Relationship between the Uncertainty of Empirical Orthogonal Function (EOF) Modes and Sampling Sizes in Climate Models

Abstract This study investigates the relationship between the uncertainty of empirical orthogonal function (EOF) modes and sampling size in climate models, using simulated results of preindustrial control (piControl) experiments in phase 6 of the Coupled Model Intercomparison Project (CMIP6), and taking the North Atlantic Oscillation (NAO) and El Niño–Southern Oscillation (ENSO) as examples. The results indicate that this relationship can be quantified by a concise fitting function [i.e., y = a/(x − b)]. Here, y is the 5%–95% confidence interval of congruence coefficient, x is the sampling size, and a and b are two parameters depending on models or observations. As compared with b, which modulates the sampling size in the fitting function, the parameter a scales the sampling size and thus plays a much more important role. Further analysis indicates that the parameter a, or the uncertainty of EOF1 mode, decreases dramatically with the increase of the difference between variance fractions of EOF1 and EOF2 modes, approximately in the form of a power function. The minimum sampling size to ensure a reliable EOF mode can also be estimated by the fitting function and shows a great diversity among models both for the NAO and ENSO. The diversity suggests the importance of estimating the minimum sampling size before model evaluations on climate variability modes and projections on the future change in modes, particularly when the EOF2 mode explains the variance close to EOF1 mode. Significance Statement Empirical orthogonal function (EOF) analysis, principal component analysis, or eigenvector analysis has been widely used in various research fields. However, it remains as an open question as to how large the sampling size is required to be to obtain reliable modes through the EOF method. In this study, we investigate the relationship between the uncertainty of EOF results and sampling size in current climate models, using adequately long simulated data, and we find that this relationship can be depicted by the fitting function y = a/(x − b). Here, y represents the uncertainty, x is the sampling size, and a and b are parameters. The parameter a is closely related to the difference between variance fractions of first and second EOF modes and plays a more important role in the fitting function. The minimum sampling sizes that are required to obtain reliable EOF modes can also be estimated by the fitting function and vary greatly from model to model. The results provide a basis for judging the reliability of EOF modes, particularly when the first and second EOF modes explain similar variance fractions.