‘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.”
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TECHNICAL APPENDIX (THE ‘RAW’ DATA!)
As I write in a comment below, once non-randomness is claimed, there is a certain obligation (even if self-imposed) to describe it. In the simple case of the three dates, there are any number of possible ways of relating them, of which the straight line is just one, and in this case not a very good one. Among the others are various polynomials, circles, exponentials, logarithmics etc. Anybody who thinks they can make a case for such a curve is very welcome to make it.e
I decided to try to find the equation of a 3D surface which would best fit the data, and selected two possible descriptions, one a bivariate radial hyperbola, and the other an elliptic logarithmic funnel, both describing a flat sheet deformed downwards through an elliptical hole, although the Shroud itself would only occupy one quarter of the model. Here is such a quarter of a model:
…and here is how it would be positioned on the Shroud:
The general equation for a bivariate radial hyperbola is:
In this case, x and y are the spatial co-ordinates defined above, and z is the Calendar Age AD, derived by converting the BP dates into AD dates using OxCal.
A is the asymptote (the true age of the cloth), and c and d are the x,y co-ordinates of the point from which the deviation from the uniform date originates. B is the slope of the dip (or more precisely a coefficient governing the rate at which the slope changes).
With the help first of Gemini (A.I.) and second an online 3D surface plotter, the following approximate values for A, B, c and d were obtained: A = 1316, B = 364, c = 44.3, d = -0.7. Note that the negative value for d indicates a point source of contamination just outside the Shroud, but on the Holland backing cloth.
Using these, dates for the twelve Damon 1989 points described above could be obtained mathematically, and compared with Damon’s own dates, thus:
This model, then, gives 1316 AD as the date of manufacture of the Shroud.
The general equation for an elliptic logarithmic funnel is:
As before, x and y are the spatial co-ordinates defined above, and z is the Calendar Age AD, derived by converting the BP dates into AD dates using OxCal.
A is the asymptote (the true age of the cloth), and c and d are the x,y co-ordinates of the point from which the deviation from the uniform date originates. B is the slope of the dip (or more precisely a coefficient governing the rate at which the slope changes). e and f are the rough length and width of the deformed surface, double their measured values so as to place the centre of deformity approximately in the centre of the hypothetical ellipse.
In the same way as before, “appropriate”best fit” values for the constants were found: A = 1320, B = 9, c = 1.9, d = 5.5, e = 100, f = 50, and the mathematically calculated dates compared with the Damon dates as before:
This model gives 1320 AD as the date of manufacture of the Shroud.
Of course, other configurations are possible, other general equations are possible, and other values for the constants are possible, but all reasonably fitting ones will result in an overall medieval date for the cloth.
No straight line model, attempting to demonstrate radiation intensity, fits the data at all well. The quote from Bob Rucker, above (“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.”) is completely contradicted by his own model. In ‘The Carbon Dating Problem for the Shroud of Turin, Part 3: The Neutron Absorption Hypothesis,’ he gives a detailed diagram of the dates predicted by his model across the whole Shroud. He also says that “The sample that was cut in 1988 for the C14 dating of the Shroud was cut from the bottom left corner of the Shroud. For the MCNP calculation, it was assumed that when the body was wrapped in the Shroud in the tomb, the bottom of the cloth near the feet was tucked under the feet, so that the sample area used for C14 dating was located under the feet along the midline of the body at the time of the radiation burst.”
Fair enough. The value under the feet, on the midline of the body is given as 1317, which is a reasonable approximation of the radiocarbon date given by Damon. But the next value along – about 14cm along – is 2452. This is a rise of 1135 years, or 80cm per year, more than double the rise Rucker himself observes of the Damon dates.
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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)
Hi Gerardo,
Thanks for that. I’m not sure about the lightening/darkening of linen, to be honest. There are numerous accounts/images of linen being bleached by laying it out in the sun, which seems to have been a standard practice, but it’s also true that people talk about it being yellowed with age. Still, I don’t think the Holland cloth went from much whiter than the Shroud to much darker than the Shroud in 500 years, while the Shroud has scarcely changed in 700 years. Its being stained is not documented anywhere, but then neither is there any reference to the staining of any putative mending of the Shroud itself. The only evidence for either is from Roger’s Thermochimica Acta paper, and his “gum dye mordant,” of which, as suggested by the rather small alteration to the radiocarbon content, only a small amount was smeared across from backing to Shroud.
And you’re also quite correct that if Rucker’s model were appropriately reconfigured, then I suppose any value for the radiocarbon date of anywhere on the cloth could be achieved. Just seems a bit ad hoc to me.
Best wishes,
Hugh
I think your objections were already answered in my previous message.
I wrote “the exposed side” of the Holland cloth became darker. That is, the side of the cloth that wasn’t protected by the Shroud (and vice versa). When the Raes corner was cut, it exposed a portion of the cloth that was previously protected, so that was lighter than the rest.
“the exposed Holland cloth in those corners was stained to darken it roughly to match the rest of the cloth.”
Is that documented anywhere, or is it your supposition? Anyway, even if that was true, that part of the Holland cloth was not in contact with the Shroud, so it’s unlikely that a significant amount of the dye would be transferred.
“undyed linen tends to get lighter, not darker, under prolonged light exposure.”
Does it? It is generally agreed that the Shroud is becoming darker with time (some fear that if that were to continue, the body image might eventually disappear). I don’t know whether that is primarily due to light exposure or to other factors. Anyway, I’d suppose the Holland cloth would do the same.
“The only value anywhere near the radiocarbon date in that position is indeed the 1317 found in the middle of the left-hand edge of the “below the body” band.”
As I wrote, that’s because Rucker normalized his results to match that number at that point. The correct normalization would be to match 1317 at the correct point, that is the left hand edge of the “to the right” or “to the left” strips, where it currently reads about 820. Taking that into account, your assertion that my statement is “transparently, and spectacularly, untrue” isn’t justified, no matter how many times you repeat it.
Hi Gerardo,
Thanks for that. I may not have been clear enough about which corner was which! When the Shroud was repaired in 1534, it was, I believe, completely rectangular, so the Holland cloth was hardly seen at all. Subsequently, pieces of the side-strip were cut away, either because they were tatty or for friends of the Savoy family, and the exposed Holland cloth in those corners was stained to darken it roughly to match the rest of the cloth. Then in 1973 the Raes sample was cut away, exposing more Holland cloth, but this time the exposed section was not stained. The crux of your comment is that you “guess that any colour differences that you see now are simply the effect of aging, with the exposed side of the cloth having become darker, not the indication that any colouring substance was applied.” I don’t think that’s true. Firstly, the Holland cloth under the Raes sample – and I guess under most of the Shroud – has not suffered your “effect of aging,” and secondly, should it be claimed that the area under the cloth was shielded from light, the Shrould has never been exposed to sufficient light to change colour in such a noticeable way. Thirdly, undyed linen tends to get lighter, not darker, under prolonged light exposure.
Rucker’s Paper 13, quite right; sorry to be misleading! Figure 13 is divided into four horizontal bands of unequal width and two ‘tabs’ hanging down from each end. The lowest horizontal band is labelled “Shroud below the body” in the key diagram (Figure 12). Earlier in the paper, Rucker says, “The sample that was cut in 1988 for the C14 dating of the Shroud was cut from the bottom left corner of the Shroud. For the MCNP calculation, it was assumed that when the body was wrapped in the Shroud in the tomb, the bottom of the cloth near the feet was tucked under the feet, so that the sample area used for C14 dating was located under the feet along the midline of the body at the time of the radiation burst.” The only value anywhere near the radiocarbon date in that position is indeed the 1317 found in the middle of the left-hand edge of the “below the body” band.
If the Shroud was not tucked under the body, but was simply flopping down over the legs, then we need to look at the end on either side of the ventral image, specifically the “Shroud to the right of the body.” […Edited out. I misread Ruckers’s diagram…] If you were correct to choose the left hand edge of the “to the right” or “to the left” strips, where the successive figures do show a growth which compares reasonably well with the calculated 36 years per inch derived from the Nature paper, the calculated dates range from about 820s to about 1260, which is completely wrong. I repeat (with added confidence!) “Your statement: “Rucker’s surface is predicted by a physical model that univocally results in that surface,” is transparently, and spectacularly, untrue.”
Best wishes,
Hugh
“The Holland cloth, at least in that area, was definitely discoloured from the pristine white we see revealed when the Raes sample was cut off, to the much darker colour of the rest of the corner.”
I’m not sure I follow you here. So when the Raes sample was cut off, the portion of the Holland cloth that had been under it showed its “pristine white”. What is the “rest of the corner” and where does one see it? And if the Holland cloth was “discoloured” why didn’t they do that also in the Raes corner? (And why would they care about the colour of the Holland cloth at all, if it was going to remain hidden beneath the Shroud?) I guess that any colour differences that you see now are simply the effect of aging, with the exposed side of the cloth having become darker, not the indication that any colouring substance was applied.
Regarding Rucker’s model, I was thinking about the shape of the surface rather than the numerical values. The model has a few adjustable parameters so it can be tuned to fit the numbers better.
Anyway, I checked the paper (by the way it’s #13 not #33). I suppose you got the figures of 1317, 2452, and 4260 from the table of Figure 13. Although Rucker himself suggests to take the points “along the midline of the body” and normalized his calculations according to that, I don’t think that’s correct, because the cloth-to-body distance is critical to his model and you’re getting very different values for that along the edge of the Shroud, that is where the sample was actually taken. So I believe it’s better to take the values on the “Shroud to the left of the body” or “Shroud to the right of the body” rows (I’m not sure on which side the sample was taken, and the numbers are similar anyway). Those indeed show a much smaller variation: neglecting that the dates should be renormalized (I don’t think that would change much) the difference between the first two points on each row is about 200 years “to the left” and 250 “to the right”, compared to 1100+ years along the midline. That seems much more in line with the 35 years/cm of the radiocarbon dates.
Hi Gerardo,
1) Oh, yes. The average Zurich ‘n’ Arizona Value is 661BP, the average Oxford value is 750BP. Shifting Oxford by 100 years would definitely place it among the others, i.e changing its radiocarbon proportion from 91.3% Fm to 92.3%. On the assumption that it was 92.3% to start with, I’m thnking that interpolated material of marine origin would be sufficient to account for the difference, although I haven’t calculated specifics. Yet.
2) Yes, certainly. If the earliest statisitically calculated Oxford date was indeed the date of that portion, and the latest statisitically calculated Arizona date was the true date of that portion, then any supposed contamination would, I’m sure, have been visible.
3) Yes, and always have done since the chronological gradient was discovered. I don’t think that flax grown at various times between, say, 1200 and 1350 made up the radiocarbon corner (or the rest of the Shroud for that matter). The variation from the true date is a matter of contamination of some kind, material or radiation or whatever.
4) No. The Holland cloth, at least in that area, was definitely discoloured from the pristine white we see revealed when the Raes sample was cut off, to the much darker colour of the rest of the corner. It must have been given its colour while it was attached to the Shroud, and some of the colour has merged into the material of the shroud itself.
5) I do concentrate on the mathematical side, because it is the foundation of the statistical analysis of the paper I was commenting on. The broader picture goes like this. We are generally agreed that the Oxford area dates older than the Zurich area, which dates older than the
Arizona area. Apart from that, it seems likely that the sub-sample dates are randomly distributed, and nothing practical can be drawn from them. Finding the best fit for them is an entertaining statistical exercise, but in real terms wholly meaningless.
But looking at the three points, 750BP, 676BP, 646BP, or as Rucker likes to convert them, 1200, 1274, 1304, (separated by 74 years, then 30 years), they don’t fit his own model well, as I mention above, whose first few points are 1317, 2452, 4260 (separated by 1135 years, then 1808 years).
According to Rucker, the dates published by Damon increase by 35 years per cm; but his own dates increase by about 90 years per cm, over the first 30cm or so of the cloth. His own statement: “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,” is belied by his own calculation, which shows a future date over about the same distance as 4260 AD. Your statement: “Rucker’s surface is predicted by a physical model that univocally results in that surface,” is transparently, and spectacularly, untrue.
It seems we agree more than it might seem.
“the 5% is often taken as a threshold. Below 5% = utterly unacceptable, above 5% = wholly uncontroversial. That’s not how things work in the real world, of course.” So you agree that the 20 years shift calculated by Schwalbe and Walsh isn’t enough to make it “wholly uncontroversial”?
I have indeed misread the passage about worst case scenario. So you agree that more than 5% would be needed for that scenario and “that would be too large to not have been noticed”?
“I don’t dispute that the overall Oxford date is older than the overall Zurich date.” So you agree that the measured radiocarbon dates don’t just match the true date of the sample (unless different parts of the sample were indeed of different ages)?
So isinglass was used as a binder for paint. But the Holland cloth was not painted and likely never even entered a painter’s studio, so I still can’t imagine how that would end up on the cloth. Suggesting that seems, well, fishy.
“Rucker’s model and my model might both fit within the error bars of each point, but the one which fits the points themselves is the better.” You keep concentrating on the mathematical side and neglecting the physical side. Rucker’s surface is predicted by a physical model that univocally results in that surface, while yours results from a hypothesis (that the Shroud sample is contaminated by a stain of a random shape) that may fit with any surface, making it essentially unverifiable. You didn’t deduce your equation from the model as Rucker did, you picked it because it was a good fit for the data (and gives the result that you want), and said “it may fit with the hypothesis” — because the hypothesis is so generic that about any surface may. Besides, as I wrote, a polynomial with N suitably crafted coefficients always fits N points perfectly, so it is a better fit than your surface, and it hasn’t any asymptote.
Hi Gerardo,
You’re quite right about Schwalbe and Walsh, but bear in mind that the 5% is often taken as a threshold. Below 5% = utterly unacceptable, above 5% = wholly uncontroversial. That’s not how things work in the real world, of course.
You have misread my passage about “the worst case scenario.”
Not being able to imagine how some “fish product” could end up on the cloth is quite understandable. Google “medieval isinglass” to release a whole new world of imagination!
It really doesn’t matter what my favourite equation is named. It describes a surface, it fits some configurations of the radiocarbon data very well. That’s all.
The uncertainties surrounding the data points do not negate the importance of the points themselves. They are calculated uncertainties, which means that each point acts as a most probable point, and the bars either side a range of increasing uncertainty. A value of 5±3 does not mean that any value between 2 and 8 is equally likely. Rucker’s model and my model might both fit within the error bars of each point, but the one which fits the points themselves is the better.
I wasn’t clear enough about my “random points.” I don’t dispute that the overall Oxford date is older than the overall Zurich date. What might be the case, however, is that all the Oxford subsample meaasurements are randomly distributed within the Oxford sample, and all the Zurich subsample measurements are randomly distributed within the Zurich subsample (and all the Arizona… etc.). In that case any best fit surface would give an illusion of structure which isn’t really there.
Best wishes,
Hugh
Hi Paolo, and through you your fellow authors of the new paper,
Thank you very much indeed for bothering to comment, with which I largely agree. Your bottom line, literally, is perfectly true. However, there are a few points which I don’t think you address.
1). I was first drawn to your paper by the extraordinary number of times the word “medieval” occurs in it. I wonder if this was deliberate or subconscious. What do you think?
2). What is the answer to your question: “Are they significantly better than the three-parameter linear model?”
3). Your Figures 7 and 8 are clearly geographical “maps” of the radiocarbon sample, and like any map, although the map has boundaries, the geography it illustrates does not. It is perfectly sensible to ask what value/s would be reasonable at point x1=90, x2= -5 for example. As we travel further and further from the known area, our speculation about what the values might be, must become more and more uncertain, but some distinction can be made between certain possible physical scenarios, such as contamination from one side of the sample or radiation from the centre of the image. Do you agree with that?
4) You do not comment on my observation that your “best” configurations for the Zurich sample are very unlikely to have been how that sample was cut up and tested, and your “best” configurations for the Arizona sample are impossible.
5) Part of the essence of my post is that I do not want anyone to predermine either a medieval contaminant or an authenticist radiation hypothesis. In an ideal world I would like every possible model fitted to every possible configuration of the data points, and the “best fitting” model/configuration combination found. My prediction would be that it would fit a medieval contaminant hypothesis better than an authenticist radiation hypothesis.
But as I say thank you very much for reading and commenting,
Best wishes,
Hugh
If I remember correctly, Schwalbe and Walsh found that if the Oxford date were lifted by 20 years, the chi-square statistics would just reach the threshold of 5% that is generally considered the minimum acceptable, that is, 1 chance out of 20 that the errors were truly random. That isn’t what I would call a “reasonably good” match, but I can see how you would. All the control samples rated much better than that.
May I please see your calculation that “in the worst case scenario” less than 5% contamination would be enough?
Mineral oil in that quantity would have been quite visible and I don’t think they would have used a Holland cloth that was obviously stained. Maybe some “fish product” could have been less visible, but I can’t imagine how that kind of stuff could end up on the cloth. Anyway, weren’t the pretreatments supposed to remove any contaminants?
Now let’s go back to the fitting. Your “favourite equation” isn’t really a quadratic surface, that is, a polynomial of degree 2, since it has a square root and a division. Ok, that curve doesn’t decrease at an ever accelerating pace. A true quadratic would.
Well, it is rather pointless to try and beat that surface. In fact, since the data points are known to have uncertainties of the order of tens of years, a surface that fits them *too well* would actually be suspect (“too good to be true”). Any surface that fits “well enough” is as good a candidate as yours, and I’d bet that includes Rucker’s linear surface.
And what does “well enough” mean? There is a well defined answer to that question in statistics: when the deviation of the data points from the surface is “statistically significant” then it doesn’t fit “well enough”. Riani et al. have calculated that the deviation from a flat surface is statistically significant, so your claim that “It may well be that random experimental error is a perfectly good explanation for the distribution of each laboratory’s points” is wrong.
Hi Hugh, please find below a comment shared by the co-authors of the published paper discussed in this post: M. Riani, A. Atkinson, A. Corbellini, and myself.
Over the years Hugh Farey has written widely and wisely about the TS.
However, as others have already commented, in this case, perhaps guided by his enthusiasm for a medieval date, he has made a mistake in model fitting.
In fitting a three-parameter model to three data points he has achieved a perfect fit to the data. If the model contains an asymptote, then all extrapolated values for higher values of the explanatory variable will be that of the asymptote. The TS is then unambiguously medieval.
However, for the raw data we have 16 observations at 12 partially known locations. For each of the 165,888 possible allocations models with three (as in our linear regression model) or more parameters could be fitted. Are they significantly better than the three-parameter linear model? Calculation of the values of R2 is one way of making this comparison. Extrapolations using all fits at some point, for example twice the values of x1 and x2, would give a comparison of the distribution of the predictions from the model.
This does require heavier computing than is described in the other contributions.
For the sake of transparency and open science, we have MATLAB routines available for general use in the GitHub repo
https://github.com/UniprJRC/Shroud
or the repo in the MathWorks file exchange
https://www.mathworks.com/matlabcentral/fileexchange/183929-partially-labelled-regression-dating-the-shroud-of-turin.
This allows further investigation of the points to do with subsample sizes and plausible sample division.
There are several comments about our Figure 7. This part of the paper has to do with identification of the effect of the small standard error for Observation 11 if weighted regression were to be used. It has no effect on the results for the dating of the TS summarized in the other Figures.
Our, it seems, misplaced hope was that our paper would end the discussion of the radiocarbon data. It remains the case that, given the inhomogeneity of the observations and the confounding of samples and laboratories, as outlined in our papers, further experimental work is required.
Finally, even if one accepts the hypothesis that the analyzed segment is medieval, this conclusion applies only to the particular portion of the cloth that was sampled. Our analysis shows that the dating process is not stationary across the cloth, and therefore no valid statistical inference can extend the date obtained for one segment to the entire Shroud. In other words, a medieval date for the sampled region cannot, by itself, be taken as evidence that the whole cloth is medieval. This is precisely why understanding the spatial heterogeneity of the material is essential, and why additional sampling from different regions of the cloth would be necessary to establish the age of the Shroud as a whole.
Hi Gerardo,
“I wasn’t aware of your explanation…” That’s quite alright; I wasn’t aware of it myself until a little while ago! It was Schwalbe and Walshe’s idea that if the Oxford date were ‘lifted’ by twenty years, it would match the Arizona and Zurich dates reasonably well. For that to happen, it had to be contaminated with something which ‘lowered’ the date, and something affected by the marine reserve, such as mineral oil or a fish product, would do the job. We recall how Richard III, died 1485, was radiocarbon dated to 1460, because he ate a lot of fish!
What Schwalbe and Walsh didn’t deal with was the possibility that all the radiocarbon dates were to a greater or lesser extent affected by contamination, so the solution was not just an “Oxford” problem, but derived from something oozing in from the Holland cloth, affecting all the samples. Rogers’s gum.
For the worst case scenario – i.e that the oldest Oxford date really is that much older than the youngest Arizona date, the proportion of contamination would be, as you suggest, too large not to have been noticed, but within a sensible margin of error, a proportion of less than 5% contaminant to over 95% ‘original’ is possible and realistic.
I’m not suggesting that the Holland cloth itself is responsible for lowering the date. It was probably made a few years, possibly decades, before 1534, and adding its fibers to the mix would make the whole thing appear younger, not older. The same applies to any invisible interpolation, unless it was affected by the marine reserve. Isinglass was a common component of medieval paint, and would serve the purpose.
I do not agree that “the “gum” is not the cause of the anomalous date,” nor that there were any repairs near the sample area. I think the staining of the Holland cloth may well be the guilty culprit, in which case the gum is indeed the cause.
As I have said before, the fitting of a mathematical model to all the dates was not my idea. As you will know, it is possible to fit a plane surface to any completely random set of x,y,z points, and to quantify, using residuals, how well it fits. At least one will be a better fit than all the others, but that does not mean that the points really do describe a plane, or that they were not completely random. It may well be that random experimental error is a perfectly good explanation for the distribution of each laboratory’s points. Other mathematically defined surfaces, hyperbolic, logarithmic, quadratic, etc. can also be experimented with to find which has the best fit, again, even if the points are in fact completely random. That’s why I began by saying that there is no a priori reason why any equation, however well it fits, must describe the true variability of the dates of the Shroud.
But if the dates really do relate to a specific effect, be it contamination from the Holland cloth or radiation from the centre of the body image, then which model describes the distribution of the points best can help us discriminate between the two – or other possibilities. We could try a variety of different planes and curved surfaces, see which one “best fits” the points given, and consider how they predict values beyond the boundaries of the radiocarbon sample area itself.
I believe – and I haven’t tried every possible surface, mostly because I don’t know how and also because I imagine it would take ages – that the best fitting surfaces are something like the ones described by the equations above. Of course you’re right that a quadradic equation can never be quite flat, but it can appear to flatten out sufficiently for our purposes. However my favourite so far is z = 1316-(364/SQRT((x-44.3)^2+(y+0.7)^2)), but as I say, anybody with time and resources is welcome to try for a better one.
It is most convenient, to avoid negative co-ordinates, to place the origin of the model at the corner of the Shroud, but as you can see from my coloured 3D model above, that does not predict that “the more you go into the inner part of the Shroud, the older the dates become,” let alone “at an ever accelerating pace.” It doesn’t really matter where you place the origin, the model looks the same.
Best wishes,
Hugh
I wasn’t aware of your explanation of Rogers’ observations as staining by the Holland cloth. It’s an interesting hypothesis. However, as an explanation for the anomalous dates, it has the same problem as any other “contamination” theory: the required amount of contaminant is too large. The Holland cloth and anything that came with it can’t be younger than 1534 (it has been replaced recently, but I assume you’re thinking of the original Holland cloth) and so it would require a large amount of material to skew the dates by the approx. 100 years of difference between Oxford and Tucson. Not as large as the 80% vs 20% needed to skew the date from 1st to 14th century, but still quite large. Furthermore the Holland cloth is younger than the Shroud, so any contamination from it would have made the Shroud appear younger, not older. By contrast, in Rogers’ hypothesis the “gum” is not the cause of the anomalous date: the date is anomalous because the sample has been repaired, and the “gum” is there because the repair was dyed in order to match the appearance of the linen of the Shroud. You don’t need 80% “gum” because there is already 100% repair.
With respect to the fitting, I’m afraid that your statement that “there is no reason why any math[e]matical model should apply to the dates” makes this hypothesis unverifiable. Still, I believe it would be rather peculiar if the distribution of contaminant was proportional to the X coordinate *multiplied* by the Y coordinate.
I take note of the difference between “all curved surfaces” and “all the curved surfaces”. That must be a nuance of English language that I am not familiar with. Anyway, I don’t understand your statement that “it really is flat”. No quadratic curve (or surface in 3D) is ever flat over any particular region (at most, a surface may be flat along a line) unless it is flat *everywhere*, that is, the quadratic and linear coefficients are zero and all that remains is the constant term. If you disagree, please show me the equation of that curve.
I failed to notice that you put the origin of the axes at the corner of the Shroud, so the Shroud actually extends towards larger positive coordinates and not towards negative coordinates. But that is even worse for your model, because then the prediction is that the more you go into the inner part of the Shroud, the older the dates become, and at an ever accelerating pace. As César already observed, you’d get a date of 33 AD only a few tens of centimeters away, and going beyond you’d reach prehistoric. If that was really the case then the only reasonable conclusion would be that the dating has gone completely astray.
Hi Gerardo,
I think I understand your point, and even partly agree with it.
Perhaps I should have stated at the outset that I think the cause of the anomalous dates could be the staining of the Holland cloth after it was applied to the Shroud, whose darkening effect compared to the Holland cloth’s original whiteness, is clearly visible in the area of the Raes sample. This stain smudged, smeared or seeped onto the Shroud itself, and, being composed partly of isinglass, had an aging effect on the radiocarbon date. It is rather accurately described by Rogers in his Thermochimica Acta paper, albeit misinterpreted as “gum.”
As such, there is no reason why any mathmatical model should apply to the dates. The contamination appears to have been slightly greater towards the end of the sheet, but beyond that general observation, patches could have been more or less random, and the resulting dates more or less random too, although it would be sensible to suppose that the further away from the visible surface of the Holland cloth, the less contaminant would be smudged, smeared or seeped onto it and the less the radiocarbon date of the ‘clean’ sheet would be affected.
However, it has been Riani and Atkinson’s contention that the 12 Damon dates could be organised statistically into a “most probable” pattern (as indeed could any collection of data), and, by association with Giulio Fanti (previous paper) and Paolo di Lazzaro (new paper), that the pattern is meaningful.
That was my first assumption, whether or not it is actually true. If the points are part of a system (mathematically speaking), then the system extends beyond the boundaries of the radiocarbon sample into into the surrounding cloth. If there are gradients of any kind, caused by the gradual reduction of contamination level, those gradients continue beyond the edges of the radiocarbon sample. To my mind, however, as illustrated above, such gradients as have been found do not suggest that the Shroud gets linearly younger and younger; they suggest that the slope flattens out onto a level plane.
That was my second assumption. From these assumptions, which are, I think “valid candidates,” which “make sense physically,” I looked for ways of interpreting the data mathematically, and settled for two general but well defined curves, as specified above, both of which fit the data. One is broadly logarithmic and one is hyperbolic. They both require a third assumption, which I’m fairly sure is incorrect, but which makes the calculations easier and does not substantially affect the ultimate result, and that is that the source of contamination was a point, not a series of daubs across the Holland cloth, as was in fact probably the case.
With that caveat, my models are both founded in real data, real observations and sensible assumptions. They are realistic curves. It is not altogether true that the data doesn’t produce models. Without the data there would be no need for any modelling at all. The data does not, of itself, produce models, but it certainly inspires them. And a mathematical model is certainly not “an explanation of how the laws of physics applied to a particular situation result in that equation,” it is, quite literally, just the equation. The laws of physics have no bearing whatsoever on how well a series of points fit a surface. However, you’re correct to the extent that having found a mathematical model, if the data actually record a physical situation, it behoves the investigator to suggest how such a model could have come about in the real world, maybe we could call it a “physical model.” In my case, as I explained above, I think it came about from decreasing contamination from the staining of the Holland cloth.
Finally (your ‘finally’) I didn’t say that “all curved surfaces result in a medieval date.” I said that all THE curved surfaces (those that that fit the data) result in a medieval date for the Shroud. Which they do. By all means experiment yourself with quadratic curves. I would have done so myself if I knew how. Find one that fits the data, and explore what results are obtained by extrapolating it beyond the limits of the radiocarbon sample itself. If the curve does not extend that far (“no date at all”), then it cannot be a valid model.
I don’t really understand your last sentence. I drew one quarter of the hyperbolic curve because the radiocarbon corner only occupies that quarter. If the Shroud extended into the other quadrants, then my model predicts what the radiocarbon dates would have been if points on those quadrants had been tested, but it doesn’t, and they weren’t. It is the Shroud which “ends there,” not the model. Models of real objects which are constricted in space do not necessarily have to be similarly constricted – and in real situations very rarely are. And where you say “beyond that it’s flat,” it really is flat. That’s what both models predict.
But please explain further if I’ve misunderstood.
Best wishes,
Hugh
You haven’t answered my main argument: you can try lots of fitting curves, but only the ones that make sense physically are valid candidates. That is, those that are predicted by a reasonable physical model. You say: “Anybody who thinks they can make a case for such a curve is very welcome to make it.” That is exactly the point: Rucker made a case for a linear curve, by proposing a physical model that predicts that (more precisely, it predicts a curve that over the few centimeters’ range of the sampled area is well approximated by linear). You didn’t make a case for a hyperbola.
Your conclusion: “Bottom line, there is enough data to produce numerous realistic mathematical models, and all of them can be ordered according to how well they fit the data. And all the curved surfaces fit the data better than any straight-line interpretation, and all of them result in a medieval date for the Shroud.”
is wrong on so many counts. To begin with, you seem to completely miss the meaning of “realistic”. It’s physics, not mathematics, that determines what is “realistic”. If a curve isn’t realistic then it doesn’t matter how well it fits. The data doesn’t “produce models”. Physical reasoning produces models, then you compare them with the data. A model isn’t just an equation, it’s an explanation of how the laws of physics applied to a particular situation result in that equation, and you didn’t give an explanation for any of the curves that you’ve tried. That is, you didn’t try any “models”. Finally, it isn’t true that all curved surfaces result in a medieval date. Most result in no date at all. For example a quadratic curve gives an unlimited range of possibilities (limited on one side, not on the other — by the way in this specific case that would exclude *younger* dates, not older!) and there’s no valid reason to pick one as “the date”. In fact, you conveniently draw only one quarter of the hyperbolic surface so as to give the impression that it “ends there” and beyond that it’s flat (and your description of “the horizontal surface from which all points decline” is also cleverly constructed to reinforce that impression). Of course it isn’t.
Hi Gerardo,
See my explanation to César above. And you’re perfectly correct about the multitudinous curves. However, bear in mind that it was not I, or Paul Damon, who started looking for relationships. One way of treating the results is that they are randomly distributed about a value (or range of values), but it is reasonable – knowing the order in which the samples were cut from – to suspect a non-random distribution. Having claimed non-randomness, however, there is a certain obligation (even if self-imposed) to describe it. In the simple case of the three dates, there are, as you suggest, any number of possible ways of relating them, of which the straight line is just one, and in this case not a very good one. Among the others are, as you suggest, are simple polynomials, circles, exponentials, logarithmics etc. Anybody who thinks they can make a case for such a curve is very welcome to make it. My choice above, which may not be correct but at least fits the points better, is for part of a hyperbola, with a horizontal asymptote. And why not? Another choice, which I’ve just been working on, is logarithmic and gives me a date of 1321.
Bottom line, there is enough data to produce numerous realistic mathematical models, and all of them can be ordered according to how well they fit the data. And all the curved surfaces fit the data better than any straight-line interpretation, and all of them result in a medieval date for the Shroud.
Hi César,
Well spotted. Unfortunately, the simple graph-drawing software on my lap-top does not include a hyperbole-calculating function, so for my first two diagrams I simply chose the “curve” drawing function and put the end points on the outer values and an internal point on the other. The curve was obviously a “best fit” because that’s where I put it! However, it was adequate to demonstrate that a straight-line relationship was far from optimal, and the concept of a 2-D hyperbolic space was valid, so that it was worth trying to find a “best fit” using all the points, using more advanced software online. In that case, as is normal, many of the points do not lie exactly on the calculated surface. I originally tried for the best fit surface with an asymptote of 1360, but it was a poor fit. By reducing the asymptote from that value, the surface became a better and better fit until 1340, after which it became worse again. To be honest, there are several equations which can be used to fit the data points given, so it’s not possible to declare 1340AD as the definitive date of manufacture, but that wasn’t the point of the exercise.
My overall point is to show that:
a) if the 12 dates given by the Damon paper are all non-random, and
b) if they are given spacial co-ordinates in line with the descriptions of how the subsamples were cut up, and
c) if it be assumed that they represent some systematic deviation from the uniform original date for the cloth,
then a mathematically sound model can be made which demonstrates a 14th century provenance for the cloth, which fits the given points much better than Bob Rucker’s radiation hypothesis.
Best wishes,
Hugh
PS. Thank you for the polynomial graph, but, as I explain above, it is not relevant to my argument. I will, however, add a detailed mathematical model to my post, which will enable anybody to experiment with ideas of their own.
Your “much better fit” than Rucker’s appears to be a quadratic curve. If so, it hasn’t an asymptote. It has a maximum, after which it goes down. You could of course also fit a constant term minus a decreasing exponential curve — that will give you an “asymptote”. Or many other types of curves. Three points are simply too few to decide which is the “best” curve, and depending on which you choose, you get wildly different results when you go outside the sampled range. That’s why extrapolation is always uncertain.
Also, the “best fit” isn’t just the curve that “fits best”. Given N points, the curve that “fits best” is a N-1 degree polynomial. That will always pass exactly through all the points, so *mathematically* you can’t beat it. But in most cases that doesn’t make sense *physically* as an interpolation curve, because there is no plausible physical model that could generate that kind of curve. That’s why the linear curve is almost always the first choice: it’s the first approximation to anything else, so, as a first approximation, it fits whatever physical model. In this specific case, a decreasing exponential may also make sense physically (if you assume that the moving dates are due to a contaminant whose concentration decreases to zero as you move away from the sampled area). A quadratic curve definitely doesn’t. And a “hyperbolic relationship” — well, I’d need to know what you mean exactly by that.
Bottom line: there isn’t enough data to decide which interpolation curve is “the best” (and there won’t be until they do another radiocarbon test). However, there is enough data to rule out one curve: the horizontal line, that is, the one that you get if you assume that the sample was homogeneous.
Dear Hugh.
I agree with many of your comment on the “regression” article regarding the possible distribution of the subsamples. However, the main conclusion is off the mark … You wrote “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 you used a polynomial fit of degree 2 that exactly fits three points. I am sure you know that this fit does not trends toward an asymptote. You surely know that this fit reaches a maximum and then decreases. The fitting you present would estimate the date 33 AD 19 cm further away!!! So, a few centimeters further away the linen would date from the time of Jesus Christ, and a few centimeters further still, it would be prehistoric.
The only reasonable fits are straight and logarithmic. Both of them provide a modern and future ages for the main part of the cloth.
Regards
César