paper review eof: the medium is the message 報告人：沈茂霖 (mao-lin shen) 2015/11/10 seminar...

Paper reviewEOF: the medium is the message

報告人：沈茂霖　　　　 (Mao-Lin Shen)

112/04/20

Seminar report

112/04/20 Mao-Lin Shen, Dept. of Atmospheric Sciences, NTUMao-Lin Shen, Dept. of Atmospheric Sciences, NTU

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 interpretat

ion of individual modes in order to distinguish the medium from the message.


Outline

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


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

TC xx

( ) ( ) ( )f f p f dx x x


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.

k k kC e e

1

( ) ( )N

n nn

t t

x e

( )n t


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)

dA B

dt

xx W

W

( ) ( ) ( )i j ijW t W t t t



Consider the simple two-dimensional system (Farrel and Ioannou)

normal for . The eigenvector of the dynamical matrix are

1 1 1

2 2 2

0.1 0.9cot 0 0

0 1 0

x x Wd

x xdt W

/ 2

1

1

0

b 2

cos

sin

b


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

2

2

[ ( )]( , ) ( ) exp

2c

o

x x tu x t U t

( )U t ( )cx t


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

( ) / ( ) 1l std U mean U ( ) / 1c oh std x


EOFs and kinematic degrees of freedom (3/3) If fluctuations in position are relatively large compare

d 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

1/ 2

21( ) ( ) ( ) ( )

2 co

t U t x t O h

1( )f x

4

2

3

8

h

l


EOFs of non-gaussian fields (1/3)

4

2

3

8

h

l


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.


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.


Thank you for your attention.

paper review eof: the medium is the message 報告人：沈茂霖 (mao-lin shen) 2015/11/10 seminar...

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