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Basic Time Series
Analyzing variable star data
for the amateur astronomer
What Is Time Series?

Single variable x that changes over time t

can be multiple variables, W W W A T
Light curve: x = brightness (magnitude)
 Each observation consists of two numbers
 Time t is considered perfectly precise
 Data (observation/measurement/estimate)
x is not perfectly precise

Two meanings of “Time Series”
TS is a process, how the variable
changes over time
 TS is an execution of the process, often
called a realization of the TS process
 A realization (observed TS) consists of
pairs of numbers (tn,xn), one such pair for
each observation

Goals of TS analysis
Use the process to define the behavior of
its realizations
 Use a realization, i.e. observed data (tn,xn),
to discover the process


This is our main goal
Special Needs
Astronomical data creates special
circumstances for time series analysis
 Mainly because the data are irregularly
spaced in time – uneven sampling
 Sometimes the time spacing (the
“sampling”) is even pathological – with big
gaps that have periods all their own

Analysis Step 1
Plot the data and look at the graph!
(visual inspection)
 Eye+brain combination is the world’s best
pattern-recognition system
 BUT – also the most easily fooled


“pictures in the clouds”
Use visual inspection to get ideas
 Confirm them with numerical analysis

Data = Signal + Noise

True brightness is a function of time f(t)


it’s probably smooth (or nearly so)
There’s some measurement error ε
it’s random
 It’s almost certainly not smooth

Additive model: data xn at time tn is sum of
signal f(tn) and noise εn

xn = f(tn) + εn

Noise is Random
That’s its definition!
 Deterministic part = signal
 Random part = noise
 Usually – the true brightness is
deterministic, therefore it’s the signal
 Usually – the noise is measurement error

Achieve the Goal
Means we have to figure out how the
signal behaves and how the noise
behaves
 For light curves, we usually just assume
how the noise behaves
 But we still should determine its
parameters

What Determines Random?
Probability distribution (pdf or pmf)
 pdf: probability that the value falls in a
small range of width dε, centered on ε is

Probability = P(ε) dε
 pmf: probability that the value is ε is P(ε)
 pdf/pmf has some mean value μ
 pdf/pmf has some standard deviation σ

Most Common Noise Model

i.i.d. = “independent identically distributed”
Each noise value is independent of others

P12(x1,x2) = P1(x1)P2(x2)

They’re all identically distributed

P1(x1) = P2(x2)

What is the Distribution?

Most common is Gaussian (a.k.a. Normal)
P( ) 
e
1
2
2
 (   ) / 
2
 2
Noise Parameters

μ = mean = <ε>

Usually assumed zero (i.e., data unbiased)
σ2 = variance = <(ε-μ)2>
 σ = √(σ2) = standard deviation

Typical value is 0.2 mag. for visual data
 Smaller for CCD/photoelectric (we hope!)
 Note: don’t diparage visual data, what they
lack in individual precision they make up by
the power of sheer numbers

Is the default noise model right?
No! We know it’s wrong
 Bias: μ values not zero
 NOT identically distributed – different
observers have different μ, σ values
 Sometimes not even independent
(autocorrelated noise)
 BUT – i.i.d. Gaussian is still a useful
working hypothesis, so W W W A T

Even if …
Even if we know the form of the noise …
 We still have to figure out its parameters

Is it unbiased (i.e. centered at zero so μ = 0)?
 How big does it tend to be (what’s σ )?

And …
We still have to separate the signal from
the noise
 And of course figure out the form of the
signal, i.e.,
 Figure out the process which determines
the signal

Whew!
Simplest Possible Signal
None at all!
f(t) = constant = βo
 This is the null hypothesis for many tests
 But we can’t be sure f(t) is constant …
 … that’s only a model of the signal

Separate Signal from Noise
We already said
data = signal + noise
 Therefore
data – signal = noise
 Approximate signal by model
 Approximate noise by residuals
data – model = residuals
xn – yn = R n
 If model is correct, residuals are all noise

Estimate Noise Parameters
Use residuals Rn to estimate noise
parameters
1
 Estimate mean μ by average R 

N
R

N
j 1

Estimate standard deviation σ by sample
standard deviation
N
s
 (R
j 1
j
 R)
N 1
2
j
Averages

When we average i.i.d. noise we expect to
get the mean
   

Standard deviation of the average
(usually called the standard error) is less
than standard deviation of the data
 ( ave)  " s.e." 
 ( raw)
N
Confidence Interval
95% confidence interval is the range in
which we expect the average to lie, 95% of
the time
 About 2 standard errors above or below
the expected value

95% C.I .  x  2 ( ave)
 x  2 ( raw) / N
Does average change?

Divide time into bins





Usually of equal time width (often 10 days)
Sometimes of equal number of data N
Compute average and standard deviation within
each bin
IF signal is constant AND noise is consistent,
THEN expected value of data average will be
constant
So: do the “bin averages” show more variation
than is expected from noise?
ANOVA test
Compare variance of averages to variance
of data (ANalysis Of VAriance = ANOVA)
 In other words… compare variance
between bins to variance within bins
 “F-test” gives a “p-value,” probability of
getting that result IF the data are just noise
 Low p-value  probably NOT just noise

Either we haven’t found all the signal
 Or the noise isn’t the simple kind

ANOVA test

50-day averages:
Fstat
df.between df.within
p
 0.315563
2
147
0.729871
 NOT significant


10-day averages:
Fstat
df.between df.within
p
 0.728138
14
135
0.743133
 NOT significant

ANOVA test

50-day averages:
Fstat
df.between df.within
 13.25758
2
147
 IS significant


p
5e-06
10-day averages:
Fstat
df.between df.within
p
 2.546476
14
135
0.002879
 IS significant

Averages Rule!

Excellent way to reduce the noise

because σ(ave) = σ(raw) / √N
Excellent way to measure the noise
 Very little change to signal



unless signal changes faster than averaging
time
So in most cases averages smooth the
data, i.e., reduce noise but not signal
Decompose the Signal
Additive model: sum of component signals
 Non-periodic part

sometimes called trend
 sometimes called secular variation


Repeating (periodic) part
or almost-periodic (pseudoperiodic) part
 can be multiple periodic parts (multiperiodic)

f(t) = S(t) + P(t)
Periodic Signal
Discover that it’s periodic!
 Find the period P

Or frequency ν
 Pν = 1 ν = 1 / P


Find amplitude A = size of variation


P=1/ν
Often use A to denote the semi-amplitude,
which is half the full amplitude
Find waveform (i.e., cycle shape)
Periodogram
Searches for periodic behavior
 Test many frequencies (i.e., many periods)
 For each frequency, compute a power



Higher power  more likely it’s periodic with
that frequency (that period)
Plot of power vs frequency is a
periodogram, a.k.a. power spectrum
Periodograms

Fourier analysis  Fourier periodogram
Don’t use DFT or FFT because of uneven
time sampling
 Use Lomb-Scargle modified periodogram OR
 DCDFT (date-compensated discrete Fourier
transform)

Folded light curve  AoV periodogram
 Many more … these are the most common

DCDFT periodogram
AoV periodogram
Lots lots more …
Non-periodic signals
 Periodic but not perfectly periodic



(parameters are changing)
What if the noise is something “different”?
Come to the next workshop!
Enjoy observing variables
See your own data used in real scientific
study (AJ, ApJ, MNRAS, A&A, PASP, …)
Participate in monitoring and observing
programs
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Make your own discoveries!
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