227 research outputs found
Some Aspects of Measurement Error in Linear Regression of Astronomical Data
I describe a Bayesian method to account for measurement errors in linear
regression of astronomical data. The method allows for heteroscedastic and
possibly correlated measurement errors, and intrinsic scatter in the regression
relationship. The method is based on deriving a likelihood function for the
measured data, and I focus on the case when the intrinsic distribution of the
independent variables can be approximated using a mixture of Gaussians. I
generalize the method to incorporate multiple independent variables,
non-detections, and selection effects (e.g., Malmquist bias). A Gibbs sampler
is described for simulating random draws from the probability distribution of
the parameters, given the observed data. I use simulation to compare the method
with other common estimators. The simulations illustrate that the Gaussian
mixture model outperforms other common estimators and can effectively give
constraints on the regression parameters, even when the measurement errors
dominate the observed scatter, source detection fraction is low, or the
intrinsic distribution of the independent variables is not a mixture of
Gaussians. I conclude by using this method to fit the X-ray spectral slope as a
function of Eddington ratio using a sample of 39 z < 0.8 radio-quiet quasars. I
confirm the correlation seen by other authors between the radio-quiet quasar
X-ray spectral slope and the Eddington ratio, where the X-ray spectral slope
softens as the Eddington ratio increases.Comment: 39 pages, 11 figures, 1 table, accepted by ApJ. IDL routines
(linmix_err.pro) for performing the Markov Chain Monte Carlo are available at
the IDL astronomy user's library, http://idlastro.gsfc.nasa.gov/homepage.htm
A Comparative Study of Two Real Root Isolation Methods
Recent progress in polynomial elimination has rendered the computation of the real roots of ill-conditioned polynomials of high degree (over 1000) with huge coefficients (several thousand digits) a critical operation in computer algebra.
To rise to the occasion, the only method-candidate that has been considered by various authors for modification and improvement has been the Collins-Akritas bisection method [1], which is a based on a variation of Vincent’s theorem [2].
The most recent example is the paper by Rouillier and Zimmermann [3], where the authors present “... a new algorithm, which is optimal in terms of memory usage and as fast as both Collins and Akritas’ algorithm and Krandick variant ...” [3]
In this paper we compare our own continued fractions method CF [4] (which is directly based on Vincent’s theorem) with the best bisection method REL described in [3]. Experimentation with the data presented in [3] showed that, with respect to time, our continued fractions method CF is by far superior to REL, whereas the two are about equal with respect to space
Linear Regression for Astronomical Data with Measurement Errors and Intrinsic Scatter
Two new methods are proposed for linear regression analysis for data with
measurement errors. Both methods are designed to accommodate intrinsic scatter
in addition to measurement errors. The first (BCES) is a direct extension of
the ordinary least squares (OLS) estimator to allow for measurement errors. It
is quite general, allowing a) for measurement errors on both variables, b) the
measurement errors for the two variables to be dependent, c) the magnitudes of
the measurement errors to depend on the measurements, and d) other `symmetric'
lines such as the bisector and the orthogonal regression can be constructed.
The second method is a weighted least squares (WLS) estimator, which applies
only in the case where the `independent' variable is measured without error and
the magnitudes of the measurement errors on the 'dependent' variable are
independent from the measurements. Several applications are made to
extragalactic astronomy: The BCES method, when applied to data describing the
color-luminosity relations for field galaxies, yields significantly different
slopes than OLS and other estimators used in the literature. Simulations with
artificial data sets are used to evaluate the small sample performance of the
estimators. Unsurprisingly, the least-biased results are obtained when color is
treated as the dependent variable. The Tully-Fisher relation is another example
where the BCES method should be used because errors in luminosity and velocity
are correlated due to inclination corrections. We also find, via simulations,
that the WLS method is by far the best method for the Tolman surface-brightness
test, producing the smallest variance in slope by an order of magnitude.
Moreover, with WLS it is not necessary to ``reduce'' galaxies to a fiducial
surface-brightness, since this model incorporates intrinsic scatter.Comment: 23 pages LaTeX, 1 postscript table, and style files (uuencoded,
gzipp'ed tar file). To appear in the Astrophysical Journal, October 10, 1996
issue, Vol. 47
The SVD-Fundamental Theorem of Linear Algebra
Given an m × n matrix A, with m ≥ n, the four subspaces associated with it are shown in Fig. 1 (see [1]).
Fig. 1. The row spaces and the nullspaces of A and AT; a1 through an and h1 through hm are abbreviations of the alignerframe and hangerframe vectors respectively (see [2]).
The Fundamental Theorem of Linear Algebra tells us that N(A) is the orthogonal complement of R(AT). These four subspaces tell the whole story of the Linear System Ax = y. So, for example, the absence of N(AT) indicates that a solution always exists, whereas the absence of N(A) indicates that this solution is unique. Given the importance of these subspaces, computing bases for them is the gist of Linear Algebra. In “Classical” Linear Algebra, bases for these subspaces are computed using Gaussian Elimination; they are orthonormalized with the help of the Gram-Schmidt method. Continuing our previous work [3] and following Uhl’s excellent approach [2] we use SVD analysis to compute orthonormal bases for the four subspaces associated with A, and give a 3D explanation. We then state and prove what we call the “SVD-Fundamental Theorem” of Linear Algebra, and apply it in solving systems of linear equations
The Ks-band Tully-Fisher Relation - A Determination of the Hubble Parameter from 218 ScI Galaxies and 16 Galaxy Clusters
The value of the Hubble Parameter (H0) is determined using the
morphologically type dependent Ks-band Tully-Fisher Relation (K-TFR). The slope
and zero point are determined using 36 calibrator galaxies with ScI morphology.
Calibration distances are adopted from direct Cepheid distances, and group or
companion distances derived with the Surface Brightness Fluctuation Method or
Type Ia Supernova. Distances are determined to 16 galaxy clusters and 218 ScI
galaxies with minimum distances of 40.0 Mpc. From the 16 galaxy clusters a
weighted mean Hubble Parameter of H0=84.2 +/-6 km s-1 Mpc-1 is found. From the
218 ScI galaxies a Hubble Parameter of H0=83.4 +/-8 km s-1 Mpc-1 is found. When
the zero point of the K-TFR is corrected to account for recent results that
find a Large Magellanic Cloud distance modulus of 18.39 +/-0.05 a Hubble
Parameter of 88.0 +/-6 km s-1 Mpc-1 is found. A comparison with the results of
the Hubble Key Project (Freedman et al 2001) is made and discrepancies between
the K-TFR distances and the HKP I-TFR distances are discussed. Implications for
Lamda-CDM cosmology are considered with H0=84 km s-1 Mpc-1. (Abridged)Comment: 37 pages including 12 tables and 7 figures. Final version accepted
for publication in the Journal of Astrophysics & Astronom
A Project Based Approach to Statistics and Data Science
In an increasingly data-driven world, facility with statistics is more
important than ever for our students. At institutions without a statistician,
it often falls to the mathematics faculty to teach statistics courses. This
paper presents a model that a mathematician asked to teach statistics can
follow. This model entails connecting with faculty from numerous departments on
campus to develop a list of topics, building a repository of real-world
datasets from these faculty, and creating projects where students interface
with these datasets to write lab reports aimed at consumers of statistics in
other disciplines. The end result is students who are well prepared for
interdisciplinary research, who are accustomed to coping with the
idiosyncrasies of real data, and who have sharpened their technical writing and
speaking skills
A Complete Catalog of Swift GRB Spectra and Durations: Demise of a Physical Origin for Pre-Swift High-Energy Correlations
We calculate durations and spectral paramaters for 218 Swift bursts detected
by the BAT instrument between and including GRBs 041220 and 070509, including
77 events with measured redshifts. Incorporating prior knowledge into the
spectral fits, we are able to measure the characteristic spectral
peak energy and the isotropic equivalent energy
(1-- keV) for all events. This complete and rather extensive catalog,
analyzed with a unified methodology, allows us to address the persistence and
origin of high-energy correlations suggested in pre-Swift observations. We find
that the - correlation is present in the Swift
sample; however, the best-fit powerlaw relation is inconsistent with the
best-fit pre-Swift relation at >5 sigma significance. Moreover, it has a factor
>~ 2 larger intrinsic scatter, after accounting for large errors on . A large fraction of the Swift events are hard and subluminous
relative to (and inconsistent with) the pre-Swift relation, in agreement with
indications from BATSE GRBs without redshift. Moreover, we determine an
experimental threshold for the BAT detector and show how the -- correlation arises artificially due to partial
correlation with the threshold. We show that pre-Swift correlations found by
Amati et al.(2002), Yonetoku et al. (2004), Firmani et al.(2006) (and
independently by others) are likely unrelated to the physical properties of
GRBs and are likely useless for tests of cosmology. Also, an explanation of
these correlations in terms of a detector threshold provides a natural and
quantitative explanation for why short-duration GRBs and events at low redshift
tend to be outliers to the correlations.Comment: 25 pages, 9 figures, 2 tables, Accepted to Ap
Evolution of the X-ray Emission of Radio-Quiet Quasars
We report new Chandra observations of seven optically faint, z \sim 4
radio-quiet quasars. We have combined these new observations with previous
Chandra observations of radio-quiet quasars to create a sample of 174 sources.
These sources have 0.1 < z < 4.7, and 10^{44} ergs s^{-1} < nu L_{nu} (2500
\AA) < 10^{48} ergs s^{-1}. The X-ray detection fraction is 90%. We find that
the X-ray loudness of radio-quiet quasars decreases with UV luminosity and
increases with redshift. The model that is best supported by the data has a
linear dependence of optical-to-X-ray ratio, alpha_{ox}, on cosmic time, and a
quadratic dependence of alpha_{ox} on log L_{UV}, where alpha_{ox} becomes
X-ray quiet more rapidly at higher log L_{UV}. We find no significant evidence
for a relationship between the X-ray photon index, Gamma_X, and the UV
luminosity, and we find marginally significant evidence that the X-ray
continuum flattens with increasing z (2 sigma). The Gamma_X-z anti-correlation
may be the result of X-ray spectral curvature, redshifting of a Compton
reflection component into the observed Chandra band, and/or redshifting of a
soft excess out of the observed Chandra band. Using the results for Gamma_X, we
show that the alpha_{ox}-z relationship is unlikely to be a spurious result
caused by redshifting of the observable X-ray spectral region. A correlation
between alpha_{ox} and z implies evolution of the accretion process. We present
a qualitative comparison of these new results with models for accretion disk
emission.Comment: Accepted by ApJ, 48 pages, 10 figures, 5 table
The origins of X-ray emission from the hotspots of FRII radio sources
We use new and archival Chandra data to investigate the X-ray emission from a
large sample of compact hotspots of FRII radio galaxies and quasars from the 3C
catalogue. We find that only the most luminous hotspots tend to be in good
agreement with the predictions of a synchrotron self-Compton model with
equipartition magnetic fields. At low hotspot luminosities inverse-Compton
predictions are routinely exceeded by several orders of magnitude, but this is
never seen in more luminous hotspots. We argue that an additional synchrotron
component of the X-ray emission is present in low-luminosity hotspots, and that
the hotspot luminosity controls the ability of a given hotspot to produce
synchrotron X-rays, probably by determining the high-energy cutoff of the
electron energy spectrum. It remains plausible that all hotspots are close to
the equipartition condition.Comment: 49 pages, 16 figures. ApJ accepted. Revised version fixes a typo in
one of the Tables and corrects a statement about 3C27
What Powers the Compact Radio Emission in Nearby Elliptical and S0 Galaxies?
Many nearby early-type (elliptical and S0) galaxies contain weak
(milli-Jansky level) nuclear radio sources on scales a few hundred parsecs or
less. The origin of the radio emission, however, has remained unclear,
especially in volume-limited samples that select intrinsically less luminous
galaxies. Both active galactic nuclei and nuclear star formation have been
suggested as possible mechanisms for producing the radio emission. This paper
utilizes optical spectroscopic information to address this issue. A substantial
fraction of the early-type galaxies surveyed with the Very Large Array by
Wrobel & Heeschen (1991) exhibits detectable optical emission lines in their
nuclei down to very sensitive limits. Comparison of the observed radio
continuum power with that expected from the thermal gas traced by the optical
emission lines implies that the bulk of the radio emission is nonthermal. Both
the incidence and the strength of optical line emission correlate with the
radio power. At a fixed line luminosity, ellipticals have stronger radio cores
than S0s. The relation between radio power and line emission observed in this
sample is consistent with the low-luminosity extension of similar relations
seen in classical radio galaxies and luminous Seyfert nuclei. A plausible
interpretation of this result is that the weak nuclear sources in nearby
early-type galaxies are the low-luminosity counterparts of more powerful AGNs.
The spectroscopic evidence supports this picture. Most of the emission-line
objects are optically classified as Seyfert nuclei or low-ionization nuclear
emission-line regions (LINERs), the majority of which are likely to be
accretion-powered sources.Comment: LaTex, 16 pages including embedded figures. Accepted for publication
in the Astrophysical Journa
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