Empirical Orthogonal Functions (PCA) Principal Components Analysis (EOF)

References:

When to Use the method ?

● The use of EOF appears when you have a combination ofspatial and temporal trends in a variable

– Ex: 2D maps of a time varying variable Z(x,y,t)● Z may be SST, SSH, P, etc ...

● You wish to find “groups” of data points that vary togetherfollowing a specified time function

● You wish to “recast” the observations into a set ofORTHOGONAL functions using a “mathematical procedure”

Basic Illustration

Basic Illustration

Terminology

The literature is quite confusing when it comes to differentiate the“principal component analysis” method from the “empirical othogonalfunctions” method. We assume there are the SAME.

The literature usually refers the EOF method when you perform theanalysis on a variable that has a combination of spatial and temporaltrends:

Ex: SST(x,y,t)

The literature usually refers the PCA method when you perform theanalysis on two or more variables that each evolve with time. You wishto rearrange the data into “modes” that evolve with time following aspecific function.

Terminology

The EOF method applied on a given Z(x,y,t) field consists indecomposing the signal into different “data” modes that are orthogonalto each other. These modes have a given “fixed” in time spatialpattern and an time evolving function:

Observed variable Z attime t and position X

i

EOF (or PC) mode nspatial pattern

Expansioncoefficients timeseries of mode n

EOF nth eigen vectors

i is the position index: i = 1, ... Mi is the position index: i = 1, ... M

Terminology

The literature usually refers the PCA method when you perform theanalysis on TWO or more variables that each evolve with time. You wishto rearrange the data into “modes” that evolve with time following aspecific function.

Expansioncoefficients timeseries of mode n

mode n PC patternnth mode eigen vector

i is the variable index: i = 1, ... M

EOF decomposition

The functions αn(t) and Φ

n(Xi) are subject to the following orthogonality

conditions:

(1):

spatial sum (over all the grid points): M

(2):

time sum over N times

Heuristic derivation of EOF (or PCA)

The previous definitions and constraints constitute an eigen valueproblem for the cross-correlation matrix.

(see demo in lecture notes)

Computational Derivation of EOF modes

The traditional approach (often referred as S-mode analysis):

1. Create the N*M data matrix X N lines, the time observationsM columns, the M grid points

2. Form the M*M spatial covariance matrix from the data matrix

3. Extract the eigen vectors phi and eigen values from this covariance matrix, arranged in decreasing eigen value magnitude. The V vectors are the EOF spatial modes.

4. Select a small number of EOFs with the largest eigen values

5. Possibly rotate these factors according to your scientific criteria

6. Examine the spatial structure and temporal variations of these selected EOF

Data MatrixN lines (N observations):length of the time seriesM columns: space samplings

Expansion coefficients:N lines (N time steps)M columns (N modes or PCs)

Principal components:M “spatial” modes

Covariance Matrix (Spatial Auto-covariance)

M rows, M columns

CONSTRAINT 1 on the expansion coefficients

TIME AVERAGEover a time seriesof length N

Orthogonality of the expansion coefficients

CONSTRAINT 2 on the PCs

Orthonormality of the “spatial” modes

Lambda are the eigen values of the covariance matrix

are are the eigen vectors of thecovariance matrix

X is the data matrixThe time series (rows)correspond to a monthlyclimatology (12 values)

Colimns correspond to all thedata points (30*30=900)