Elliptical Population Research Articles

Adjusted location‐invariant U‐tests for the covariance matrix with elliptically high‐dimensional data

AbstractThis paper analyzes several covariance matrix U‐tests, which are constructed by modifying the classical John‐Nagao and Ledoit‐Wolf tests, under the elliptically distributed data structure. We study the limiting distributions of these location‐invariant test statistics as the data dimension may go to infinity in an arbitrary way as the sample size does. We find that they tend to have unsatisfactory size performances for general elliptical population. This is mainly because such population often possesses high‐order correlations among their coordinates. Taking such kind of dependency into consideration, we propose necessary corrections for these tests to cope with elliptically high‐dimensional data. For computational efficiency, alternative forms of the new test statistics are also provided. We derive the universal asymptotic null distributions of the proposed test statistics under elliptical distributions and beyond. The powers of the proposed tests are further investigated. The accuracy of the tests is demonstrated by simulations and an empirical study.

Tests for Large-Dimensional Shape Matrices via Tyler’s M Estimators

Tyler’s M estimator, as a robust alternative to the sample covariance matrix, has been widely applied in robust statistics. However, classical theory on Tyler’s M estimator is mainly developed in the low-dimensional regime for elliptical populations. It remains largely unknown when the parameter of dimension p grows proportionally to the sample size n for general populations. By using the eigenvalues of Tyler’s M estimator, this article develops tests for the identity and equality of shape matrices in a large-dimensional framework where the dimension-to-sample size ratio p/n has a limit in (0, 1). The proposed tests can be applied to a broad class of multivariate distributions including the family of elliptical distributions (see model (2.1) for details). To analyze both the null and alternative distributions of the proposed tests, we provide a unified theory on the spectrum of a large-dimensional Tyler’s M estimator when the underlying population is general. Simulation results demonstrate good performance and robustness of our tests. An empirical analysis of the Fama-French 49 industrial portfolios is carried out to demonstrate the shape of the portfolios varying. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Separable sample covariance matrices under elliptical populations with applications

This paper is to investigate the spectral properties of separable covariance matrices under elliptical populations. The separable covariance matrix model can handle both cross-row and cross-column correlations thus gain more popularity recently. Under the high-dimensional setting where the dimension p p and the sample size n n tend to infinity proportionally, we find the limit of the empirical spectral distribution and establish the central limit theorems (CLT) for linear spectral statistics of such kinds of sample covariance matrices. Some applications of our established CLT are also given.

Relaxed blue ellipticals: accretion-driven stellar growth is a key evolutionary channel for low mass elliptical galaxies

ABSTRACT How elliptical galaxies form is a key question in observational cosmology. While the formation of massive ellipticals is strongly linked to mergers, the low mass (M⋆/M⊙ < 109.5) regime remains less well explored. In particular, studying elliptical populations when they are blue, and therefore rapidly building stellar mass, offers strong constraints on their formation. Here, we study 108 blue low-mass ellipticals (which have a median stellar mass of 108.7 M⊙) at z < 0.3 in the COSMOS field. Visual inspection of extremely deep optical HSC images indicates that less than 3 per cent of these systems have visible tidal features, a factor of 2 less than the incidence of tidal features in a control sample of galaxies with the same distribution of stellar mass and redshift. This suggests that the star formation activity in these objects is not driven by mergers or interactions but by secular gas accretion. We combine accurate physical parameters from the COSMOS2020 catalogue, with measurements of local density and the locations of galaxies in the cosmic web, to show that our blue ellipticals reside in low-density environments, further away from nodes and large-scale filaments than other galaxies. At similar stellar masses and environments, blue ellipticals outnumber their normal (red) counterparts by a factor of 2. Thus, these systems are likely progenitors of not only normal ellipticals at similar stellar mass but, given their high star formation rates, also of ellipticals at higher stellar masses. Secular gas accretion therefore likely plays a significant (and possibly dominant) role in the stellar assembly of elliptical galaxies in the low-mass regime.

A Robust Test for Elliptical Symmetry

Most signal processing and statistical applications heavily rely on specific data distribution models. The Gaussian distributions, although being the most common choice, are inadequate in most real world scenarios as they fail to account for data coming from heavy-tailed populations or contaminated by outliers. Such problems call for the use of Robust Statistics. The robust models and estimators are usually based on elliptical populations, making the latter ubiquitous in all methods of robust statistics. To determine whether such tools are applicable in any specific case, goodness-of-fit (GoF) tests are used to verify the ellipticity hypothesis. Ellipticity GoF tests are usually hard to analyze and often their statistical power is not particularly strong. In this work, assuming the true covariance matrix is unknown we design and rigorously analyze a robust GoF test consistent against all alternatives to ellipticity on the unit sphere. The proposed test is based on Tyler's estimator and is formulated in terms of easily computable statistics of the data. For its rigorous analysis, we develop a novel framework based on the exchangeable random variables calculus introduced by de Finetti. Our findings are supported by numerical simulations comparing them to other popular GoF tests and demonstrating the significantly higher statistical power of the suggested technique.

Lessons learned from the two largest Galaxy morphological classification catalogues built by convolutional neural networks

ABSTRACT We compare the two largest galaxy morphology catalogues, which separate early- and late-type galaxies at intermediate redshift. The two catalogues were built by applying supervised deep learning (convolutional neural networks, CNNs) to the Dark Energy Survey data down to a magnitude limit of ∼21 mag. The methodologies used for the construction of the catalogues include differences such as the cutout sizes, the labels used for training, and the input to the CNN – monochromatic images versus gri-band normalized images. In addition, one catalogue is trained using bright galaxies observed with DES (i < 18), while the other is trained with bright galaxies (r < 17.5) and ‘emulated’ galaxies up to r-band magnitude 22.5. Despite the different approaches, the agreement between the two catalogues is excellent up to i < 19, demonstrating that CNN predictions are reliable for samples at least one magnitude fainter than the training sample limit. It also shows that morphological classifications based on monochromatic images are comparable to those based on gri-band images, at least in the bright regime. At fainter magnitudes, i > 19, the overall agreement is good (∼95 per cent), but is mostly driven by the large spiral fraction in the two catalogues. In contrast, the agreement within the elliptical population is not as good, especially at faint magnitudes. By studying the mismatched cases, we are able to identify lenticular galaxies (at least up to i < 19), which are difficult to distinguish using standard classification approaches. The synergy of both catalogues provides an unique opportunity to select a population of unusual galaxies.

Starbursts with suppressed velocity dispersion revealed in a forming cluster at z = 2.51

One of the most prominent features of galaxy clusters is the presence of a dominant population of massive ellipticals in their cores. Stellar archaeology suggests that these gigantic beasts assembled most of their stars in the early Universe via starbursts. However, the role of dense environments and their detailed physical mechanisms in triggering starburst activities remain unknown. Here we report spatially resolved Atacama Large Millimeter/submillimeter Array (ALMA) observations of the CO J = 3−2 emission line, with a resolution of about 2.5 kpc, toward a forming galaxy cluster core with starburst galaxies at z = 2.51. In contrast to starburst galaxies in the field often associated with galaxy mergers or highly turbulent gaseous disks, our observations show that the two starbursts in the cluster exhibit dynamically cold (rotation-dominated) gas-rich disks. Their gas disks have extremely low velocity dispersion (σ0 ∼ 20−30 km s−1), which is three times lower than their field counterparts at similar redshifts. The high gas fraction and suppressed velocity dispersion yield gravitationally unstable gas disks, which enables highly efficient star formation. The suppressed velocity dispersion, likely induced by the accretion of corotating and coplanar cold gas, might serve as an essential avenue to trigger starbursts in massive halos at high redshifts.

Galaxy properties as revealed by MaNGA – II. Differences in stellar populations of slow and fast rotator ellipticals and dependence on environment

ABSTRACT We present estimates of stellar population (SP) gradients from stacked spectra of slow rotator (SR) and fast rotator (SR) elliptical galaxies from the MaNGA-DR15 survey. We find that (1) FRs are ∼5 Gyr younger, more metal rich, less α-enhanced and smaller than SRs of the same luminosity Lr and central velocity dispersion σ0. This explains why when one combines SRs and FRs, objects which are small for their Lr and σ0 tend to be younger. Their SP gradients are also different. (2) Ignoring the FR/SR dichotomy leads one to conclude that compact galaxies are older than their larger counterparts of the same mass, even though almost the opposite is true for FRs and SRs individually. (3) SRs with σ0 ≤ 250 km s−1 are remarkably homogeneous within ∼Re: they are old, α-enhanced, and only slightly supersolar in metallicity. These SRs show no gradients in age and M*/Lr, negative gradients in metallicity, and slightly positive gradients in [α/Fe] (the latter are model dependent). SRs with σ0 ≥ 250 km s−1 are slightly younger and more metal rich, contradicting previous work suggesting that age increases with σ0. They also show larger M*/Lr gradients. (4) Self-consistently accounting for M*/L gradients yields Mdyn ≈ M* because gradients reduce Mdyn by ∼0.2 dex while only slightly increasing the M* inferred using a Kroupa (not Salpeter) initial mass function. (5) The SR population starts to dominate the counts above $M_*\ge 3\times 10^{11}\, \mathrm{M}_\odot$; this is the same scale at which the size–mass correlation and other scaling relations change. Our results support the finding that this is an important mass scale that correlates with the environment and above which mergers matter.

High-dimensional covariance matrices in elliptical distributions with application to spherical test

This paper discusses fluctuations of linear spectral statistics of high-dimensional sample covariance matrices when the underlying population follows an elliptical distribution. Such population often possesses high order correlations among their coordinates, which have great impact on the asymptotic behaviors of linear spectral statistics. Taking such kind of dependency into consideration, we establish a new central limit theorem for the linear spectral statistics in this paper for a class of elliptical populations. This general theoretical result has wide applications and, as an example, it is then applied to test the sphericity of elliptical populations.

Cold gas in a complete sample of group-dominant early-type galaxies

We present IRAM 30 m and APEX telescope observations of CO(1-0) and CO(2-1) lines in 36 group-dominant early-type galaxies, completing our molecular gas survey of dominant galaxies in the Complete Local-volume Groups Sample. We detect CO emission in 12 of the galaxies at > 4σ significance, with molecular gas masses in the range ∼ 0.01 − 6 × 108 M⊙, as well as CO in absorption in the non-dominant group member galaxy NGC 5354. In total 21 of the 53 CLoGS dominant galaxies are detected in CO and we confirm our previous findings that they have low star formation rates (0.01–1 M⊙yr−1) but short depletion times (>1 Gyr) implying rapid replenishment of their gas reservoirs. Comparing molecular gas mass with radio luminosity, we find that a much higher fraction of our group-dominant galaxies (60 ± 16%) are AGN-dominated than is the case for the general population of ellipticals, but that there is no clear connection between radio luminosity and the molecular gas mass. Using data from the literature, we find that at least 27 of the 53 CLoGS dominant galaxies contain H I, comparable to the fraction of nearby non-cluster early type galaxies detected in H I and significantly higher that the fraction in the Virgo cluster. We see no correlation between the presence of an X-ray detected intra-group medium and molecular gas in the dominant galaxy, but find that the H I-richest galaxies are located in X-ray faint groups. Morphological data from the literature suggests the cold gas component most commonly takes the form of a disk, but many systems show evidence of galaxy-galaxy interactions, indicating that they may have acquired their gas through stripping or mergers. We provide improved molecular gas mass estimates for two galaxies previously identified as being in the centres of cooling flows, NGC 4636 and NGC 5846, and find that they are relatively molecular gas poor compared to our other detected systems.

Differential Sampling of Visual Space in Ventral and Dorsal Early Visual Cortex.

A fundamental feature of cortical visual processing is the separation of visual processing for the upper and lower visual fields. In early visual cortex (EVC), the upper visual field is processed ventrally, with the lower visual field processed dorsally. This distinction persists into several category-selective regions of occipitotemporal cortex, with ventral and lateral scene-, face-, and object-selective regions biased for the upper and lower visual fields, respectively. Here, using an elliptical population receptive field (pRF) model, we systematically tested the sampling of visual space within ventral and dorsal divisions of human EVC in both male and female participants. We found that (1) pRFs tend to be elliptical and oriented toward the fovea with distinct angular distributions for ventral and dorsal divisions of EVC, potentially reflecting a radial bias; and (2) pRFs in ventral areas were larger (∼1.5×) and more elliptical (∼1.2×) than those in dorsal areas. These differences potentially reflect a tendency for receptive fields in ventral temporal cortex to overlap the fovea with less emphasis on precise localization and isotropic representation of space compared with dorsal areas. Collectively, these findings suggest that ventral and dorsal divisions of EVC sample visual space differently, likely contributing to and/or stemming from the functional differentiation of visual processing observed in higher-level regions of the ventral and dorsal cortical visual pathways.SIGNIFICANCE STATEMENT The processing of visual information from the upper and lower visual fields is separated in visual cortex. Although ventral and dorsal divisions of early visual cortex (EVC) are commonly assumed to sample visual space equivalently, we demonstrate systematic differences using an elliptical population receptive field (pRF) model. Specifically, we demonstrate that (1) ventral and dorsal divisions of EVC exhibit diverging distributions of pRF angle, which are biased toward the fovea; and (2) ventral pRFs exhibit higher aspect ratios and cover larger areas than dorsal pRFs. These results suggest that ventral and dorsal divisions of EVC sample visual space differently and that such differential sampling likely contributes to different functional roles attributed to the ventral and dorsal pathways, such as object recognition and visually guided attention, respectively.

Tests for the parallelism and flatness hypotheses of multi-group profile analysis for high-dimensional elliptical populations

This paper is concerned with tests for the parallelism and flatness hypotheses in multi-group profile analysis for high-dimensional data. We extend to elliptical distributions the procedures developed for normal populations by Harrar and Kong (2016). Specifically, we prove that their statistics continue to be asymptotically normal when the underlying population is elliptical, and we obtain new tests by improving their estimator of the asymptotic variance. Using asymptotic normality, we show that the asymptotic size of the proposed tests is equal to the nominal significance level, and we also derive the asymptotic power. Finally, we present simulation results and find that the power of the new tests is superior to that of the original Harrar–Kong procedure.

Cluster and field elliptical galaxies atz~ 1.3

We compared the properties of 56 elliptical galaxies selected from three clusters at $1.2<z<1.4$ with those of field galaxies in the GOODS-S (~30), COSMOS (~180) and CANDELS (~220) fields. We studied the relationships among effective radius, surface brightness, stellar mass, stellar mass density $\Sigma_{Re}$ and central mass density $\Sigma_{1kpc}$ within 1 kpc radius. We find that cluster ellipticals do not differ from field ellipticals: they share the same structural parameters at fixed mass and the same scaling relations. On the other hand, the population of field ellipticals at $z\sim1.3$ shows a significant lack of massive ($M_*> 2\times 10^{11}$ M$_\odot$) and large (R$_e > 4-5$ kpc) ellipticals with respect to the cluster. Nonetheless, at $M*<2\times 10^{11}$ M$_\odot$, the two populations are similar. The size-mass relation of ellipticals at z~1.3 defines two different regimes, above and below a transition mass $m_t\sim 2-3\times10^{10}$ M$_\odot$: at lower masses the relation is nearly flat (R$_e\propto M_*^{-0.1\pm 0.2}$), the mean radius is constant at ~1 kpc and $\Sigma_{Re}\sim \Sigma_{1kpc}$ while, at larger masses, the relation is R$_e\propto M*^{0.64\pm0.09}$. The transition mass marks the mass at which galaxies reach the maximum $\Sigma_{Re}$. Also the $\Sigma_{1kpc}$-mass relation follows two different regimes, $\Sigma_{1kpc}\propto M*^{0.64\ >m_t}_{1.07\ <m_t}$, defining a transition mass density $\Sigma_{1kpc}\sim 2-3\times10^3$ M$_\odot$ pc$^{-2}$. The mass density $\Sigma_{Re}$ does not correlate with mass, dense/compact galaxies can be assembled over a wide mass regime, independently of the environment. The central mass density, $\Sigma_{1kpc}$, besides to be correlated with the mass, is correlated to the age of the stellar population: the higher the central stellar mass density, the higher the mass, the older the age of the stellar population. [Abridged]

The growth of discs and bulges during hierarchical galaxy formation – I. Fast evolution versus secular processes

We present a theoretical model for the evolution of mass, angular momentum and size of galaxy disks and bulges, and we implement it into the semi-analytic galaxy formation code SAGE. The model follows both secular and violent evolutionary channels, including smooth accretion, disk instabilities, minor and major mergers. We find that the combination of our recipe with hierarchical clustering produces two distinct populations of bulges: merger-driven bulges, akin to classical bulges and ellipticals, and instability-driven bulges, akin to secular (or pseudo-)bulges. The model mostly reproduces the mass-size relation of gaseous and stellar disks, the evolution of the mass-size relation of ellipticals, the Faber-Jackson relation, and the magnitude-colour diagram of classical and secular bulges. The model predicts only a small overlap of merger-driven and instability-driven components in the same galaxy, and predicts different bulge types as a function of galaxy mass and disk fraction. Bulge type also affects the star formation rate and colour at a given luminosity. The model predicts a population of merger-driven red ellipticals that dominate both the low-mass and high-mass ends of the galaxy population, and span all dynamical ages; merger-driven bulges in disk galaxies are dynamically old and do not interfere with subsequent evolution of the star-forming component. Instability-driven bulges dominate the population at intermediate galaxy masses, especially thriving in massive disks. The model green valley is exclusively populated by instability-driven bulge hosts. Through the present implementation the mass accretion history is perceivable in the galaxy structure, morphology and colours.

A steep slope and small scatter for the high-mass end of the L–σ relation atz∼ 0.55

We measure the intrinsic relation between velocity dispersion ($\sigma$) and luminosity ($L$) for massive, luminous red galaxies (LRGs) at redshift $z \sim 0.55$. We achieve unprecedented precision by using a sample of 600,000 galaxies with spectra from the Baryon Oscillation Spectroscopic Survey (BOSS) of the third Sloan Digital Sky Survey (SDSS-III), covering a range of stellar masses $M_* \gtrsim 10^{11} M_{\odot}$. We deconvolve the effects of photometric errors, limited spectroscopic signal-to-noise ratio, and red--blue galaxy confusion using a novel hierarchical Bayesian formalism that is generally applicable to any combination of photometric and spectroscopic observables. For an L-$\sigma$ relation of the form $L \propto \sigma^{\beta}$, we find $\beta = 7.8 \pm 1.1$ for $\sigma$ corrected to the effective radius, and a very small intrinsic scatter of $s = 0.047 \pm 0.004$ in $\log_{10} \sigma$ at fixed $L$. No significant redshift evolution is found for these parameters. The evolution of the zero-point within the redshift range considered is consistent with the passive evolution of a galaxy population that formed at redshift $z=2-3$, assuming single stellar populations. An analysis of previously reported results seems to indicate that the passively-evolved high-mass L-$\sigma$ relation at $z\sim0.55$ is consistent with the one measured at $z=0.1$. Our results, in combination with those presented in Montero-Dorta et al. (2014), provide a detailed description of the high-mass end of the red sequence (RS) at $z\sim0.55$. This characterization, in the light of previous literature, suggest that the high-mass RS distribution corresponds to the "core" elliptical population.

Group Symmetric Robust Covariance Estimation

In this paper, we consider Tyler's robust covariance M-estimator under group symmetry constraints. We assume that the covariance matrix is invariant to the conjugation action of a unitary matrix group, referred to as group symmetry. Examples of group symmetric structures include circulant, perHermitian, and proper quaternion matrices. We introduce a group symmetric version of Tyler's estimator (STyler) and provide an iterative fixed point algorithm to compute it. The classical results claim that at least n=p+1 sample points in general position are necessary to ensure the existence and uniqueness of Tyler's estimator, where p is the ambient dimension. We show that the STyler requires significantly less samples. In some groups, even two samples are enough to guarantee its existence and uniqueness. In addition, in the case of elliptical populations, we provide high probability bounds on the error of the STyler. These, too, quantify the advantage of exploiting the symmetry structure. Finally, these theoretical results are supported by numerical simulations.

Asymptotics of the two-stage spatial sign correlation

The spatial sign correlation (D\"urre, Vogel and Fried, 2015) is a highly robust and easy-to-compute, bivariate correlation estimator based on the spatial sign covariance matrix. Since the estimator is inefficient when the marginal scales strongly differ, a two-stage version was proposed. In the first step, the observations are marginally standardized by means of a robust scale estimator, and in the second step, the spatial sign correlation of the thus transformed data set is computed. D\"urre et al. (2015) give some evidence that the asymptotic distribution of the two-stage estimator equals that of the spatial sign correlation at equal marginal scales by comparing their influence functions and presenting simulation results, but give no formal proof. In the present paper, we close this gap and establish the asymptotic normality of the two-stage spatial sign correlation and compute its asymptotic variance for elliptical population distributions. We further derive a variance-stabilizing transformation, similar to Fisher's z-transform, and numerically compare the small-sample coverage probabilities of several confidence intervals.

Geodesic Hypothesis Testing for Comparing Location Parameters in Elliptical Populations

In this paper we study the geometry of the differentiable manifold associated with two samples of symmetric distributions in the real line equipped with the Fisher information as Riemannian metric. Expressions for the entries of the information matrix are obtained under different assumptions. The geodesic or Rao distance induced by this geometry is used to construct asymptotic parametrization-invariant testing procedures for comparing location parameters. As special cases, we obtain new asymptotic tests for the two sample Behrens-Fisher and Fieller-Creasy problems. Testing equality of several location parameters is also considered. It is shown that when scale parameters are equal, the geodesic test statistic is a strictly monotone increasing function of the Wald statistic. Empirical results for the Student-t distribution provide evidence that the geodesic test statistic has good sampling properties in terms of level and power.

The random matrix regime of Maronna’s M-estimator with elliptically distributed samples

This article demonstrates that the robust scatter matrix estimator CˆN∈CN×N of a multivariate elliptical population x1,…,xn∈CN originally proposed by Maronna in 1976, and defined as the solution (when existent) of an implicit equation, behaves similar to a well-known random matrix model in the limiting regime where the population N and sample n sizes grow at the same speed. We show precisely that CˆN∈CN×N is defined for all n large with probability one and that, under some light hypotheses, ‖CˆN−SˆN‖→0 almost surely in spectral norm, where SˆN follows a classical random matrix model. As a corollary, the limiting eigenvalue distribution of CˆN is derived. This analysis finds applications in the fields of statistical inference and signal processing.

Segmented classification analysis with a class of rectangle-screened elliptical populations

In many practical situations, a statistical practitioner often faces a problem of classifying an object from one of the segmented (or screened) populations where the segmentation was conducted by a set of screening variables. This paper addresses this problem, proposing and studying yet another optimal rule for classification with segmented populations. A class of q-dimensional rectangle-screened elliptically contoured (RSEC) distributions is considered for flexibly modeling the segmented populations. Based on the properties of the RSEC distributions, a parametric procedure for the segmented classification analysis (SCA) is proposed. This includes motivation for the SCA as well as some theoretical propositions regarding its optimal rule and properties. These properties allow us to establish other important results which include an efficient estimation of the rule by the Monte Carlo expectation–conditional maximization algorithm and an optimal variable selection procedure. Two numerical examples making use of utilizing a simulation study and a real dataset application and advocating the SCA procedure are also provided.