Degrees of freedom in Northern Hemisphere circulation data

Abstract

Estimates of the number of degrees of freedom (dof) of the 700 hPa Northern Hemisphere extratropical wintertime daily and time average circulation patterns are presented using two statistical methods. In the first method, it is assumed that the distribution of the mean square distances between the circulation patterns and the climate mean follows a χ2 distribution. The number of dof of the best χ2 fit to the empirical data is the most probable number of dof of the circulation data. The other, newly developed hyperspheres method compares the local density properties in the empirical data to density in independently and identically distributed (iid) normal distributions with different number of dof. The best fit, again, determines the most probable number of dof for the observed data. Though the hyperspheres method uses the smallest scales available in the data, just as the Grassberger-Procaccia method (GPM) and other “local” dimension estimation methods, it has distinct features. Most importantly, the GPM assumes that the empirical data is unbounded and distributed uniformly in the multivariate space. In contrast, the hyperspheres method, in accordance with many geophysical applications, assumes a bounded distribution in which density is a strong function of distance from the mean. Additionally, the hyperspheres method directly uses density information, that is neglected by GPM. The advantages of the hyperspheres method as compared to GPM are: (1) no systematic negative bias is present in the dof estimates; (2) the edge effect is eliminated; (3) no scaling is necessary. The results from the two methods overlap, thus indicating that at the 10% significance level the number of dof of the daily circulation data set is 24. This result pertains to a finite dataset in which the definitional requirement for dynamical dimension that the scales measured go to zero is not satisfied. We interpret our dof results as the dimension of a hypothetical subset of the atmospheric circulation that governs the large scale motions. Since the database of this study covers only part of the global atmosphere, the dimension of the whole atmosphere would be larger than our estimate. Our qualitative dimension estimate for the global circulation (60–90) refers to the minimum number of independent variables needed to model the large scale, low frequency variability of the atmosphere. In the present study grid point values served to describe the dynamical system of the atmosphere (spatial embedding, instead of time-delay or temporal embedding.) An argument is made that due to weak coupling between remote processes, estimates with temporal embedding measure only the dimension of a regional subset of the large-scale global circulation DOI: 10.1034/j.1600-0870.1995.t01-3-00005.x