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EOF Analysis |
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Principal Component |
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Rotated EOF |
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Complex EOF |
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Empirical Orthogonal Function (EOF) analysis
attempts to find a relatively small number of independent variables
(predictors; factors) which convey as much of the original information as
possible without redundancy. |
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EOF analysis can be used to explore the
structure of the variability within a data set in a objective way, and to
analyze relationships within a set of variables. |
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EOF analysis is also called principal component
analysis or factor analysis. |
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In brief, EOF analysis uses a set of orthogonal
functions (EOFs) to represent a time series in the following way: |
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Z(x,y,t) is the original time series as a
function of time (t) and space (x, y). |
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EOF(x, y) show the spatial structures (x, y) of the major factors
that can account for the temporal variations of Z. |
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PC(t) is the principal component that tells you how the amplitude of
each EOF varies with time. |
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Suppose the Pacific SSTs are described by values
at grid points: x1, x2, x3, ...xN.
We know that the xi’s are probably correlated with each other. |
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Now, we want to determine a new set of
independent predictors zi to describe the state of Pacific SST,
which are linear combinations of xi: |
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Mathematically, we are rotating the old set of
variable (x) to a new set of variable (z) using a projection matrix (e): |
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The EOF analysis asks that the projection
coefficients are determined in such a way that: |
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(1)
z1 explains the maximum possible amount of the variance of the x’s; |
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(2) z2
explains the maximum possible amount of the remaining variance |
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of the x’s; |
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(3)
so forth for the remaining z’s. |
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With these requirements, it can be shown
mathematically that the projection coefficient functions (eij)
are the eigenvectors of the covariance matrix of x’s. |
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The fraction of the total variance explained by
a particular eigenvector is equal to the ratio of that eigenvalue to the
sum of all eigenvalues. |
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The EOF analysis has to start from calculating
the covariance matrix. |
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For our case, the state of the Pacific SST is
described by values at model grid points Xi. |
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Let’s assume the observational network in the
Pacific has 10 grids in latitudinal direction and 20 grids in longitudinal
direction, then there are 10x20=200 grid points to describe the state of
pacific SST. So we have 200 state variables: |
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Xm(t), m =1, 2, 3, 4, …,
200 |
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In our case, there are monthly observations of
SSTs over these 200 grid points from 1900 to 1998. So we have N
(12*99=1188) observations at each Xm: |
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Xmn = Xm(tn),
m=1, 2, 3, 4, …., 200 |
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n=1, 2, 3, 4,
….., 1188 |
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The covariance between two state variables Xi
and Xj is: |
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The covariance matrix is as following: |
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Any symmetric matrix R can be decomposed in the
following way through a diagonalization, or eigenanalysis: |
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Where E is the matrix with the eigenvectors ei
as its columns, and L is the matrix with the eigenvalues li,
along its diagonal and zeros
elsewhere. |
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The set of eigenvectors, ei, and
associated eigenvalues, li, represent a coordinate
transformation into a coordinate space where the matrix R becomes diagonal. |
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There are orthogonal constrains been build in in
the EOF analysis: |
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The
principal components (PCs) are orthogonal in time. |
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There are no simultaneous temporal correlation between any two
principal components. |
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(2) The EOFs are orthogonal in space. |
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There are no spatial correlation between any two EOFs. |
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The second orthogonal constrain is removed in
the rotated EOF analysis. |
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I don’t want to go through the mathematical
details of EOF analysis. Only some basic concepts are described in the
following few slids. |
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Through mathematic derivations, we can show that
the empirical orthogonal functions (EOFs) of a time series Z(x, y, t) are
the eigenvectors of the covarinace matrix of the time series. |
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The eigenvalues of the covariance matrix tells
you the fraction of variance explained by each individual EOF. |
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A two-dimensional data matrix X: |
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The transpose of this matrix is XT: |
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The inner product of these two matrices: |
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Sometime, we use the correlation matrix, in
stead of the covariance matrix, for EOF analysis. |
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For the same time series, the EOFs obtained from
the covariance matrix will be different from the EOFs obtained from the
correlation matrix. |
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The decision to choose the covariance matrix or
the correlation matrix depends on how we wish the variance at each grid
points (Xi) are weighted. |
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In the case of the covariance matrix
formulation, the elements of the state vector with larger variances will be
weighted more heavily. |
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With the correlation matrix, all elements
receive the same weight and only the structure and not the amplitude will
influence the principal components. |
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The correlation matrix should be used for the
following two cases: |
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The state vector is a combination of things with
different units. |
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(2) The variance of the state vector varies from
point to point so much that this distorts the patterns in the data. |
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If we want to get the principal component, we
project a single eigenvector onto the data and get an amplitude of this
eigenvector at each time, eTX: |
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For example, the amplitude of EOF-1 at the first
measurement time is calculated as the following: |
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There are several ways to present EOFs. The
simplest way is to plot the values of EOF itself. This presentation can not
tell you how much the real amplitude this EOF represents. |
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One way to represent EOF’s amplitude is to take
the time series of principal components for an EOF, normalize this time
series to unit variance, and then regress it against the original data set. |
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This map has the shape of the EOF, but the
amplitude actually corresponds to
the amplitude in the real data with which this structure is associated. |
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If we have other variables, we can regress them
all on the PC of one EOF and show the structure of several variables with
the correct amplitude relationship, for example, SST and surface vector
wind fields can both be regressed on PCs of SST. |
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Another way to present EOF is to correlate the
principal component of an EOF with the original time series at each data
point. |
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This way, present the EOF structure in a
correlation map. |
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In this
way, the correlation map tells you what are the co-varying part of the
variable (for example, SST) in the spatial domain. |
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In this presentation, the EOF has no unit and is
non-dimensional. |
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We can use Singular Value Decomposition (SVD) to
get EOFs, eigenvalues, and PC’s
directly from the data matrix, without the need to calculate the covariance
matrix from the data first. |
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If the data set is relatively small, this may be
easier than computing the covariance matrices and doing the eigenanalysis
of them. |
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If the sample size is large, it may be
computationally more efficient to use the eigenvalue method. |
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Any m by n matrix A can be factored into |
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The columns of U (m by m) are the EOFs |
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The columns of V (n by n) are the PCs. |
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The diagonal values of S are the eigenvalues
represent the amplitudes of the EOFs, but not the variance explained by the
EOF. |
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The square of the eigenvalue from the SVD is
equal to the eigenvalue from the eigen analysis of the covariance matrix. |
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There are no definite ways to decide this.
Basically, we look at the eigenvalue spectrum and decide: |
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The 95% significance errors in the estimation of
the eigenvalues is: |
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If
the eigenvalues of adjacent EOF’s are closer together than this standard
error, then it is unlikely that their particular structures are
significant. |
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(2) Or we can just look at the slope of the eigenvalue spectrum. |
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We
would look for a place in the eigenvalue spectrum where it levels off so
that successive eigenvalues are indistinguishable. We would not consider
any eigenvectors beyond this point as being special. |
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The orthogonal constrain on EOFs sometime cause
the spatial structures of EOFS to have significant amplitudes all over the
spatial domain. |
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č We can
not get localized EOF structures. |
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č
Therefore, we want to relax the spatial orthogonal constrain |
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on EOFs (but still keep the temporal orthogonal constrain). |
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č We apply
the Rotated EOF analysis. |
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To perform the rotated EOF analysis, we still
have to do the regular EOF first. |
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č We then
only keep a few EOF modes for the rotation. |
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č We
“rotated” these selected few EOFs to form new EOFs (factors). |
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based on some criteria. |
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č These
criteria determine how “simple” the new factors are. |
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Basically, the criteria of rotating EOFs is to
measure the “simplicity” of the EOF structure. |
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Basically, simplicity of structure is supposed
to occur when most of the elements of the eigenvector are either of order
one (absolute value) or zero, but not in between. |
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There are two popular rotation criteria: |
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Quartimax Rotation |
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Varimax
Rotation |
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Ouartimax Rotation |
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It
seeks to rotate the original EOF matrix into a new EOF matrix for which the
variance of squared elements of the eigenvectors is a maximum. |
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Varimax Rotation (more popular than the
Quartimax rotation) |
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It
seeks to simplify the individual EOF factors. |
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The
criterion of simplicity of the complete factor matrix is defined as the
maximization of the sum of the simplicities of the individual factors. |
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The following few examples are from a recent
paper published on Journal of Climate: |
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