Significance of trends in long-term correlated records

Abstract

We study the distribution $P(x;\ensuremath{\alpha},L)$ of the relative trend $x$ in long-term correlated records of length $L$ that are characterized by a Hurst exponent $\ensuremath{\alpha}$ between 0.5 and 1.5. The relative trend $x$ is the ratio between the strength of the trend $\ensuremath{\Delta}$ in the record measured by linear regression and the standard deviation $\ensuremath{\sigma}$ around the regression line. We consider $L$ between 400 and 2200, which is the typical length scale of monthly local and annual reconstructed global climate records. Extending previous work by Lennartz and Bunde [S. Lennartz and A. Bunde, Phys. Rev. E 84, 021129 (2011)] we show explicitly that $x$ follows the Student's $t$ distribution $P\ensuremath{\propto}{[1+{(x/a)}^{2}/l]}^{\ensuremath{-}(l+1)/2}$, where the scaling parameter $a$ depends on both $L$ and $\ensuremath{\alpha}$, while the effective length $l$ depends, for $\ensuremath{\alpha}$ below 1.15, only on the record length $L$. From $P$ we can derive an analytical expression for the trend significance $S(x;\ensuremath{\alpha},L)={\ensuremath{\int}}_{\ensuremath{-}x}^{x}P({x}^{\ensuremath{'}};\ensuremath{\alpha},L)d{x}^{\ensuremath{'}}$ and the border lines of the $95%$ significance interval. We show that the results are nearly independent of the distribution of the data in the record, holding for Gaussian data as well as for highly skewed non-Gaussian data. For an application, we use our methodology to estimate the significance of central west Antarctic warming.