arXiv subjects

Daniela M. Witten

Publications and source records attributed to Daniela M. Witten.

9 recordsLinked to original sources

arXiv2022

Tree-Values: selective inference for regression trees

We consider conducting inference on the output of the Classification and Regression Tree (CART) [Breiman et al., 1984] algorithm. A naive approach to inference that does not account for the fact that the tree was estimated from the data will not achieve standard guarantees, such as Type 1 error rate control and nominal coverage. Thus, we propose a selective inference framework for conducting inference on a fitted CART tree. In a nutshell, we condition on the fact that the tree was estimated from the data. We propose a test for the difference in the mean response between a pair of terminal nodes that controls the selective Type 1 error rate, and a confidence interval for the mean response within a single terminal node that attains the nominal selective coverage. Efficient algorithms for computing the necessary conditioning sets are provided. We apply these methods in simulation and to a dataset involving the association between portion control interventions and caloric intake.

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arXiv2022

Selective inference for k-means clustering

We consider the problem of testing for a difference in means between clusters of observations identified via k-means clustering. In this setting, classical hypothesis tests lead to an inflated Type I error rate. To overcome this problem, we take a selective inference approach. We propose a finite-sample p-value that controls the selective Type I error for a test of the difference in means between a pair of clusters obtained using k-means clustering, and show that it can be efficiently computed. We apply our proposal in simulation, and on hand-written digits data and single-cell RNA-sequencing data.

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arXiv2022

More powerful selective inference for the graph fused lasso

The graph fused lasso -- which includes as a special case the one-dimensional fused lasso -- is widely used to reconstruct signals that are piecewise constant on a graph, meaning that nodes connected by an edge tend to have identical values. We consider testing for a difference in the means of two connected components estimated using the graph fused lasso. A naive procedure such as a z-test for a difference in means will not control the selective Type I error, since the hypothesis that we are testing is itself a function of the data. In this work, we propose a new test for this task that controls the selective Type I error, and conditions on less information than existing approaches, leading to substantially higher power. We illustrate our approach in simulation and on datasets of drug overdose death rates and teenage birth rates in the contiguous United States. Our approach yields more discoveries on both datasets.

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arXiv2021

Quantifying uncertainty in spikes estimated from calcium imaging data

In recent years, a number of methods have been proposed to estimate the times at which a neuron spikes on the basis of calcium imaging data. However, quantifying the uncertainty associated with these estimated spikes remains an open problem. We consider a simple and well-studied model for calcium imaging data, which states that calcium decays exponentially in the absence of a spike, and instantaneously increases when a spike occurs. We wish to test the null hypothesis that the neuron did not spike -- i.e., that there was no increase in calcium -- at a particular timepoint at which a spike was estimated. In this setting, classical hypothesis tests lead to inflated Type I error, because the spike was estimated on the same data used for testing. To overcome this problem, we propose a selective inference approach. We describe an efficient algorithm to compute finite-sample p-values that control selective Type I error, and confidence intervals with correct selective coverage, for spikes estimated using a recent proposal from the literature. We apply our proposal in simulation and on calcium imaging data from the spikefinder challenge.

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arXiv2019

Adaptive Non-Parametric Regression With the $K$-NN Fused Lasso

The fused lasso, also known as total-variation denoising, is a locally-adaptive function estimator over a regular grid of design points. In this paper, we extend the fused lasso to settings in which the points do not occur on a regular grid, leading to an approach for non-parametric regression. This approach, which we call the $K$-nearest neighbors ($K$-NN) fused lasso, involves (i) computing the $K$-NN graph of the design points; and (ii) performing the fused lasso over this $K$-NN graph. We show that this procedure has a number of theoretical advantages over competing approaches: specifically, it inherits local adaptivity from its connection to the fused lasso, and it inherits manifold adaptivity from its connection to the $K$-NN approach. We show that excellent results are obtained in a simulation study and on an application to flu data. For completeness, we also study an estimator that makes use of an $ε$-graph rather than a $K$-NN graph, and contrast this with the $K$-NN fused lasso.

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arXiv2016

Network Reconstruction From High Dimensional Ordinary Differential Equations

We consider the task of learning a dynamical system from high-dimensional time-course data. For instance, we might wish to estimate a gene regulatory network from gene expression data measured at discrete time points. We model the dynamical system non-parametrically as a system of additive ordinary differential equations. Most existing methods for parameter estimation in ordinary differential equations estimate the derivatives from noisy observations. This is known to be challenging and inefficient. We propose a novel approach that does not involve derivative estimation. We show that the proposed method can consistently recover the true network structure even in high dimensions, and we demonstrate empirical improvement over competing approaches.

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arXiv2012

The joint graphical lasso for inverse covariance estimation across multiple classes

We consider the problem of estimating multiple related but distinct graphical models on the basis of a high-dimensional data set with observations that belong to distinct classes. A motivating example occurs in the analysis of gene expression data for tissue samples with and without cancer. In this case, we might wish to estimate a gene expression network for the normal tissue and a gene expression network for the tumor tissue. We expect the two gene expression networks to be similar but not identical to each other, and so more accurate estimation of these two networks may be possible using a joint approach. We propose the joint graphical lasso for this purpose. Rather than estimating a graphical model for each class separately, or a single graphical model across all classes, we borrow strength across the classes in order to estimate multiple graphical models that share certain characteristics, such as the locations or weights of nonzero edges. Our approach is based upon maximizing a penalized log likelihood. We employ fused lasso or group lasso penalties, and implement a very fast computational approach that solves the joint graphical lasso problem. In a simulation study we demonstrate that our proposed approach leads to more accurate estimation of networks and covariance structure than competing approaches. We further illustrate our proposal on a publicly-available lung cancer gene expression data set.

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arXiv2012

Classification and clustering of sequencing data using a Poisson model

In recent years, advances in high throughput sequencing technology have led to a need for specialized methods for the analysis of digital gene expression data. While gene expression data measured on a microarray take on continuous values and can be modeled using the normal distribution, RNA sequencing data involve nonnegative counts and are more appropriately modeled using a discrete count distribution, such as the Poisson or the negative binomial. Consequently, analytic tools that assume a Gaussian distribution (such as classification methods based on linear discriminant analysis and clustering methods that use Euclidean distance) may not perform as well for sequencing data as methods that are based upon a more appropriate distribution. Here, we propose new approaches for performing classification and clustering of observations on the basis of sequencing data. Using a Poisson log linear model, we develop an analog of diagonal linear discriminant analysis that is appropriate for sequencing data. We also propose an approach for clustering sequencing data using a new dissimilarity measure that is based upon the Poisson model. We demonstrate the performances of these approaches in a simulation study, on three publicly available RNA sequencing data sets, and on a publicly available chromatin immunoprecipitation sequencing data set.

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arXiv2008

Testing significance of features by lassoed principal components

We consider the problem of testing the significance of features in high-dimensional settings. In particular, we test for differentially-expressed genes in a microarray experiment. We wish to identify genes that are associated with some type of outcome, such as survival time or cancer type. We propose a new procedure, called Lassoed Principal Components (LPC), that builds upon existing methods and can provide a sizable improvement. For instance, in the case of two-class data, a standard (albeit simple) approach might be to compute a two-sample $t$-statistic for each gene. The LPC method involves projecting these conventional gene scores onto the eigenvectors of the gene expression data covariance matrix and then applying an $L_1$ penalty in order to de-noise the resulting projections. We present a theoretical framework under which LPC is the logical choice for identifying significant genes, and we show that LPC can provide a marked reduction in false discovery rates over the conventional methods on both real and simulated data. Moreover, this flexible procedure can be applied to a variety of types of data and can be used to improve many existing methods for the identification of significant features.

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