Multivariate Time Series Analysis Charles D. Camp MSRI July 18, 2008

PCA: 2 variable example Weakly Correlated VariablesStrongly Correlated Variables

PCA algorithm

PCA algorithm, cont.

An example using Column Ozone Data

Brewer-Dobson circulation and Planetary Waves  Upwelling planetary waves break in shaded region  Drives the Brewer- Dobson circulation  Transports heat to the polar vortex  Effect on the strength of the polar night vortex Courtesy of M. Salby

2D CTM Model  The Caltech/JPL two-dimensional chemistry and transport model (2D CTM) is used to investigate interannual variability of the total ozone column.  Forced by the monthly mean meridional circulation (isentropic circulation) and eddy diffusivity calculated from the NCEP/DOE Reanalysis2 data (NCEP2).  Compared to the MOD observations.

Isentropic Mass Stream Function Seasonal Cycle derived from NCEP Units: 10 9 kg/s

IAV of Isentropic Stream Function EOFsPCsPower SpectraCumul. Variance 70% 88% 94%

Part II: TOMS and MOD data sets  TOMS: 1°×1.25° lat-lon grid, Nov Apr.1993, monthly means  Merged Ozone Data (MOD) combines TOMS and SBUV data: 5°×10° lat-lon grid; Nov Dec.2000, monthly means

MOD decomposition & PCA  data is deseasonalized: mean for each month removed  then detrended: linear trend removed.  Anomaly field is spectrally filtered to remove intra-annual variability.  Principal Component Analysis (PCA) is performed to get EOF patterns and PC time series.

MOD EOF patterns and PC time series 42% 75% 90% 93% EOFs PCs Amplitude SpectraCumul. Var.

MOD EOF 1: QBO (and Decadal) Captures 42% of the interannual variance. R=0.80 ( 1% )  <= 0

MOD EOF2: Decadal and QBO Captures 33% of the interannual variance. R= ( 5% ) <= 0

Filtered PCs 1 & 2: Separating the QBO and Decadal signals PC1 PC2 [15, 72] mo.[72, max] mo.

Linear Combinations of EOFs 1 & 2: Patterns for QBO and Decadal Signals  Using the standard deviations of the filtered PCs as weights, take weighted sum and difference of EOFs 1 & 2. (Zonal averages shown) EOF 1 EOF2 sum => QBO diff => decadal

EOF 3: interaction between QBO and annual cycles (QBO-annual beat) Captures 15% of the interannual variance. <= 0

QBO-annual beat (analysis with intra-annual variability)  Quadratic nonlinearity between the QBO and annual cycles creates signals with periods of 20 and 8.6 months (for a average QBO period of 30 months):

EOF 4: ENSO Captures 3% of the interannual variance. R=0.71 ( < 0.1% ) <= 0

ENSO in TOMS R=0.76 ( 0.1% ) <= 0