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ASTRONOMICAL TIME SERIES
ANALYSIS USING INFORMATION
THEORETIC LEARNING
Pablo A. Estévez, DIE, Universidad de Chile
Workshop on CI Challenges, September 2012
Joint work with
Pavlos Protopapas, University of Harvard
Pablo Zegers, Universidad de los Andes, Chile
Pablo Huijse, PhD Student, Universidad de Chile
Jose C. Principe, University of Florida, Gainesville
Astronomical Time Series: Light Curves
Light Curve: Stellar brightness (magnitude or flux)
versus time.
Variable stars: stars whose luminosity varies over
time (3% of the stars in the universe are variables,
and 1% are periodic variable stars)
Light Curve Analysis: Useful for period detection,
event detection, stellar classification, extra solar
planet discovery, measure distance to earth, etc.
An Example of a Light Curve
Challenges
Light curves are unevenly spaced or irregularly
sampled, with gaps of different sizes. This is due to:
Time constraints on the observation time
Day-night cycle, weather conditions
Equipment operability
Light curves are noisy due to photometric errors,
atmospheric and sky background
Astronomical surveys generate tens of millions of
light curves. Light curve generation rate will continue
growing during the next years.
Variable stars
Eclipsing binary stars Pulsating star
Problem Statement
Discriminate periodic versus non-periodic light
curves in astronomical survey databases
Estimate the underlying period of periodic light
curves.
Goal: To develop an automated method for
periodic detection and estimation based on
information theoretic learning.
Information theoretic learning (ITL)
Apply concepts of information theory such as entropy
and mutual information to machine learning
Renyi’s quadratic entropy, with Gaussian kernel
Renyi´s entropy is a generalization of Shannon’s entropy
IP: Information potential is the argument of the logarithm
CORRENTROPY (Generalized Correlation): It measures
similarity between feature vectors separated by a
certain time delay
Proposed discrimination metric
It combines correntropy (generalized correlation)
with a periodic kernel
The periodic kernel measures similarity among
samples separated by a given period
The new metric provides a periodogram, whose
peaks are associated with the fundamental
frequencies present in the data
It is computed directly from the available samples
Correntropy Kernelized Periodogram (CKP)
Correntropy Kernelized Periodogram
Synthetic data example: sin(2 pi t /P) + noise in time and
magnitude
Noise in time simulates uneven sampling
True period 2.456 days
The CKP reaches a global maximum at the corresponding
true period (left figure).
Statistical Test Using CKP
99%
90%
Degree of
confidence
Receiver Operator Characteristic
ROC curves for CKP, and alternative methods: LS-periodogram and AoV-
periodogram. Dataset: 750 periodic light curves and 1500 aperiodic light
curves from the MACHO survey. Due to the natural classs imbalance, very low
false positive rates are required (0.1%).
EROS Survey
Survey of the Magellanic Clouds and the Galactic bulge
Data taken from ESO Observatory, in La Silla, Chile
EROS main goal: search for the dark matter of the
Galactic halo
EROS survey is a goldmine for stellar variability studies:
Cepheids, RR-Lyrae, Eclipsing Binaries, and Supernovas.
Each EROS field has ~17,300 light curves.
There are 88x32 fields of the Large Magellanic Cloud
(LMC), i.e.
48.744.522 light curves =>48.7M light curves
Computational Time Requirements
for EROS Survey
Computational time measured using NVIDIA Tesla C2070
GPU (448 cores)
Sweeping 20000 trial periods with CKP, the total time
per light curve (~650 samples): 1.5 [s]
For 48.7M light curves: ~845 days!
Evaluating 600 precomputed trial periods (by using
correntropy and other methods) and optimizing the code:
0.2 [s] per light curve
For 48.7M million light curves: ~113 days!
NCSA Dell/NVIDIA Cluster: FORGE
National Center for Supercomputing Applications (NCSA)
at the University of Illinois at Urbana-Champaign
We are using a queue with 12 machines each having 8
cores with Tesla C2070 GPUs => 96 GPUs
Computing eight EROS fields using a machine with 8 cores
takes 1 hour
So far we have processed 1.2M light curves in 40 mins
At this rate for computing 48.7M light curves: ~30 hours!
FORGE has 44 machines with 288 GPUs in total. Using
the whole cluster we might process 1 BILLION light curves
in 10 days
Conclusions & Future Work
A framework for light curve analysis based on ITL
and kernel methods has been introduced.
CKP allows discriminating between periodic and
non-periodic light curves with high accuracy and
low number of false positives.
Required: Efficient computation of ITL based
methods
Challenge: Applying our methods to large
untested astronomical databases.
ALMA Site in Northern Chile
THE END
P.Huijse, P. Estevez, P. Zegers, P. Protopapas, J.
Principe, “Period Estimation in Astronomical Time
Series using Slotted Correntropy”, IEEE Signal
Processing Letters, Vol. 18, n°6, pp. 371-374,
2011.
P.Huijse, P. Estevez, P. Protopapas, P. Zegers, J.
Principe, “An Information Theoretic Algorithm for
Finding Periodicities in Stellar Light Curves”, IEEE
Transactions on Signal Processing, Vol. 60, n°10, pp.
5135-5145, 2012.
Computational Intelligence Applied to Time Series Analysis
Pablo A. Estévez
Department of Electrical Engineering
University of Chile
University of Cyprus, Cyprus
September 14, 2012
Outline
First Topic
Introduction to Self-Organizing Maps (SOM)
SOMs for temporal sequence processing
Short-term Gamma Memories
Experimental Results
Conclusions
Second Topic
Analysis of Astronomical Time Series
Information Theoretic Learning Approach
Kohonen’s Map
Self-Organizing Feature Map (SOM)
Unsupervised Neural Networks
Vector Quantization of Feature Space
Topological Ordered Mapping
Main Applications:
Dimensionality reduction
Visualization of high-dimensional data in 2D or 3D maps
Clustering
Knowledge discovery in Databases
Topological Ordered Map
SOM defines a fixed grid in output space
Each node in the output grid is associated with a prototype (codebook) vector in input space
Neighborhood is measured in the output space
This neighborhood is used for updating codebook vectors in input space
Example: Kohonen’s Map in 2D
It uses a 2D output grid for visualization of high-dimensional data
Example of Neural Gas
Connections are created between the best matching unit and the second closest Connections are allowed aging and are removed eventually if not refreshed
SOMs for data temporal processing
Several recent extensions of SOM for processing data sequences that are temporally or spatially connected
For example: words, DNA sequences, time series
Models differ on the notion of context, i.e. the way they store sequences
Each neuron is represented by a weight (codebook) vector and a context (several) vector(s)
diw
dc i
Gamma Memories
The Gamma filter is defined in the time domain as
where is the input signal, is the filter output, and are the filter parameters
Parameter controls the tradeoff between depth and resolution of the filter
1
1( 1) (1 ) 1
K
k k
k
k k k
y n c n
c n c n c n
0( ) ( )c n x n ( )y n
,k k
Cascade of K-stages
A recursive rule for context descriptor of order-K can be constructed
The K context descriptors are described as
1 1
1 1
1
0
1
( ) 1 ,
: previous winner
n n
n n
I I
k k k
I I
n
c n c c k
c w
I
Gamma SOM Map
Delay Coordinate Embedding
Takens´embedding theorem allows us to reconstruct the dynamics of an n-dimensional space state starting by a one-dimensional time series, e.g. strange attractor.
To embed a time series, the following delay coordinate vector is constructed:
Embedding parameters (t,m) are found by
using ad-hoc methods
First minimum of the average mutual information (t)
False nearest neighbor algorithm (m)
( ) ( ), ( ), , ( ( 1) )i i is t x t x t x t mt t
Gamma Filtering Embedding
Gamma SOM construct a Gamma filtered embedding, as follows:
Where is the weight vector and are the contexts
Embedding parameters are determined by sweeping an array of (, K) values
Find the top 10 combinations of parameters with lower temporal quantization errors
Project to the principal direction by using PCA
Search for the 1D-PCA projection (allowing for shift delays) having maximal mutual information with the original time series
1( ) ( ), ( ), , ( )i i i i
Ku t w t c t c t
iw c i
tu
Experiments
Chaotic Lorenz System: state variable
NH3-Far Infrared Laser:
Data set A in the Santa Fe Time Series Competition
( )x t
Phase Portrait for Lorenz original dataset
Bicup 2006 challenge time series
Phase Portrait for noisy Lorenz dataset
1D projection of Gamma SOM for noisy Lorenz dataset
2D projection of Gamma SOM for Laser Time Series
Conclusions
Gamma SOM models can reconstruct the state space by using Gamma filtering embedding
Useful tools for non-linear time series analysis
Advantage of noise reduction
Future work: Time series prediction
References
Estevez, P.A., Hernandez, R.: Gamma SOM for Temporal Sequence Processing. In: Advances in Self-Organizing Maps, WSOM 2009, LNCS 5629, St. Augustine, FL, pp. 63-71 (2009)
Estevez, P.A., Hernandez, R., Perez, C.A., Held, C.M.: Gamma-filter Self-organizing Neural Networks for Unsupervised Sequence Processing. Electronics Letters (2011)-
Estevez, P.A., Hernandez, R.: Gamma –filter Self-Organizing Neural Networks for Time Series Analysis. In: Advances in Self-Organizing Maps, WSOM 2011, LNCS 5629, Espoo, Finland, pp. 63-71 (2011)
Estevez, P.A. and Vergara, J.: Nonlinear Time Series Analysis by Using Gamma Growing Neural Gas, WSOM 2012, Santiago, Chile (in press)
ALMA Site in Northern Chile