Paper review EOF: the medium is the message

1. Seminar report Paper reviewEOF: the medium is the message 報告人：沈茂霖 (Mao-Lin Shen) 2014/8/28

2. 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. Outline • EOF analysis • EOFs and dynamical modes • EOFs and kinematic degrees of freedom • EOFs of non-gaussian fields • Non-locality of EOFs • Conclusions

4. 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. EOF Analysis (2/2) • The EOFs can be defined as the eigenvectors ek of C: • In particular, we can write • where the expansion coefficient time series are the principal components.

6. 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. 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. EOFs and Dynamical Modes (3/5)

9. EOFs and Dynamical Modes (4/5)

10. EOFs and Dynamical Modes (5/5)

11. 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. 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. 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. EOFs of non-gaussian fields (1/3)

15. EOFs of non-gaussian fields (2/3)

16. EOFs of non-gaussian fields (3/3)

17. 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. 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. Thank you for your attention.