The standard classical statistics approach to isochron calculation assumes that the distribution of uncertainties on the data arising from isotopic analysis is strictly Gaussian. This effectively excludes datasets that have more scatter from consideration, even though many appear to have age significance. A new approach to isochron calculations is developed in order to circumvent this problem, requiring only that the central part of the uncertainty distribution of the data defines a “spine” in the trend of the data. This central spine can be Gaussian but this is not a requirement. This approach significantly increases the range of datasets from which age information can be extracted but also provides seamless integration with well-behaved datasets and thus all legacy age determinations. The approach is built on the robust statistics of

The ability to fit a straight line through a body of isotope ratio data in order to form an isochron is the cornerstone of many geochronological methods. In detail, however, this is a non-trivial task, since uncertainties are usually associated with all variables, and these are often correlated, precluding simple “least squares” line-fitting techniques. Most of the research in this area was conducted in the late 1960s and early 1970s, being dominated by a classical statistics approach in which data uncertainties, derived from the analytical methods, are taken to be strictly Gaussian distributed

In this contribution, we examine some of the problems inherent in these techniques and suggest an alternative approach. Our primary focus here will be on general-purpose isochron calculations that determine the age of an “event” that established the isotopic compositions of samples in a dataset. This involves what are called model 1 and 2 calculations in

In order to show that there are significant problems in using

Age uncertainty (age

What if

Excess scatter occurs when the data distribution has higher variability in the tails than is indicated by the variability of the central part of the distribution, for example, if the distribution is Gaussian in the centre but having “fat” tails.

. At this point,In summary, then, in

Given that

Rather than being truly Gaussian, data uncertainties may well be Gaussian distributed in their centres but slightly fat-tailed, distant from the centres. An isotopic dataset looks intuitively acceptable if the data have a central linear “spine” in which scatter is commensurate with stated analytical uncertainty, but this spine is flanked by data of somewhat larger scatter (i.e. excess scatter from the “fat tail”). This excess scatter may originate in the isotopic analysis or as a result of geological disturbance. Age significance in such data manifests primarily via the position of the spine. In the following, we focus on this spine in the data.

Adopting this spine approach, a successful calculation method for a dataset that may not have strictly Gaussian-distributed uncertainties must, firstly, ascertain whether or not such a spine exists in the data – and hence whether calculations yield an isochron or an errorchron. Secondly, in the case of an isochron calculation, the successful method must reliably locate the spine without being perturbed by vagaries in the more scattered data. Classical statistical methods can do neither of these things, tending to be excessively influenced by the data at the extremes of the scatter. However, the field of robust statistics offers calculation methods that can. When a dataset has no excess scatter, reflected in

An algorithm is sought that finds a robust straight line through a two-dimensional linear data trend, while converging with the classical statistical approach of

determines a preliminary fit of the data, not dependent on vagaries of the data scatter;

determines the spine width in relation to this preliminary fit

if the spine width is in an acceptable range: isochron, or

if the spine width is not in an acceptable range: errorchron;

determines a robust fit of the data, starting from the preliminary fit, with this fit converging with

Geochronological datasets are collected on the presumption that the isotopic compositions were established via an “event” the age of which is to be estimated. Given the focus here on data with linear trends, even if the effect of the event is recorded perfectly by the samples analysed – the isotopic compositions lying on a line – the actual data are measured with finite precision and so the data inevitably scatter about the trend. An uncertainty probability distribution can be used to describe the form of the data scatter.

Classical statistical methods assume that the underlying uncertainty distribution of a dataset is known, typically taken to be Gaussian. Under the Gaussian assumption, if the analytical uncertainty on the measurements has been appropriately inferred,

While there are many possible non-Gaussian uncertainty distributions, this paper is concerned with a situation commonly occurring in datasets, in which the data points form a linear spine with Gaussian-like scatter, but additional scatter is seen in the tails of the distribution. Such a dataset still encodes meaningful age information in its spine, yet it will typically fail an

In

If, instead, data uncertainties are

The normalisation constant makes

The 95 % confidence intervals for

Whereas one-sided confidence intervals are advocated in Appendix A (columns marked with an asterisk in the table), two-sided confidence intervals are also given in the table. Using the column with the asterisk, for example, for a dataset with 10 data points (

We seek a statistical approach to isochron calculation that is robust

In both the Huber approach in

The function that is minimised to find the best-fit line can be written

For an example data point,

Plots of

In

The value used here,

The iteration developed in Appendix B minimises

The

determine a preliminary fit of the data using

determine the spine width using

if the spine width is in an acceptable range, from col. 6 in Table 1: isochron, or

if the spine width is not in the acceptable range: errorchron;

determine a robust fit of the data, minimising

Assessing algorithms for data fitting is best done using simulated datasets, so that the true “age” represented by the data is known. In this case, datasets were generated by drawing data points from a range of uncertainty distributions, all centred on a linear trend reflecting an age of 4 Ma. Full details are provided in Appendix D. Two features of the datasets are varied: the number of data points in the dataset and the uncertainty structure adopted, the latter via varying

Given that the datasets investigated have fat-tailed contaminated Gaussian uncertainty distributions, the focus is on the effect of excess scatter in the data or, in other words, data scatter over and above what is expected for Gaussian data uncertainties. Nevertheless, a small proportion of datasets do have small scatter, giving

The analysis below compares the results of

Percentage of simulated datasets excluded by the

Note that, for example, for

A 95 % confidence interval on age can be found by ordering the list of ages calculated for the datasets, selecting the lower limit at the 2.5 % point in the list, and the upper limit at the 97.5 % point. For datasets that lie within the

Considering age uncertainty, it might be expected that the uncertainties suffer from the excess scatter in the data in datasets that yield an errorchron with

Kernel density estimates (

Kernel density estimates for age uncertainty calculated with

Whereas

Kernel density estimates for age uncertainty calculated with

In order to show the real-world utility of

Laser ablation example 0708. See text.

With 0708, the

Quantile–quantile plot for sample 0708, using the

Generally, geochronological datasets do not have obvious outliers – they might be “cleaned” of isolated data points before an age calculation, or the dataset might even be discarded. But many of the simulated datasets from contaminated Gaussian distributions, as used above, do contain outliers. In Appendix E, an example of a simulated dataset with outliers (from 25 %3N) is used to show how

This work was motivated by the belief that many isotopic datasets contain meaningful age information that cannot be identified using classical statistical methods and may therefore be discarded or discounted. In such cases, the age information is contained in a linear spine in the data, but the data also contain scatter that is inconsistent with a Gaussian uncertainty distribution, having fatter tails than Gaussian. A statistical test based on the spine width is devised, akin to using

In most robust statistics data-fitting approaches, the formal uncertainties output during data measurement are ignored. Instead, the scale used in the data fitting is derived from the scatter in the data, an approach adopted by

A problem with

Here, some results of calculations for sample 0708 are collected, used in Fig.

The thought experiment sketched in Fig.

Age uncertainty (age

In the figure, extending from the left, through

In the top part of Table A1, the results for

Summary of results used in constructing Fig. A1 (see text).

Additional algorithmic details for

In the

The so-called error-in-variables (

In the

In

The

Writing the

The minimisation of

At the minimum of

In the first scheme, at iteration

The second iteration scheme is the iteration implemented in the Python code. Such iterations are known to be stable, e.g.

Iteration is needed as

Accepting that an isochron has been calculated, the covariance matrix of

Assuming that

In

If, in addition, all

The second iteration in Appendix B is coded in the Python function,

Datafile for sample 0708, data0708.txt, see Fig. 6

Example output, running on the command line

This work was originally motivated by the dating of speleothems using the lower intercept with a U–Pb concordia in Tera–Wasserburg-style plots

Overall, 10 000 simulated datasets, each containing 5, 6, 8, 10, and 15 data points, respectively, were used to assess

Results are presented in terms of kernel density estimates using an Epanechnikov kernel

To see the consequence of outliers stemming from residuals from a contaminated Gaussian distribution, a dataset from simulations is shown in Fig.

A 25 %3N simulation with

Quantile–quantile plot, for example, in Fig.

In general, with larger contaminated Gaussian datasets, the differences between the

RP created the approach and coded the Python script; EG helped validate the math/statistics and write the paper; TMS helped with the simulations; and JW oversaw the applicability of the approach.

The authors declare that they have no conflict of interest.

We would like to thank the anonymous reviewers for their work, particularly reviewer 4 (twice). Tim Pollard has materially helped in our understanding of the

This paper was edited by Pieter Vermeesch and reviewed by Douglas Martin and three anonymous referees.