‘Partially Labelled Regression Analysis: Dating the Shroud of Turin Using the ‘Raw’ Data,’
Marco Riani, Anthony C. Atkinson, Aldo Corbellini, Paolo Di Lazzaro,
Statistical Methods & Applications, 2026
Since the publication of the report on the radiocarbon dating of the Shroud, there have been several attempts to discredit its claim that the Shroud is medieval on statistical grounds. None of them have succeeded, but they have all, reasonably successfully, at least cast doubt on the precision of the report’s conclusion: “The results of radiocarbon measurements at Arizona, Oxford and Zurich yield a calibrated calendar age range with at least 95% confidence for the linen of the Shroud of Turin of AD 1260 – 1390 (rounded down/up to nearest 10 yr).”1 Some authenticists have claimed that these studies have wholly discredited the competence and/or honesty of the technicians and/or statisticians involved in the original tests, but in fact the papers they refer to have been considerably more circumspect. Riani, Atkinson, Fanti and Crosilla, in an earlier analysis, ended, “our results indicate that, for whatever reasons, the structure of the TS is more complicated than that of the three fabrics with which it was compared,”2 and Casabianca, Marinelli, Pernagallo and Torrisi admitted that “our statistical results do not imply that the medieval hypothesis of the age of the tested sample should be ruled out.”3 Schwalbe and Walsh said “we do not assess nor challenge the general medieval dating conclusions reached in Damon.”4
In this most recent paper to analyse the so-called ‘raw data,’ the same conclusions are made: that “given the heterogeneity that has been revealed, it is clearly erroneous to expect that a narrow sample cut from one corner of the fabric can provide an unambiguous dating of the whole of the TS.” No surprises there, but the question has always been whether the sample area is sufficiently unrepresentative of the rest of the cloth for its dates to be meaningless, or whether the proposed ambiguity is simply a wider range than proposed by Damon et al. I have explored this before, in ‘The Chronological Gradient,’ and the latest paper hardly effects it at all. What is unusual is the emphasis on the medieval age of the sample, which is mentioned no less than five times.
— “Whichever allocation is chosen, the age range is not far from the medieval values suggested by Damon et al.”
——–“In line with the analysis of Damon et al. (1989), all allocations suggest a medieval age for the TS.”
——–“Whatever allocation is chosen for the subsamples, the TS has a medieval date.”
——–“The ordered plots of Fig. 4, show that, although the upper and lower estimated ages of the TS are functions of the allocations analysed, all suggest a medieval date.”
——–“The conclusion of our analysis of the age of the TS, […] is that all maximum and minimum ages from the 165,888 permutations support a medieval date.”
The extraordinary emphasis on the word medieval puts one in mind of Alfred Hitchcock’s movie Blackmail, where the word “knife” is repeated with increasing emphasis, illustrating the guilty conscience of a murderer. No doubt unintentionally, it weakens the authors’ subsequent rather forlorn aspiration that the gradients they found along and across the strip make it “impossible to provide a scientifically based dating for the complete TS.” Their own analysis shows that this isn’t true. If it is possible to derive a “best configuration” for a set of data about an area, then it is entirely possible to extrapolate that data into the hinterland around the area. Robert Rucker’s neutron enrichment hypothesis depends entirely on exactly that sort of extrapolation, misguided though it be. Rather simplistically, he spots a 36 years per centimetre gradient along the sample and supposes that “at this rate, if the sample point is moved by 10 inches then the carbon date would change by 910 years, i.e., from the uncorrected carbon date of 1260 AD to a future date of 2170 AD.”5 But that gradient doesn’t exist. Even using his own version of how the data was to be arranged, a straight-line interpretation of the gradient is clearly sub-optimal. Here is his diagram:
Obviously not. Here is a much better fit:
Rucker has forgotten the first rule of finding the best fit line in such cases, which is to ask whether a straight line is even appropriate, and not to assume it. In this case, clearly not. In the ‘better’ case above, the best fit line appears to reach an asymptote at about 1310, which, continued indefinitely, would suggests that date for the whole cloth.
But Rucker uses only three ‘average’ points, and models in one spatial and one temporal dimension, while Riani et al. have modelled a more precise configuration, using twelve data points and two spacial dimensions. Without knowing exactly how the individual laboratories cut up their pieces, it is impossible to know which configuration actually applied, so they tried to model every one possible, to find out which was most statistically likely. Well, I say “possible,” but they sensibly restricted the way a piece of fabric could be cut up to a few more probable configurations rather than the infinite number that were literally possible. The Oxford sample for example, could have been cut up in any of these ways:
…and the three values for the dates can be entered into each of these in six ways, so that there are 36 possible ways in which the Oxford data could be represented on a diagram.
The Zurich sample seems to have been cut in two, and those two pieces sub-divided into three and two respectively, like this:
The two values for the larger subdivisions can be entered in two ways, and the three values for the smaller subdivisions can be entered in six ways, so each of the four configurations above could carry any of 12 configurations, and there are 48 possible ways in which the Zurich data could be represented on a diagram.
Arizona’s sample has historically been more difficult to configure, but is in fact the easiest. In their earlier paper, Riani et al. didn’t know if the extra sliver Arizona was given to make up a total of 50mg was used or not, so it had to be included in their consideration. Since then, it has become known that not only was the ‘extra sliver’ not used, but neither was a sizeable chunk of the larger piece. This, it seems, was not taken into account by the authors of the new paper. Given that knowledge though, it seems to me that only one configuration is at all likely:
Even so, the four dates which must be fitted into these squares can be arranged in 24 different ways. This means that there are 36 x 48 x 24 possible configurations = 41472.
Riani et al. have, however, included the highly unlikely
and its 90° rotation as two of the possible Zurich configurations, and the completely impossible
and its 90° rotation as two of the Arizona configurations.
This is unfortunate, as just these configurations are among the top six “best configurations” out of all the 165,888 configurations that Riani et al. analysed. They are illustrated in the new paper, and frankly, none of them seem at all likely.
In the paper, the axes of these graphs are labelled x1 and x2, and represent distances in millimetres. Although the distance along the x-axis has a clear zero at the edge of the Shroud, for the y-axis could represent either distances from the “bottom left’ of the sample, as is apparent from the diagrams above, or, in reflection, from the “top left.” Nevertheless, it can be seen that in all the cases above, the oldest date is in the top left hand and the younger in the bottom right.
However, although both the ‘Oxford’ variations above are credible, none of the Zurich sets are; and now that we know – as in fact we knew before this paper was published – that an upper portion of the Arizona set was not used for dating, neither of the two Arizona sets is possible, let alone credible.
Still, let’s take what Riani and Atkinson call the “best configuration,” and a strip along the middle of the sample. The middle Oxford value is 745, the middle Zurich value is 635 and the middle Arizona value (the average of 701 and 608) is 655. Following Bob Rucker’s “C14 Date AD” calculation (i.e. subtracting these B.P. numbers from 1950, we get 1205, 1315 and 1322, at distances of about 50, 65 and 75mm from the end of the Shroud respectively. Like this:
Once again, the best fit line seems to level off, this time at about 1325. If the Zurich and Arizona configurations fitted what Zurich and Arizona actually did, a similar levelling off would occur, but at a different age.
I explored something similar to this in another post (‘The Chronological Gradient’) and came up with this, the older dates on the top and the younger on the bottom. It made a certain sense at the time, but other permutations are certainly possible.
Accordingly, for this post, I chose a similar geometrical configuration, but placed the values slightly differently. Not only that, but by using OxCal, Christopher Ramsey’s online radiocarbon calibrator, I converted the dates from Before Present (in Damon et al. 1989) to calendar dates. Note that these are quite different from what you get simply by subtracting them from 1950, à la Rucker.
Following the ‘levelling off’ appearance of the data illustrated above, these data were fed into graphical software together with four extra points, to the right and below this diagram, whose dates were not specified, but specified to be equal, so that the software could calculate not only the best-fitting mathematical 2D-surface, for these points, but also the ‘levelled off’ value, which would represent the date of the cloth which produced the closest fit.
(For the curious, I also switched the axes’ zeros to the very corner of the radiocarbon corner of the Shroud, so that the distance into the Shroud did not become negative. The resulting plot is a mirror-image of the actual results – reflected about the Y-axis – but this does not effect the values of the results.)
Here is the result:
And the “best fit” date of the horizontal surface from which all these points declne is: 1340AD.
CONCLUSION
Riani, Atkinson et al. decide that that they “have provided highly significant evidence that there is a trend in estimated age over the sample, making it impossible to provide a scientifically based dating for the complete TS.” But I disagree. Their “best configuration” is not, in fact, a possible configuration at all, and a more credible configuration clearly suggests a hyperbolic relationship between several sets of points. I have no doubt that analysing all my 41472 ‘credible’ arrangements would produce a great many possible dates for the entire Shroud, but that if all the surfaces were arranged in order, the “best fitting” surfaces would never stray far from giving a 1340AD date to the manufacture of the cloth. Contrary to Riani et al.’s conclusion, extrapolating their own methods to fit the radiocarbon data to a curved surface does indeed “provide a scientifically based dating for the complete TS,” which turns out, like the radiocarbon corner, to be “medieval … medieval … medieval.”
:
1). ‘Radiocarbon Dating of the Shroud of Turin,’ Paul Damon et al., Nature, 1989
2). ‘Regression Analysis with Partially Labelled Regressors: Carbon Dating of the Shroud of Turin,’ Marco Riani et al., Statistics and Computing, 2013
3). ‘The Radiocarbon Dating of the Turin Shroud: New Evidence from Raw Data,’ Tristan Casabianca et al., Archaeometry, 2019
4). ‘On Cleaning Methods and the Raw Radiocarbon Data from the Shroud of Turin,’ Larry Schwalbe and Bryan Walsh, International Journal of Archaeology, 2021
5). ‘Solving the Carbon Dating Problem for the Shroud of Turin,’ Robert Rucker, 2022 (Paper 33 at shroudresearch.net)