Transcript Principal Component Analysis of Variable Star Light Curves

PRINCIPAL COMPONENT ANALYSIS OF VARIABLE STAR LIGHT CURVES
Ruka Murugan1, Shashi Kanbur2
1University of Rochester, Rochester, NY
2Oswego, Oswego, NY
Principal Component Analysis (PCA)
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We have tried several different input matrices to
conduct PCA on, and each has yielded interesting
results:
Method developed by Karl Pearson in 1901
Primarily used as a statistical tool in exploratory data
analysis
Linearly transforms the data matrix into a space where each
orthogonal basis vector is ordered in decreasing variance
along its direction
In this case, we want our data matrix to represent the light
curve
o AkBk Method: each star is a row, reading:
Log(P) A1 B1 A2 B2 … Ak Bk
Where the A’s and B’s are the Fourier coefficients
of the light curve.
Why use PCA to analyze variable stars?
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Reduces the dimensionality of a problem by providing very
good approximations for fewer terms used
Can we link it to a physical property of the star, such as
metallicity?
Data Set
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We have studied stars from two data sets in this project:
o 19 Non-Blazhko RR Lyraes from Kepler
o 1770 LMC Cepheids from OGLE III survey
METHODS
Fig. 1: Reconstructed light curve comparing
a 20th order Fourier fit to 5th order PCA fits.
Light Curve Reconstruction
Nth order PCA approximations to light curves
can be reconstructed by using the formula:
𝑁
𝑉 =
o Interpolation Method: each star is a row, with each
column holding the value of the light curve’s
amplitude at a given time. We obtain the data by
estimating an average from the Fourier fit.
o Standardization: we employ this is as a scaling for
both the AkBk and Interpolation methods, by taking
Mij = (Mij – Aj )/Sj, where Aj is the average value of
the column, and Sj is the standard deviation of the
column..
𝑃𝐶𝑘 𝑥𝑘
𝑘=1
Where PC is the principal component value
and x is the eigenvector.
Analysis
Fig 2. Kepler AkBk
Fig 3. Kepler Interpolation
References:
Kanbur et. al., MNRAS 10 October 2002
Kanbur et. al., MNRAS 27 June 2004
Nemec et. al., MNRAS 01 July 2011
Deb & Singh, Astronomy and Astrophysics 08 August 2011
NSF Office of International Science and
Engineering award number 1065093
The next step in the project is to analyze the
output of the PCA and determine what data
can be used reliably to create an empirical
formula. This will involve using further
statistical analysis of our principal
component values to find correlation
between PC, period, magnitude and
metallicity. We must also work out a reliable
way to scale the data, see how principal
components vary in individual stars from
cycle to cycle, and determine how sensitive
PC values are to small changes in the
dataset, such as removing one or several
stars from the pool.
Fig 4. OGLE AkBk unstandardized
Fig 6. OGLE Interp. unstandardized
Fig 5. OGLE AkBk standardized
Fig 7. OGLE Interp. standardized