In polar meteorology, researchers often seek links between local winds and global climate modes such as El Niño or the Southern Annular Mode. The work by Spears et al. (2012) claimed a significant correlation between the frequency of foehn days and the SAM index. However, the basic statistical properties of the wind data had not been checked before. Nearby in the archive are papers on weather and climate, hurricane forecasts, and solar flares, but this article stands apart: it does not build a forecast but tests whether standard correlations can be applied to such series at all.
The author analyzes wind speed data from automatic weather stations in the McMurdo Dry Valleys, the same data used by Spears and colleagues. Instead of calculating means and correlations, he first estimates the distribution function of wind speeds. Kolmogorov–Smirnov and Shapiro–Wilk tests are applied to check normality. Then the power-law exponent is estimated. The key step is checking whether the first moment of the distribution is finite, that is, whether the expected wind speed exists.
The wind speed distributions at stations near Taylor Glacier, Lakes Bonney, Hoare, Fryxell, and Explorers Cove fail normality tests. They follow a power law with an exponent less than 2. In that case, the integral defining the mean diverges: the mean wind speed is mathematically undefined. Any arithmetic mean computed from a finite sample does not converge to a true mean but varies randomly from sample to sample. Consequently, the Pearson correlations between mean values and climate indices presented in Table II of Spears et al. have no statistical meaning. The claim of significance at the 95% level is erroneous because no rigorous definition of significance exists for correlations of means.
This work shows that before any correlation analysis of geophysical series, one must check the type of distribution. If the data follow a power law with an exponent less than 2, standard methods based on means and variances are inapplicable. This distinguishes this article from neighboring works that use machine learning for prediction but do not question the existence of distribution moments. Here, for the first time, the divergence of the mean is demonstrated for winds in the Dry Valleys, casting doubt on all previous conclusions about links to ENSO and SAM.
The topic could develop toward applying methods for analyzing non-Gaussian processes with heavy tails. Instead of mean values, one could use medians or quantiles, and instead of Pearson correlation, rank correlations or measures of dependence that do not require finite moments. Also promising is the study of physical mechanisms generating power-law wind distributions, such as self-organized criticality.
The results are important for polar meteorology and climatology because they force a rethinking of statistical processing of wind data. They are also relevant to any field where extreme events with heavy-tailed distributions occur.
The next step will be to reanalyze wind time series using robust statistics and to check whether the claimed links hold after proper treatment.
In meteorology, the problem of statistical processing of series with extreme events remains unresolved. Many atmospheric variables, including wind speed, precipitation, and temperature, can have heavy-tailed distributions, making classical methods based on normality inappropriate.
If a distribution has an exponent less than 2, the mean is infinite, but the median remains finite. So in such cases, it is better to describe the typical wind speed with the median rather than the arithmetic mean.