Multivariate Time Series Analysis

Charles D. Camp

MSRI

July 18, 2008

PCA: 2 variable example

Weakly Correlated Variables Strongly Correlated Variables

PCA algorithm

PCA algorithm, cont.

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 FunctionSeasonal Cycle derived from NCEP

Units: 109 kg/s

Part II: TOMS and MOD data sets

TOMS: 1°×1.25° lat-lon grid, Nov.1978 - Apr.1993, monthly means

Merged Ozone Data (MOD) combines TOMS and SBUV data: 5°×10° lat-lon grid; Nov.1978 - 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= -0.73 ( 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

EOF2sum => 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):

tttt 212111 sinsinsinsin2

EOF 4: ENSO

Captures 3% of the interannual variance.

R=0.71 ( < 0.1% )

<= 0

ENSO in TOMS

R=0.76 ( 0.1% )

<= 0