Transcript 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
• 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