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Principal Component Analysis for SPAT PG course

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. Purpose/applications.

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Principal Component Analysis for SPAT PG course

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  1. Principal Component Analysisfor SPAT PG course Joanna D. Haigh

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

  3. 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

  4. 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

  5. Correlations PC2 value at point 2 1 PC1 2 value at point 1 • to calculate PCs we need to rotate axes • with M points just rotate in M dimensions 11 Nov 2013 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 

  6. Approach 11 Nov 2013 E.g. data measured N times at M spatial points In M-dimensional space Find axis of greatest correlation, i.e. main variability, this is PC1. Find axis orthogonal to this of next highest variability, this is PC2. 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.

  7. 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

  8. ENSO as EOF1 of SST data http://www.esrl.noaa.gov/psd/enso/impacts/currentclimo.html EOF1 of tropical Pacific SSTs: 576 monthly anomalies Jan 1950 - Dec 1997 EOF1 explains 45% of the total SST variance over this domain.  11 Nov 2013

  9. 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

  10. Examples of applications 11 Nov 2013

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

  12. Southern Annular Modegeopotential height of 1000hPa surface 11 Nov 2013

  13. Examples of applications 11 Nov 2013

  14. Landsat Thematic Mapper (Wageningen) 11 Nov 2013 0.5 0.6 0.7 µm 0.8 1.6 2.2

  15. µm 0.5 0.6 0.7 0.8 1.6 2.2 11.5 • eigenvalues: 1011 131 38 7 4 2 1 • EOF: 1 2 3 4 5 6 7 example of TM EOFs (unnormalised) [NB not for Wageningen images] 11 Nov 2013

  16. Examples of applications 11 Nov 2013

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

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

  19. Examples of applications 11 Nov 2013

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

  21. 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 eigenvaluesλi & eigenvectors vi EOFs in MxM matrix of eigenvectors E MxN matrix of PCs P = ETD NB can rewrite D = (ET)-1 P = EP (EHermitian) i.e. PCs give weighting of EOFs in data 11 Nov 2013

  22. 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

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

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