1. Principal Component Analysis: Preliminary Studies Émille E. O. Ishida IF - UFRJ First Rio-Saclay Meeting: Physics Beyond the Standard Model Rio de Janeiro - dec/2006

2. The main objective of: Physics Statistics Science Simplification Statistics is the art of extracting simple comprehensible facts that tell us what we want to know for practical reasons Principal Component Analysis (PCA) is a tool for simplifying one particular class of data...... –astro-ph/9905079

3. For example... np n objects and p things we know about them... -height; -n° publications; -flier miles; -fuel consumption; -height; -n° publications; -flier miles; -fuel consumption; -height; -n° publications; -flier miles; -fuel consumption; -height; -n° publications; -flier miles; -fuel consumption; -height; -n° publications; -flier miles; -fuel consumption; -height; -n° publications; -flier miles; -fuel consumption; n=6p=4 n=6 objects and p=4 things we know about them... How this parameters are related to each other? –astro-ph/9905079

4. For example... Do people who spend most of their lives in airports publish more? Do people with inefficient cars fly more..... or just the ones with lots of publications do? Do these correlations represent any real causal connection? or..... once you buy a car, stop publishing and give lots of talks in exotic foreign locations? –astro-ph/9905079

5. First try: Plot everything against everything else......as the number of parameters increases this becomes impossibly complicated! PCA looks for sets of parameters that always correlate togheter The first application of PCA was in social science.... Ex :give a sample of n people a set of p exams testing their creativity, memory, math skills.... And look for correlations..... Result : nearly all tests correlates to each other, indicating that one underlying variable could predict the performances in all tests IQ.....an infamous begginig...!! –astro-ph/9905079

6. General Idea: Given a sample of: n objects; p measured quantities - x i (i=1,2,3,...., p ) Find a new set of p orthogonal variables  i,...  p  each a linear combination of the original ones Determine a ij such that the smallest number of new variables account for as much of the sample variance as possible. Principal Components –astro-ph/9905079

7. Basic Statistics Mean Value: Covariance: Variance: http://csnet.otago.ac.nz/cosc453/student_tutorials/principal_components.pdf

8. Covariance Matrix in 2-D Eigenvectors  New axes (new uncorrelated variables) Eigenvalues  variances in the direction of the Principal Components The largest eigenvalue  First Principal Component http://csnet.otago.ac.nz/cosc453/student_tutorials/principal_components.pdf

9. But.....that´s not our case.... We want to make inferences about a model using a sample of data.... Parameter Estimation Consistency: Bias: Efficiency: Robusteness: http://pdg.lbl.gov/)

10. The Method of Maximum Likelihood http://pdg.lbl.gov/)

11. For an unbiased estimator.... We can calculate the covariance between the parameters of the theory Fisher Matrix http://pdg.lbl.gov/)

12. What about Cosmology? Direct evidence for an accelerated expansion: Can we get information out of SN Ia observations without the assumption of General Relativity?

13. Definitions....

14. As proposed by Shapiro & Turner (2006)... Modulus Distance  z = 0.05; Data from Gold Sample (Riess et al. ) Gaussian probability distribution in each bin...

15. The Fisher Matrix Observation about ...

16. PC1 PC2 PC3 PC4 PC5 PC6

17. Reconstruction of q(z) We need more data! –arXiv:astro-ph/0512586

18. Next Steps.... Small corrections in the present code (optimization); Change the observable; Get used to this procedure and be able to handle large data sets in a model independent way

19. References - D. Huterer e G. Starkman, Parametrization of dark energy properties: A Principal-Component Approach, Physical Review Letters, 90 (3), Janeiro/2003 –C. Shapiro e M. S. Turner, What do we really know about cosmic acceleration?, arXiv:astro-ph/0512586 –G. Cowan, Statistical Data Analysis, Clarendon Press, Oxford (1998) –P. J. Francis and B. J. Wills, Introduction to Principal Component Analysis, arXiv: astro-ph/9905079 –W.-M. Yao et al., Journal of Physics G 33, 1 (2006) available on the PDG WWW pages (URL: http://pdg.lbl.gov/)

20. –arXiv:astro-ph/0512586 Shapiro & Turner (2006) Principal Components