arXiv subjects

Christopher Genovese

Publications and source records attributed to Christopher Genovese.

15 recordsLinked to original sources

arXiv2016

Finding Singular Features

We present a method for finding high density, low-dimensional structures in noisy point clouds. These structures are sets with zero Lebesgue measure with respect to the $D$-dimensional ambient space and belong to a $d<D$ dimensional space. We call them "singular features." Hunting for singular features corresponds to finding unexpected or unknown structures hidden in point clouds belonging to $\R^D$. Our method outputs well defined sets of dimensions $d<D$. Unlike spectral clustering, the method works well in the presence of noise. We show how to find singular features by first finding ridges in the estimated density, followed by a filtering step based on the eigenvalues of the Hessian of the density.

stat.ME↗

arXiv2014

The functional mean-shift algorithm for mode hunting and clustering in infinite dimensions

We introduce the functional mean-shift algorithm, an iterative algorithm for estimating the local modes of a surrogate density from functional data. We show that the algorithm can be used for cluster analysis of functional data. We propose a test based on the bootstrap for the significance of the estimated local modes of the surrogate density. We present two applications of our methodology. In the first application, we demonstrate how the functional mean-shift algorithm can be used to perform spike sorting, i.e. cluster neural activity curves. In the second application, we use the functional mean-shift algorithm to distinguish between original and fake signatures.

stat.ME↗

arXiv2014

Estimating the distribution of Galaxy Morphologies on a continuous space

The incredible variety of galaxy shapes cannot be summarized by human defined discrete classes of shapes without causing a possibly large loss of information. Dictionary learning and sparse coding allow us to reduce the high dimensional space of shapes into a manageable low dimensional continuous vector space. Statistical inference can be done in the reduced space via probability distribution estimation and manifold estimation.

astro-ph.GA↗

arXiv2014

Functional Regression for Quasar Spectra

The Lyman-alpha forest is a portion of the observed light spectrum of distant galactic nuclei which allows us to probe remote regions of the Universe that are otherwise inaccessible. The observed Lyman-alpha forest of a quasar light spectrum can be modeled as a noisy realization of a smooth curve that is affected by a `damping effect' which occurs whenever the light emitted by the quasar travels through regions of the Universe with higher matter concentration. To decode the information conveyed by the Lyman-alpha forest about the matter distribution, we must be able to separate the smooth `continuum' from the noise and the contribution of the damping effect in the quasar light spectra. To predict the continuum in the Lyman-alpha forest, we use a nonparametric functional regression model in which both the response and the predictor variable (the smooth part of the damping-free portion of the spectrum) are function-valued random variables. We demonstrate that the proposed method accurately predicts the unobservable continuum in the Lyman-alpha forest both on simulated spectra and real spectra. Also, we introduce distribution-free prediction bands for the nonparametric functional regression model that have finite sample guarantees. These prediction bands, together with bootstrap-based confidence bands for the projection of the mean continuum on a fixed number of principal components, allow us to assess the degree of uncertainty in the model predictions.

stat.ME↗

arXiv2013

Nonparametric Inference For Density Modes

We derive nonparametric confidence intervals for the eigenvalues of the Hessian at modes of a density estimate. This provides information about the strength and shape of modes and can also be used as a significance test. We use a data-splitting approach in which potential modes are identified using the first half of the data and inference is done with the second half of the data. To get valid confidence sets for the eigenvalues, we use a bootstrap based on an elementary-symmetric-polynomial (ESP) transformation. This leads to valid bootstrap confidence sets regardless of any multiplicities in the eigenvalues. We also suggest a new method for bandwidth selection, namely, choosing the bandwidth to maximize the number of significant modes. We show by example that this method works well. Even when the true distribution is singular, and hence does not have a density, (in which case cross validation chooses a zero bandwidth), our method chooses a reasonable bandwidth.

stat.ME↗

arXiv2011

Minimax Manifold Estimation

We find the minimax rate of convergence in Hausdorff distance for estimating a manifold M of dimension d embedded in R^D given a noisy sample from the manifold. We assume that the manifold satisfies a smoothness condition and that the noise distribution has compact support. We show that the optimal rate of convergence is n^{-2/(2+d)}. Thus, the minimax rate depends only on the dimension of the manifold, not on the dimension of the space in which M is embedded.

stat.ML↗

arXiv2010

Image Coaddition with Temporally Varying Kernels

Large, multi-frequency imaging surveys, such as the Large Synaptic Survey Telescope (LSST), need to do near-real time analysis of very large datasets. This raises a host of statistical and computational problems where standard methods do not work. In this paper, we study a proposed method for combining stacks of images into a single summary image, sometimes referred to as a template. This task is commonly referred to as image coaddition. In part, we focus on a method proposed in previous work, which outlines a procedure for combining stacks of images in an online fashion in the Fourier domain. We evaluate this method by comparing it to two straightforward methods through the use of various criteria and simulations. Note that the goal is not to propose these comparison methods for use in their own right, but to ensure that additional complexity also provides substantially improved performance.

astro-ph.IM↗

arXiv2009

Revisiting Marginal Regression

The lasso has become an important practical tool for high dimensional regression as well as the object of intense theoretical investigation. But despite the availability of efficient algorithms, the lasso remains computationally demanding in regression problems where the number of variables vastly exceeds the number of data points. A much older method, marginal regression, largely displaced by the lasso, offers a promising alternative in this case. Computation for marginal regression is practical even when the dimension is very high. In this paper, we study the relative performance of the lasso and marginal regression for regression problems in three different regimes: (a) exact reconstruction in the noise-free and noisy cases when design and coefficients are fixed, (b) exact reconstruction in the noise-free case when the design is fixed but the coefficients are random, and (c) reconstruction in the noisy case where performance is measured by the number of coefficients whose sign is incorrect. In the first regime, we compare the conditions for exact reconstruction of the two procedures, find examples where each procedure succeeds while the other fails, and characterize the advantages and disadvantages of each. In the second regime, we derive conditions under which marginal regression will provide exact reconstruction with high probability. And in the third regime, we derive rates of convergence for the procedures and offer a new partitioning of the ``phase diagram,'' that shows when exact or Hamming reconstruction is effective.

math.ST↗

arXiv2009

Inference for the dark energy equation of state using Type IA supernova data

The surprising discovery of an accelerating universe led cosmologists to posit the existence of "dark energy"--a mysterious energy field that permeates the universe. Understanding dark energy has become the central problem of modern cosmology. After describing the scientific background in depth, we formulate the task as a nonlinear inverse problem that expresses the comoving distance function in terms of the dark-energy equation of state. We present two classes of methods for making sharp statistical inferences about the equation of state from observations of Type Ia Supernovae (SNe). First, we derive a technique for testing hypotheses about the equation of state that requires no assumptions about its form and can distinguish among competing theories. Second, we present a framework for computing parametric and nonparametric estimators of the equation of state, with an associated assessment of uncertainty. Using our approach, we evaluate the strength of statistical evidence for various competing models of dark energy. Consistent with current studies, we find that with the available Type Ia SNe data, it is not possible to distinguish statistically among popular dark-energy models, and that, in particular, there is no support in the data for rejecting a cosmological constant. With much more supernova data likely to be available in coming years (e.g., from the DOE/NASA Joint Dark Energy Mission), we address the more interesting question of whether future data sets will have sufficient resolution to distinguish among competing theories.

astro-ph↗

arXiv2007

Mapping the Cosmological Confidence Ball Surface

We present a new technique to compute simultaneously valid confidence intervals for a set of model parameters. We apply our method to the Wilkinson Microwave Anisotropy Probe's (WMAP) Cosmic Microwave Background (CMB) data, exploring a seven dimensional space (tau, Omega_DE, Omega_M, omega_DM, omega_B, f_nu, n_s). We find two distinct regions-of-interest: the standard Concordance Model, and a region with large values of omega_DM, omega_B and H_0. This second peak in parameter space can be rejected by applying a constraint (or a prior) on the allowable values of the Hubble constant. Our new technique uses a non-parametric fit to the data, along with a frequentist approach and a smart search algorithm to map out a statistical confidence surface. The result is a confidence ``ball'': a set of parameter values that contains the true value with probability at least 1-alpha. Our algorithm performs a role similar to the often used Markov Chain Monte Carlo (MCMC), which samples from the posterior probability function in order to provide Bayesian credible intervals on the parameters. While the MCMC approach samples densely around a peak in the posterior, our new technique allows cosmologists to perform efficient analyses around any regions of interest: e.g., the peak itself, or, possibly more importantly, the 1-alpha confidence surface.

astro-ph↗

arXiv2004

A stochastic process approach to false discovery control

This paper extends the theory of false discovery rates (FDR) pioneered by Benjamini and Hochberg [J. Roy. Statist. Soc. Ser. B 57 (1995) 289-300]. We develop a framework in which the False Discovery Proportion (FDP)--the number of false rejections divided by the number of rejections--is treated as a stochastic process. After obtaining the limiting distribution of the process, we demonstrate the validity of a class of procedures for controlling the False Discovery Rate (the expected FDP). We construct a confidence envelope for the whole FDP process. From these envelopes we derive confidence thresholds, for controlling the quantiles of the distribution of the FDP as well as controlling the number of false discoveries. We also investigate methods for estimating the p-value distribution.

math.ST↗

arXiv2004

Multi-Tree Methods for Statistics on Very Large Datasets in Astronomy

Many fundamental statistical methods have become critical tools for scientific data analysis yet do not scale tractably to modern large datasets. This paper will describe very recent algorithms based on computational geometry which have dramatically reduced the computational complexity of 1) kernel density estimation (which also extends to nonparametric regression, classification, and clustering), and 2) the n-point correlation function for arbitrary n. These new multi-tree methods typically yield orders of magnitude in speedup over the previous state of the art for similar accuracy, making millions of data points tractable on desktop workstations for the first time.

astro-ph↗

arXiv2001

A Non-parametric Analysis of the CMB Power Spectrum

We examine Cosmic Microwave Background (CMB) temperature power spectra from the BOOMERANG, MAXIMA, and DASI experiments. We non-parametrically estimate the true power spectrum with no model assumptions. This is a significant departure from previous research which used either cosmological models or some other parameterized form (e.g. parabolic fits). Our non-parametric estimate is practically indistinguishable from the best fit cosmological model, thus lending independent support to the underlying physics that governs these models. We also generate a confidence set for the non-parametric fit and extract confidence intervals for the numbers, locations, and heights of peaks and the successive peak-to-peak height ratios. At the 95%, 68%, and 40% confidence levels, we find functions that fit the data with one, two, and three peaks respectively (0 <= l <= 1100). Therefore, the current data prefer two peaks at the 1 sigma level. However, we also rule out a constant temperature function at the >8 sigma level. If we assume that there are three peaks in the data, we find their locations to be within l_1 = (118,300), l_2 = (377,650), and l_3 = (597,900). We find the ratio of the first peak-height to the second (Delta T_1)/(Delta T_2)^2= (1.06, 4.27) and the second to the third (Delta T_2)/(Delta T_3)^2= (0.41, 2.5). All measurements are for 95% confidence. If the standard errors on the temperature measurements were reduced to a third of what they are currently, as we expect to be achieved by the MAP and Planck CMB experiments, we could eliminate two-peak models at the 95% confidence limit. The non-parametric methodology discussed in this paper has many astrophysical applications.

astro-ph↗

arXiv2001

Controlling the False Discovery Rate in Astrophysical Data Analysis

The False Discovery Rate (FDR) is a new statistical procedure to control the number of mistakes made when performing multiple hypothesis tests, i.e. when comparing many data against a given model hypothesis. The key advantage of FDR is that it allows one to a priori control the average fraction of false rejections made (when comparing to the null hypothesis) over the total number of rejections performed. We compare FDR to the standard procedure of rejecting all tests that do not match the null hypothesis above some arbitrarily chosen confidence limit, e.g. 2 sigma, or at the 95% confidence level. When using FDR, we find a similar rate of correct detections, but with significantly fewer false detections. Moreover, the FDR procedure is quick and easy to compute and can be trivially adapted to work with correlated data. The purpose of this paper is to introduce the FDR procedure to the astrophysics community. We illustrate the power of FDR through several astronomical examples, including the detection of features against a smooth one-dimensional function, e.g. seeing the ``baryon wiggles'' in a power spectrum of matter fluctuations, and source pixel detection in imaging data. In this era of large datasets and high precision measurements, FDR provides the means to adaptively control a scientifically meaningful quantity -- the number of false discoveries made when conducting multiple hypothesis tests.

astro-ph↗

arXiv1999

Transition Matrix Monte Carlo

Although histogram methods have been extremely effective for analyzing data from Monte Carlo simulations, they do have certain limitations, including the range over which they are valid and the difficulties of combining data from independent simulations. In this paper, we describe an complementary approach to extracting information from Monte Carlo simulations that uses the matrix of transition probabilities. Combining the Transition Matrix with an N-fold way simulation technique produces an extremely flexible and efficient approach to rather general Monte Carlo simulations.

cond-mat.stat-mech↗