Abstract
In the absence of annual laminations, time series generated from lake sediments or other similar stratigraphic sequences are irregularly spaced in time, which complicates formal analysis using classical statistical time series models. In lieu, statistical analyses of trends in palaeoenvironmental time series, if done at all, have typically used simpler linear regressions or (non-) parametric correlations with little regard for the violation of assumptions that almost surely occurs due to temporal dependencies in the data or that correlations do not provide estimates of the magnitude of change, just whether or not there is a linear or monotonic trend. Alternative approaches have used LOESS-estimated trends to justify data interpretations or test hypotheses as to the causal factors without considering the inherent subjectivity of the choice of parameters used to achieve the LOESS fit (e.g. span width, degree of polynomial). Generalized additive models (GAMs) are statistical models that can be used to estimate trends as smooth functions of time. Unlike LOESS, GAMs use automatic smoothness selection methods to objectively determine the complexity of the fitted trend, and as formal statistical models, GAMs, allow for potentially complex, non-linear trends, a proper accounting of model uncertainty, and the identification of periods of significant temporal change. Here, I present a consistent and modern approach to the estimation of trends in palaeoenvironmental time series using GAMs, illustrating features of the methodology with two example time series of contrasting complexity; a 150-year bulk organic matter δ15N time series from Small Water, UK, and a 3000-year alkenone record from Braya-So, Greenland. I discuss the underlying mechanics of GAMs that allow them to learn the shape of the trend from the data themselves and how simultaneous confidence intervals and the first derivatives of the trend are used to properly account for model uncertainty and identify periods of change. It is hoped that by using GAMs greater attention is paid to the statistical estimation of trends in palaeoenvironmental time series leading to more a robust and reproducible palaeoscience.
Highlights
Palaeoecology and palaeolimnology have moved away from being descriptive disciplines, rapidly adopting new statistical developments in the 1990s and beyond (Smol et al, 2012)
The example analyses were all performed using the mgcv package and R, and the Supplementary Material contains a fully annotated document showing the R code used to replicate all the analyses described in the remainder of the paper
Can we do better than these polynomial fits? In the remainder, I hope to demonstrate that the answer to that question is emphatically “yes”! Below I describe a coherent and consistent approach to modelling palaeoenvironmental time series using generalised additive models that builds upon the linear regression framework
Summary
Palaeoecology and palaeolimnology have moved away from being descriptive disciplines, rapidly adopting new statistical developments in the 1990s and beyond (Smol et al, 2012). The Pearson correlation coefficient, r, is often used to detect trends in palaeo time series (Birks, 2012a), despite the fact that r provides no information as to the magnitude of the estimated trend, and the same temporal autocorrelation problem that dogs ordinary least squares plagues significance testing for r (Tian et al, 2011) Both the simple least squares trend line and r are tests for linear trends only, and yet we typically face data sets with potentially far more complex trends than can be identified by these methods. I briefly discuss the application of GAM trend analysis to multivariate species abundance and compositional data
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