Palaeolimnological diatom data comprise counts of many species expressed as percentages for each sample. Reconstruction of past lake-water pH from such data involves two steps; (i) regression, where responses of modern diatom abundances to pH are modelled and (ii) calibration where the modelled responses are used to infer pH from diatom assemblages preserved in lake sediments. In view of the highly multivariate nature of diatom data, the strongly nonlinear response of diatoms to pH, and the abundance of zero values in the data, a compromise between ecological realism and computational feasability is essential. The two numerical approaches used are (i) the computationally demanding but formal statistical approach of maximum likelihood (ML) Gaussian logit regression and calibration and (ii) the computationally straightforward but heuristic approach of weighted averaging (WA) regression and calibration. When the Surface Water Acidification Project (SWAP) modern training set of 178 lakes is reduced by data-screening to 167 lakes, WA gives superior results in terms of lowest root mean squared errors of prediction in cross-validation. Bootstrapping is also used to derive prediction errors, not only for the training set as a whole but also for individual pH reconstructions by WA for stratigraphic samples from Round Loch of Glenhead, southwest Scotland covering the last 10 000 years. These reconstructions are evaluated in terms of lack-of-fit to pH and analogue measures and are interpreted in terms of rate of change by using bootstrapping of the reconstructed pH time-series.

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Diatoms and pH Reconstruction

Summary This paper reviews the wide variety of approaches to both univariate and multivariate calibration. In particular, in univariate calibration, it considers the classical, inverse, Bayesian and non-parametric approaches together with the approaches via tolerance regions. In multivariate calibration, it considers the ridge regression, multiple linear regression, partial least squares regression and principal components regression approaches together with the Bayesian and profile likelihood approaches. An extensive bibliography is provided.

Statistical Calibration: A Review

Many applied problems in physical anthropology involve estimation of an unobservable quantity (such as age at death or stature) from quantities that are observable. Two of the more disparate subdisciplines of our discipline, paleoanthropology and forensic anthropology, routinely make use of various estimation methods on a case-by-case basis. We discuss the rationales for making estimations on isolated cases, taking stature estimation from femoral and humerus lengths as an example. We show that the entirety of our discussion can be placed within the context of calibration problems, where a large calibration sample is used to estimate an unobservable quantity for a single skeleton. Taking a calibration approach to the problem highlights the essentially Bayesian versus maximum likelihood nature of the question of stature estimation. On the basis of both theoretical arguments and practical examples, we show that inverse calibration (regression of stature on bone length) is generally preferred when the stature distribution for a reference sample forms a reasonable prior, while classical calibration (regression of bone length on stature followed by solving for stature) is preferred when there is reason to suspect that the estimated stature will be an extrapolation beyond the useful limits of the reference sample statures. The choice between these two approaches amounts to the decision to use either a Bayesian or a maximum likelihood method.

Stature estimation and calibration: Bayesian and maximum likelihood perspectives in physical anthropology

WACALIB version 2.1 — a computer program to reconstruct environmental variables from fossil assemblages by weighted averaging

A LANDSAT-based rangelands monitoring system has been designed for semi-arid chenopod shrublands in southern Australia. Simultaneous ground and LANDSAT measurements were used to test multivariate calibration methods for estimating vegetation cover. Of three methods compared, the Lwin-Maritz and inverse estimators outperformed the classical approach. Data were analysed by rangeland type and as a combined set. Calibration curves, with errors of estimation, are presented for five major rangeland types.

Calibration of LANDSAT data for sparsely vegetated semi-arid rangelands

The competing classical and inverse point estimators in statistical calibration can be obtained by direct or inverse regression and are to some extent supported by the maximum likelihood and Bayesian approaches, respectively. Both of these approaches depend on specific distributional assumptions, but these assumptions confuse the main issue because both estimators can be justified without reference to them. By using a compound estimation approach, it is shown that the classical estimator can be derived as a linear compound estimator satisfying the criterion of asymptotic unbiasedness, while the inverse estimator is a linear compound estimator without the unbiasedness constraint. This formulation requires neither specific distributional assumptions nor reference to direct or inverse regression. Assessments of the two estimators are made in terms of their performance in estimating the current x value. It is shown that superiority of the inverse estimator can only be guaranteed if the current x value is samp...

An Analysis of the Linear-Calibration Controversy From the Perspective of Compound Estimation

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An Analysis of the Linear-Calibration Controversy From the Perspective of Compound Estimation