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 test for partial correlation with censored astronomical data

A new procedure is presented, which allows, based on Kendall's \tau, to test for partial correlation in the presence of censored data. Further, a significance level can be assigned to the partial correlation -- a problem which hasn't been addressed in the past, even for uncensored data. The results of various tests with simulated data are reported. Finally, we apply this newly developed methodology to estimate the influence of selection effects on the correlation between the soft X--ray luminosity and both total and core radio luminosity in a complete sample of Active Galactic Nuclei

Estimating Black Hole Masses in Active Galaxies Using the Halpha Emission Line

It has been established that virial masses for black holes in low-redshift active galaxies can be estimated from measurements of the optical continuum strength and the width of the broad Hbeta line. Under various circumstances, however, both of these quantities can be challenging to measure or can be subject to large systematic uncertainties. To mitigate these difficulties, we present a new method for estimating black hole masses. From analysis of a new sample of broad-line active galactic nuclei, we find that Halpha luminosity scales almost linearly with optical continuum luminosity and that a strong correlation exists between Halpha and Hbeta line widths. These two empirical correlations allow us to translate the standard virial mass system to a new one based solely on observations of the broad Halpha emission line.Comment: to appear in Apj; 8 pages; 5 figures; uses emulateapj5.st

The black hole fundamental plane from a uniform sample of radio and X-ray emitting broad line AGNs

We derived the black hole fundamental plane relationship among the 1.4GHz radio luminosity (L_r), 0.1-2.4keV X-ray luminosity (L_X), and black hole mass (M) from a uniform broad line SDSS AGN sample including both radio loud and radio quiet X-ray emitting sources. We found in our sample that the fundamental plane relation has a very weak dependence on the black hole mass, and a tight correlation also exists between the Eddington luminosity scaled X-ray and radio luminosities for the radio quiet subsample. Additionally, we noticed that the radio quiet and radio loud AGNs have different power-law slopes in the radio--X-ray non-linear relationship. The radio loud sample displays a slope of 1.39, which seems consistent with the jet dominated X-ray model. However, it may also be partly due to the relativistic beaming effect. For radio quiet sample the slope of the radio--X-ray relationship is about 0.85, which is possibly consistent with the theoretical prediction from the accretion flow dominated X-ray model. We briefly discuss the reason why our derived relationship is different from some previous works and expect the future spectral studies in radio and X-ray bands on individual sources in our sample to confirm our result.Comment: 23 pages, 7 figures, ApJ accepte

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

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

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

The X-ray Energy Dependence of the Relation between Optical and X-ray Emission in Quasars

We develop a new approach to the well-studied anti-correlation between the optical-to-X-ray spectral index, alpha_ox, and the monochromatic optical luminosity, l_opt. By cross-correlating the SDSS DR5 quasar catalog with the XMM-Newton archive, we create a sample of 327 quasars with X-ray S/N > 6, where both optical and X-ray spectra are available. This allows alpha_ox to be defined at arbitrary frequencies, rather than the standard 2500 Angstroms and 2 keV. We find that while the choice of optical wavelength does not strongly influence the alpha_ox-l_opt relation, the slope of the relation does depend on the choice of X-ray energy. The slope of the relation becomes steeper when alpha_ox is defined at low (~ 1 keV) X-ray energies. This change is significant when compared to the slope predicted by a decrease in the baseline over which alpha_ox is defined. The slopes are also marginally flatter than predicted at high (~ 10 keV) X-ray energies. Partial correlation tests show that while the primary driver of alpha_ox is l_opt, the Eddington ratio correlates strongly with alpha_ox when l_opt is taken into account, so accretion rate may help explain these results. We combine the alpha_ox-l_opt and Gamma -L_bol/L_Edd relations to naturally explain two results: 1) the existence of the Gamma-l_x relation as reported in Young et al. (2009) and 2) the lack of a Gamma-l_opt relation. The consistency of the optical/X-ray correlations establishes a more complete framework for understanding the relation between quasar emission mechanisms. We also discuss two correlations with the hard X-ray bolometric correction, which we show correlates with both alpha_ox and Eddington ratio. This confirms that an increase in accretion rate correlates with a decrease in the fraction of up-scattered disk photons.Comment: 27 pages preprint style, 10 figures. New material added to section 4.3 (Fig. 11) regarding hard X-ray bolometric corrections. Accepted for publication in Astrophysical Journa

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

Improved maximum likelihood estimators in a heteroskedastic errors-in-variables model

This paper develops a bias correction scheme for a multivariate heteroskedastic errors-in-variables model. The applicability of this model is justified in areas such as astrophysics, epidemiology and analytical chemistry, where the variables are subject to measurement errors and the variances vary with the observations. We conduct Monte Carlo simulations to investigate the performance of the corrected estimators. The numerical results show that the bias correction scheme yields nearly unbiased estimates. We also give an application to a real data set.Comment: 12 pages. Statistical Paper