Variability and Confidence Intervals for the Mean of Climate Data with Short- and Long-Range Dependence

This work presents an adaptive procedure for estimating the variability and determining error bars as confidence intervals for climate mean states by accounting for both short- and long-range dependence. While the prevailing methods for quantifying the variability of climate means account for short-range dependence, they ignore long memory, which is demonstrated to lead to underestimated variability and hence artificially narrow confidence intervals. To capture both short- and long-range correlation structures, climate data are modeled as fractionally integrated autoregressive moving-average processes. The preferred model can be selected adaptively via an information criterion, and the estimated variability of the climate mean state can be computed directly from the chosen model. The procedure was demonstrated by determining error bars for four 30-year means of surface temperatures observed at Potsdam, Germany, from 1896 to 2015. These error bars are roughly twice the width as those obtained using prevailing methods which disregard long memory, leading to a substantive reinterpretation of differences among mean states of this particular dataset. Despite their increased width, the new error bars still suggest that a significant increase occurred in the mean temperature state of Potsdam from the 1896--1925 period to the most recent period, 1986--2015. The new wider error bars, therefore, communicate greater uncertainty in the mean state yet present even stronger evidence of a significant temperature increase. These results corroborate a need for more meticulous consideration of the correlation structures of climate data---especially of their long-memory properties---in assessing the variability and determining confidence intervals for their mean states.