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

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

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 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 $\nu F_{\nu}$ spectral peak energy $E_{\rm pk,obs}$ and the isotropic equivalent energy $E_{\rm iso}$ (1--$10^4$ 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 $E_{\rm pk,obs}$-$E_{\rm iso}$ 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 $E_{\rm pk,obs}$. 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 $E_{\rm pk,obs}$--$E_{\rm iso}$ 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 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

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