THE CHRONOLOGY OF GROUPS OF DIES FOR LARGE ISSUES OF ANCIENT COINS USING MAHALANOBIS DISTANCES*

Discriminant analyses of distances and angles obtained from 25 coins struck by P. Crepusius in 82 BC permitted the correct chronology for five groups of closely-related dies to be obtained by the method of Carter and Frurip. Twenty distances and 15 angles were measured on 10 x projections of the coins which represented five groups of dies separated in time from each other. The use of all or many variables gave poorer results than the use of five to ten variables having a high ‘figure of merit’, defined as the maximum range of mean values for a given variable in the five groups of dies divided by the mean standard deviation of the variable in the five groups. It is recommended that chronologies of ancient coins be determined from Mahalanobis distances by using only variables of relatively high figure of merit.     

MODIFIED MAHALANOBIS FILTERS FOR GROUPING POTTERY BY CHEMICAL COMPOSITION*

A statistical procedure for grouping pottery in provenance studies by chemical data is presented, which now is routinely in use in our laboratory. It is based on the Mahalanobis filter method and X² -statistics, and can be used for both establishing groups and assigning single sherds to already known groups, thus replacing principal components analysis or cluster analysis and avoiding their problems in grouping pottery. The new method is able to consider correlations, uncertainties of measurement and constant shifts of the data in case of dilution. In particular, considering dilution effects results in both a better assignment to and separation of reference groups and is also equivalent to the compositional data approach, if log-transformed data are used. Other distortions of data (e.g., mixing of clays) can also be considered.     

THE EFFICIENT COMPUTATION OF UNBIASED MAHALANOBIS DISTANCES AND THEIR INTERPRETATION IN ARCHAEOMETRY*

This paper argues for cross-validation to reduce bias in estimating Mahalanobis distances of individuals to groups, a particular problem with small sample sizes. Formulae for the efficient computation of cross-validated distances, which avoid excessive matrix inversion, are given. The use of Mahalanobis distances as measures of consistency, or as indications of outlying values, is described. The little-known technique of gamma plotting is outlined and discussed as an aid to interpreting distances in these terms. Both cross-validation and gamma plotting are illustrated in an example on marble composition.     

DETECTING MULTIVARIATE OUTLIERS IN ARTEFACT COMPOSITIONAL DATA*

Multivariate statistical analysis of artefact compositional data, usually undertaken to investigate structure in the data, often incidentally reveals the presence of multivariate outliers. Much statistical methodology dealing with the detection of such outliers is not well suited to archaeometric data that, in the event, consist of two or more groups. The paper provides examples to illustrate the importance of detecting and dealing with outliers, and critically examines a range of different approaches to outlier detection. The examples show that cluster analysis, the technique most widely used for this purpose, can fail to reveal outliers clearly identified by other methods.