1. Paper review EOF: the medium is the message 報告人：沈茂霖 (Mao-Lin Shen) 2015/11/10 Seminar report

2. Mao-Lin Shen, Dept. of Atmospheric Sciences, NTU Page 2 2015/11/10 A. H. Monahan, J. C. Fyfe, M. H. P. Ambaum, D. B. Stephenson, and G. R. North (2009) “Empirical Orthogonal Functions: The Medium is the Message,” Journal of Climate. To demonstrate the care that must be taken in the interpretation of individual modes in order to distinguish the medium from the message.

3. Mao-Lin Shen, Dept. of Atmospheric Sciences, NTU Page 3 2015/11/10 Outline EOF analysis EOFs and dynamical modes EOFs and kinematic degrees of freedom EOFs of non-gaussian fields Non-locality of EOFs Conclusions

4. Mao-Lin Shen, Dept. of Atmospheric Sciences, NTU Page 4 2015/11/10 EOF Analysis (1/2) Some relevant facts Empirical Orthogonal Function (EOF) analysis (also known as Principal Component Analysis (PCA), or Proper Orthogonal Decomposition ) A N-dimensional vector time series x(t), as a continuous field sampled at N discrete points in space. The covariance matrix of x is given by The brackets denote probabilistic expectation: if p(x) is the probability density function of x, then

5. Mao-Lin Shen, Dept. of Atmospheric Sciences, NTU Page 5 2015/11/10 EOF Analysis (2/2) The EOFs can be defined as the eigenvectors e k of C: In particular, we can write where the expansion coefficient time series are the principal components.

6. Mao-Lin Shen, Dept. of Atmospheric Sciences, NTU Page 6 2015/11/10 EOFs and Dynamical Modes (1/5) Sustained small-amplitude variability in a broad range of physical situations in the atmosphere and ocean can be described by linear dynamics subject to random forcing representing the effects of unresolved physical scales: is a vector of independent white noise processes (uncorrelated in both space and time)

7. Mao-Lin Shen, Dept. of Atmospheric Sciences, NTU Page 7 2015/11/10 EOFs and Dynamical Modes (2/5) Consider the simple two-dimensional system (Farrel and Ioannou) normal for. The eigenvector of the dynamical matrix are

8. Mao-Lin Shen, Dept. of Atmospheric Sciences, NTU Page 8 2015/11/10 EOFs and Dynamical Modes (3/5)

9. Mao-Lin Shen, Dept. of Atmospheric Sciences, NTU Page 9 2015/11/10 EOFs and Dynamical Modes (4/5)

10. Mao-Lin Shen, Dept. of Atmospheric Sciences, NTU Page 10 2015/11/10 EOFs and Dynamical Modes (5/5)

11. Mao-Lin Shen, Dept. of Atmospheric Sciences, NTU Page 11 2015/11/10 EOFs and kinematic degrees of freedom (1/3) A simple model of a fluctuating jet in zonal-mean wind for which the EOF problem is analytically solvable. The model with Gaussian profile and fluctuating in strength and position: The jet strength and position are the natural kinematic variables, what we will call the kinematic degrees of freedom

12. Mao-Lin Shen, Dept. of Atmospheric Sciences, NTU Page 12 2015/11/10 EOFs and kinematic degrees of freedom (2/3) Base on observations of the extratropical zonal-mean eddy-driven jet, we will assume that From the expansions the leading EOFs can be determined in terms of the normalised basis vectors

13. Mao-Lin Shen, Dept. of Atmospheric Sciences, NTU Page 13 2015/11/10 EOFs and kinematic degrees of freedom (3/3) If fluctuations in position are relatively large compared to those of strength, then the leading EOF is the dipole.The leading PC time series is given by The spatial pattern of the second EOF is a monopole/tripole hybrid where the degree of hybridization is determined by the quantity

14. Mao-Lin Shen, Dept. of Atmospheric Sciences, NTU Page 14 2015/11/10 EOFs of non-gaussian fields (1/3)

15. Mao-Lin Shen, Dept. of Atmospheric Sciences, NTU Page 15 2015/11/10 EOFs of non-gaussian fields (2/3)

16. Mao-Lin Shen, Dept. of Atmospheric Sciences, NTU Page 16 2015/11/10 EOFs of non-gaussian fields (3/3)

17. Mao-Lin Shen, Dept. of Atmospheric Sciences, NTU Page 17 2015/11/10 Non-locality of EOFs The scale of the EOF spatial structures will be determined by the size of the domain. In particular, the leading EOF mode will be the gravest mode with a wavelength determined not by the properties of the field but by the size of the domain. Dommenget (2007) uses this fact to suggest a “stochastic null hypothesis” for determining if EOF structures more reflect the variability of the field or the geometry of the domain.

18. Mao-Lin Shen, Dept. of Atmospheric Sciences, NTU Page 18 2015/11/10 Conclusions EOF analysis is a powerful and versatile tool for dimensionality reduction, but it is not free from this “bias”. In general EOF modes cannot be expected to be of individual dynamical, kinematic, or statistical meaning. ICA, NLPCA, SVD, etc.

19. Mao-Lin Shen, Dept. of Atmospheric Sciences, NTU Page 19 2015/11/10 Thank you for your attention.