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Time Series Analysis
Astro193 April 13, 2015


Characterizing Variability
· · · Variability
- typically a change of some observed quantity with time:
· flux, spectrum, emission or absorption line properties

Variability
- gives direct link to physical processes

Today:
· Periodogram · Noise

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Further 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

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Time Series Analysis
· · Time-series data Models have trends with time
· Not all the values are independent, for example yi+1 can depend on y
i

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Goals of analysis:
· Characterize the temporal correlations including its significance:
Example: Differentiate between pulsating and eclipsing stars.

· Forecast future values of y:
Example: solar activity, or orbit predictions for a comet or asteroid.


Time Series Analysis: Deterministic process - predictable, for example perfectly periodic star Random process - not perfectly predictable, even if the rule for generating variations known, time series has a stochastic nature


PSD: Power Spectral Density
Represents the amount of variability power (mean of the squared amplitude) as a function of temporal frequency (time-scale-1). For evenly sampled light curve PSD can be calculated via classic periodogram.


Classic Periodogram
Detecting periodic signal in noisy data Xi = X(ti) = Xs(ti) + R(ti) Classical periodogram Using Discrete FT:

For evenly spaced data t = 1 , t j = j , X j = X (t j ) Evaluated at N=N0/2


Classic Periodogram
Hard to use in noisy data need to collect a lot of data to reduce the noise:
signal noise

S/N


Classic Periodogram
Spectral leakage:
for a sinusoidal signal at 0 the power in the periodogram leaks to other frequencies, especially in the finite amount of data: - to nearby frequencies - sidelobes - due to finite interval of the sampled data. - leakage to distant frequencies - due to the finite size of the interval between samples.

Aliasing:
leakage of power from high frequencies to much lower frequencies. - sensitive to the unevennes of the sampling, and can be reduced with irregular sampling.


Lomb-Scargle Periodogram


Noise low

Classic vs. L-S

Noise high

No detection