Eigen analysis of tree-ring records: part 3, taking heteroscedasticity and sampling effects into consideration

Abstract

This paper reports on the further development of a new technique for standardization of tree-ring records called the eigen analysis of tree-ring records. The data are from the same sample set of 56 long-lived Qilian junipers (Sabina przewalskii Kom.) from the Dulan region in western China as was used in our previous paper (Yang et al. 2011b). To assess the heteroscedasticity of individual record deviations from the sample set regional curve (RC), we tested five different definitions of those deviations. Direct computations of eigenvectors of all relevant intrarecord covariation matrices turned out to be greatly affected by observational and computational noise; an analytic approximation of these vectors was therefore desirable. The Bessel function of the first kind and the zero order proved suitable for such an approximation, especially because the deviations were defined via subtraction of the RC from raw ring width records. Exclusion of the contributions of the first segment of the Bessel approximation, corresponding to the extremely large first eigenvalue, rendered individual record deviations from RC homoscedastic. Therefore, the routine Fourier basis became applicable to extract climate-dependent components of the residual deviations. A Fourier expansion of the Dulan chronology revealed the quasi-200-year-long solar activity cycle to be the main factor affecting Dulan tree growth.