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Armin Schwartzman

Publications and source records attributed to Armin Schwartzman.

At least 37 records · Page 2Linked to original sources
arXiv2019

On critical points of Gaussian random fields under diffeomorphic transformations

Let $\{X(t), t\in M\}$ and $\{Z(t'), t'\in M'\}$ be smooth Gaussian random fields parameterized on Riemannian manifolds $M$ and $M'$, respectively, such that $X(t) = Z(f(t))$, where $f: M \to M'$ is a diffeomorphic transformation. We study the expected number and height distribution of the critical points of $X$ in connection with those of $Z$. As an important case, when $X$ is an anisotropic Gaussian random field, then we show that its expected number of critical points becomes proportional to that of an isotropic field $Z$, while the height distribution remains the same as that of $Z$.

math.PR↗
arXiv2019

Point Source Detection and False Discovery Rate Control on CMB Maps

We discuss a new procedure to search for point sources in Cosmic Microwave background maps; in particular, we aim at controlling the so-called False Discovery Rate, which is defined as the expected value of false discoveries among pixels which are labelled as contaminated by point sources. We exploit a procedure called STEM, which is based on the following four steps: 1) needlet filtering of the observed CMB maps, to improve the signal to noise ratio; 2) selection of candidate peaks, i.e., the local maxima of filtered maps; 3) computation of \emph{p-}values for local maxima; 4) implementation of the multiple testing procedure, by means of the so-called Benjamini-Hochberg method. Our procedures are also implemented on the latest release of Planck CMB maps.

astro-ph.CO↗
arXiv2019

Simultaneous Confidence Bands for Functional Data Using the Gaussian Kinematic Formula

This article constructs simultaneous confidence bands (SCBs) for functional parameters using the Gaussian Kinematic formula of $t$-processes (tGKF). Although the tGKF relies on Gaussianity, we show that a central limit theorem (CLT) for the parameter of interest is enough to obtain asymptotically precise covering rates even for non-Gaussian processes. As a proof of concept we study the functional signal-plus-noise model and derive a CLT for an estimator of the Lipschitz-Killing curvatures, the only data dependent quantities in the tGKF SCBs. Extensions to discrete sampling with additive observation noise are discussed using scale space ideas from regression analysis. Here we provide sufficient conditions on the processes and kernels to obtain convergence of the functional scale space surface. The theoretical work is accompanied by a simulation study comparing different methods to construct SCBs for the population mean. We show that the tGKF works well even for small sample sizes and only a Rademacher multiplier-$t$ bootstrap performs similarily well. For larger sample sizes the tGKF often outperforms the bootstrap methods and is computational faster. We apply the method to diffusion tensor imaging (DTI) fibers using a scale space approach for the difference of population means. R code is available in our Rpackage SCBfda.

math.ST↗
arXiv2018

Unexpected Topology of the Temperature Fluctuations in the Cosmic Microwave Background

We study the topology generated by the temperature fluctuations of the Cosmic Microwave Background (CMB) radiation, as quantified by the number of components and holes, formally given by the Betti numbers, in the growing excursion sets. We compare CMB maps observed by the Planck satellite with a thousand simulated maps generated according to the LCDM paradigm with Gaussian distributed fluctuations. The survey of the CMB over $\mathbb{S}^2$ is incomplete due to obfuscation effects by bright point sources and other extended foreground objects like our own galaxy. To deal with such situations, where analysis in the presence of "masks" is of importance, we introduce the concept of relative homology. The parametric $χ^2$-test shows differences between observations and simulations, yielding $p$-values at per-cent to less than per-mil levels roughly between 2 to 7 degrees. The highest observed deviation for $b_0$ and $b_1$ is approximately between $3σ$-4$σ$ at scales of 3 to 7 degrees. There are reports of mildly unusual behaviour of the Euler characteristic at 3.66 degrees in the literature, computed from independent measurements of the CMB temperature fluctuations by Planck's predecessor WMAP satellite. The mildly anomalous behaviour of Euler characteristic is related to the strongly anomalous behaviour of components and holes. These are also the scales at which the observed maps exhibit low variance compared to the simulations. Non-parametric tests show even stronger differences at almost all scales. Regardless, beyond the trivial possibility that this may still be a manifestation of an extreme Gaussian case, these observations, along with the super-horizon scales involved, may motivate to look at primordial non-Gaussianity. Alternative scenarios worth exploring may be models with non-trivial topology.

astro-ph.CO↗
arXiv2018

Standardization of multivariate Gaussian mixture models and background adjustment of PET images in brain oncology

In brain oncology, it is routine to evaluate the progress or remission of the disease based on the differences between a pre-treatment and a post-treatment Positron Emission Tomography (PET) scan. Background adjustment is necessary to reduce confounding by tissue-dependent changes not related to the disease. When modeling the voxel intensities for the two scans as a bivariate Gaussian mixture, background adjustment translates into standardizing the mixture at each voxel, while tumor lesions present themselves as outliers to be detected. In this paper, we address the question of how to standardize the mixture to a standard multivariate normal distribution, so that the outliers (i.e., tumor lesions) can be detected using a statistical test. We show theoretically and numerically that the tail distribution of the standardized scores is favorably close to standard normal in a wide range of scenarios while being conservative at the tails, validating voxelwise hypothesis testing based on standardized scores. To address standardization in spatially heterogeneous image data, we propose a spatial and robust multivariate expectation-maximization (EM) algorithm, where prior class membership probabilities are provided by transformation of spatial probability template maps and the estimation of the class mean and covariances are robust to outliers. Simulations in both univariate and bivariate cases suggest that standardized scores with soft assignment have tail probabilities that are either very close to or more conservative than standard normal. The proposed methods are applied to a real data set from a PET phantom experiment, yet they are generic and can be used in other contexts.

stat.ME↗
arXiv2017

Uniqueness questions in a scaling-rotation geometry on the space of symmetric positive-definite matrices

Jung et al. (2015) introduced a geometric structure on ${\rm Sym}^+(p)$, the set of $p \times p$ symmetric positive-definite matrices, based on eigen-decomposition. Eigenstructure determines both a stratification of ${\rm Sym}^+(p)$, defined by eigenvalue multiplicities, and fibers of the "eigen-composition" map $F:M(p):=SO(p)\times{\rm Diag}^+(p)\to{\rm Sym}^+(p)$. When $M(p)$ is equipped with a suitable Riemannian metric, the fiber structure leads to notions of scaling-rotation distance between $X,Y\in {\rm Sym}^+(p)$, the distance in $M(p)$ between fibers $F^{-1}(X)$ and $F^{-1}(Y)$, and minimal smooth scaling-rotation (MSSR) curves, images in ${\rm Sym}^+(p)$ of minimal-length geodesics connecting two fibers. In this paper we study the geometry of the triple $(M(p),F,{\rm Sym}^+(p))$, focusing on some basic questions: For which $X,Y$ is there a unique MSSR curve from $X$ to $Y$? More generally, what is the set ${\cal M}(X,Y)$ of MSSR curves from $X$ to $Y$? This set is influenced by two potential types of non-uniqueness. We translate the question of whether the second type can occur into a question about the geometry of Grassmannians $G_m({\bf R}^p)$, with $m$ even, that we answer for $p\leq 4$ and $p\geq 11$. Our method of proof also yields an interesting half-angle formula concerning principal angles between subspaces of ${\bf R}^p$ whose dimensions may or may not be equal. The general-$p$ results concerning MSSR curves and scaling-rotation distance that we establish here underpin the explicit $p=3$ results in Groisser et al. (2017). Addressing the uniqueness-related questions requires a thorough understanding of the fiber structure of $M(p)$, which we also provide.

math.MG↗
arXiv2017

Geometric foundations for scaling-rotation statistics on symmetric positive definite matrices: minimal smooth scaling-rotation curves in low dimensions

We investigate a geometric computational framework, called the "scaling-rotation framework", on ${\rm Sym}^+(p)$, the set of $p \times p$ symmetric positive-definite (SPD) matrices. The purpose of our study is to lay geometric foundations for statistical analysis of SPD matrices, in situations in which eigenstructure is of fundamental importance, for example diffusion-tensor imaging (DTI). Eigen-decomposition, upon which the scaling-rotation framework is based, determines both a stratification of ${\rm Sym}^+(p)$, defined by eigenvalue multiplicities, and fibers of the "eigen-composition" map $SO(p)\times{\rm Diag}^+(p)\to{\rm Sym}^+(p)$. This leads to the notion of scaling-rotation distance [Jung et al. (2015)], a measure of the minimal amount of scaling and rotation needed to transform an SPD matrix, $X,$ into another, $Y,$ by a smooth curve in ${\rm Sym}^+(p)$. Our main goal in this paper is the systematic characterization and analysis of minimal smooth scaling-rotation (MSSR) curves, images in ${\rm Sym}^+(p)$ of minimal-length geodesics connecting two fibers in the "upstairs" space $SO(p)\times{\rm Diag}^+(p)$. The length of such a geodesic connecting the fibers over $X$ and $Y$ is what we define to be the scaling-rotation distance from $X$ to $Y.$ For the important low-dimensional case $p = 3$ (the home of DTI), we find new explicit formulas for MSSR curves and for the scaling-rotation distance, and identify ${\cal M}(X,Y)$ in all "nontrivial" cases. The quaternionic representation of $SO(3)$ is used in these computations. We also provide closed-form expressions for scaling-rotation distance and MSSR curves for the case $p = 2$.

math.MG↗
arXiv2016

Expected Number and Height Distribution of Critical Points of Smooth Isotropic Gaussian Random Fields

We obtain formulae for the expected number and height distribution of critical points of smooth isotropic Gaussian random fields parameterized on Euclidean space or spheres of arbitrary dimension. The results hold in general in the sense that there are no restrictions on the covariance function of the field except for smoothness and isotropy. The results are based on a characterization of the distribution of the Hessian of the Gaussian field by means of the family of Gaussian orthogonally invariant (GOI) matrices, of which the Gaussian orthogonal ensemble (GOE) is a special case. The obtained formulae depend on the covariance function only through a single parameter (Euclidean space) or two parameters (sphere), and include the special boundary case of random Laplacian eigenfunctions.

math.PR↗
arXiv2016

Multiple testing of local maxima for detection of peaks on the (celestial) sphere

We present a topological multiple testing scheme for detecting peaks on the sphere under isotropic Gaussian noise, where tests are performed at local maxima of the observed field filtered by the spherical needlet transform. Our setting is different from the standard Euclidean/large same asymptotic framework, yet highly relevant to realistic experimental circumstances for some important areas of application in astronomy. More precisely, we focus on cases where a single realization of a smooth isotropic Gaussian random field on the sphere is observed, and a number of well-localized signals are superimposed on such background field. The proposed algorithms, combined with the Benjamini-Hochberg procedure for thresholding p-values, provide asymptotic strong control of the False Discovery Rate (FDR) and power consistency as the signal strength and the frequency of the needlet transform get large. This novel multiple testing method is illustrated in a simulation of point-source detection in Cosmic Microwave Background radiation (CMB) data.

math.ST↗
arXiv2015

Scaling-rotation distance and interpolation of symmetric positive-definite matrices

We introduce a new geometric framework for the set of symmetric positive-definite (SPD) matrices, aimed to characterize deformations of SPD matrices by individual scaling of eigenvalues and rotation of eigenvectors of the SPD matrices. To characterize the deformation, the eigenvalue-eigenvector decomposition is used to find alternative representations of SPD matrices, and to form a Riemannian manifold so that scaling and rotations of SPD matrices are captured by geodesics on this manifold. The problems of non-unique eigen-decompositions and eigenvalue multiplicities are addressed by finding minimal-length geodesics, which gives rise to a distance and an interpolation method for SPD matrices. Computational procedures to evaluate the minimal scaling--rotation deformations and distances are provided for the most useful cases of $2 \times 2$ and $3 \times 3$ SPD matrices. In the new geometric framework, minimal scaling--rotation curves interpolate eigenvalues at constant logarithmic rate, and eigenvectors at constant angular rate. In the context of diffusion tensor imaging, this results in better behavior of the trace, determinant and fractional anisotropy of interpolated SPD matrices in typical cases.

math.MG↗
arXiv2015

Confidence regions for excursion sets in asymptotically Gaussian random fields, with an application to climate

The goal of this paper is to give confidence regions for the excursion set of a spatial function above a given threshold from repeated noisy observations on a fine grid of fixed locations. Given an asymptotically Gaussian estimator of the target function, a pair of data-dependent nested excursion sets are constructed that are sub- and super-sets of the true excursion set, respectively, with a desired confidence. Asymptotic coverage probabilities are determined via a multiplier bootstrap method, not requiring Gaussianity of the original data nor stationarity or smoothness of the limiting Gaussian field. The method is used to determine regions in North America where the mean summer and winter temperatures are expected to increase by mid 21st century by more than 2 degrees Celsius.

stat.ME↗
arXiv2014

Distribution of the Height of Local Maxima of Gaussian Random Fields

Let $\{f(t): t\in T\}$ be a smooth Gaussian random field over a parameter space $T$, where $T$ may be a subset of Euclidean space or, more generally, a Riemannian manifold. For any local maximum of $f(t)$ located at $t_0$ in the interior of $T$, we provide general formulae and asymptotic approximations for both the tail distribution of the height of a local maximum $\mathbb{P}\{f(t_0)>u | t_0 \text{is a local maximum of} f(t) \}$ and the overshoot distribution of a local maximum $\mathbb{P}\{f(t_0)>u+v | t_0 \text{is a local maximum of} f(t) \text{and} f(t_0)>v\}$. Assuming further that $f$ is isotropic, we apply techniques from random matrix theory related to the Gaussian orthogonal ensemble to compute such conditional probabilities explicitly when $T$ is Euclidean or a sphere of arbitrary dimension. Such calculations are motivated by the statistical problem of detecting peaks in the presence of smooth Gaussian noise.

math.PR↗
arXiv2014

Lognormal Distributions and Geometric Averages of Positive Definite Matrices

This article gives a formal definition of a lognormal family of probability distributions on the set of symmetric positive definite (PD) matrices, seen as a matrix-variate extension of the univariate lognormal family of distributions. Two forms of this distribution are obtained as the large sample limiting distribution via the central limit theorem of two types of geometric averages of i.i.d. PD matrices: the log-Euclidean average and the canonical geometric average. These averages correspond to two different geometries imposed on the set of PD matrices. The limiting distributions of these averages are used to provide large-sample confidence regions for the corresponding population means. The methods are illustrated on a voxelwise analysis of diffusion tensor imaging data, permitting a comparison between the various average types from the point of view of their sampling variability.

stat.ME↗
arXiv2014

Multiple Testing of Local Maxima for Detection of Peaks in Random Fields

A topological multiple testing scheme is presented for detecting peaks in images under stationary ergodic Gaussian noise, where tests are performed at local maxima of the smoothed observed signals. The procedure generalizes the one-dimensional scheme of Schwartzman et al. (2011) to Euclidean domains of arbitrary dimension. Two methods are developed according to two different ways of computing p-values: (i) using the exact distribution of the height of local maxima (Cheng and Schwartzman, 2014), available explicitly when the noise field is isotropic; (ii) using an approximation to the overshoot distribution of local maxima above a pre-threshold (Cheng and Schwartzman, 2014), applicable when the exact distribution is unknown, such as when the stationary noise field is non-isotropic. The algorithms, combined with the Benjamini-Hochberg procedure for thresholding p-values, provide asymptotic strong control of the False Discovery Rate (FDR) and power consistency, with specific rates, as the search space and signal strength get large. The optimal smoothing bandwidth and optimal pre-threshold are obtained to achieve maximum power. Simulations show that FDR levels are maintained in non-asymptotic conditions. The methods are illustrated in a nanoscopy image analysis problem of detecting fluorescent molecules against the image background.

stat.ME↗
arXiv2013

Multiple testing of local maxima for detection of peaks in ChIP-Seq data

A topological multiple testing approach to peak detection is proposed for the problem of detecting transcription factor binding sites in ChIP-Seq data. After kernel smoothing of the tag counts over the genome, the presence of a peak is tested at each observed local maximum, followed by multiple testing correction at the desired false discovery rate level. Valid p-values for candidate peaks are computed via Monte Carlo simulations of smoothed Poisson sequences, whose background Poisson rates are obtained via linear regression from a Control sample at two different scales. The proposed method identifies nearby binding sites that other methods do not.

stat.AP↗
arXiv2012

Multiple testing of local maxima for detection of peaks in 1D

A topological multiple testing scheme for one-dimensional domains is proposed where, rather than testing every spatial or temporal location for the presence of a signal, tests are performed only at the local maxima of the smoothed observed sequence. Assuming unimodal true peaks with finite support and Gaussian stationary ergodic noise, it is shown that the algorithm with Bonferroni or Benjamini--Hochberg correction provides asymptotic strong control of the family wise error rate and false discovery rate, and is power consistent, as the search space and the signal strength get large, where the search space may grow exponentially faster than the signal strength. Simulations show that error levels are maintained for nonasymptotic conditions, and that power is maximized when the smoothing kernel is close in shape and bandwidth to the signal peaks, akin to the matched filter theorem in signal processing. The methods are illustrated in an analysis of electrical recordings of neuronal cell activity.

math.ST↗
arXiv2010

Peak Detection as Multiple Testing

This paper considers the problem of detecting equal-shaped non-overlapping unimodal peaks in the presence of Gaussian ergodic stationary noise, where the number, location and heights of the peaks are unknown. A multiple testing approach is proposed in which, after kernel smoothing, the presence of a peak is tested at each observed local maximum. The procedure provides strong control of the family wise error rate and the false discovery rate asymptotically as both the signal-to-noise ratio (SNR) and the search space get large, where the search space may grow exponentially as a function of SNR. Simulations assuming a Gaussian peak shape and a Gaussian autocorrelation function show that desired error levels are achieved for relatively low SNR and are robust to partial peak overlap. Simulations also show that detection power is maximized when the smoothing bandwidth is close to the bandwidth of the signal peaks, akin to the well-known matched filter theorem in signal processing. The procedure is illustrated in an analysis of electrical recordings of neuronal cell activity.

stat.ME↗