Much evidence has accumulated that temperature extremes and variability are changing. Accurately diagnosing such changes is of vital societal interest, not least because human induced climate change is often expected to materialise primarily through changes in the extreme tails.
Quantifying these features of climate time series statistically in climate models and observations is not straightforward. To a large extent, that is because extreme events are rare by definition, a fact that seems hardly surprising. This fact implies, however, that conventional methods quickly break down when it comes to the tails. This blog post serves s a cautionary note, in which we discuss how apparently very simple methods can result in severely biased estimates, and how this can be avoided1,2.
Many studies that take on global or continental-scale perspectives circumvent this “small-sample” issue by counting extremes across space, which results in a metric such as the “global land area affected by temperature extremes” (or similar ones). However, an ‘objective comparison’ across space requires a transformation based on the local statistical properties of the time series, because locations differ in their climatic means and variability. A conventional approach that has been termed ‘reference period normalization’ is to subtract the long-term mean and divide by the standard deviation of a time series estimated in a ‘climatic reference period’ (assuming Gaussianity), yielding standardized (‘comparable’), bell-shaped estimates for climatic variables across space.
However, a simple experiment with artificial and stationary Gaussian data shows that the approach breaks down when it comes to the extremes (Fig. 1a, extremes defined as exceeding a certain level of variability, i.e. ‘sigma-extremes’): The number of extreme events increases drastically from the period that was used as reference period to the non-reference (‘independent’) period. These artefacts are more severe for short reference periods and more extreme extremes (e.g. an artificial increase in observed extremes by 48.2% for 2-sigma events in a n=30 year reference period), but more importantly the issue holds generically.
The reasons for these apparent methodological problems are statistically surprisingly simple: the estimators of mean and standard deviation have been calculated from values taken from the reference period – therefore the normalization transformation is conducted using dependent estimators in the reference period, and independent estimators outside. From a statistical point of view, it is well known that such operation yields different ‘normalized’ distributions (Fig. 1b and 1c) – in this case, it would result in a so-called beta- and t-distribution within and outside the reference period, respectively. Importantly, these distributions differ considerably in the shape of their tails – leading to the observed inconsistencies in the number of extremes across time and space. These findings have led to a downward correction of previous studies1 quantifying the number of summer temperature extremes across the Northern hemisphere2 (Fig. 2).
Furthermore, this statistical issue extends to estimates of the normalized climatic variability, because different statistical distributions entail differences in their variances (Fig. 3a). Therefore, accounting and correcting for the normalization issue shows that in contrast to previous reports3, both global and normalized temperature variability has not increased at the global scale (Fig. 3b).
To summarize, caution is required when estimating properties of climatic time series in one ‘reference’ part of the series, and subsequent application and analyses on the entire series. This finding is not new4 – but given abundant recent interest in climatic extremes, data analytic tools and indices require careful reconsideration in light of this issue.
1. Hansen, J., M. Sato, and R. Ruedy (2012), Perception of climate change, Proc. Natl. Acad. Sci. U.S.A., 109, E2415.
2. Sippel, S., J. Zscheischler, M. Heimann, F. E. L. Otto, J. Peters, and M. D. Mahecha (2015), Quantifying changes in climate variability and extremes: Pitfalls and their overcoming, Geophys. Res. Lett., 42.
3. Huntingford, C., P. D. Jones, V. N. Livina, T. M. Lenton, and P. M. Cox (2013), No increase in global temperature variability despite changing regional patterns, Nature, 500, 327.
4. Zhang, X. B., G. Hegerl, F. W. Zwiers, and J. Kenyon (2005), Avoiding inhomogeneity in percentile-based indices of temperature extremes, J. Climate, 18, 1641.