Quantitative paleo-estimation: hypothetical experiments with extrapolation and the no-analog problem

We experiment with artificial data to test the response of five numerical techniques in extrapolating paleo-environments for no-analog conditions. No-analog conditions are those beyond the technique calibration (modern) data set and will be encountered in applications to the geologic past, though they may not be easy to recognize. In the ideal, a numerical technique will correctly extrapolate to no-analog conditions. Failing this, the technique will have a consistent, predictable error response to increasing no-analog conditions, as these are measured by a reliable index. The no-analog conditions that we used are a natural extension of the calibration conditions we created. Thus we test techniques for their response to shifting environmental conditions rather than for factors unrelated to the ecology of the taxa (e.g. post-depositional fossil preservation). Five numerical techniques we test with our hypothetical data are (1) multivariate regression of species percents, (2) correlation-based principal components with linear regression, (3) covariance-based principal components with linear regression, (4) correlation-based principal components with non-linear regression, and (5) the Imbrie and Kipp technique. All the techniques show increasing estimation error as conditions depart from those of the calibration data set. There are two main causes of error in our estimates: (1) the distorting effects of matrix closure on taxon abundances; and (2) generation of ratio no-analogs among species abundances because of non-linear responses to conditions departing progressively from the calibration range. With all the techniques, the distribution of error for no-analog conditions is complex. Non-linear regression with factors shows the least predictable error response. We found that currently developed no-analog indicators do not have a good correlation to estimation error. This means that better indicators, more closely linked to the accuracy of estimates, need to be developed.