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Ryan J. Tibshirani

Publications and source records attributed to Ryan J. Tibshirani.

At least 73 records · Page 4Linked to original sources
arXiv2017

Excess Optimism: How Biased is the Apparent Error of an Estimator Tuned by SURE?

Nearly all estimators in statistical prediction come with an associated tuning parameter, in one way or another. Common practice, given data, is to choose the tuning parameter value that minimizes a constructed estimate of the prediction error of the estimator; we focus on Stein's unbiased risk estimator, or SURE (Stein, 1981; Efron, 1986) which forms an unbiased estimate of the prediction error by augmenting the observed training error with an estimate of the degrees of freedom of the estimator. Parameter tuning via SURE minimization has been advocated by many authors, in a wide variety of problem settings, and in general, it is natural to ask: what is the prediction error of the SURE-tuned estimator? An obvious strategy would be simply use the apparent error estimate as reported by SURE, i.e., the value of the SURE criterion at its minimum, to estimate the prediction error of the SURE-tuned estimator. But this is no longer unbiased; in fact, we would expect that the minimum of the SURE criterion is systematically biased downwards for the true prediction error. In this paper, we formally describe and study this bias.

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arXiv2016

Approximate Recovery in Changepoint Problems, from $\ell_2$ Estimation Error Rates

In the 1-dimensional multiple changepoint detection problem, we prove that any procedure with a fast enough $\ell_2$ error rate, in terms of its estimation of the underlying piecewise constant mean vector, automatically has an (approximate) changepoint screening property---specifically, each true jump in the underlying mean vector has an estimated jump nearby. We also show, again assuming only knowledge of the $\ell_2$ error rate, that a simple post-processing step can be used to eliminate spurious estimated changepoints, and thus delivers an (approximate) changepoint recovery property---specifically, in addition to the screening property described above, we are assured that each estimated jump has a true jump nearby. As a special case, we focus on the application of these results to the 1-dimensional fused lasso, i.e., 1-dimensional total variation denoising, and compare the implications with existing results from the literature. We also study extensions to related problems, such as changepoint detection over graphs.

stat.ME↗
arXiv2016

The Multiple Quantile Graphical Model

We introduce the Multiple Quantile Graphical Model (MQGM), which extends the neighborhood selection approach of Meinshausen and Buhlmann for learning sparse graphical models. The latter is defined by the basic subproblem of modeling the conditional mean of one variable as a sparse function of all others. Our approach models a set of conditional quantiles of one variable as a sparse function of all others, and hence offers a much richer, more expressive class of conditional distribution estimates. We establish that, under suitable regularity conditions, the MQGM identifies the exact conditional independencies with probability tending to one as the problem size grows, even outside of the usual homoskedastic Gaussian data model. We develop an efficient algorithm for fitting the MQGM using the alternating direction method of multipliers. We also describe a strategy for sampling from the joint distribution that underlies the MQGM estimate. Lastly, we present detailed experiments that demonstrate the flexibility and effectiveness of the MQGM in modeling hetereoskedastic non-Gaussian data.

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arXiv2016

Exact Post-Selection Inference for Changepoint Detection and Other Generalized Lasso Problems

We study tools for inference conditioned on model selection events that are defined by the generalized lasso regularization path. The generalized lasso estimate is given by the solution of a penalized least squares regression problem, where the penalty is the l1 norm of a matrix D times the coefficient vector. The generalized lasso path collects these estimates for a range of penalty parameter (λ) values. Leveraging a sequential characterization of this path from Tibshirani & Taylor (2011), and recent advances in post-selection inference from Lee et al. (2016), Tibshirani et al. (2016), we develop exact hypothesis tests and confidence intervals for linear contrasts of the underlying mean vector, conditioned on any model selection event along the generalized lasso path (assuming Gaussian errors in the observations). By inspecting specific choices of D, we obtain post-selection tests and confidence intervals for specific cases of generalized lasso estimates, such as the fused lasso, trend filtering, and the graph fused lasso. In the fused lasso case, the underlying coordinates of the mean are assigned a linear ordering, and our framework allows us to test selectively chosen breakpoints or changepoints in these mean coordinates. This is an interesting and well-studied problem with broad applications, our framework applied to the trend filtering and graph fused lasso serves several applications as well. Aside from the development of selective inference tools, we describe several practical aspects of our methods such as valid post-processing of generalized estimates before performing inference in order to improve power, and problem-specific visualization aids that may be given to the data analyst for he/she to choose linear contrasts to be tested. Many examples, both from simulated and real data sources, are presented to examine the empirical properties of our inference methods.

stat.ME↗
arXiv2016

Trend Filtering on Graphs

We introduce a family of adaptive estimators on graphs, based on penalizing the $\ell_1$ norm of discrete graph differences. This generalizes the idea of trend filtering [Kim et al. (2009), Tibshirani (2014)], used for univariate nonparametric regression, to graphs. Analogous to the univariate case, graph trend filtering exhibits a level of local adaptivity unmatched by the usual $\ell_2$-based graph smoothers. It is also defined by a convex minimization problem that is readily solved (e.g., by fast ADMM or Newton algorithms). We demonstrate the merits of graph trend filtering through examples and theory.

stat.ML↗
arXiv2016

Nonparametric modal regression

Modal regression estimates the local modes of the distribution of $Y$ given $X=x$, instead of the mean, as in the usual regression sense, and can hence reveal important structure missed by usual regression methods. We study a simple nonparametric method for modal regression, based on a kernel density estimate (KDE) of the joint distribution of $Y$ and $X$. We derive asymptotic error bounds for this method, and propose techniques for constructing confidence sets and prediction sets. The latter is used to select the smoothing bandwidth of the underlying KDE. The idea behind modal regression is connected to many others, such as mixture regression and density ridge estimation, and we discuss these ties as well.

stat.ME↗
arXiv2015

Exact Post-Selection Inference for Sequential Regression Procedures

We propose new inference tools for forward stepwise regression, least angle regression, and the lasso. Assuming a Gaussian model for the observation vector y, we first describe a general scheme to perform valid inference after any selection event that can be characterized as y falling into a polyhedral set. This framework allows us to derive conditional (post-selection) hypothesis tests at any step of forward stepwise or least angle regression, or any step along the lasso regularization path, because, as it turns out, selection events for these procedures can be expressed as polyhedral constraints on y. The p-values associated with these tests are exactly uniform under the null distribution, in finite samples, yielding exact type I error control. The tests can also be inverted to produce confidence intervals for appropriate underlying regression parameters. The R package "selectiveInference", freely available on the CRAN repository, implements the new inference tools described in this paper.

stat.ME↗
arXiv2015

Fast and Flexible ADMM Algorithms for Trend Filtering

This paper presents a fast and robust algorithm for trend filtering, a recently developed nonparametric regression tool. It has been shown that, for estimating functions whose derivatives are of bounded variation, trend filtering achieves the minimax optimal error rate, while other popular methods like smoothing splines and kernels do not. Standing in the way of a more widespread practical adoption, however, is a lack of scalable and numerically stable algorithms for fitting trend filtering estimates. This paper presents a highly efficient, specialized ADMM routine for trend filtering. Our algorithm is competitive with the specialized interior point methods that are currently in use, and yet is far more numerically robust. Furthermore, the proposed ADMM implementation is very simple, and importantly, it is flexible enough to extend to many interesting related problems, such as sparse trend filtering and isotonic trend filtering. Software for our method is freely available, in both the C and R languages.

stat.ML↗
arXiv2015

A General Framework for Fast Stagewise Algorithms

Forward stagewise regression follows a very simple strategy for constructing a sequence of sparse regression estimates: it starts with all coefficients equal to zero, and iteratively updates the coefficient (by a small amount $ε$) of the variable that achieves the maximal absolute inner product with the current residual. This procedure has an interesting connection to the lasso: under some conditions, it is known that the sequence of forward stagewise estimates exactly coincides with the lasso path, as the step size $ε$ goes to zero. Furthermore, essentially the same equivalence holds outside of least squares regression, with the minimization of a differentiable convex loss function subject to an $\ell_1$ norm constraint (the stagewise algorithm now updates the coefficient corresponding to the maximal absolute component of the gradient). Even when they do not match their $\ell_1$-constrained analogues, stagewise estimates provide a useful approximation, and are computationally appealing. Their success in sparse modeling motivates the question: can a simple, effective strategy like forward stagewise be applied more broadly in other regularization settings, beyond the $\ell_1$ norm and sparsity? The current paper is an attempt to do just this. We present a general framework for stagewise estimation, which yields fast algorithms for problems such as group-structured learning, matrix completion, image denoising, and more.

stat.ML↗
arXiv2015

High-Dimensional Longitudinal Classification with the Multinomial Fused Lasso

We study regularized estimation in high-dimensional longitudinal classification problems, using the lasso and fused lasso regularizers. The constructed coefficient estimates are piecewise constant across the time dimension in the longitudinal problem, with adaptively selected change points (break points). We present an efficient algorithm for computing such estimates, based on proximal gradient descent. We apply our proposed technique to a longitudinal data set on Alzheimer's disease from the Cardiovascular Health Study Cognition Study, and use this data set to motivate and demonstrate several practical considerations such as the selection of tuning parameters, and the assessment of model stability.

stat.AP↗
arXiv2014

Degrees of Freedom and Model Search

Degrees of freedom is a fundamental concept in statistical modeling, as it provides a quantitative description of the amount of fitting performed by a given procedure. But, despite this fundamental role in statistics, its behavior not completely well-understood, even in some fairly basic settings. For example, it may seem intuitively obvious that the best subset selection fit with subset size k has degrees of freedom larger than k, but this has not been formally verified, nor has is been precisely studied. In large part, the current paper is motivated by this particular problem, and we derive an exact expression for the degrees of freedom of best subset selection in a restricted setting (orthogonal predictor variables). Along the way, we develop a concept that we name "search degrees of freedom"; intuitively, for adaptive regression procedures that perform variable selection, this is a part of the (total) degrees of freedom that we attribute entirely to the model selection mechanism. Finally, we establish a modest extension of Stein's formula to cover discontinuous functions, and discuss its potential role in degrees of freedom and search degrees of freedom calculations.

math.ST↗
arXiv2014

The Falling Factorial Basis and Its Statistical Applications

We study a novel spline-like basis, which we name the "falling factorial basis", bearing many similarities to the classic truncated power basis. The advantage of the falling factorial basis is that it enables rapid, linear-time computations in basis matrix multiplication and basis matrix inversion. The falling factorial functions are not actually splines, but are close enough to splines that they provably retain some of the favorable properties of the latter functions. We examine their application in two problems: trend filtering over arbitrary input points, and a higher-order variant of the two-sample Kolmogorov-Smirnov test.

stat.ML↗
arXiv2014

Flexible Modeling of Epidemics with an Empirical Bayes Framework

Seasonal influenza epidemics cause consistent, considerable, widespread loss annually in terms of economic burden, morbidity, and mortality. With access to accurate and reliable forecasts of a current or upcoming influenza epidemic's behavior, policy makers can design and implement more effective countermeasures. We developed a framework for in-season forecasts of epidemics using a semiparametric Empirical Bayes framework, and applied it to predict the weekly percentage of outpatient doctors visits for influenza-like illness, as well as the season onset, duration, peak time, and peak height, with and without additional data from Google Flu Trends, as part of the CDC's 2013--2014 "Predict the Influenza Season Challenge". Previous work on epidemic modeling has focused on developing mechanistic models of disease behavior and applying time series tools to explain historical data. However, these models may not accurately capture the range of possible behaviors that we may see in the future. Our approach instead produces possibilities for the epidemic curve of the season of interest using modified versions of data from previous seasons, allowing for reasonable variations in the timing, pace, and intensity of the seasonal epidemics, as well as noise in observations. Since the framework does not make strict domain-specific assumptions, it can easily be applied to other diseases as well. Another important advantage of this method is that it produces a complete posterior distribution for any desired forecasting target, rather than mere point predictions. We report prospective influenza-like-illness forecasts that were made for the 2013--2014 U.S. influenza season, and compare the framework's cross-validated prediction error on historical data to that of a variety of simpler baseline predictors.

q-bio.PE↗
arXiv2014

A significance test for the lasso

In the sparse linear regression setting, we consider testing the significance of the predictor variable that enters the current lasso model, in the sequence of models visited along the lasso solution path. We propose a simple test statistic based on lasso fitted values, called the covariance test statistic, and show that when the true model is linear, this statistic has an $\operatorname {Exp}(1)$ asymptotic distribution under the null hypothesis (the null being that all truly active variables are contained in the current lasso model). Our proof of this result for the special case of the first predictor to enter the model (i.e., testing for a single significant predictor variable against the global null) requires only weak assumptions on the predictor matrix $X$. On the other hand, our proof for a general step in the lasso path places further technical assumptions on $X$ and the generative model, but still allows for the important high-dimensional case $p>n$, and does not necessarily require that the current lasso model achieves perfect recovery of the truly active variables. Of course, for testing the significance of an additional variable between two nested linear models, one typically uses the chi-squared test, comparing the drop in residual sum of squares (RSS) to a $χ^2_1$ distribution. But when this additional variable is not fixed, and has been chosen adaptively or greedily, this test is no longer appropriate: adaptivity makes the drop in RSS stochastically much larger than $χ^2_1$ under the null hypothesis. Our analysis explicitly accounts for adaptivity, as it must, since the lasso builds an adaptive sequence of linear models as the tuning parameter $λ$ decreases. In this analysis, shrinkage plays a key role: though additional variables are chosen adaptively, the coefficients of lasso active variables are shrunken due to the $\ell_1$ penalty. Therefore, the test statistic (which is based on lasso fitted values) is in a sense balanced by these two opposing properties - adaptivity and shrinkage - and its null distribution is tractable and asymptotically $\operatorname {Exp}(1)$.

math.ST↗
arXiv2014

Adaptive piecewise polynomial estimation via trend filtering

We study trend filtering, a recently proposed tool of Kim et al. [SIAM Rev. 51 (2009) 339-360] for nonparametric regression. The trend filtering estimate is defined as the minimizer of a penalized least squares criterion, in which the penalty term sums the absolute $k$th order discrete derivatives over the input points. Perhaps not surprisingly, trend filtering estimates appear to have the structure of $k$th degree spline functions, with adaptively chosen knot points (we say ``appear'' here as trend filtering estimates are not really functions over continuous domains, and are only defined over the discrete set of inputs). This brings to mind comparisons to other nonparametric regression tools that also produce adaptive splines; in particular, we compare trend filtering to smoothing splines, which penalize the sum of squared derivatives across input points, and to locally adaptive regression splines [Ann. Statist. 25 (1997) 387-413], which penalize the total variation of the $k$th derivative. Empirically, we discover that trend filtering estimates adapt to the local level of smoothness much better than smoothing splines, and further, they exhibit a remarkable similarity to locally adaptive regression splines. We also provide theoretical support for these empirical findings; most notably, we prove that (with the right choice of tuning parameter) the trend filtering estimate converges to the true underlying function at the minimax rate for functions whose $k$th derivative is of bounded variation. This is done via an asymptotic pairing of trend filtering and locally adaptive regression splines, which have already been shown to converge at the minimax rate [Ann. Statist. 25 (1997) 387-413]. At the core of this argument is a new result tying together the fitted values of two lasso problems that share the same outcome vector, but have different predictor matrices.

math.ST↗
arXiv2012

The solution path of the generalized lasso

We present a path algorithm for the generalized lasso problem. This problem penalizes the $\ell_1$ norm of a matrix D times the coefficient vector, and has a wide range of applications, dictated by the choice of D. Our algorithm is based on solving the dual of the generalized lasso, which greatly facilitates computation of the path. For $D=I$ (the usual lasso), we draw a connection between our approach and the well-known LARS algorithm. For an arbitrary D, we derive an unbiased estimate of the degrees of freedom of the generalized lasso fit. This estimate turns out to be quite intuitive in many applications.

math.ST↗
arXiv2012

The Lasso Problem and Uniqueness

The lasso is a popular tool for sparse linear regression, especially for problems in which the number of variables p exceeds the number of observations n. But when p>n, the lasso criterion is not strictly convex, and hence it may not have a unique minimum. An important question is: when is the lasso solution well-defined (unique)? We review results from the literature, which show that if the predictor variables are drawn from a continuous probability distribution, then there is a unique lasso solution with probability one, regardless of the sizes of n and p. We also show that this result extends easily to $\ell_1$ penalized minimization problems over a wide range of loss functions. A second important question is: how can we deal with the case of non-uniqueness in lasso solutions? In light of the aforementioned result, this case really only arises when some of the predictor variables are discrete, or when some post-processing has been performed on continuous predictor measurements. Though we certainly cannot claim to provide a complete answer to such a broad question, we do present progress towards understanding some aspects of non-uniqueness. First, we extend the LARS algorithm for computing the lasso solution path to cover the non-unique case, so that this path algorithm works for any predictor matrix. Next, we derive a simple method for computing the component-wise uncertainty in lasso solutions of any given problem instance, based on linear programming. Finally, we review results from the literature on some of the unifying properties of lasso solutions, and also point out particular forms of solutions that have distinctive properties.

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