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Notes on Principal Component Analysis

Notes on Principal Component Analysis. Used in: Moore, S.K., N.J. Mantua, J.P. Kellogg and J.A. Newton, 2008: Local and large-scale climate forcing of Puget Sound oceanographic properties on seasonal to interdecadal timescales. Limnol . Oceanogr ., 53, 1746-1758. Vocabulary.

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Notes on Principal Component Analysis

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  1. Notes on Principal Component Analysis Used in: Moore, S.K., N.J. Mantua, J.P. Kellogg and J.A. Newton, 2008: Local and large-scale climate forcing of Puget Sound oceanographic properties on seasonal to interdecadal timescales. Limnol. Oceanogr., 53, 1746-1758.

  2. Vocabulary • PCA = Principal Component Analysis, same as • EOF = Empirical Orthogonal Functions (typical term used in Physical Oceanography), and equivalent to • Factor Analysis, term used by social scientists • Reference: Emery, W.J., and R.E. Thomson, 1997: Data analysis methods in physical oceanography. PergamonPress, 634 pp. See in particular Section 4.3).

  3. What is PCA? • A method to represent the patterns of co-variability of a number of different time series • E.g. say you have monthly values of salinity at 50 locations through Puget Sound, over a time span of 10 years (120x50 data values) • The PCA represents these as the sum of 50 “maps” each multiplied by its own time series • The map is an eigenvector, and its time series is the corresponding eigenvalue • Each map (and its time series) account for a certain amount of the variance of the original signal • ** The first few components usually account for most of the total variance • ** Hopefully the time series of the first few components correspond to known forcing functions (like riverflow, or the PDO)

  4. Notes of caution • In preparing fields for the analysis you: • Make all time series the same length, with same start and end times (may involve interpolation to fill in data gaps) • Remove the mean and the linear trend of each time series (Moore et al. remove the monthly means – so they get rid of the annual cycle as well) • Normalize each time series by its own standard deviation • What is lost?

  5. Example: EOF analysis of currents on a section in Puget Sound • Bretschneider, D.E., G.A. Cannon, J. R. Holbrook, and D.J. Pashinski, (1985) Variability of subtidal current structure in a fjord estuary: Puget Sound, Washington. J. Geophys. Res., 90(C6), 11,949-11,958.

  6. Moorings & Mean Current • From a 31-day time series

  7. EOF 1 • Due to wind

  8. EOF 2 • Due to deep water intrusions coming over Admiralty Inlet Sill

  9. EOF 3 • Due to tidal pumping, which forces greater net transport through Colvos Passage

  10. Summary: % Variance Explained

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