Conficence level and sigma

[hjja...@arcor.de]

The en.wikipedia writes under "Calibration", the sentence "For example, 'cal 1220–1281 AD (1σ)' means a calibrated date for which the true date lies between 1220 AD and 1281 AD, with the confidence level given as 1σ, or one standard deviation."

This looks very convincing, interpreting the two dates as expressing ±1σ. In contrast, calibrating the age of Ötzi (the iceman) with given 4550 ±19 (= 1σ) BP, OxCal yields for all three confidence levels (68.3; 95.4; 99.7) naturally the same mean μ, however and surprisingly, also the same sigma =89, in spite of naturally different overall dates. For me, the OxCal output is logically contradicting the cited sentence.

Hans

[MILLARD, ANDREW R.]

The cited sentence is wrong. The range given is a 68% probability region. A 1-sigma range can be a 68% confidence range but ONLY if the distribution is a normal distribution. Calibrated dates are (a) not normally distributed, (b) not expressing confidence intervals [they are the results of a Bayesian calculation and not likelihood distributions from which a confidence interval can be calculated].

 

The calibrated probability distribution has a standard deviation, this is a fixed property of the specific distribution. It does not change whichever probability level is used to summarise the distribution, which is why OxCal only reports one value.

 

I don’t see that sentence in the current English Wikipedia article. https://en.wikipedia.org/w/index.php?title=Radiocarbon_calibration&oldid=1021352733

 

Best wishes

Andrew

--

Dr. Andrew Millard

Associate Professor of Archaeology, and

Designated Individual under the Human Tissue Act,

Durham University, UK

Email: A.R.M...@durham.ac.uk 

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Scottish Soldiers Project: https://www.dur.ac.uk/scottishsoldiers

Dunbar 1650 MOOC: https://www.futurelearn.com/courses/battle-of-dunbar-1650

 

 

From: ox...@googlegroups.com <ox...@googlegroups.com> On Behalf Of hjja...@arcor.de
Sent: 15 June 2021 15:49
To: OxCal <ox...@googlegroups.com>
Subject: Conficence level and sigma

 

[EXTERNAL EMAIL]

The en.wikipedia writes under "Calibration", the sentence "For example, 'cal 1220–1281 AD (1σ)' means a calibrated date for which the true date lies between 1220 AD and 1281 AD, with the confidence level given as 1σ, or one standard deviation."

This looks very convincing, interpreting the two dates as expressing ±1σ. In contrast, calibrating the age of Ötzi (the iceman) with given 4550 ±19 (= 1σ) BP, OxCal yields for all three confidence levels (68.3; 95.4; 99.7) naturally the same mean μ, however and surprisingly, also the same sigma =89, in spite of naturally different overall dates. For me, the OxCal output is logically contradicting the cited sentence.

Hans

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[Christopher Ramsey]

There are two methods of calibration in use. The intercept method which was originally used is classical and normally quoted in terms of 1σ or 2σ. This can be chosen in OxCal under [Options > Analysis > Intercept method]. The more often used method is probability based and takes the highest probability density range which for comparability are usually 68.3 or 95.4%; this method is ultimately Bayesian in its derivation though it is possible to derive it a number of different ways.

In either case you can look at the 3σ or 99.7% range but that is rarely used.

In both methods this particular date leads to 2 or 3 very distinct separate ranges because of the fluctuations in the calibration curve. The probability distribution is very far from being normally distributed.

For any probability distribution you can calculate a mean and variance (or standard deviation) - but this is not usually used as it can be very misleading. In this particular case the mean lies on a point which is unlikely to be the true age - though at least the 2σ range does include all likely dates. The mean and standard deviation are most often used when the dates are required for some other form of analysis which requires this simplified input but is never ideal.

Christopher

[hjja...@arcor.de]

 Dear Andrew,
1. thanks for the clarification and convincing explanations.
Find the cited sentence under "https://en.wikipedia.org/wiki/Radiocarbon_dating" - Calibration.
The cited sentence is obviously meant as a terminological example, however, obviously very misleading.
I would like to suggest that YOU correct it, with much higher competence than mine.

2. A related item: We know that the total cosmic flux is virtually irregularly modulated by solar activity and the Earth's magnetic field. This leads to unreliable means in the calibrations. Now we can approximate a more true mean by extending the sigma of the BP-input, what I exercised for the Ötzi example. The result is

---------------------

Example of Ötzi calibrations R_dat (4450,originally 19) with OxCal 4.4 in years

---------------
Thus, the OxCal BP-input covers the full range of the standard deviation with over ±3,5 sigma.
    Regarding the output, we see that all means (expectations) μ circle around 3251 calBC which should now approximate the intersection with the hypothetical strait line of the unmodulated cosmic flux and thus the most probable or "true" mean, allowing now not only a standard deviation, but also a narrower range in wiggle areas. (Alternative means of reconstructing that line would be by drawing a spline envelope over say, 1000 years of the calibration curve, and than find the middle by a Bezier line. Or, having the numerical table, we could do all this by an appropriate smoothing algorithm, what I successfully did with the Holcene icecore data). Find attached a simplified first model via a freely constructed cosmic flux example.
    It must be allowed to ask what would have happened without (or in areas with neglectable) modulation. This is shown in a simpler reconstruction (modulated/unmodulated, attached): Without the modulation, the 14C measurement would likely intersect the straight unmodulated flux line at a point perpendicular to the mode.

[MILLARD, ANDREW R.]

1. I see Joe Roe has already made an edit to that article to clarify it.
2. I can’t see the use of a hypothetical unmodulated calibration curve. It does not reflect past reality. We simply have to accept that calibrated radiocarbon dates have probability distributions that are not summarised well by a mean, and therefore we have to work with the full distribution.

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[hjja...@arcor.de]

Andrew,

I expected this answer. However, I have long enough worked with the (only seemingly) irregular curves of the S/A and the icecore temperatures (see my papers). And the art or trick is to individually find out the optimal smoothing between an unusuable "spaghetti graph" and a worthless straight line, which lies in choosing the optimal length or base. In the case of "Ötzi" (4550,19) the mean convincingly converges around 3252 ±70 a calBC, demonstrated in the numerical example I uploaded. It is purely philosophical to call a mathematical mean of any modulated curve "unrealistic". In this provisional form it surely only gives an approximation, but closer to the "true" or "unmodulated" age, with a clear standard deviation.

Hans