The aim of Empirical Orthogonal Function (EOF) analysis is to determine the principal modes of variability of the studied data. Once the eigenvectors are found, it is important to separate those eigenvectors (or EOFs) which contain data that represent physically relevant patterns from those that represent only noisy variations. It has also been noticed that in analysis of this type, the eigenvalues fall exponentially and most of the variability is captured only by very few vectors (as compared to the dimensionality of the space considered). This paper aims to provide a statistical basis for the choice of only a certain set of vectors over others using techniques borrowed from random vector theory that has its origins in quantum physics of complex systems. The results of our analysis are also compared with the results of standard selection rules like Rule N.

This method assumes that the components of the eigenvectors of purely noisy data follow a Gaussian distribution and it follows that the squares of these components follow the Porter-Thomas distribution, a special case of the Chi-square distribution. The cumulative distribution function of the components is compared with the cumulative distribution function of the assumed distribution and the deviation from the expected is used as a criterion to separate vectors with information from those that are purely noisy.

The data sets used include the wind stress data over the tropical pacific region (20S-20N,130E-70W), sea level pressure over the North Atlantic region (20N-90N,120W-30E) and geopotential height at pressure levels ranging from 1000mb to 100mb over the North Atlantic region (20N-90N,120W-30E) - all considered from 1948 to 2000 - using monthly and weekly data from the NOAA-CIRES reanalysis data sets.

The North Atlantic Oscillation (NAO) is a large scale seesaw in atmospheric mass between the subtropical high and the polar low, first identified by Walker and Bliss. Our EOF analysis of sea level pressure and geopotential height data captures this NAO pattern. Further, only the dominant vectors which represent the NAO are also considered significant by our selection test.

Geopotential height data ranging from 1000mb to 100mb was used and composite EOFs in 3-D were obtained. The results yield interesting teleconnection patterns in all the three space dimensions - capturing the opposing behaviour in the subtropical and polar regions. The principal components of the 3D EOF structure agree closely with the documented NAO indices.

The wind-stress data set gives dominant eigenvectors that depict the equatorial Pacific wind stress patterns. This has been documented and agrees with our results. Our selection rules choose only the first three eigenvectors which correspond to the findings of Legler(1983) and they carry physical significance.

These results were compared with Rule N . In general, Rule N provides a very conservative estimate of the number of significant eigenvectors while our analysis provides a more realistic estimate and corresponds closely with patterns that have physical significance.