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\textit{Oh, an empty article!} \documentclass[12pt]{article}  You can get started \usepackage{hyperref}  \usepackage{amsmath}  \usepackage{graphicx}  \usepackage[font=small]{subcaption}  \usepackage[font=small,labelfont=bf]{caption}  \newcommand{\makesubfig}[3]{  \begin{subfigure}{#2\textwidth}  \centering  \includegraphics[width=#3\linewidth]{#1}  \end{subfigure}}  \newcounter{fignum}  \setcounter{fignum}{0}  \newcommand{\makefig}[2]{  \begin{figure}  \makesubfig{#1}{1.0}{1.0}  \caption{\label{fig:\arabic{fignum}}#2}  \end{figure}  \stepcounter{fignum}}  \begin{document}  \title{CARMA Analysis of SDSS Quasars in Stripe 82}  \author{Jack O'Brien}  \date{\today}  \maketitle  \section{Data}  Our sample consists of 56 spectroscopically confirmed quasar light curves from the seventh data release of the Sloan Digital Sky Survey (Schneider et al. 2010). These quasars range in redshift from $z = 0.19$ to $z = 3.83$ and in luminosity from $L = 10^{15} L_{\odot}$ to $L = 10^{20} L_{\odot}$. %  These samples were taken from the Southern Equitorial Stripe known as Stripe 82. Stripe 82 is a $275^{2}$ degree area of sky with repeated sampling centered on the celestial equator. It reaches 2 magnitudes deeper than SDSS single pass data going as deep as magnitude 23.5 in the r-band for galaxies with median seeing of 1.1'' (Annis et al. 2014). %  These samples were chosen because they exist in the field of the Kepler K2 mission's campaign 8. This will eventually give us a much denser data set with shorter cadence allowing us to probe far deeper into the short-term variability properties of these objects than would be possible with SDSS alone.%  Each light curve contains photometric information from two bands (g and r) with as much as 10 years of data. The number of epochs of data range from 29 observations to 81 observations in all photometric bands with sampling intervals ranging from one day to two years. This inconsistent sampling will lead to issues with our analysis which will be discussed later. %  PSF magnitudes are calibrated using a set of standard stars (Ivezic et al 2007) to reduce the error in our data down to 1\%. We then convert these magnitudes to fluxes for our analysis and convert observed time sampling intervals to the rest frame of the quasar. %  We use asinh magnitudes (also referred to as "Luptitudes") for flux conversion (York et al. 2000) (Lupton, Gunn, & Szalay 1999) as is standard for SDSS.   \makefig{https://www.authorea.com/users/3982/articles/112661/master/file/figures/K2S82FOV/K2S82FOV.png}{The positions of the 56 SDSS quasars (red) overlaid on the K2 campaign 8 field of view (blue) and the Stripe 82 region (green). The combined long term variability information from SDSS and short term variability information from Kepler will allow us to more tightly constrain out models in the future. }  \section{CARMA}  A continuous-time ARMA (CARMA) process is the continuous case of the discrete ARMA process. A system described  by \textbf{double clicking} a CARMA process obeys the stochastic differential equation     $$d^{P}f(t) + \alpha_{P-1} d^{P-1}f(t) + ... + \alpha_{0} f(t) = \beta_{Q}d^{Q}w(t) + \beta_{Q-1} d^{Q-1}w(t) + ... + w(t)$$    Where $d$ represents a change in a variable between times $t$ and $t + dt$, $f$ represents the state of the system minus the mean, and $w \sim IID\ N(0, \sigma^{2})$ continuous-time white noise random process representing the driving noise in the system due to non-linear effects. In  this text block case, it will represent temperature fluctuations due to magnetohydrodynamic instabilities in the accretion disk. The most well known example is the case where $p=1$  and begin editing. You $q=0$ CAR(1) process also known in the astronomical community as a damped random walk (DRW).    We construct the auto-regressive polynomial as the characteristic polynomial of the auto-regressive side of the CARMA process.     $$A(z) = \sum_{k=0}^{P} \alpha_{k}z^{k}$$    The roots of the auto-regressive polynomial, $\rho_{p}$, indicate timescales on which variability presents itself. The auto-correlation function of a CARMA(p,q) process goes as    $$ $$    \section{The Kalman Filter}    \section{Canonical Light Curves}  \subsection{SDSS J134342.5-004243.8 (sample 36 in r-band)}  This sample acts as a good example of a light curve well fit by a CARMA(2,1) model. We'll be using this example for a good amount of the discussion as since most of the plots are relatively clean and easy to understand compared to some of the poorer fits. The PSD estimate is very similar to that of a CARMA(1,0) model even though this CARMA model contains complex roots. This is interesting to us because we can clearly demonstrate an example of where we have a light-curve with quasi-periodic behavior disguised to look like a damped-random walk by visual inspection. This light-curve also has a relatively low measurement noise level which means we can infer more properties about the light curve, as well as relate it's sampling to it's accuracy.  \makefig{https://www.authorea.com/users/3982/articles/112661/master/file/figures/LC_Sample36CalibratedrBand/LC_Sample36CalibratedrBand.png}{Interpolated light-curve of SDSS J134342.6-004243.8 in the r-band. The relatively low errors, high rate of sampling, and large time density of data (due to high z ~ 2.228), make this an ideal candidate as canonical example of an SDSS light curve well fit by a CARMA process. The purple regions indicate the 95th percentile interpolation of the light-curve between observations as determined by our model through the Kalman filter.}  At first inspection of the light-curve, we  can make out that there may be some sort of low-frequency oscillatory behavior with timescales on the order of 600 days. Since we converted the time differences between cadences to the rest-frame of the quasar (z \~ 2.228), we've more than doubled the time-density of our data, allowing us to sample much shorter timescales than with nearby objects. This means that we are able to push the depth of the frequency domain of our power spectrum much deeper and consequently, better distinguish this light-curve from other model orders.   We use the Green's function to describe the auto-regressive behavior of this light-curve. The Green's function,  also click the \textbf{Text} button below to add new block elements. Or you known as the impulse-response function tells us how the brightness of the accretion disk changes with an impulse and is described by setting the auto-regressive side of the CARMA process to a delta-function.  $$\frac{d^{P}f(t)}{dt^{P}} + \alpha_{P-1} \frac{d^{P-1}f(t)}{dt^{P-1}} + ... + \alpha_{0} f(t) = \delta(t)$$  The time it takes for the Green's function to reach it's maximum tells us how long it takes for a specific driving impulse to a maximum effect on the brightness of the light curve. Below we plot the distribution of possible Green's functions with 1, 2, and 3 $\sigma$ confidence intervals was well as the distribution of the possible maximization timescales. The Green's function for this CARMA(2,1) model is  $$G(t) = \frac{e^{\rho_{1}t} - e^{\rho_{2}t}}{\rho_{1} - \rho_{2}}$$  where $\rho_{p}$ is the $p^{th}$ root of the auto-regressive polynomial. %Talk about the AR polynomial earlier adn reference the ACF  (...talk about numerical results)  \makefig{https://www.authorea.com/users/3982/articles/112661/master/file/figures/MasterGreensFunc_Sample36CalibratedrBand/MasterGreensFunc_Sample36CalibratedrBand.png}{Distribution of Green’s function for sample 36 r-band at 1, 2, and 3 σσ. Distribution of timescales at which the Green’s functions are maximized are plotted on the top with corresponding values of the right. We  can \textbf{drag see that the maximum value of the Green’s function increases with longer timescales. The median Green’s function maximizing timescale is on the order of 100 days with an e-folding time of around 300 days.}  Another useful tool for analyzing the variable properties of a light-curve is it's power spectrum. This relates the variance between observations to the frequency at which they are observed. For SDSS data, estimating the true power-spectrum from the periodogram is difficult because of the irregularity  and drop infrequency of the sampling. The CARMA model however allows us to make a theoretical prediction of what the power spectrum should look like fairly easily.  \makefig{https://www.authorea.com/users/3982/articles/112661/master/file/figures/PSD_Sample36CalibratedrBand/PSD_Sample36CalibratedrBand.png}{The relatively short sampling of this light-curve allows us to probe the power spectrum down to timescale as little as 3 days. Though the roots of the auto-regressive polynomial are complex, we don’t observe any strong PSD features as the widths of their Lorentzians are too large, smearing them out. Because of this, from inspection alone, it would be hard to distinguish this from  an image} right onto over-damped oscillator.}  \subsection{Sample 13/24 in g and r bands}  In this section, we look at the differences in the g and r results. For sample 13, the model order changes between the two bands from 3,0 in g to 2,1 in r. In sample 24, the model order remains the same (2,1 in  this text. Happy writing! case). We will investigate possible explanations for similarities and differences in the selected model orders for these light curves in the context of accretion disk geometry and statistical uncertainties in the data.  \end{document}