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,

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. Pergamon Press, 634 pp. See in particular Section 4.3).

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)

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?

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.

Moorings & Mean Current From a 31-day time series

EOF 1 Due to wind

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

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

Summary: % Variance Explained