R}$ (not necessarily independent) are available. Some of the questions we deal with are as follows: under what conditions, does the OLS estimator converge and what is the limit? What happens if the dimension is allowed to grow with $n$? What happens if the observations are dependent with dependence possibly strengthening with $n$? How to do statistical inference under these kinds of misspecification? What happens to the OLS estimator under variable selection? How to do inference under misspecification and variable selection? We answer all the questions raised above with one simple deterministic inequality which holds for any set of observations and any sample size. This implies that all our results are a finite sample (non-asymptotic) in nature. In the end, one only needs to bound certain random quantities under specific settings of interest to get concrete rates and we derive these bounds for the case of independent observations. In particular, the problem of inference after variable selection is studied, for the first time, when $d$, the number of covariates increases (almost exponentially) with sample size $n$. We provide comments on the ``right'' statistic to consider for inference under variable selection and efficient computation of quantiles.

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arXiv2019

Models as Approximations I: Consequences Illustrated with Linear Regression

In the early 1980s Halbert White inaugurated a "model-robust'' form of statistical inference based on the "sandwich estimator'' of standard error. This estimator is known to be "heteroskedasticity-consistent", but it is less well-known to be "nonlinearity-consistent'' as well. Nonlinearity, however, raises fundamental issues because in its presence regressors are not ancillary, hence can't be treated as fixed. The consequences are deep: (1)~population slopes need to be re-interpreted as statistical functionals obtained from OLS fits to largely arbitrary joint $\xy$~distributions; (2)~the meaning of slope parameters needs to be rethought; (3)~the regressor distribution affects the slope parameters; (4)~randomness of the regressors becomes a source of sampling variability in slope estimates; (5)~inference needs to be based on model-robust standard errors, including sandwich estimators or the $\xy$~bootstrap. In theory, model-robust and model-trusting standard errors can deviate by arbitrary magnitudes either way. In practice, significant deviations between them can be detected with a diagnostic test.

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arXiv2019

Models as Approximations II: A Model-Free Theory of Parametric Regression

We develop a model-free theory of general types of parametric regression for iid observations. The theory replaces the parameters of parametric models with statistical functionals, to be called "regression functionals'', defined on large non-parametric classes of joint $\xy$ distributions, without assuming a correct model. Parametric models are reduced to heuristics to suggest plausible objective functions. An example of a regression functional is the vector of slopes of linear equations fitted by OLS to largely arbitrary $\xy$ distributions, without assuming a linear model (see Part~I). More generally, regression functionals can be defined by minimizing objective functions or solving estimating equations at joint $\xy$ distributions. In this framework it is possible to achieve the following: (1)~define a notion of well-specification for regression functionals that replaces the notion of correct specification of models, (2)~propose a well-specification diagnostic for regression functionals based on reweighting distributions and data, (3)~decompose sampling variability of regression functionals into two sources, one due to the conditional response distribution and another due to the regressor distribution interacting with misspecification, both of order $N^{-1/2}$, (4)~exhibit plug-in/sandwich estimators of standard error as limit cases of $\xy$ bootstrap estimators, and (5)~provide theoretical heuristics to indicate that $\xy$ bootstrap standard errors may generally be more stable than sandwich estimators.

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arXiv2018

Model-free Study of Ordinary Least Squares Linear Regression

Ordinary least squares (OLS) linear regression is one of the most basic statistical techniques for data analysis. In the main stream literature and the statistical education, the study of linear regression is typically restricted to the case where the covariates are fixed, errors are mean zero Gaussians with variance independent of the (fixed) covariates. Even though OLS has been studied under misspecification from as early as the 1960's, the implications have not yet caught up with the main stream literature and applied sciences. The present article is an attempt at a unified viewpoint that makes the various implications of misspecification stand out.

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arXiv2018

Using Recursive Partitioning to Find and Estimate Heterogenous Treatment Effects In Randomized Clinical Trials

Heterogeneous treatment effects can be very important in the analysis of randomized clinical trials. Heightened risks or enhanced benefits may exist for particular subsets of study subjects. When the heterogeneous treatment effects are specified as the research is being designed, there are proper and readily available analysis techniques. When the heterogeneous treatment effects are inductively obtained as an experiment's data are analyzed, significant complications are introduced. There can be a need for special loss functions designed to find local average treatment effects and for techniques that properly address post selection statistical inference. In this paper, we tackle both while undertaking a recursive partitioning analysis of a randomized clinical trial testing whether individuals on probation, who are low risk, can be minimally supervised with no increase in recidivism.

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arXiv2018

Assumption Lean Regression

It is well known that models used in conventional regression analysis are commonly misspecified. A standard response is little more than a shrug. Data analysts invoke Box's maxim that all models are wrong and then proceed as if the results are useful nevertheless. In this paper, we provide an alternative. Regression models are treated explicitly as approximations of a true response surface that can have a number of desirable statistical properties, including estimates that are asymptotically unbiased. Valid statistical inference follows. We generalize the formulation to include regression functionals, which broadens substantially the range of potential applications. An empirical application is provided to illustrate the paper's key concepts.

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arXiv2018

Valid Post-selection Inference in Assumption-lean Linear Regression

Construction of valid statistical inference for estimators based on data-driven selection has received a lot of attention in the recent times. Berk et al. (2013) is possibly the first work to provide valid inference for Gaussian homoscedastic linear regression with fixed covariates under arbitrary covariate/variable selection. The setting is unrealistic and is extended by Bachoc et al. (2016) by relaxing the distributional assumptions. A major drawback of the aforementioned works is that the construction of valid confidence regions is computationally intensive. In this paper, we first prove that post-selection inference is equivalent to simultaneous inference and then construct valid post-selection confidence regions which are computationally simple. Our construction is based on deterministic inequalities and apply to independent as well as dependent random variables without the requirement of correct distributional assumptions. Finally, we compare the volume of our confidence regions with the existing ones and show that under non-stochastic covariates, our regions are much smaller.

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arXiv2017

Calibrated Percentile Double Bootstrap For Robust Linear Regression Inference

We consider inference for the parameters of a linear model when the covariates are random and the relationship between response and covariates is possibly non-linear. Conventional inference methods such as z-intervals perform poorly in these cases. We propose a double bootstrap-based calibrated percentile method, perc-cal, as a general-purpose CI method which performs very well relative to alternative methods in challenging situations such as these. The superior performance of perc-cal is demonstrated by a thorough, full-factorial design synthetic data study as well as a real data example involving the length of criminal sentences. We also provide theoretical justification for the perc-cal method under mild conditions. The method is implemented in the R package `perccal', available through CRAN and coded primarily in C++, to make it easier for practitioners to use.

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arXiv2016

Smoothing Effects of Bagging: Von Mises Expansions of Bagged Statistical Functionals

Bagging is a device intended for reducing the prediction error of learning algorithms. In its simplest form, bagging draws bootstrap samples from the training sample, applies the learning algorithm to each bootstrap sample, and then averages the resulting prediction rules. We extend the definition of bagging from statistics to statistical functionals and study the von Mises expansion of bagged statistical functionals. We show that the expansion is related to the Efron-Stein ANOVA expansion of the raw (unbagged) functional. The basic observation is that a bagged functional is always smooth in the sense that the von Mises expansion exists and is finite of length 1 + resample size $M$. This holds even if the raw functional is rough or unstable. The resample size $M$ acts as a smoothing parameter, where a smaller $M$ means more smoothing.

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arXiv2015

Kernel Additive Principal Components

Additive principal components (APCs for short) are a nonlinear generalization of linear principal components. We focus on smallest APCs to describe additive nonlinear constraints that are approximately satisfied by the data. Thus APCs fit data with implicit equations that treat the variables symmetrically, as opposed to regression analyses which fit data with explicit equations that treat the data asymmetrically by singling out a response variable. We propose a regularized data-analytic procedure for APC estimation using kernel methods. In contrast to existing approaches to APCs that are based on regularization through subspace restriction, kernel methods achieve regularization through shrinkage and therefore grant distinctive flexibility in APC estimation by allowing the use of infinite-dimensional functions spaces for searching APC transformation while retaining computational feasibility. To connect population APCs and kernelized finite-sample APCs, we study kernelized population APCs and their associated eigenproblems, which eventually lead to the establishment of consistency of the estimated APCs. Lastly, we discuss an iterative algorithm for computing kernelized finite-sample APCs.

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arXiv2014

Rate Optimal Denoising of Simultaneously Sparse and Low Rank Matrices

We study minimax rates for denoising simultaneously sparse and low rank matrices in high dimensions. We show that an iterative thresholding algorithm achieves (near) optimal rates adaptively under mild conditions for a large class of loss functions. Numerical experiments on synthetic datasets also demonstrate the competitive performance of the proposed method.

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arXiv2013

Improved Precision in Estimating Average Treatment Effects

The Average Treatment Effect (ATE) is a global measure of the effectiveness of an experimental treatment intervention. Classical methods of its estimation either ignore relevant covariates or do not fully exploit them. Moreover, past work has considered covariates as fixed. We present a method for improving the precision of the ATE estimate: the treatment and control responses are estimated via a regression, and information is pooled between the groups to produce an asymptotically unbiased estimate; we subsequently justify the random X paradigm underlying the result. Standard errors are derived, and the estimator's performance is compared to the traditional estimator. Conditions under which the regression-based estimator is preferable are detailed, and a demonstration on real data is presented.

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arXiv2013

Penalized Split Criteria for Interpretable Trees

This paper describes techniques for growing classification and regression trees designed to induce visually interpretable trees. This is achieved by penalizing splits that extend the subset of features used in a particular branch of the tree. After a brief motivation, we summarize existing methods and introduce new ones, providing illustrative examples throughout. Using a number of real classification and regression datasets, we find that these procedures can offer more interpretable fits than the CART methodology with very modest increases in out-of-sample loss.

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arXiv2013

Valid post-selection inference

It is common practice in statistical data analysis to perform data-driven variable selection and derive statistical inference from the resulting model. Such inference enjoys none of the guarantees that classical statistical theory provides for tests and confidence intervals when the model has been chosen a priori. We propose to produce valid ``post-selection inference'' by reducing the problem to one of simultaneous inference and hence suitably widening conventional confidence and retention intervals. Simultaneity is required for all linear functions that arise as coefficient estimates in all submodels. By purchasing ``simultaneity insurance'' for all possible submodels, the resulting post-selection inference is rendered universally valid under all possible model selection procedures. This inference is therefore generally conservative for particular selection procedures, but it is always less conservative than full Scheffe protection. Importantly it does not depend on the truth of the selected submodel, and hence it produces valid inference even in wrong models. We describe the structure of the simultaneous inference problem and give some asymptotic results.

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arXiv2011

A Sparse SVD Method for High-dimensional Data

We present a new computational approach to approximating a large, noisy data table by a low-rank matrix with sparse singular vectors. The approximation is obtained from thresholded subspace iterations that produce the singular vectors simultaneously, rather than successively as in competing proposals. We introduce novel ways to estimate thresholding parameters which obviate the need for computationally expensive cross-validation. We also introduce a way to sparsely initialize the algorithm for computational savings that allow our algorithm to outperform the vanilla SVD on the full data table when the signal is sparse. A comparison with two existing sparse SVD methods suggests that our algorithm is computationally always faster and statistically always at least comparable to the better of the two competing algorithms.

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