1. Principal Components Analysis
Τεχνικές Παρατήρησης και Επεξεργασίας Δεδομένων στην Αστροφυσική
Τομέας Αστροφυσικής, Αστρονομίας & Μηχανικής
Principal Components Analysis
Antonios Karampelas, PhD

2. Astrostatistics Big Data Data Mining Machine Learning Glossary
A discipline used to process the vast amount of astronomical data.
Big Data
Large or complex data sets difficult to process using traditional data processing applications.
Data Mining
The computational process of discovering patterns in large data sets.
Machine Learning
A scientific discipline that explores the construction and study of algorithms that can learn from data.

3. Big Data: An alternative for Astrophysicists?
McKinsey Global Institute Report1
Big data: The next frontier for innovation, competition, and productivity
Harvard Business Review2
Data Scientist: The Sexiest Job of the 21st Century
Fortune3
Big Data could generate millions of new jobs
1http://
1https://hbr.org/2012/10/data-scientist-the-sexiest-job-of-the-21st-century/
1http://fortune.com/2013/05/21/big-data-could-generate-millions-of-new-jobs/

4. Data Mining/Machine Learning categories
1. Descriptive Data Mining/ Unsupervised Machine Learning
2. Predictive Data Mining/ Supervised Machine learning

5. Data Mining/Machine Learning categories
1. Descriptive Data Mining/ Unsupervised Machine Learning
Discover patterns, trends, clusters, outliers
Principal Components Analysis (PCA)
Self-Organizing Map (SOM)
K-means Clustering
2. Predictive Data Mining/ Supervised Machine learning
Parameterize, Classify, Recognize Patterns
Decision Trees (DT)
Support Vector Machine (SVM)
Artificial Neural Network (ANN)

6. Artwork by Sandbox Studio, Chicago with Kimberly Boustead
Astrostatistics
Artwork by Sandbox Studio, Chicago with Kimberly Boustead

7. Principal Components Analysis (PCA)
(Karhounen-Loeve transformation)
Linear orthogonal transformation in a new base, in which the data variance is highlighted.
New axes = Principal Components (PCs)
Data = linear combination of PCs

8. Fingerprint Recognition
Very effective in: Data compression, Dimensionality reduction, Noise extraction
Applications
Astronomy
Biology
Graphology
Face Recognition
Fingerprint Recognition

9. Indicative Astronomy Articles
SDSS Yip et al. 2004, AJ
Gaia Karampelas et al. 2012, A&A
Spitzer Wang et al. 2011, MNRAS
2dF Folkes et al. 1999, MNRAS

10. PCA procedure Standardize the original data, if necessary.
2. Construct the variance-covariance matrix or the correlation matrix.
3. Determine the eigenvalues (λi) and eigenvectors (PCi) of the matrix.
Data covariance has been eliminated.
Eigenvalues λ represent the variances of the transformed data.
PC1 corresponds to the biggest λ (λ1) and summarizes the majority of the data variance.
PC2 corresponds to λ2 and summarizes the majority of the rest of the data variance etc.
4. Determine the admixture coefficients αi (data projection on the new axes).
Full data reconstruction: Data = α1PC1 + α2PC2 + … + αkPCk
Partial reconstruction is usually sufficient: Data ≈ α1PC1 + α2PC2 + … + α5PC5

11. No widespread Information
PCA procedure
Full reconstruction
Data = α1PC1 + α2PC2 + α3PC3 + … + αk-1PC(k-1) + αkPCk
Widespread information
Noise
No widespread Information
Partial reconstruction
Data ≈ α1PC1 + α2PC2 + α3PC3

12. PCA implementation (Data)
Data set
Synthetic galaxy spectra1
used for the Gaia Mission.
Size
7160 spectra X 801 flux values
Waveband
300 – 1,100 nm
Redshift No
Spectral types
4 (E, S, I, QSFG)
1 Karampelas et al. 2012, Fioc & Rocca-Volmerange 1997, 1999, Le Borgne & Rocca-Volmerange 2002

13. PCA implementation (IDL)
7160 spectra X 801 pixels
(admixture coefficients)
7160 spectra X 801 pixels
(original data)
result=PCOMP(data, COEFFICIENTS = coefficients, EIGENVALUES = eigenvalues, VARIANCES = variances, /COVARIANCE)
m=801 & eigenvectors = coefficients/REBIN(eigenvalues, m, m)
801 spectra X 801 pixels (PCs)

14. PCA implementation (PC1, PC2, PC3)
OIII
SIII
OII Ha
Resembles the average spectrum
Strong emission lines
Dominant emission lines

15. PCA implementation (PC4, PC5, PC6)
Strong emission lines
Dominant emission lines

16. PCA implementation (Reconstruction)
Spectrum = α1PC1 + α2PC2 + α3PC3 + α4PC4 + … =
= α1
+ α2
+ α3
+ …
+ α4

17. Admixture coefficients α2 Admixture coefficients α1
PCA implementation (Projection to PC1/PC2)
Irregular
Admixture coefficients α2
Spiral
QSFG
Early-type
Admixture coefficients α1

18. PCA & Astrostatistics - 1
Yip et al. 2004, AJ
≈ 170, 000 SDSS galaxy spectra
PC1
PC2
Outliers
Red galaxies
PC3
PC4
Blue galaxies
Post starburst galaxies

19. PCA & Astrostatistics - 2
Folkes et al. 1999, MNRAS
≈ 6, 000 2dF galaxy spectra

20. Bailer-Jones et al. 1998, MNRAS
PCA & Astrostatistics - 3
Bailer-Jones et al. 1998, MNRAS
5, 000 Michigan Spectral Survey stellar spectra

21. PCA & Astrostatistics - 4
Karampelas et al. 2012, A&A
≈ 30, 000 PEGASE synthetic galaxy spectra

22. Karampelas et al. in preparation
PCA & Astrostatistics - 5
Karampelas et al. in preparation
≈ 7, 000 PEGASE synthetic galaxy spectra
Classification with PCA + Decision Trees

23. PCA & Face Recognition Eigenfaces