periodicities in variable stars: a few issues chris koen dept. statistics university of the western...
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Periodicities in variable stars: a few issues
Chris Koen
Dept. Statistics
University of the Western Cape
2
1
2
1
sincos1
)(N
kkk
N
kkk tyty
NI
jkallforttII
jk ,0)(sin)()(
*
**
0)(sin)(21
1 1
N
j
N
jkjk ttS
Significance testing of the largest peak
• For regularly spaced data:
- statistical distribution of ordinates known
- ordinates independent in Fourier frequencies
• For irregularly spaced data:
- ordinates can be transformed to known distribution – ordinates not independent
Correlation between periodogram ordinates for increasing separation between frequencies
(irregularly spaced data)
Complex Demodulation
• Transform data so that frequency of interest is near zero
• Apply a low pass filter to the transformed data
Time Domain Modelling
Gaussiant
t
tB
tA
tB
tA
tetB
tAtt
tetttCtY
,)(
)(
)1(
)1(
)(
)(
)()(
)(sincos
)())((cos)()(
00
0
O-C (Observed – Calculated)
• Equivalent to CUSUMS• Sparsely observed process:
)()( 0*
1
1
PNTTCO
countcyclecumulativenN
TandTbetweenelapsedcyclesofnumbern
jjj
j
iij
jjj
The Light-time Effect
)(2
)(sin)(
2
)(tan
1
1arctan2)(
sin])([sin)(cos1
1)(
0
2
ttP
tEetE
tE
e
et
ette
eAtT
b
A stochastic period-change model
),0(~
)(
21
110
1 1*
jjjj
N
kkk
j
ijiij
j
i
N
kkkiij
jjj
j
j
ePNnCO
enTT
P
State Space Formulation:
jN
k
j
kk
j
iiik
j
iiij
j
j
j
j
j
jj
nnU
GaussianUnU
eTU
nT
1 1
1
1
1
1
1
1
*
,10
1
1
General form of Information Criteria:
IC = -2 log(likelihood)+penalty(K)
• Akaike : penalty=2K
• Bayes: penalty=K log(N)
• Model with minimum IC preferred
Order Sigma_error BIC
3 1.1921 153.57
4 1.1036 142.74
#5 0.51673 -4.4166*
6 0.51335 -1.1247
7 0.51519 4.1961
RW 0.43166 41.661
IRW 0.51412 55.247
Order sigma_error BIC 1 0.24656 -170.82
4 0.23132 -169.76
5 0.21551 -179.32
6 0.21558 -174.65
7 0.21589 -169.76
# RW 0.19477 -185.97*
IRW 0.21756 -171.33
Order sigma_error BIC
4 0.29048 -124.22
5 0.27773 -128.59
6 0.24941 -145.5
7 0.24809 -141.95
8 0.24678 -138.41
RW 0.17886 -119.37
#IRW 0.2194 -149.06*