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Time Series Analysis
Astro193 April 20, 2015


· Today:
· `Noise' · Correlations, Structure Function · CAR model - applications

· Reading:
· Scargle Jeff, `Studies in Astronomical Time Series Analysis', Part I ApJS, 45,1 - analysis of random processes Part II, ApJ, 263, 835 - periodogram Press 1978, `Flicker noises in Astronomy' and Elsewhere', Comments on Astrophysics, 7, 103 Kelly et al 2009, Are the Variations in Quasar Optical Flux Driven by Thermal Fluctuations? , ApJ. 698, 895.

· ·


White Noise - class of random processes R:

Different kinds of white noise process exist according to the probability distribution of R. Gaussian, normally distributed noise is very common.


Different realization of a random process:


White noise - flat PSD

Random walk PSD ~ f -2


Flicker ~ f

-1

3C273 80 years

time


Moving Average X = C*R Autoregressive Process R = A*X

X R C A

-

realization of a process uncorrelated white noise convolution filter autoregressive filter


White Noise

Random Walk
White noise (flat PSD)

A convolution with the response function filter

Pink noise (1/f)

Red noise (1/f2)


Autocorrelation and Structure Function


SDSS Quasars - Structure Function Analysis

Vanden Berk et al 2004, ApJ.601, 692


Autoregressive Process
Retains memory of previous states - such as random walk: xi = xi-1 + i - based on the last previous value AR(k) - depends on k past values

i - random variable with mean xi measured at discrete times.
AR , 2
AR

= 0 and variance 2

AR

- parameters of the model.

Generalization to uneven data = CAR(k)

continuous autoregressive process


CAR(1)
Process described by

- relaxation time

(t) - white noise with mean = 0 and variance = 1

Initial value

Stochastic component



Stochastic View of the Accretion Disk
Dexter and Agol 2011 ApJ 727 L24

n=2200

n=550

n=140

Temperature maps assuming that Temp(, r, time) follows a damped random walk in each independent zone n assuming the local temperature characteristic timescale of 200 days.


Modeling X-ray Variability
MCG-6-30-15
data

model XTE ~10 years XMM ~1.5 days

Time [days]
Kelly, Sobolewska & Siemiginowska, 2011 ApJ 730 52

Time [sec]


Modeling X-ray Variability
100 realizations of the PSD given the observed lightcurves
One break

median Two breaks best-fit

MCG-6-30-15

Akn 564



Simulations

Power Spectrum
flat

Light crossing time

Orbital time red noise leak

Thermal

MBH = 108 Msun and with different timescales 7 years, sampled every 5 days