On studying relations between time series in climatology

Abstract

Abstract. Relationships between time series are often studied on the basis of cross-correlation coefficients and regression equations. This approach is generally incorrect for time series, irrespective of the cross-correlation coefficient value, because relations between time series are frequency-dependent. Multivariate time series should be analyzed in both time and frequency domains, including fitting a parametric (preferably, autoregressive) stochastic difference equation to the time series and then calculating functions of frequency such as spectra and coherent spectra, coherences, and frequency response functions. The example with a bivariate time series "Atlantic Multidecadal Oscillation (AMO) -sea surface temperature in Niño area 3.4 (SST3.4)" proves that even when the cross correlation is low, the time series' components can be closely related to each other. A full time and frequency domain description of this bivariate time series is given. The AMO-SST3.4 time series is shown to form a closed-feedback loop system with a 2-year memory. The coherence between AMO and SST3.4 is statistically significant at intermediate frequencies where the coherent spectra amount up to 55 % of the total spectral densities. The gain factors are also described. Some recommendations are offered regarding time series analysis in climatology.

Multi-annual mass variations from GRACE monthly solutions_preliminary results

“…Note finally, that the correlation of the derived multi-annual variations of mass with the ENSO events may be theoretically a difficulty. Accoding to Privalsky (2015), relationships between time series studied on the basis of cross-correlation coefficients and regression equations may result in incomplete or erroneous conclusion. Instead, multivariate linear autoregressive methods (Box et al, 1994;Kashyap and Rao, 1976;Privalsky and Jensen, 1995;Privalsky, 2015) seem to provide more reliable comparison.…”

“…Accoding to Privalsky (2015), relationships between time series studied on the basis of cross-correlation coefficients and regression equations may result in incomplete or erroneous conclusion. Instead, multivariate linear autoregressive methods (Box et al, 1994;Kashyap and Rao, 1976;Privalsky and Jensen, 1995;Privalsky, 2015) seem to provide more reliable comparison. Comparison of the multi-annual variations of the ENSO and of the mass variations by an appropriate method should be the next step of the present research.…”

On reconstruction of time series in climatology

“…(see Bendat and Piersol, 2010). The task of fitting a proper autoregressive model to a bivariate time series is discussed, for example, in Box et al (2015), while some recommendations for the case of climate data analysis can be found in Privalsky (2015). A key point in the parametric time series analysis is choosing a proper order p for the model Eq.…”

“…In those cases, the correlation coefficient between GST and ENSO or between AMO and ENSO would be very close to zero (−0.06 between AMO and the sea surface temperature in the ENSO area 3.4) while the coherence function estimates will significantly differ from zero in the frequency band between approximately 0.15 and 0.40 year −1 . In this latter case, the linear-regression contribution of ENSO to GST will be less than 0.4 % while the proper autoregressive approach will show a contribution of 25 to more than 50 % of spectral energy within the respective frequency band (see Privalsky, 2015). In the case of GST and ENSO, the linear regression contribution is less than 10 % while the autoregressive approach gives from 25 to 66 % between approximately 0.1 and 0.4 year −1 .…”