Principal Component Analysis for SPAT PG course Joanna D. Haigh

PCA also known as… Empirical Orthogonal Function (EOF) Analysis Singular Value Decomposition Hotelling Transform Karhunen-Loève Transform 11 Nov 2013

Purpose/applications To identify internal structure in a dataset (e.g. “modes of variability”) Data compression – by identifying redundancy, reducing dimensionality Noise reduction Feature identification, classification…. 11 Nov 2013

Basic approach Data measured as function of two variables E.g. surface pressure (space, time) If measurements at two points in space are highly correlated in time then we only need one measure (not two) as a function of time to identify their behaviour. How many measures we need overall depends on correlations between each point and every other. 11 Nov 2013

Correlations 11 Nov 2013 value at point 1 value at point 2 measurements at point 1 and point 2 highly correlated main (average) signal is measure in direction of PC1 deviations (the interesting bit?) are in PC2 PC1 PC2 to calculate PCs we need to rotate axes with M points just rotate in M dimensions 1 2

11 Nov 2013 Approach E.g. data measured N times at M spatial points In M-dimensional space i.Find axis of greatest correlation, i.e. main variability, this is PC1. ii.Find axis orthogonal to this of next highest variability, this is PC2. iii.Continue until M new axes, i.e. M PCs. Each PC is composed of a weighted average of the original axes. The weightings are the EOFs.

Concept Often it is possible to identify a particular mode/feature with an EOF. Each PC indicates the variation with time (in our example) of the mode identified with its EOF. Once EOFs established can project other datasets (e.g. different time periods) onto them to compare behaviours. 11 Nov 2013

ENSO as EOF1 of SST data EOF1 of tropical Pacific SSTs: 576 monthly anomalies Jan Dec 1997 EOF1 explains 45% of the total SST variance over this domain. 11 Nov

Maths Calculate MxM covariance matrix Find eigenvectors and eigenvalues EOFs are the M eigenvectors, ranked in order of decreasing eigenvalue Eigenvalues give measure of variance PCs from decomposition of data onto EOFs. 11 Nov 2013

Examples of applications 11 Nov 2013 ApplicationMNVisualise dataEOFs: weightings of PCs MeteorologyspacetimeTime series at each place (or map at each time) places (maps) Time series of EOFs maps Earth obs (e.g. land cover) spectral bands spaceMap in each wavelength band bandsMaps of band combos Earth obs (e.g. cloud) caseswave- length Spectrum for each case casesSpectra of case combos Polarity of IMF Solar longitude timeIMF polarity f(longitude) at each time longitudesTime series of lon. distbn

High cloud E. Asia Kang et al (1997) 11 Nov 2013

Southern Annular Mode geopotential height of 1000hPa surface 11 Nov 2013

Examples of applications 11 Nov 2013 ApplicationMNVisualise dataEOFs: weightings of PCs MeteorologyspacetimeTime series at each place (or map at each time) places (maps) Time series of EOFs maps Earth obs (e.g. land cover) spectral bands spaceMap in each wavelength band bandsMaps of band combos Earth obs (e.g. cloud) caseswave- length Spectrum for each case casesSpectra of case combos Polarity of IMF Solar longitude timeIMF polarity f(longitude) at each time longitudesTime series of lon. distbn

Landsat Thematic Mapper ( Wageningen) 11 Nov µm

example of TM EOFs (unnormalised) [NB not for Wageningen images] 11 Nov 2013 µm eigenvalues: EOF:

Examples of applications 11 Nov 2013 ApplicationMNVisualise dataEOFs: weightings of PCs MeteorologyspacetimeTime series at each place (or map at each time) places (maps) Time series of EOFs maps Earth obs (e.g. land cover) spectral bands spaceMap in each wavelength band bandsMaps of band combos Earth obs (e.g. cloud) caseswave- length Spectrum for each case casesSpectra of case combos Polarity of IMF Solar longitude timeIMF polarity f(longitude) at each time longitudesTime series of lon. distbn

Modelled IR spectra of cirrus cloud Bantges et al (1999) 11 Nov 2013

PC0: Average PC1: Ice water path PC2: Effective radius PC3: Aspect ratio Bantges et al (1999) 11 Nov 2013

Examples of applications 11 Nov 2013 ApplicationMNVisualise dataEOFs: weightings of PCs MeteorologyspacetimeTime series at each place (or map at each time) places (maps) Time series of EOFs maps Earth obs (e.g. land cover) spectral bands spaceMap in each wavelength band bandsMaps of band combos Earth obs (e.g. cloud) caseswave- length Spectrum for each case casesSpectra of case combos Polarity of IMF Solar longitude timeIMF polarity f(longitude) at each time longitudesTime series of lon. distbn

Polarity of Interplanetary Magnetic Field 11 Nov 2013 Cadavid et al 2007

Maths – a little more detail Represent data by MxN matrix D MxM covariance matrix is C = (D – D)(D – D) T Calculate i=1,M eigenvalues λ i & eigenvectors v i EOFs in MxM matrix of eigenvectors E MxN matrix of PCs P = E T D NB can rewrite D = (E T ) -1 P = E P (E Hermitian) i.e. PCs give weighting of EOFs in data 11 Nov 2013

Data reduction/noise removal Higher order PCs are composed of lowest correlations so uncorrelated noise lies in these. Can reconstruct data omitting higher order EOFs to reduce noise. Can reduce data by keeping only PCs of lowest order EOFs. 11 Nov 2013

Books R W Priesendorfer 1988 PCA in meteorology and oceanography Elsevier I T Jolliffe 2002 Principal component analysis Springer 11 Nov 2013