arXiv subjects

Armin Schwartzman

Publications and source records attributed to Armin Schwartzman.

At least 19 recordsLinked to original sources

arXiv2026

False Discovery Controlled Regions for Excursion Sets

Identifying areas where the signal is prominent is an important task in image analysis, with particular applications in brain mapping. In this work, we develop False Discovery Controlled Regions (FDCRs) for excursion sets, the set of locations where the values are above or below a given level. We achieve this by treating the confidence procedure as a testing problem at the given level, allowing control of the False Discovery Rate (FDR). Methods are developed to control the FDR, separately for positive and negative excursions, as well as jointly over both. Furthermore, power is increased by incorporating a two-stage adaptive procedure. Simulation results in a variety of different settings show that the FDCRs successfully control the FDR under the nominal alpha level. We showcase our methods with an application to functional magnetic resonance imaging (fMRI) data from the Human Connectome Project illustrating the improvement in statistical power over existing approaches.

stat.ME↗

arXiv2026

Principal Components Decomposition of Fraction of Variance Explained in High Dimensional Linear Models with Strong Correlation

The fraction of variance explained (FVE) in a linear model quantifies the extent to which predictors account for outcome variability. In high-dimensional settings, where traditional FVE estimators do not apply, modern FVE estimators such as GWASH or linear mix-effect model estimated through the restricted maximum likelihood (LMM-REML) struggle with strong correlation among predictors, often found, for example, in brain imaging data. We propose a decomposition framework that partitions the FVE into two components: a low-dimensional component capturing the strong correlation, estimable by low dimensional methods, and a high-dimensional component with remaining weak correlation, estimable by high dimensional methods. Simulations demonstrate that decomposing dominant principal components (PCs) and estimating the high-dimensional FVE using GWASH or LMM-REML leads to improved bias reduction compared to directly applying standard approaches such as GWASH and LMM-REML. Our method shows consistent performance asymptotically as both the number of predictors and the number of samples increase. We illustrate the method in an analysis of the Adolescent Brain Cognitive Development (ABCD) brain imaging dataset, capturing nuanced heritability signals in the FVE of cognitive measures predicted by high-resolution brain imaging data.

stat.ME↗

arXiv2025

SCoRES: An R Package for Simultaneous Confidence Region Estimates

The identification of domain sets whose outcomes belong to predefined subsets can address fundamental risk assessment challenges in climatology and medicine. Existing approaches for inverse domain estimates require restrictive assumptions, including domain density and continuity of function near thresholds, and large-sample guarantees, which limit the applicability. Besides, the estimation and coverage depend on setting a fixed threshold level, which is difficult to determine. Recently, Ren et al. (2024) proved that confidence sets of multiple levels can be simultaneously constructed with the desired confidence non-asymptotically through inverting simultaneous confidence bands. Here, we present the SCoRES R package, which implements Ren's approach for both the estimation of the inverse region and the corresponding simultaneous outer and inner confidence regions, along with visualization tools. Besides, the package also provides functions that help construct SCBs for regression data, functional data and geographical data. To illustrate its broad applicability, we present three rigorous examples that demonstrate the SCoRES workflow.

stat.CO↗

arXiv2025

Model uncertainty quantification using feature confidence sets for outcome excursions

When implementing prediction models for high-stakes real-world applications such as medicine, finance, and autonomous systems, quantifying prediction uncertainty is critical for effective risk management. Traditional approaches to uncertainty quantification, such as confidence and prediction intervals, provide probability coverage guarantees for the expected outcomes $f(\boldsymbol{x})$ or the realized outcomes $f(\boldsymbol{x})+ε$. Instead, this paper introduces a novel, model-agnostic framework for quantifying uncertainty in continuous and binary outcomes using confidence sets for outcome excursions, where the goal is to identify a subset of the feature space where the expected or realized outcome exceeds a specific value. The proposed method constructs data-dependent inner and outer confidence sets that aim to contain the true feature subset for which the expected or realized outcomes of these features exceed a specified threshold. We establish theoretical guarantees for the probability that these confidence sets contain the true feature subset, both asymptotically and for finite sample sizes. The framework is validated through simulations and applied to real-world datasets, demonstrating its utility in contexts such as housing price prediction and time to sepsis diagnosis in healthcare. This approach provides a unified method for uncertainty quantification that is broadly applicable across various continuous and binary prediction models.

stat.ML↗

arXiv2025

Consistency of heritability estimation from summary statistics in high-dimensional linear models

In Genome-Wide Association Studies (GWAS), heritability is defined as the fraction of variance of an outcome explained by a large number of genetic predictors in a high-dimensional polygenic linear model. This work studies the asymptotic properties of the most common estimator of heritability from summary statistics called linkage disequilibrium score (LDSC) regression, together with a simpler and closely related estimator called GWAS heritability (GWASH). These estimators are analyzed in their basic versions and under various modifications used in practice including weighting and standardization. We show that, with some variations, two conditions which we call weak dependence (WD) and bounded-kurtosis effects (BKE) are sufficient for consistency of both the basic LDSC with fixed intercept and GWASH estimators, for both Gaussian and non-Gaussian predictors. For Gaussian predictors it is shown that these conditions are also necessary for consistency of GWASH (with truncation) and simulations suggest that necessity holds too when the predictors are non-Gaussian. We also show that, with properly truncated weights, weighting does not change the consistency results, but standardization of the predictors and outcome, as done in practice, introduces bias in both LDSC and GWASH if the two essential conditions are violated. Finally, we show that, when population stratification is present, all the estimators considered are biased, and the bias is not remedied by using the LDSC regression estimator with free intercept, as originally suggested by the authors of that estimator.

math.ST↗

arXiv2025

On the peak height distribution of non-stationary Gaussian random fields: 1D general covariance and scale space

We study the peak height distribution of certain non-stationary Gaussian random fields. The explicit peak height distribution of smooth, non-stationary Gaussian processes in 1D with general covariance is derived. The formula is determined by two parameters, each of which has a clear statistical meaning. For multidimensional non-stationary Gaussian random fields, we generalize these results to the setting of scale space fields, which play an important role in peak detection by helping to handle peaks of different spatial extents. We demonstrate that these properties not only offer a better interpretation of the scale space field but also simplify the computation of the peak height distribution. Finally, two efficient numerical algorithms are proposed as a general solution for computing the peak height distribution of smooth multidimensional Gaussian random fields in applications.

stat.ME↗

arXiv2025

Peak Inference for Gaussian Random Fields on a Lattice

In this work we develop a Monte Carlo method to compute the height distribution of local maxima of a stationary Gaussian or Gaussian-related random field that is observed on a regular lattice. We show that our method can be used to provide valid peak based inference in datasets with low levels of smoothness, where existing formulae derived for continuous domains are not accurate. We also extend the methods in Worsley (2005) and Taylor et al. (2007) to compute the peak height distribution and compare them with our approach. Lastly, we apply our method to a task fMRI dataset to show how it can be used in practice.

stat.ME↗

arXiv2024

SCoRE Sets: A Versatile Framework for Simultaneous Inference

We study asymptotic statistical inference in the space of bounded functions endowed with the supremums norm over an arbitrary metric space $S$ using a novel concept: Simultaneous COnfidence Region of Excursion (SCoRE) Sets. They simultaneously quantify the uncertainty of several lower and upper excursion sets of a target function. We investigate their connection to multiple hypothesis tests controlling the familywise error rate in the strong sense and show that they grant a unifying perspective on several statistical inference tools such as simultaneous confidence bands, quantification of uncertainties in level set estimation, for example, CoPE sets, and multiple hypothesis testing over $S$, for example, finding relevant differences or regions of equivalence within $S$. In particular, our abstract setting allows us to refine and reduce the assumptions in recent articles on CoPE sets and relevance and equivalence testing using the supremums norm.

math.ST↗

arXiv2024

Object detection under the linear subspace model with application to cryo-EM images

Detecting multiple unknown objects in noisy data is a key problem in many scientific fields, such as electron microscopy imaging. A common model for the unknown objects is the linear subspace model, which assumes that the objects can be expanded in some known basis (such as the Fourier basis). In this paper, we develop an object detection algorithm that under the linear subspace model is asymptotically guaranteed to detect all objects, while controlling the family wise error rate or the false discovery rate. Numerical simulations show that the algorithm also controls the error rate with high power in the non-asymptotic regime, even in highly challenging regimes. We apply the proposed algorithm to experimental electron microscopy data set, and show that it outperforms existing standard software.

math.ST↗

arXiv2023

Robust FWER control in Neuroimaging using Random Field Theory: Riding the SuRF to Continuous Land Part 2

Historically, applications of RFT in fMRI have relied on assumptions of smoothness, stationarity and Gaussianity. The first two assumptions have been addressed in Part 1 of this article series. Here we address the severe non-Gaussianity of (real) fMRI data to greatly improve the performance of voxelwise RFT in fMRI group analysis. In particular, we introduce a transformation which accelerates the convergence of the Central Limit Theorem allowing us to rely on limiting Gaussianity of the test-statistic. We shall show that, when the GKF is combined with the Gaussianization transformation, we are able to accurately estimate the EEC of the excursion set of the transformed test-statistic even when the data is non-Gaussian. This allows us to drop the key assumptions of RFT inference and enables us to provide a fast approach which correctly controls the voxelwise false positive rate in fMRI. We employ a big data \cite{Eklund2016} style validation in which we process resting state data from 7000 subjects from the UK BioBank with fake task designs. We resample from this data to create realistic noise and use this to demonstrate that the error rate is correctly controlled.

stat.AP↗

arXiv2023

A genericity property of Fréchet sample means on Riemannian manifolds

Let $(M,g)$ be a Riemannian manifold. If $μ$ is a probability measure on $M$ given by a continuous density function, one would expect the Fréchet means of data-samples $Q=(q_1,q_2,\dots, q_N)\in M^N$, with respect to $μ$, to behave ``generically''; e.g. the probability that the Fréchet mean set $\mbox{FM}(Q)$ has any elements that lie in a given, positive-codimension submanifold, should be zero for any $N\geq 1$. Even this simplest instance of genericity does not seem to have been proven in the literature, except in special cases. The main result of this paper is a general, and stronger, genericity property: given i.i.d. absolutely continuous $M$-valued random variables $X_1,\dots, X_N$, and a subset $A\subset M$ of volume-measure zero, $\mbox{Pr}\left\{\mbox{FM}(\{X_1,\dots,X_N\})\subset M\backslash A\right\}=1.$ We also establish a companion theorem for equivariant Fréchet means, defined when $(M,g)$ arises as the quotient of a Riemannian manifold $(\widetilde{M},\tilde{g})$ by a free, isometric action of a finite group. The equivariant Fréchet means lie in $\widetilde{M}$, but, as we show, project down to the ordinary Fréchet sample means, and enjoy a similar genericity property. Both these theorems are proven as consequences of a purely geometric (and quite general) result that constitutes the core mathematics in this paper: If $A\subset M$ has volume zero in $M$ , then the set $\{Q\in M^N : \mbox{FM}(Q) \cap A\neq\emptyset\}$ has volume zero in $M^N$. We conclude the paper with an application to partial scaling-rotation means, a type of mean for symmetric positive-definite matrices.

math.PR↗

arXiv2023

Inverse set estimation and inversion of simultaneous confidence intervals

Motivated by the questions of risk assessment in climatology (temperature change in North America) and medicine (impact of statin usage and COVID-19 on hospitalized patients), we address the problem of estimating the set in the domain of a function whose image equals a predefined subset. Existing methods that construct confidence sets require strict assumptions. We generalize the estimation of such sets to dense and non-dense domains with protection against "data peeking" by proving that confidence sets of multiple levels can be simultaneously constructed with the desired confidence non-asymptotically through inverting simultaneous confidence bands. A non-parametric bootstrap algorithm and code are provided.

stat.ME↗

arXiv2023

Averaging symmetric positive-definite matrices on the space of eigen-decompositions

We study extensions of Fréchet means for random objects in the space ${\rm Sym}^+(p)$ of $p \times p$ symmetric positive-definite matrices using the scaling-rotation geometric framework introduced by Jung et al. [\textit{SIAM J. Matrix. Anal. Appl.} \textbf{36} (2015) 1180-1201]. The scaling-rotation framework is designed to enjoy a clearer interpretation of the changes in random ellipsoids in terms of scaling and rotation. In this work, we formally define the \emph{scaling-rotation (SR) mean set} to be the set of Fréchet means in ${\rm Sym}^+(p)$ with respect to the scaling-rotation distance. Since computing such means requires a difficult optimization, we also define the \emph{partial scaling-rotation (PSR) mean set} lying on the space of eigen-decompositions as a proxy for the SR mean set. The PSR mean set is easier to compute and its projection to ${\rm Sym}^+(p)$ often coincides with SR mean set. Minimal conditions are required to ensure that the mean sets are non-empty. Because eigen-decompositions are never unique, neither are PSR means, but we give sufficient conditions for the sample PSR mean to be unique up to the action of a certain finite group. We also establish strong consistency of the sample PSR means as estimators of the population PSR mean set, and a central limit theorem. In an application to multivariate tensor-based morphometry, we demonstrate that a two-group test using the proposed PSR means can have greater power than the two-group test using the usual affine-invariant geometric framework for symmetric positive-definite matrices.

stat.ME↗

arXiv2023

An approximation to peak detection power using Gaussian random field theory

We study power approximation formulas for peak detection using Gaussian random field theory. The approximation, based on the expected number of local maxima above the threshold $u$, $\mathbb{E}[M_u]$, is proved to work well under three asymptotic scenarios: small domain, large threshold, and sharp signal. An adjusted version of $\mathbb{E}[M_u]$ is also proposed to improve accuracy when the expected number of local maxima $\mathbb{E}[M_{-\infty}]$ exceeds 1. Cheng and Schwartzman (2018) developed explicit formulas for $\mathbb{E}[M_u]$ of smooth isotropic Gaussian random fields with zero mean. In this paper, these formulas are extended to allow for rotational symmetric mean functions, so that they are suitable for power calculations. We also apply our formulas to 2D and 3D simulated datasets, and the 3D data is induced by a group analysis of fMRI data from the Human Connectome Project to measure performance in a realistic setting.

stat.ME↗

arXiv2022

Functional delta residuals and applications to simultaneous confidence bands of moment based statistics

Given a functional central limit (fCLT) for an estimator and a parameter transformation, we construct random processes, called functional delta residuals, which asymptotically have the same covariance structure as the limit process of the functional delta method. An explicit construction of these residuals for transformations of moment-based estimators and a multiplier bootstrap fCLT for the resulting functional delta residuals are proven. The latter is used to consistently estimate the quantiles of the maximum of the limit process of the functional delta method in order to construct asymptotically valid simultaneous confidence bands for the transformed functional parameters. Performance of the coverage rate of the developed construction, applied to functional versions of Cohen's d, skewness and kurtosis, is illustrated in simulations and their application to test Gaussianity is discussed.

math.ST↗

arXiv2022

Spatial Confidence Regions for Combinations of Excursion Sets in Image Analysis

The analysis of excursion sets in imaging data is essential to a wide range of scientific disciplines such as neuroimaging, climatology and cosmology. Despite growing literature, there is little published concerning the comparison of processes that have been sampled across the same spatial region but which reflect different study conditions. Given a set of asymptotically Gaussian random fields, each corresponding to a sample acquired for a different study condition, this work aims to provide confidence statements about the intersection, or union, of the excursion sets across all fields. Such spatial regions are of natural interest as they directly correspond to the questions "all random fields exceed a predetermined threshold?", or "Where does at least one random field exceed a predetermined threshold?". To assess the degree of spatial variability present, we develop a method that provides, with a desired confidence, subsets and supersets of spatial regions defined by logical conjunctions (i.e. set intersections) or disjunctions (i.e. set unions), without any assumption on the dependence between the different fields. The method is verified by extensive simulations and demonstrated using a task-fMRI dataset to identify brain regions with activation common to four variants of a working memory task.

stat.ME↗

arXiv2020

Estimation of Expected Euler Characteristic Curves of Nonstationary Smooth Gaussian Random Fields

The expected Euler characteristic (EEC) curve of excursion sets of a Gaussian random field is used to approximate the distribution of its supremum for high thresholds. Viewed as a function of the excursion threshold, the EEC is expressed by the Gaussian kinematic formula (GKF) as a linear function of the Lipschitz-Killing curvatures (LKCs) of the field, which solely depend on the domain and covariance function of the field. So far its use for non-stationary Gaussian fields over non-trivial domains has been limited because in this case the LKCs are difficult to estimate. In this paper, consistent estimators of the LKCs are proposed as linear projections of "pinned" observed Euler characteristic curves and a linear parametric estimator of the EEC curve is obtained, which is more efficient than its nonparametric counterpart for repeated observations. A multiplier bootstrap modification reduces the variance of the estimator, and allows estimation of LKCs and EEC of the limiting field of non-Gaussian fields satisfying a functional CLT. The proposed methods are evaluated using simulations of 2D fields and illustrated in thresholding of 3D fMRI brain activation maps and cosmological simulations on the 2-sphere.

math.ST↗

arXiv2019

Multiple Testing of Local Extrema for Detection of Change Points

A new approach to detect change points based on differential smoothing and multiple testing is presented for long data sequences modeled as piecewise constant functions plus stationary ergodic Gaussian noise. As an application of the STEM algorithm for peak detection developed in \citet{schwartzman2011multiple} and \citet{cheng2017multiple}, the method detects change points as significant local maxima and minima after smoothing and differentiating the observed sequence. The algorithm, combined with the Benjamini-Hochberg procedure for thresholding p-values, provides asymptotic strong control of the False Discovery Rate (FDR) and power consistency, as the length of the sequence and the size of the jumps get large. Simulations show that FDR levels are maintained in non-asymptotic conditions and guide the choice of smoothing bandwidth. The methods are illustrated in magnetometer sensor data and genomic array-CGH data. An R package named "dSTEM" is available in R cran.

math.ST↗