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ChronoModel User`s manual

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2. 00 Save project as Save As My project 1 a Tags 4 jl s zm um G Bureau j Q FAVORITES LI Dropbox A Applications LS Bur cau m Documents T l chargements DEVICES Disque distant SHARED TAGS M Hide extension New Folder Cancel Save Once done a dialog box appears and ask you to define the study period see Figure 3 2 This is actually the first thing to do otherwise no other action will be allowed by this software The study period is the period under which you assume that the unknown calendar dates of the events are likely to be These pieces of information To start your new project you first have to define a study period and click the a Apply button OK Figure 3 2 Dialog box to define the study period may be modified by filling the following boxes on the right hand side part of the win dow See Figure 3 3 The Apply button will stay red as long as the study period STUDY PERIOD Apply Figure 3 3 Study period definition window is undefined For this example let s use a study period from 0 to 2 000 Now the interface looks like Figure 3 4 the left hand side part represents the events scene and the right hand side part gives different types of information that will be further detailed in the following sections The tab gives information about csv events and associated datations The EM tab allows to import a CSV file containi
3. Figure 6 2 presents the marginal posterior densities of each date parameter the event and the calendar dates of the calibrated measurements In this example 9576 inter vals CI and HPD are represented We can see that all calendar dates seem to be contemporary Numerical values displayed in Figure 6 4 show that the MAP and the mean values were quite close as well as HPD et CI intervals The event is dated with at 1370 mean value associated with its 95 HPD interval 1417 1314 Figure 6 3 bouquet1 SacA15966 SacA18758 SacA15967 SacA18759 al SacAl5968 SacALB8760 No Phase Figure 6 1 Modelling of Bouquet 1 with ChronoModel shows the history plots or the trace of the Markov chains of each date parameter During the acquisition period all chains seem to have good mixing properties We may assume that all chains have reach their equilibrium before the acquisition period Figure 6 5 presents the autocorrelation functions of each date parameter We can see that all autocorrelation functions decrease exponentially and fall under the 9596 confidence interval after a lag of 30 for calibrated dates and after a lag of 50 for the event This autocorrelation between successive values may be reduced by increasing the thinning interval at 10 for example In order to keep 10 000 observations in the acquire period we ask for 100 000 but only 1 out of 10 values were kept for the analy sis The
4. 2200 2300 2400 2800 2900 3100 3200 3300 3400 Data SacA15971 0 018 o Ueeeseeeeepeeennrr 180 Data SacA18763 0 011 aidati T HA d m 3 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 AE D 2300 400 00 2 2 i 3600 3700 00 3900 Al 2800 2900 3100 3200 3300 3400 3500 Data SacA15972 0 0006 0 Rn MEE DE od RET Terre e ea pre Re Eph rure PITE e rn een pe TREE EE Pe IEEE QE IEEE THEE EU ETE EIE EE EE E EUN pa EFFET UTE OO D 1 200 3 460 3b0 70 abo 980 1000 1100 12i 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500 3600 3700 3800 3900 4000 Data SacA18764 0 00059 0 eee TO rer eeeeren rr ree ree repere reei ER rer perreereen pere een rer perereereepeeenns eeeeree ae paren eer perree reet pere rereni pee ree rer prier eere ere e EE prre E Er prae RT pP EN eU pU LEO perenne y vw 1 200 3 Al 5 70 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500 3600 3700 3800 3900 400 Figure 6 15 Marginal posterior densities of indivudal standard deviations related to the modellisa tion of Bouquet 2 Figure 6 16 Numerical values of indivudal standard deviations related to the modellisation of Bouquet 2 6 1 3 Modellisation of bouquets 1 and 2 simultaneously Now let s say we want to date Bouquet 1 and Bouquet 2 simultaneously See bou quets1
5. 015 0 0 1 1380 1370 1360 1350 1340 1330 1320 Mio Moo _1290 ai 1380 1370 Event bouquet1 Event bouquet1 0 034 0 033 EN 0 z A J peerevrteperet erpereereerepeerrtrred LULA 1380 1370 1300 1290 AL 1380 70 1300 _1390 al Event bouquet2 Event bouquet2 0 029 0 029 SEE LARA UN URL ES PAR LEURS 0 2 1380 1370 1360 0 lt bes AREER REE N DD 1380 Ayo 1360 B 1300 1390 al Event accession date Tutankamun 1 016 Figure 6 20 Marginal posterior densities related to the modellisation of Bouquets 1 and 2 with bounds fixed bounds on the left handside figure with uniform bounds on the right handside figure Modellisation of Bouquets 1 and 2 Without bounds With fixed bounds With uniform bounds Event Bouquet 1 Mean 1371 1338 1338 HPD region 1418 1315 1357 1316 1358 1316 Event Bouquet 2 Mean 1344 1336 1336 HPD region 1406 1278 1356 1313 1357 1313 Table 6 1 Numerical values related to the modellisations of Bouquets 1 and 2 6 1 4 Modellisation of the phase including bouquets 1 and 2 Estimation of the duration of the phase Let ssay that now we want to estimate the duration of the phase including both bouquets This phase might be seen as the duration of Sennefer s burial Three different modellings are possible and presented here in turn For all of them the study period
6. ChronoModel User s manual Version 2 Date June 23 2015 Copyright CNRS Universit de Nantes Director of project Philippe Lanos CNRS Co director Anne Philippe Univ Nantes Authors Affiliation Email Marie Anne Vibet Univ Nantes amp CHL marie anne vibet Quniv nantes fr Anne Philippe Univ Nantes anne philippe univ nantes fr Philippe Lanos CNRS philippe lanosQuniv rennes1 fr Philippe Dufresne CNRS philippe dufresneQu bordeaux montaigne fr Licence Creative Commons Attribution 4 0 International http creativecommons org licenses by nc nd 4 0 EY We HD To cite Chronomodel in publications use Ph Lanos A Philippe H Lanos Ph Dufresne 2015 Chronomodel Chronological Modelling of Archaeological Data using Bayesian Statistics Version 1 1 Available from http www chronomodel fr A BibTeX entry for LaTeX users is Manual chronomodel title Chronomodel Chronological Modelling of Archaeological Data using Bayesian Statistics author Ph Lanos and A Philippe and H Lanos and Ph Dufresne year 2015 http www chronomodel fr url Contents 1 Introduction 2 Bayesian modelling 2 1 Eventmodel 2 2 2 2 2 2 5 2 5 24 41 24 2 2 1 3 2 1 4 24 0 210 Observations Definition of an event Prior information about 0 and 94 04 Particular case of C measurements the wi
7. TT FT TTT T T T T T T T T T T I T T T T T T T T T I T T T T T n eS E LL LL LES M I PIE T T T I T T T T T T T T T I T T T T T T T T T I T T T T T 1700 1600 1500 1400 1300 1200 1100 1000 Event bouquet1 0 017 N 0 MEE Get VL ZEE ES EAS I T T T T T T T T T I T T T T T T T T T I T T T mr I T T T T T T T YT il T qp T T I T T T T T T T T T I T T T T T T T T T I T T T T T 1700 1600 1500 1400 1300 1200 1100 1000 Event bouquet2 0 018 E N D Pete er eee I M re I Y T HM p g Pere ler Cl Pate ete te ED EY rer rte 1700 1600 1500 1400 1300 1200 1100 1000 Chronomodel 0 4 Figure 6 27 Marginal posterior densities related to the modellisation of the phase including Bou quets 1 and 2 having a fixed duration The densities of the minimum and the maximum are drawn in red the density of Bouquet 1 is drawn in green the density of Bouquet 2 is drawn in purple In fact the prior information about the maximum duration of a phase gives less information than two constraining bounds 6 2 Toy scenario In this example we present a fictitious archaeological excavation with stratigraphy on several structures This toy scenario is used to give an idea different modellings that may be designed with ChronoModel Figure 6 28 shows the stratigraphy of the archaeological field For all modellings the study period is 2 000 to 2 000 Not documented line d Not documented line Protohistoric enclosure Figure
8. under the Results options section 43 5 1 1 Is the equilibrium reached Look at the history plots History plot tab Unfortunately there is no theoretical way to determine how long a burn in period needs to be The first thing to do is to observe the trace or history plots of the chain and to inspect it for signs of convergence Traces should have good mixing properties they should not show tendencies or constant stages Figure 5 1 displays an example of good mixing properties History plots of dates and variances should be checked Posterior distrib History plots Acceptation rate Autocorrelation Data SacA 15966 4e 03 l T t T L Y T t t L T r L Y T T r Y Y T 50000 60000 70000 Data SacA18758 4e 03 1 do 60800 doo Data SacA 15967 do 60000 m Data SacA18759 03 do 60000 doo Data SacA 15968 4e 03 L_ 50000 M doo Data SacA 18760 4e 03 Y T T L Y Y Y T Y T Y T Y T T T T d T Y 50000 60000 70000 Figure 5 1 Examples of history plots with good mixing propoerties Producing parallel Markov chains all with different starting values can help de ciding whether a chain has reach its equilibrium If the equilibrium is reached then the posterior distributions of each chain should be similar What should be done when equilibrium is not reached First MCMC settings should be changed by asking for a longer burn in period or a longer adapt period in that case a higher numb
9. Event Eglise T A Event Chapelle ji Event Sepulture 2 ji Event Sepulture 1 ji A Event Sepulture 3 T Phase Thermes gt Event Caldarium Event Tepidarium Event Forum A Phase Villae a Event Atelier SS a Event Vilae ZF Phase Proto E ra Event Enclos prot 42 Figure 6 32 Marginal posterior densities of all events and phases modelised by the second model 6 2 3 Grouping events into two kinds of phases Now keeping the same sequences we may add new phases corresponding to typo chronological criteria STUDY PERIODES Scies 2000 Sue 2000 w Occupation MED Eglise Sepulture 1 Chapelle Sepulture 2 Sepulture 3 ies Occupation GR Caldarium Tepidarium Forum Atelier Vilae Bi Necropole Sepulture 1 Sepulture 2 Sepulture 3 5 Thermes Caldarium Tepidarium Forum oo Endos protohistoire Figure 6 33 Design of the third modelling by ChronoModel Posterior distrib History plots Acceptation rate Autocorrelation 4 1 I l 100 1000 0 1000 mum ou Event Eglise 0 0065 01 10 os ave f 0 Event Chapelle 0 0066 Event Sepulture 0 016 01 ORE B s SELENA 0 0076 amp 5 o5 J dvec COPY UD Text SURS EI 0 0087 do omn a aM PEINE 0 0096 amp od Event Forum 0
10. 0068 Figure 6 34 Marginal posterior densities of the events Third modelling vents Posterior distrib History plots Acceptation rate Autocorrelation 4 p I Li I L Li I LI LI Li Li 100 1000 0 1000 Phase Occupation MED 0 012 Phase Occupation Phase Necropole Phase Vilae d od od cs Save Copy Copy Phase Thermes Phase Proto UNT 2ave save Copy Third mod Figure 6 35 Marginal posterior densities of the beginning and the end of the phases elling Bibliography 1 2 3 4 5 6 7 8 Christen JA Summarizing a set of radiocarbon determinations a robust ap proach Applied Statistics 1994 43 3 489 503 Lunn D Jackson C Spiegelhalter D The BUGS Book A Practical Introduction to Bayesian Analysis Chapman and Hall CRC 2012 Daniels MJ prior for the variance in hierarchical models Canad J Statist 1999 27 3 567 578 Spiegelhalter DJ Abrams KR Myles JP Bayesian Approaches to Clinical Trials and Health Care Evaluation Chichester Wiley 2004 Stuiver M Reimer P Bard E Beck J Burr G Hughen K et al INTCAL98 radiocarbon age calibration 24 000 0 cal BP Radiocarbon 2006 40 3 Avail able from https journals uair arizona edu index php radiocarbon article view 3781 Reimer P Baillie M Bard E Bayliss A Beck J Bertrand C et al Int Cal04
11. 3 4 2 2 Archeomagnetism dating AM 3 4 2 3 Luminescence dating TL OSL 3 4 2 4 Typo chronological reference Typo ref 3 4 2 5 Gaussian dating Gauss 3 4 2 6 Calibration process 3 4 2 7 Options 3 4 2 8 Deleting Restoring a measurement Creating new bound Deleting Restoring an event or a bound Creating Deleting a constraint Using the grid ws ss ew or hoy om or hy m o9 pese Using the overview Exporting the image of the events scene 3 9 Creating phases and constraints between phases 3 0 1 3 0 2 3 0 3 3 0 4 3 0 0 3 0 0 9 0 1 Creating anew phase Modifying Deleting a phase Including Removing events or bounds Creating Deleting a constraint between two phases Using the grid 64938 43Rc o4 9 9 R94 EE HE ES Using the overview Exporting the image of the phases scene 4 Numerical methods 4 1 Choice of the MCMC algorithm 4 1 1 4 1 2 4 1 3 4 1 4 Drawings from the conditional posterior distribution of the event Drawings from the full conditional posterior distribution of the calibrated date t Drawings from the conditio
12. 6 28 Field model lable 6 3 listes all the measurements and the corresponding structures Each structure will be considered as an event in ChronoModel and so there are only one radiocarbon specimen for each event Table 6 3 Radiocarbon datations 6 2 1 A sequential model without phases In this first modelling no phase is used Each measurement is included in one event and stratigraphic constraints between events are added to the modelling Figure 6 29 shows the first modelling of this toy scenario by ChronoModel Tepidarium Mo Phase Foum ri N Figure 6 29 Design of the sequential model by ChronoModel Markov chains were checked convergence was reached before the Acquire period au tocorrelation functions felt under the 95 confidence interval after a lag of 2 and acceptance rates were all about 4396 Figure 6 30 shows the marginal posterior den sities of the events of this sequential model Posterior distrib History plots Acceptation rate Autocorrelation I I 1 I 100 1000 0 1000 Event Eglise tH go om c Save Save Copy Copy Event Chapelle if A fi m Event e O9 OLA Event Sepulture 2 go om c Save Save Copy Copy Event Sepulture 3 T SSS d gt Event Atelier So amp cs ave Save Copy Copy Event Tepidarium 01 ORE E o Save Save C Co Copy Copy Event Vilae
13. CSV file and dragging information lines to the event node Using the tab Select the event in which you want to add a measurement An event is selected if it is bordered by a red line Then you may select measurement type and include information measurement by measurement csv Using the MMS tab you may also import measurements from a CSV file by clicking on Load CSV file However the CSV file has to be organised according to the type of measurements included more details are given in the followings Then you may drag a selected line to the corresponding event Be aware that by default ChronoModel reads a CSV file using a coma as cell separator and a dot as a decimal separator To change these options use the ChronoModel drop down menu and select Preferences Application Settings Auto save project E Auto save interval in minutes 5 CSV decimal separator Open last project at launch x Reset OK Cancel 3 4 2 1 Radiocarbon dating 14C 1 From tab Clicking on the radiocarbon extension window will be opened See Figure 3 6 Within this window you can insert the reference name of the mea surement the age value given by the laboratory Age and its associated stan dard error Error and you can choose a reference curve from the drop down menu If the reference curve you need is not included in that list you may add it in th
14. M L Fe p 2 7 where e N 0 1 u may represent for instance the true radiocarbon date or the true archeomagnetism date We assume that u follows a normal distribution with mean g t and variance 9 T where g is the function of calibration associated with the type of measurement of Mi pi gilti ot 6r 2 8 Hence pooling 2 7 and 2 8 together M gilts sie agite Gilti Sie 2 9 where c1 N 0 1 and S s2 B us So conditionally on t M follows a normal distribution with mean g t and variance E 2 1 5 3 Particular case of C measurements Calibration from multiple measurements Let s say we have K measurements M from a unique sample For example a sample may be sent to K laboratories that give radiocarbon datations All these measure ments refer to the same true radiocarbon date u In that case the Bayesian model first gathers all information about u before calibrating Hence Vk 1 K My pi t S ex 2 10 where et N 0 1 Vk 1 K and ef Et are independent Let s M M 1 s y wm and s LL Now as all Mj refer to the same u we have Sk NERA k 1 s K M yu ts aq 2 11 k 1 Hi gilti a 2 12 where 02 N 0 ty gi is the function of calibration Hence K M g t LS 4 p A 4 cLabCal Mult 2 13 k 1 where e C M N 0 57 and S 3 o ti So conditionally on t the calibrated measurement has a normal distr
15. Name My event 1 coo UJ OA Method AR proposal Double Exponential My radiocarbon date 1 Type 14C Method MH proposal distribution of calibrated date Age 1200 50 Ref curve intcal13 14c Wiggle 0 Manual Website Typo Ref Calibrate HPD 35 95 Export Image Figure 3 19 Calibration process of a radiocarbon datation 3 4 2 7 Options o The MEAE icon on the right hand side of the window gives the possibility to change the reference curve for all radiocarbon datations included in the current project in one click 3 4 2 8 Deleting Restoring a measurement lo delete measurement from an event first select the event in the event s scene by clicking on its name The event will be selected only if it is bordered by a red line Then using the tab you may select the measurement to be deleted X Delete and then you may click on the icon situated on the right hand side of the window to delete it from this event o deleted measurement may be restored in a selected event Click on the icon situated on the right hand side of the window and choose the measurement to be restored from the list of deleted measurements 3 4 3 Creating a new bound on the left hand side of the window new window will be opened asking you to name this new bound For the To create a new bound in the events scene select auis exampl
16. autocorrelation functions obtained decrease exponentially and fall under the 95 confidence interval after a lag of 6 for each parameter see Figure Figure 6 6 However with this new MCMC settings all other results are similar to those already given Now let s look at the individual standard deviations results The marginal poste rior densities of each individual standard deviations presented Figure 6 7 seem to be of similar behaviour with a mean about 50 and a standard deviation about 48 numerical values displayed in Figure 6 8 History plots of these individual standard deviations presented Figure 6 9 seem to have good mixing properties Hence the equilibrium is assumed to be reached Each acceptance rates presented Figure 6 10 are close to the optimal rate of 43 And finally each autocorrelation function dis played in Figure 6 10 shows an exponential decrease and all values fall under the 95 interval of signification after a lag of 10 In conclusion the modelling of Bouquet 1 seems consistent All individual stan Event bouquet 0 017 Data SacA15966 1540 Data SacA18758 Zn rere T T T T z T 1540 1340 1330 Data SacA15967 1540 Data SacA18759 0 1450 1540 5807 1520 io Data SacA15968 0 014 0 epnerseern n EZER GERE rgroverrrrrgyrrr 1450 1540 1530 1320 iio 400 Iso 1480 1470 Data SacA 18760 0 01
17. b g p umm Figure 6 30 Marginal posterior densities obtained from the sequential model using ChronoModel 6 2 2 Grouping events into phases Another equivalent way to build the chronology is to introduce phases In our exam ple we can see 4 sequences nested in 4 phases Each phase corresponds to a group structures Figure 6 31 displays the design of this second model including phases For this modelling constraints between events were only kept within phases Constraints between events of different phases were replaced by constraints between phases How ever both modellings lead to the same results This second modelling is also a way to simplify the design of the model limiting the number of constraints between events Figure 6 32 shows the marginal posterior densities computed with the second mod elling These denstiities are similar to those obtained from the first modelling How ever now information about phases the density if the beginning and the end of each phase may also be seen STUDY PERIOIM Aba re SEE 2000 Seg 2000 Properties RL Eglise Sepulture 1 Chapelle Sepulture 2 Sepulture 3 L Thermes Caldarium Tepidarium Forum a Proto Encbs protohistorique Figure 6 31 Design of the second modelling by ChronoModel introduction of phases TEE Events Posterior distrib History plots Acceptation rate Autocorrelation Phase MED i
18. is chosen to start at 2000 and end at 2000 using a step of 1 year All other parameters are those used for the modelling of Bouquet 1 and of Bouquet 2 6 1 4 1 Phase without constraints The modelling of the phase including both bouquets is displayed in Figure 6 21 In this modelling no further constraints are included The phase s duration is kept unknown bouquet1 bouquet2 SacA15969 Fou fA SacA15970 1t SacA18762 1 A f TON xm ns y cA15 FEAISISS SacA15971 t A 7 NII SacA18763 i I SacA15972 L SacA18764 bouquet2 bouquetl Figure 6 21 Modelling of the phase including Bouquets 1 and 2 including a phase Figure 6 22 displays the marginal posterior densities of both events and those of the beginning and the end of the phase Statistical results regarding Bouquet 1 and Bouquet 2 are unchanged by the introduction of the phase the results are similar to those presented in the last section when no bounds were introduced See Table 6 1 In addition this modelling allows to estimate the mean duration of the phase 35 as well as its credibility interval 0 101 loo loo Figure 6 22 Marginal posterior densities related to the modellisation of Bouquets 1 and 2 including a phase and without bounds The densities of the minimum and the maximum are drawn in red the density of Bouquet 1 is drawn in green the density of Bouquet 2 is drawn in purple 6 1 4 2 Phase with bounds Now le
19. of a fixed value or known within a range of values In order to include prior information about a hiatus double click on the constraint the black arrow to be modified and then fill the dialog box shown in Figure 3 32 Using the drop down menu the hiatus may be fixed or known within a range of values This dialog box should also be used in order to delete the constraint Hiatus min Unknown Delete constraint OK Cancel Figure 3 32 Dialog box to modify a hiatus between two phases 3 5 5 Using the grid H By default the phases scene is white A grid may be added using the icon situated on the right hand side of the window 3 0 6 Using the overview e The phases scene may also be seen from an overview using the ESS icon situated on the right hand side of the window 3 5 7 Exporting the image of the phases scene The image of the phases scene may be saved by clicking on the S icon placed on the right hand side of the window You may save it either in PNG format or in Scalable Vector Graphics SVG format In both cases you will need to name the image and to choose the directory where to save the image see Figure 3 25 If you choose the PNG format you will be ask for the image size factor and the number of dots per inch Chapter 4 Numerical methods In general the posterior distribution does not have an analytical form Elaborated algorithms are then required to approximate this posterior di
20. phase T 6b a The posterior distribution of all these elements may be approximated by MCMC methods See section 4 for more details and statistical results such as the mean the standard deviation and so on may be estimated 2 2 3 Prior information about the duration of a phase The duration of a phase 7 may be known by prior information This kind of infor mation may be included in ChronoModel by two different ways e Fixed duration T Tfised e Duration known through a range of values 7 Tgin TMaxl Consequently all the events of the phase 0 j 1 r have to verify the constraint of duration according to the following equation miax 8 3 mes min 8 i4 x lt T 2 2 4 Prior information about a succession of phases Succession or stratigraphic constraints between two phases may be included These constraint of succession called hiatus may be of a known duration These constraints act on the all events included in both phases imposing a temporal order between the two groups of events 2 2 4 1 Simple stratigraphic constraints or unknown hiatus Let s say that phase P containing rp events is constrained to happen before phase P 1 containing rp 4 events Then in ChronoModel this means that all events of phase P are constrained to happen before the events included in phase P 4 1 The following equations should be verified by all the events included in both phases Vi 1 rp Vj e Uber p l
21. posterior densities may be seen for each chain using the MCMC options on the right hand side of the window By default only the posterior distribution of events or bounds are shown To see the fa Untold posterior distribution of all calendar dates use the icon on the right hand side of the window For parameter event dates individual standard deviations parameters of phases the marginal posterior density is associated with its HPD region and its credibility interval By default HPD regions and credibility intervals are given at 95 This might be changed using the posterior distribution options on the right hand side of the window Credibility intervals might be hidden If a phase is modelled then the posterior density of the beginning dashed line and the end plain line of the phase are displayed on the same graphic The density of the duration of the phase may also be seen by clicking on Show duration on the left hand side of the window in the part corresponding to the associated phase 5 2 2 Statistical results E ChronoModel gives also a list of statistical results Click on the ZZ icon on the right hand side of the window to see those results For each parameter the following results are given e Maximum a posteriori MAP the highest mode of the posterior density e Mean the mean of the posterior density function e Std deviation the standard deviation of the chain e QI the numerical va
22. settings and to keep values from the Markov chains with a higher interval between them 5 1 3 Look at the acceptance rates Acceptation rates tab The Metropolis Hastings algorithm generates a candidate value from a proposal den sity This candidate value is accepted with a probability An interesting point is the acceptance rate of this proposal density The theoretical optimal rate for the adapta tive Gaussian random walk is 4396 17 In ChronoModel this algorithm is used at least for each individual variance and for TL OSL measurements This method might also be used for other parameter Figure 5 3 displays an example of acceptance rates that are close to 4376 within the interval of 40 and 46 Acceptance rates of each adaptative Gaussian random walk should be checked Be careful if all batches are used in the adapt period this might tell that all adapta tive Gaussian random walks did not reach the optimal interval To see that use the Log icon on the top of the window From there in the MCMC part the number of batches used is stated Figure 5 4 displays an example where only 18 batches out of Event bouquet1 I 10 20 30 Data 5acA15966 l l 10 20 30 Data 5acA18758 I 10 20 30 Data 5ac 15967 i 1 20 30 Data 5ac 18759 I 1 20 30 Data 5ac 15968 10 20 30 Data 5acA18760 I d j 1 20 30 Figure 5 2 Examples of autocorrelation fun
23. terrestrial radiocarbon age calibration 0 26 cal kyr BP Radiocarbon 2004 46 3 Available from https journals uair arizona edu index php radiocarbon article view 4167 Hughen K Baillie M Bard E Beck J Bertrand C Blackwell P et al Ma rine04 marine radiocarbon age calibration 0 26 cal kyr BP Radiocarbon 2004 46 3 Available from https journals uair arizona edu index php radiocarbon article view 4168 McCormac FG Hogg A Blackwell P Buck C Higham T Reimer P SHCal04 Southern Hemisphere calibration 0 11 0 cal kyr BP Radiocarbon 2004 46 3 Available from https journals uair arizona edu index php radiocarbon article view 4169 87 9 10 11 12 13 14 15 16 17 18 Reimer P Baillie M Bard E Bayliss A Beck J Blackwell P et al IntCal09 and Marine09 Radiocarbon Age Calibration Curves 0 50 000 Years cal BP Ra diocarbon 2009 51 4 Available from https journals uair arizona edu index php radiocarbon article view 3569 Reimer P Bard E Bayliss A Beck J Blackwell P Ramsey CB et al IntCal13 and Marinel3 Radiocarbon Age Calibration Curves 0 50 000 Years cal BP Ra diocarbon 2013 55 4 Available from https journals uair arizona edu index php radiocarbon article view 16947 Hogg A Hua Q Blackwell P Niu M Buck C Guilderson T et al SHCal13 Southern Hemisphere Calibration 0 50 000 Years cal BP Radiocarbon 2013 55 4 Available from https j
24. the calibrated inclina tion and declination values before they are statistically combined to produce a single age estimate Curves implemented in ChronoModel are listed in Table 2 2 http www brad ac uk archaeomagnetism archaeomagnetic dating Name Reference Gal20028ph2014 D Herv et al 12 13 Gal20028ph2014 I Herv et al 12 13 Gws2003uni F Herv et al 12 13 Table 2 2 Calibration curves implemented in chronoModel e Paleodose datation The calibration curve is g t t e Gaussian datation For Gaussian measurements the calibration curve may is a quadratic polyno mial g t 2 a t b t 4 c By default the calibration curve is g t t 2 1 5 2 Calibration from one measurement If the information about t come from only one measurement that needs to be cali brated then the following DAG applies Hi 2 Si Figure 2 3 DAG representation of an individual calibration Directed edges represent stochastic relationships between two variables blue circles represent model parameters pink rectangulars nodes represent stochastic observed data pink triangles represent observed and deterministic data M is the observation data the measurement made by the laboratory and s is its variance error In ChronoModel M is assumed to follow a normal distribution with mean u latent variable and with variance sz the laboratory error This may be expressed by the following equation
25. 00 105 1 18 05 Data SacA15968 1e 03 pu qq er qq tamra 0 10000 20000 30800 40000 50800 60000 70000 80000 90000 1E 05 1 18405 Data SacA18760 le 03 Chen ar a L L T T 5 r 10000 20000 30800 40000 50800 60000 70000 80600 90000 1E 05 1 18405 Figure 6 3 History plots of date parameters related to the modellisation of Bouquet 1 Figure 6 4 Numerical values of each date parameter related to the modellisation of Bouquet 1 Event bouquet1 Data SacA15966 Data SacA18758 Data SacA15967 Data SacA18759 Data SacA15968 Data SacA18760 Figure 6 5 Autocorrelation functions related to the modellisation of Bouquet 1 Thinning interval 1 Event bouquet1 Data SacA15966 Data SacA18758 Data SacA15967 Data SacA18759 Data SacA15968 Data SacA18760 Figure 6 6 Autocorrelation functions related to the modellisation of Bouquet 1 Thinning interval 10 Figure 6 7 Marginal posterior densities of indivudal standard deviations related to the modellisation of Bouquet 1 dard deviations take values close to 50 compared to 1400 for the event That is to say standard deviations are rather small compared to the event s posterior mean Hence according to ChronoModel all datations seem to be contemporary Now we can dr
26. 00 maximum batches in the Adapt period and 100 000 iterations in the Acquire period using thinning intervals of 10 The marginal posterior densities presented in Figure 6 13 are of two sorts Althought the first 6 datations seem to be contemporary the two last ones seem to be some kind of outliers Indeed their density function takes values about 1600 whereas the other densities take values between 1500 and 1200 All history plots have good mixing properties and autocorrelation functions are correct results not shown Looking at individual standard deviations the marginal posterior densities displayed Figure 6 15 show three distinct standard deviations The first 5 samples are associated with a standard deviation density function that takes small values with mean values about 50 The next sample s individual standard deviation has a mean posterior density at 100 And the 2 last samples are associated with individual standard deviations with a mean higher than 2 000 See Figure 6 16 for numerical values Hence these two last datations give a piece of information that has a reduced importance in the construc tion of the posterior density function of the event Bouquet 2 All individual standard deviations have a history plot with good properties an acceptance rate about 43 or higher and a correct autocorrelation function results not shown As a conclusion the first 6 samples seem to be contemporary but the two last ones seem to be some
27. 2 chr The study period was chosen to start at 2000 and end at 2000 using a step of 1 year All other parameters are those used for the modellisation of Bouquet 1 and of Bouquet 2 Three different modellisations are compared here In the first modellisation no fur ther constraints are included See Figure 6 17 In the two next modellisations two bounds are introduced to constrain the beginning and the end of the burial of Sen nefer See Figure 6 18 Indeed the burial of Sennefer is assumed to have happened between the accession date of Tutankamun and the accession date of Horemheb See 18 These accesssion dates are considered as bounds in ChronoModel There are two different ways to introduce a bound A bound may be fixed Accession date of Tutankamun 1356 and Accession date of Horemheb 1312 or a bound may have a uniform distribution Accession date of Tutankamun uniform on 1360 1352 Ac cession date of Horemheb uniform on 1316 1308 Figure 6 19 displays the marginal posterior densities of both bouquets when the model lisation does not include bounds Figure 6 20 displays the marginal posterior densities of both bouquets when the modellisation includes bounds using fixed bounds figure on the left hand side and using uniform bounds figure on the right hand side From these results we can see that the introduction of bounds helps restrain the posterior densities and the HPD interval of both events However using fi
28. 5 Now when the event is selected in the event s scene that is when the event is circled by a red line the event s properties may be seen on the right hand side of the window in the Qo Properties tab From there the name the color and the MCMC method might be changed See Section 4 for more details on MCMC methods eoo Chronomodel 1 1 My project 1 chr New Open Save Undo MCMC Run Log L2 a Li Q Manual Website Search event name You have selected an element You can now STUDY PERIOD Apply Calib Resol e csv q Edit its properties from the right panel Create a constraint by holding the Option key down and clicking on Start o o S O End 2000 Properties Dans AE another element Merge it with another element by holding the Shift key down and aa My wane 1 z dragging the selected element onto another one Method AR proposal Double Exponential estor Qn Save Image My event 1 Overview H Grid No Phase Typo Ref Calibrate Figure 3 5 Chronomodel window showing the new event in the events scene 3 4 2 Including measurements An event may be associated with data information such as measurements or typo chronological references There are two ways to insert data with ChronoModel either by clicking on a measurement icon from the tab and filling it directly or by importing a
29. 6 Figure 6 2 Marginal posterior densities related to the modellisation of Bouquet 1 The dark lines correspond to distribution of calibrated dates the green lines correspond to posterior density functions Highest posterior density HPD intervals are represented by the green shadow area under the green lines Credibility intervals are represented by thick lines drawn above the green lines Event bouquet 2e 02 1 1E 05 118405 TU c TO SEND TN A Lo CET ts nm les CR a cn ea me Data SacA15966 1 2e 03 oo 1 0 10000 Data SacA18758 1 2e 03 20000 30000 ELE RE 40000 Gp p pc a 50800 SRE 60000 poet we ee 7000 TA eee NE 80000 9000 16405 1 1405 7 1 5e 03 f T 0 10000 20000 Data SacA15967 1 2e 03 nr nn Tn nnn pn n qn rn Pd RS nr pr RS RS rq nr ane A r3 eee ES 2 OE ELE BER n meee mmm o gt ES D NER ERR RR m xm m um 1 5e 03 f T 0 10000 20000 Data SacA18759 1 1e 03 K 1 Q H P A E J K H M ee nr rn nr prn qr nnd rer 30000 i 40800 i 50000 60000 i 70000 80000 90800 16405 1 16405 1 4e 03 M EEEN BASO A ASA AE BEEE M AEE M ee M 10000 20000 30000 40800 sodoo 60800 70600 80800 900
30. Max duration 4 Unknown Fixed Figure 3 28 Dialog box to include prior information about the maximum duration of the phase 1 3 5 2 Modifying Deleting a phase lo modify a phase double click on its name and the the dialog box presenetd in Figure 3 27 will be reopened To delete phase select the phase to be deleted by clicking on its name and then click on the d icon There is no option for restoring a phase either than using the und icon situated on the left hand side of the window or creating a new phase 3 5 3 Including Removing events or bounds Events may be included in a phase You may select one or several events or bounds from the events scene and then click in the white square on the left of the phase s box The Figure 3 29 presents the inclusion of a bound in a phase As a result the color of the phase appears on the bottom of the event s or the bound s box and the name and the color of the event or the bound appears at the bottom of the phase s box Several events or bounds may be included in a same phase An event or a bound may also belong to several phases Figure 3 50 illustrates these cases We may see that My event 1 No Phase My phase 1 My bound 1 Figure 3 29 Including a bound in a phase My phase 1 contains two elements My event 1 and My bound 1 My bound 1 belongs to both phases as both colors appear at the bottom of its box and a
31. W Advanced MH proposal prior distribution MH proposal distribution of calibrated date Method EVA MH proposal adapt Gaussian random walk Figure 3 13 Insert a paleodose measurement in luminescence with advanced options csv 2 From the ME tab TL CLER 203 1280 TL OSL TL CLER 2 1170 140 1990 TL OSL TL CLER 2 987 120 1990 Figure 3 14 Organisation of the CSV file containing luminescence measurements 3 4 2 4 Typo chronological reference Typo ref 1 Clicking on will open the Typo reference extension window See Fig ure 3 15 You have to enter the Name of this reference and the Lower date and the Upper date corresponding to this The Lower date and the Upper date must to be different Here the likelihood is computing according to this rule A Create Modify Data Chronomodel 9 x Name New Date gt Lower date 0 Upper date 100 OK M Cancel Figure 3 15 Insert a Typo reference L t le Upper csv 2 From the BR tab Name Lower date Upper date comment TYPO REF My date from a Typo ref 250 456 Figure 3 16 Organisation of the CSV file containing typo chronological references 3 4 2 5 Gaussian dating Gauss Qo Properties 1 From Clicking on will open the Gauss extension window See Figure 3 17 You need to give a name a Measurement and its error You can change the calib
32. aw conclusions about the calendar date of the event The event Bouquet 1 may be dated at 1370 mean value with a 95 interval of 1417 1314 HPD interval Figure 6 8 Numerical values of indivudal standard deviations related to the modellisation of Bou quet 1 Event bouquet 2e 02 ES L Pr qq T T Q 10000 20000 30000 40800 50000 60800 70600 80800 90000 1E405 118405 Data SacA15966 1 2e 03 4 1 5e 03 a 7 A a 7 RG aaa Aaa T Aaa T 10000 20000 30000 40000 50000 60000 70000 80000 90000 1E 05 1 1E 05 Data SacA18758 13e 03 4 1 5e 03 T9599 n rr q nr S SA S PSS SDS I rn rn rn rq rq 0 10000 20000 30000 40000 50000 60800 70000 80000 9000 1 os 118405 Data SacA15967 1 2e 03 1 5e 03 r r am T v r 1 r T v LS AS ARS RS a A A r r r M 7 iy 10000 i 20000 30000 Es 40800 50000 60000 70600 80800 90600 d 1E405 1 18405 Data SacA18759 1 1e 03 1 4e 03 10800 20000 30000 40800 sodo0 60000 70600 80800 90000 16105 1 18 05 Data SacA15968 1e 03 4 l hs qq qq MN T L r T st a rr a raaraa meaa aaas a aiaa rr T 10000 20000 30000 40800 50000 60000 70000 80800 90600 16105 1 18 05 Data SacA18760 le 03 1 4e 03 te r rt tt rr T t e a tr maaa T rt a
33. cross appears in the white square of each phase s box My phase 1 6 My event 1 My bound 1 d My phase 2 C My bound 1 Figure 3 30 Including several events in a phase and an event in different phases To remove an element from a phase first select the element from the events scene then click on the cross associated in the phase from which this element should be remove Once the element removed from a phase its name appears no more at the bottom of the phase s box and the phase s color does not appear any longer in the elements box 3 5 4 Creating Deleting a constraint between two phases constraint may be put between two phases in the same way as between two elements from the events scene First select the oldest phase Press and keep pressing the Alt key from your keyboard Then move the arrow up to the youngest phase and click on its name Now you may release the Alt key and a black arrow should be seen between both phases heading from the oldest phase to the youngest one a My phase 1 My event 1 My phase 2 My bound 1 Figure 3 31 Example of the phases scene showing a constraint between two phases In ChronoModel a hiatus expresses the minimum time elapsed between two phases By default the hiatus between these phases is unknown However if prior informa tion about the hiatus are available it might be included in the model by two different ways a hiatus might be
34. ctions that fall quickly enough under the 95 confidence interval Data SacA15966 1e 02 Data SacA18758 1e 02 Data SacA15967 le 02 Data SacA18759 le 02 Data SacA15968 le 02 Data SacA18760 Figure 5 3 Examples of acceptation rates close to 43 100 were used That means that all adaptative Gaussian random walks reached the optimal interval within these 13 batches e HO VU v 9 b L New Open Save Unda teda Model MCMC Run Results m Model Results MCMC Chains Log Chain 1 Seed 466 Adapt OK at batch 13 100 All Seeds to be used in MCMC Settings dialog Figure 5 4 Reading the number of batches used What should be done if the acceptance rates are not close to 4396 From the MCMC settings ask for a longer number of iterations per batch 5 2 Interpretation If all Markov chains have reached their equilibrium and are not autocorrelated then the statistical results may be interpreted These results are estimated using the values of the Markov chains drawn in the Acquire period Results are estimated using all values from all chains if several chains were requested chains concatenation 5 2 1 Marginal posterior densities Posterior densities are in fact marginal densities These densities should be interpreted parameter by parameter By default results are given from the concatenation of all chains but
35. date Calib Resol thod AR proposal Double Exponential lt lt New Date gt Type 14C Method MH proposal distribution of calibrated date Age 1200 50 Ref curve intcal13 14c Wiggle 0 My event 1 Figure 3 8 Measurement and properties 2 From the The CSV file has to be organised as shown in Figure 3 9 dac SacA15966 14C SacA18758 intcal09 14c intcal09 14c gaussian 3000 14C 14C SacA15967 intcal09 14c fixed 52 SacA18759 intcal09 14C 14C SacA15968 intcal09 14C 14C SacA18760 intcal09 14C Figure 3 9 Organisation of the CSV file containing radiocarbon measurements 3 4 2 2 Archeomagnetism dating AM 1 From Clicking on the itii button will open the AM extension window See Figure 3 10 Within this window you can insert the reference name of the measurement you can choose the magnetic parameter and can insert the value and the associated error alpha 95 A Create Modify Data Chronomodel Figure 3 10 Insert AM measurement e Inclination enter the Inclination value the alpha 95 value and choose the calibration curve With this parameter the extension is calculating the error with the formula 95 7i 9 448 e Declination enter both Declination and Inclination values and alpha 95 and choose the reference curve to calibrate The assoc
36. der to see what a radiocarbon determination means in terms of a true age we need to know how the atmospheric concentra tion has changed with time This is why calibration curves are needed Calibration curves implemented in ChronoModel are listed in Table 2 1 e Archeomagnetism datation The process of calibration translates the measured magnetic vector into calen dar years record of how the Earth s magnetic field has changed over time is required to do this and is referred to as a calibration curve A date is obtained by comparing the mean magnetic vector defined by the declination and incli Name Reference Uwsy98 Stuiver et al 1998 5 IntCal04 Reimer et al 2004 6 Marine04 Hughen et al 2004 7 ShCal04 McCormac et al 2004 8 IntCal09 Reimer et al 2009 9 Marine09 Reimer et al 2009 9 IntCall3 Reimer et al 2013 10 Marinel3 Reimer et al 2013 10 ShCall3 McCormac et al 2013 11 Table 2 1 Calibration curves implemented in chronoModel nation values with the secular variation curve the potential age of the sampled feature corresponds to the areas where the magnetic vector overlaps with the calibration curve Unfortunately the Earth s magnetic poles have reoccupied the same position on more than one occasion and can result in multiple age ranges being produced calibrated date is obtained using the separate inclination and declination cali bration curves Probability distributions are produced for
37. described above generate a Markov chain for each parameter that is a sample of values derived from the full conditional distribution These chains described the posterior distribution of each parameter Hence the values sampled give information about the posterior densities In ChronoModel three period are distinguished 4 2 1 Burn These Markov chains start from initial values randomly selected for each parameter and need a sufficient time to reach their equilibrium The burn in period is used to forget these initial values 4 2 2 Adapt In ChronoModel adaptative Gaussian random walks are used at least for each indi vidual variances The adapt period is the period needed to calibrate all variances of adaptative Gaussian random walks Variances are estimated using all values drawn during the first batch and the accep tance rate is estimated l he adaptation period goes on with another batch using the variance estimated on this last batch unless the acceptance rate estimated is included between 4096 and 4696 or the maximum number of batches is reached 4 2 3 Acquire In this period all Markov chains are assumed to have reached their equilibrium dis tribution Of course this has to be checked and the next section provides useful tools that can help controlling if the equilibrium is actually reached If so Markov chains may be sampled and information about conditional posterior distributions may be extracted Sampling fro
38. e let s call it My bound 1 New Bound Name My bound 1 Color ERR Cancel OK After validation the bound appears in the events scene See Figure 3 20 Now when the bound is selected in the event s scene the bound s properties may be seen on the Qo right hand side of the window in the MEME tab From there its name and its color might be changed and values may be added A bound may either have a fixed value or have a uniform distribution within a range of values These options may be changed from that window as well eoo Chronomodel 1 1 test chr ere B Model MCMC Run Help Manual Website STUDY PERIOD Apply re csv 99 t dL PN Name My bound 1 oo e Fixed Value 0 y Uniform Start 0 My bound 1 My event 1 Figure 3 20 Creation of a new bound in the events scene 3 4 4 Deleting Restoring an event or a bound An event or a bound may be deleted from the events scene first by selecting the event X and then by clicking on the icon placed on the left hand side of the events scene An event is deleted with all its measurements Any element deleted may be restored by clicking on the MEAS icon on the left hand side of the events scene Then the element to be restored may be picked from the new window presenting all deleted elements bounds or events See Figure 3 21 An event containing measurements is r
39. e folder Applications Chronomodel app Contents Resources Calib 14C for MAC users A Chronomodel 0 0 9 My project 1 chr Name lt NewDatem 14C Measurements Age 0 Error 50 Reference curve intcall3 14c Li Folder Applications Chronomodel app Contents Resources Calib 14C P Advanced OK Cancel Figure 3 6 Inserting a radiocarbon measurement Now you may need to include a wiggle matching or you may want to change the MCMC method used for that datation To do that click on the Advanced menu at the bottom of the same window See Figure 3 7 The default method for datation from radiocarbon measurement is the Metropo los Hastings algorithm using the distribution of the calibrated date as a proposal Two other proposal may also be selected the prior distribution or a Gaussian adaptative random walk See Section 4 for more details on MCMC methods The wiggle matching may be fixed or included in a range or even have a Gaus sian distribution defined by its mean value and its standard error By default the wiggle matching is set to 0 Once the measurement is validated its details appear in the Properties tab and might be changed by double clicking on it in the list See Figure 3 8 A Chronomodel 0 0 9 My project 1 chr Name My radiocarbon date 1 14C Measurements Age 12700 teference curve intcall3 l4c MH proposal distribution of calibrated
40. er of iterations per batch should be asked for Then if the equilibrium is still not reached changing the algorithm used to draw from full conditional posterior distributions might be of help 5 1 2 Correlation between successive values Look at the au tocorrelation functions Autocorrelation tab A Markov chain is a sequence of random variables 6 9 for which for any t the distribution of 0 given all previous 6 s depend only on the recent value 4470 14 15 Hence a high correlation between two consecutive values is expected How ever correlation will biased the estimations done from those values That is why thinning the Markov chains is an a good thing to do before drawing estimations By default the thinning interval is 10 that is only one value of the Markov chains out of 10 is kept This thinning interval should be long enough to reduce the correlation between successive values If not it should be increased lo check whether the chain is correlated observe the autocorrelation plot Only the first correlations should be high at lag 0 the remaining correlations should be neg ligible Autocorrelations should have an exponential decrease The autocorrelations of events dates and variances should be checked Figure 5 2 displays several autocor relation functions having a good behaviour What should be done if correlation is high A good thing to do is to increase the Thinning interval from the MCMC
41. estored with all its measurements 3 4 5 Creating Deleting a constraint From the events scene stratigraphic or succession constraint may be added between any two elements events and bounds To create such a constraint first select the A Chronomodel 0 0 1 My project 1 chr Select the item to be restored L Figure 3 21 Window presenting all events or bounds that may be restored first element that is usually the oldest element of the succession This selected element should now be bordered by a red line Now click and keep clicking on the Alt key from your keyboard A black arrow should now be seen in the events scene Now why your mouse selected the youngest element of the succession To validate the constraint you need to click on the name of the second element before releasing the Alt key The arrow should go from the oldest element to the youngest one My event 1 My bound 1 No Phase Figure 3 22 Events scene showing a constraint between a bound and an event lo delete a constraint move the mouse over the corresponding black arrow This arrow should become red and a cross should appear in the middle of the arrow You may now click on the cross to begin the deleting process confirmation box should appear to validate the deleting action My event 1 No Fhase My bound 1 No Phase zs Figure 3 23 Events scene showing a cons
42. eters Now we need to define prior information about 0 and o1 0 this is done in the next section Then the wiggle matching case specific to radiocarbon datations is explained According to the likelihood three main types of data information may be implemented into ChronoModel a single measurement with its laboratory error a combination of multiple measurements or an interval referring to a typo chronological reference These different types are explained in section 2 1 5 2 1 3 Prior information about 0 and ci 0 Without any other constraint that the beginning Tm and the end Tm of the study period Tm and Ty are be fixed parameters the unknown calendar date 0 is assumed to have a uniform distribution on the study period p 8 jm ru 9 2 2 TM Tm The variances o7 for i 1 to n are assumed to have a shrinkage uniform distribution See 3 2 50 p aj s2 ji o2 2 3 where ME Tr bp e 2 4 s on 8 eal with the variance of the posterior distribution of t obtained by the individual calibration See 4 2 1 4 Particular case of C measurements the wiggle matching Case This case is specific to radiocarbon datations Let s say that we have m radiocarbon datations referring to the unknown calendar date 0 shifted by a quantity called 6 Then the stochastic relationship between t and 0 is given by the following equation where e amp M N 0 1 for i 1 t
43. ggle matching case POU 5x24 92x X49 wo HS Ure b o SE 2 1 5 1 Calibration curves 2 1 5 2 Calibration from one measurement 2 1 5 3 Particular case of C measurements Calibration from multiple measurements 2 1 5 4 Particularity of archeomagnetism measurements com 0 an a a a a e 2 1 5 5 Typo chronological information Stratigraphic constraints 2 2 Event model including phases 2254 22 4 2 45 2 2 4 Definition of a phase ls Beginning end and duration of aphase Prior information about the duration of a phase Prior information about a succession of phases 2 2 4 1 Simple stratigraphic constraints or unknown hiatus 2 2 4 2 Prior information about the duration of the hiatus 3 Use of ChronoModel 3 1 Installation ue ox yo 3 2 Creating a new project Opening a project 3 3 Description of the icons of the main window ill ID c Ct Ct CO U 10 10 10 11 11 11 EL 12 12 12 13 13 14 LT 3 4 Creating events bounds and constraints 3 4 1 3 4 2 3 4 3 3 4 4 3 4 5 3 4 6 3 4 7 3 4 8 Creating a new event Including measurements 3 4 3 1 Radiocarbon dating 14C
44. hip between t and 6 where N 0 1 for 1 to n and e CM are independent 0 is the unknown parameter of interest and o1 0 are the unknown standard deviation pa rameters Such a model means that each parameter t can be affected by errors o that can come from different sources See Lanos amp Philippe and see ChronoModel is based on a bayesian hierarchical model Such a model can easily be represented by a directed acyclic graph DAG See 2 A DAG is formed by nodes and edges node can either represent an observation data or a parameter that can be stochastic or deterministic An edge is a directed arc that represents depen dencies between two nodes The edge starts at the parent node and heads to the child node This relationship is often a stochastic one single arc but it may also be a deterministic one double arc The DAG can be read as follow each node of the DAG is conditionally on all its parent nodes independent of all other nodes except of its child nodes The following DAG is a representation of the event model Conditionally on 0 and on o that are the parameters of interest t is independent of all other parameters Figure 2 1 DAG representation of the event model Directed edges represent stochastic relation ships between two variables blue circles represent model unknown parameters Rectangular plates are used to show repeated conditionally independent param
45. iated error on the declination is calculated with alpha 95 and the inclination with the formula pc O_o cos Inclination 2 448 e Intensity When you choose intensity the label of alpha 95 box change o 1 to Error Then insert the Intensity value and its Error And finally you choose the corresponding reference curve In this case directly S error Usually the directional calibration curves are in degrees so nclination Declina tion and alpha 95 must be in degrees two By clicking the Advanced arrow you find a roll box offering the type of MCMC sampler See section 4 1 2 for more details The default type of sampler is MH Proposal distribution of calibrated date A Create Modify Data Chronomodel Figure 3 11 Insert AM measurement csv 2 From the W tab AM Bo TEMP EO GAL2002sph2014_D ref AM Inc inclination 69 2 0 0 1 2 GAL2002sph2014 l ref Figure 3 12 Organisation of the CSV file containing archeomagnetism measurements 3 4 2 3 Luminescence dating TL OSL o Properties 1 From Clicking on will open the TL OSL extension window See Figure 3 13 You need to give a name an Age and its error as well as a reference year Now if you wish to change the default MCMC method used you can unfold the Advanced section of the window and use the drop down menu Name Eten Date TL Measurements Age 1000 Error 30 NENNEN Ref year 2015 _
46. ibution with mean g t and variance S Figure 2 4 represents the corresponding DAG Hi Figure 2 4 DAG representation of a calibration from multiple measurements Arrows represent stochastic relationships between two variables blue circles represent model parameters pink rectan gulars represent stochastic observed data pink triangles represent deterministic observed data 2 1 5 4 Particularity of archeomagnetism measurements combine 2 1 5 5 Typo chronological information Let s say that a typo chronological information is a period defined by two calendar dates tim and t y with the constraint tim lt ti y The distribution of tim tim conditional on t is given by the following equation Pltim t m t Jg a ee ee ee 2 14 where A is a positive constant Figure 2 5 represents the corresponding DAG ti lim ti M Figure 2 5 DAG representation of a typo chronological information Arrows represent stochas tic relationships between two variables blue circles represent model parameters pink rectangulars represent stochastic observed data 2 1 6 Stratigraphic constraints Several events may be in stratigraphic constraints Let s say that three events are assumed to happen successively in time then their true calendar date is assumed to verify the following relationship 0i lt 0 lt 05 In ChronoModel constraints links events and not calibrated dates bounds may also be introduced in order t
47. is drawn in green the density of Bouquet 2 is drawn in purple 6 1 4 3 Phase with fixed duration The phase of Sennefer s burial is assumed to have happened between the accession date of Tutankamun 1356 and the accession date of Horemheb 1312 Hence the duration of the phase is smaller than 44 years In this last modellisation no bounds are included but the maximum duration of the phase is fixed at 44 years the delay between the accession date of Tutankamun and the accession date of Horemheb bouquet1 SacA15966 Ji SacA18758 bouquet2 SacA15969 SacA15967 a h SacA18761 SacA18759 P u SacA15970 A I DX bouquet2 SacA15968 A SacA18762 SacA18760 LA A SacA15971 k y P bouquet1 duration lt 44 Le m Figure 6 26 Modelling of the phase including Bouquets 1 and 2 having a fixed duration This modellisation leads to the following results Duration of the phase Mean 20 Credibility interval 0 41 Event Bouquet 1 Bouquet 2 Mean 1364 1357 HPD region 1408 1317 1403 1306 Table 6 2 Numerical values related to the phase including Bouquets 1 and 2 and having a fixed duration In that case although the duration of the phase is reduced compared to the mod elling without bounds however the estimation of the calendar dates of bouquet 1 and bouquet 2 are less precise than those estimated with the modelling with bounds Phase 0 02 3 X 4 x 0 244 I i
48. ithms are implemented the rejection sampling 39 method also called acceptance rejection and the Metropolis Hastings algo rithm Both algorithms required proposal density function that should be easily sampled from in order to generate new candidate values For the rejection sampling algorithm it is common to use if possible the prior function or the likelihood as a proposal function For the Metropolis Hastings algorithm a common choice is to use a symmetric density such as the Gaussian density Depending on the type of the parameter the event the mean of a calibrated measure the variance of a calibrated measure or a bound different methods are proposed at each step of the Gibbs sampler These methods are described here in turn 4 1 1 Drawings from the conditional posterior distribution of the event 0 Three different methods can be chosen e Rejection sampling with a Gaussian proposal 16 e Rejection sampling with a Double exponential proposal 16 e Metropolis Hastings algorithm with an adaptative Gaussian random walk 1 The first two methods are exact methods We recommend to use one of these two methods except when the event is involved in stratigraphic constraints In that case the last method should rather be used 4 1 2 Drawings from the full conditional posterior distribution of the calibrated date t In this case three different methods can be chosen e Metropolis Hastings algorithm using the poste
49. kind of outliers Then the event Bouquet 2 may be dated at 1344 mean value with a 95 HPD interval 1405 1278 The example shows that the modelling is robust to outliers Indeed even if two outliers were included in the analysis the datation of the event was not affected by them ChronoModel do not need any particular manipulation of outliers before analysing the datations Indeed there is no need to withdraw any datation or to use a special treatment to them Figure 6 13 Marginal posterior densities related to the modellisation of Bouquet 2 Figure 6 14 Numerical values related to the modellisation of Bouquet 2 100 Data SacA15969 0 017 a IET pue peque UTERE m E ES EEUU EMITE TEETH Data SacA18761 0 014 ee ae ec tet M a T hater RET n bl T t nae li d Obs titii alo lo alb NECI RNC Es t 77 v ian Data SacA15970 0 017 arie pure rrrepesieeenen rere ree pere rere aperi en ren pere TEE 1 00 1400 1500 1600 eee eee eee ee iiaii iiki ane eee eee eee Eee rene iiai iaidd hiiit nee eee enr rer eu iiiki iaiia iiaia iiaiai BRIE bie S iiiki iaiki ade alan hiii 1700 800 900 00 2 2 00 00 3900 4 2000 2100 2200 2300 2400 2800 2900 3100 3200 3300 3400 3500 3600 3 Data SacA18762 0 018 EF ek iiic i Y RENE HART MUT TNR STU ORC NC EC HAE ECT CC HSE RT aa Ho Me MET OEC CONO CR RET ds ORC OR 1300 1400 15 700 1900 1000 2 600 2700 3 joo 3700 00 3900 Al
50. lue separating the lower 2596 of the data from the higher 75 e Q2 the numerical value separating the lower 50 of the data from the higher 5096 e Q3 the numerical value separating the lower 75 of the data from the higher 25 e Crediblity interval CI or Bayesian confidence interval the smallest credible interval e Highest probability density region HPD the region with the highest proba bility density CI and HPD regions may be of a chosen percentage By default the 95 inter Posterior distrib vals are estimated To change that go on the Posterior Distribution tab and change the percent from the options on the right hand side of the window Graph thichness 8 MCMC Chains Chains concatenation Chain 1 Results options e Calendar dates Distrib of calib dates L Wiggle shifted Q Individual std deviations Post distrib options Show credibility HPD Credibility X FFT length 1024 Bandwidth factor 1n Chapter 6 Examples ol 6 1 Radiocarbon datation in Sennefer s tomb Egypt For this illustrative example we used data published in the article of Anita Quiles 18 regarding Sennefer s tomb Several bouquets of flowers were found in Sennefer s tomb at Deir el Medineh As they were found at the entrance of the tomb they were assumed to date the same archeological event one of the three phases of the burial of sennefer The objective of this st
51. m these Markov chains need to be carefully made Indeed successive value of a Markov chain are not independent In order to limit the correlation of the sample we can choose to thin the sample by only keeping equally spaced values Chapter 5 hesults and Interpretations After having gone through the running process the results tab appears res Now before any interpretations the Markov chains have to be checked in order to known whether they have reach their equilibrium before the acquire period 5 1 Checking the Markov chains When Markov chains are generated two points have to be verified the convergence of the chains and the absence of correlation between successive values If the Markov chain has not reach its equilibrium values extracted from the chains will give inap propriate estimates of the posterior distribution If high correlation remains between successive values of the chain then variance of the posterior distribution will be bi ased Here are some tools to detect whether a chain has reach its equilibrium and whether successive values are correlated We also give indications about what can be done in these unfortunate situations By default results shown correspond to the date density functions To see results corresponding to individual standard deviations o select Individual variances Unfold Stats Zoom X Zoom Y Rendering Standard fas Update display MCMC Chains
52. nal posterior distribution of the vari 2 D 9 X do o CNE HN d c 4 d 4 ance of a calibrated measure Drawings from the full conditional posterior distribution of a bound ze coo wee EE OER Wo as ROW 4 2 MCMC settings 18 18 20 20 23 29 26 26 27 29 29 29 30 30 32 32 32 34 34 39 39 37 38 38 38 39 99 AQ AQ Al 4 2 1 4 2 2 4 2 3 BP x 33 92x EX T 5axxs ed qe 3 PUB ul x9 D Acquire on ik Sow a hoo wm sed nb DE mo X EO on 5 Results and Interpretations 5 1 Checking the Markov chains 5 1 1 Is the equilibrium reached Look at the history plots History BUS e ee REED EKER ED EDs 5 1 2 Correlation between successive values Look at the autocorre lation functions Autocorrelation tab 5 1 3 Look at the acceptance rates Acceptation rates tab 5 2 nterpretation 224264646 a8 REE REE 4 ROS e 9 4 Ro 5 2 1 Marginal posterior densities 5 2 2 Statistical results 2 2 0 0 ee 6 Examples 6 1 Radiocarbon datation in Sennefer s tomb Egypt DI BOW cae web ae Rho eee ee eae Od ew ee x s DLA POSE 2c ae ee ee ee eee be See AE ri 6 1 3 Modellisation of bouquets 1 and 2 simultaneously 6 1 4 Modellisation of the phase including bouquets 1 and 2 Esti 6 2 mation of the duration of the phase 6 1 4 1 Phase without constrai
53. ng informations about datations and to export all datations used in the mod elisation of the current project The MEL m8 tab shows the phases scene e Chronomodel 1 1 My project 1 chr mEHOO were as New Open Save Undo Redo 1 MCMC Run Results Log Help Manual Website t Search event name Define a study period on the right panel apply it and start creating your STUDY PERIOD Apply e csv model by clicking on New Event Properties Data Color Method AR proposal Double Exponential Typo Ref Calibrate Figure 3 4 ChronoModel interface when starting a new project To open a project already saved use n the left hand side of the window and choose your project ChronoModel projects are saved with the extension CHR 3 3 Description of the icons of the main window All the design of ChronoModel is oriented to the idea Information are in the middle and tools are around This section gives a description of all icons contained in the top of the window On the left hand side of the main window three icons help managing projects and two icons help managing modelling actions New To create a new project Open To open an existing model ChronoModel projects are saved using the extension file CHR m H Save To save the current project U Undo To undo the last actions Redo To redo the last actions On the middle of the wind
54. nts 6 1 4 2 Phase with bounds 6 1 4 3 Phase with fixed duration Toy SCenatlo a ss ed HE SS 6 2 1 0 2 2 6 2 3 A sequential model without phases Grouping events into phases Grouping events into two kinds of phases A 42 42 43 43 44 45 45 48 48 49 ol 02 02 62 67 12 2 74 T7 19 Sl 82 89 Chapter 1 Introduction Chronological modelling with ChronoModel e Events ChronoModel is based on the concept of Event An Event is a point in time for which we can define a hierarchical Bayesian statistical model It is estimated by laboratory dating 14C TL OSL AM or by reference dating i e typo chronology e Phases A Phase is a group of Events It is defined on the basis of archaeological geological environmental criteria we want to locate in time Unlike Event model the Phase does not respond to a statistical model indeed we do not know how Events can be a priori distributed in a phase However we may question the beginning end or duration of a phase from the Events that are observed there query A level of a priori information can be added the Events from one phase may be constrained by a more or less known duration and a hiatus between two phases can be inserted this imposes a temporal order between the two groups of Events e time order relations Events and
55. o constrain one or several events Let s say that the three events are assumed to happen after a special event with true cal endar date 0P Then the following relationship holds 0P lt 04 lt 05 lt 05 2 2 Event model including phases 2 2 1 Definition of a phase phase is a group of Events defined on the basis of objective criteria such as archae ological geological or environmental criteria We may want to locate a phase in time Unlike the event model a phase does not respond to a statistical model indeed we do not know how events can a priori be distributed in a phase Moreover a phase can only reflect the information given by the events included So if a real phase is not entirely covered by archaeological data included is the analysis the phase as implemented in ChronoModel can only illustrate the period covered by this data However we may have information about the beginning the end or the duration of a phase or even a hiatus between two successive phases 2 2 2 Beginning end and duration of a phase As said before a phase is estimated according to the events included in it The following information are given for each phase The beginning of a phase o reflects the minimum of the r events included in the phase Q min j 1 r The end of a phase 6 reflects the maximum of the r events included in the phase p Max 4 The duration 7 is the time between the beginning and the end of a
56. o n and ef VM are independent may either be a deterministic or a stochastic parameter Then t 0 follows a 2 normal distribution with mean 0 and variance o If 0 is stochastic then its prior distribution function is a uniform distribution on Id3 dy 1 6j du RR NECS lias d 0 2 6 Figure 2 2 DAG representation of the event model Directed edges represent stochastic relation ships between two variables blue circles represent model unknown parameters Rectangular plates are used to show repeated conditionally independent parameters In that case the associated DAG is presented in Figure 2 2 2 1 5 Likelihood As said before different types of measurement may be included in ChronoModel in order to estimate unknown calendar dates t These different types are the following ones a C age in radiocarbon a paleodose measurement in luminescence an in clination a declination or an intensity of the geomagnetic field in archeaomanetism a typo chronological reference or a Gaussian measurement Except for the typo chronological reference all other measurement may be associated with a calibration curve Hence for the typo chronological reference only the last section may be ap plied 2 1 5 1 Calibration curves e Radiocarbon datations Radiocarbon measurements are always reported in terms of years before present BP that is before 1950 In or
57. or Phases can check order relations These order relations are de fined in different ways by the stratigraphic relationship physical relationship observed in the field or by criteria of stylistic technical architectural etc de velopment which may be a priori known These constraints act between facts constraint of succession between phases is equivalent to putting order con straints between groups of Events Chapter 2 Bayesian modelling 2 1 Event model 2 1 1 Observations The measurement may represent a 14C age in radiocarbon a paleodose measurement in luminescence e an inclination a declination or an intensity of the geomagnetic field in archeao Or magnetism Lil e a typo chronological reference for instance an interval of ceramic dates BALS e Gaussian measurement with known variance If needed these measurements M may be converted by ChronoModel into calendar dates using appropriate calibration curves See section 2 1 5 1 2 1 2 Definition of an event Let s say that an event is determined by its unknown calendar date 0 Assuming that this event can reliably be associated with one or several suitable samples out of which measurements can be made the event model implemented in ChronoModel combines contemporary dates t with individual errors 904 0 in order to estimate the unknown calendar date 6 The following equation shows the stochastic relations
58. ournals uair arizona edu index php radiocarbon article view 16783 Herv G Chauvin A Lanos P Geomagnetic field variations in Western Europe from 1500BC to 200AD Part I Directional secular variation curve Physics of the Earth and Planetary Interiors 2013 May 218 1 13 Available from http www Sciencedirect com science article pii 80031920113000265 Herv G Chauvin A Lanos P Geomagnetic field variations in Western Europe from 1500BC to 200A D Part II New intensity secular variation curve Physics of the Earth and Planetary Interiors 2013 May 218 51 65 Available from http www sciencedirect com science article pii S0031920113000277 Buck CE Litton CD Cavanagh W G The Bayesian Approach to Interpreting Archaeological Data England Chichester J Wiley and Son 1996 Robert CP Casella G Monte Carlo Statistical Methods 2nd ed Springer 2004 Robert CP Simulation of truncated normal variables Statistics and Computing 1995 5 2 121 125 Available from http dx doi org 10 1007 BF00143942 Roberts GO Rosenthal JS Optimal Scaling for Various Metropolis Hastings Algorithms Statistical Science 2001 16 4 351 367 Available from http dx doi org 10 2307 3182776 Quiles A Aubourg E Berthier B Delque Koli E Pierrat Bonnefois G Dee MW et al Bayesian modelling of an absolute chronology for Egypt s 18th Dynasty by astrophysical and radiocarbon methods Journal of Archaeological Science 2013 40 423 432
59. ow the icons refer to the different steps of the modelling with ChronoModel Model Tab used to design the model MCMC Open the MCMC settings window Run Run the bayesian modelling L Results Tab showing all results of the modelling Log Tab presenting summaries of the MCMC methods used for this modelling Then on the right of the window Help Context sensitive tips seen in yellow bubbles By defaults Help is active Por Manual ownload the user manual Website Open the ChronoModel website 3 4 Creating events bounds and constraints 3 4 1 Creating a new event G To create a new event select Mi on the left hand side of the events scene A window will be opened asking you to name that new event For the example let s call it My event 1 A Chronomodel 0 0 9 My project 1 chr New Event L Log eui Ue er model Color Cancel OK default color is given to this new event You may wish to change the color Click on the color chosen by default and select a new color in the Colors window a oo Colors Semel T Q SS Palette Apple ur A Chronomodel 0 0 9 My project 1 chr Magenta EN Orange Mew Event L B Purple gm Search Define Name mmm c SC RN E UT model Color Cancel cance Es After validation the event appears in the events scene See Figure 3
60. phases scene are separated in order to keep the design of the model comprehensible Indeed an event may belong to several phases However the phases scene may be placed side by side with the events scene in order to have a complete look at the design qj The phases scene may be seen from the MI tab placed on the right hand side of the window See Figure 3 26 eoo A Chronomodel 1 1 1 test chr w e HO UL 0 D L amp n Rec s MCMC Run Result c Help Manual Website STUDY PERIOD Apply My event 1 My bound 1 a S 5 tio o Phase Figure 3 26 View of the events scene and the phases scene 3 5 1 Creating a new phase To create a phase click on the MASS icon on the left hand side of the window This action will open a new dialog box presented in Figure 3 27 This dialog box asks for the name and the color given to the phase and its maximum duration By default the duration of the phase is unknown If prior information about the maximum duration are available it may be included in the model by two Max duration Unknown al OK Cancel Figure 3 27 Dialog box to create a new phase different ways The maximum duration may be fixed or known within a range of values To include this kind of information use the drop down menu next to Maz duration and choose the way adapted to your information See Figure 3 28 Phase name My phase 1
61. r f 10000 20000 30000 40800 50000 60d00 70600 80800 90000 16105 1 18405 Figure 6 9 History plots of indivudal standard deviations related to the modellisation of Bouquet 1 Data SacA15966 le 02 Data SacA18758 le 02 Data SacA15967 le 02 Data SacA18759 le 02 Data SacA15968 1e 02 Data SacA18760 1e 02 Figure 6 10 Acceptation rates of indivudal standard deviations related to the modellisation of Bouquet 1 Figure 6 11 Autocorrelation functions of indivudal standard deviations related to the modellisation of Bouquet 1 6 1 2 Bouquet 2 Now let s say we want to estimate the calendar date of bouquet 2 8 samples were extracted from Bouquet 2 and radiocarbon dated The modelling of this bouquet by ChronoModel is represented by Figure 6 12 bouquet SacA15969 SacA18 761 SacA15970 SacA18 62 5acA15971 5acA18763 5acA15972 SacA18764 No Phase Figure 6 12 Modelling of Bouquet 2 with ChronoModel Each radiocarbon measurement is calibrated using IntCal09 curve No reservoir offset is taken into account The study period is chosen to start at 2000 and end at 2000 using a step of 1 year This study period is chosen so that every distribution of the calibrated date is included in this study period The default MCMC methods are used We use 1 000 iterations in the Burn in period 1 000 iterations in each of the 1
62. ration curve Now if you wish to change the default MCMC method used you can unfold the Advanced section of the window and use the drop down menu csv 2 From the MILIA tab For Gaussian measurements a b c refer to the calibration curve Indeed the MH proposal prior distribution MH proposal distribution of calibrated date MH proposal adapt Gaussian random walk Figure 3 17 Insert a Gaussian measurement and its variance with advanced options equation of the calibration curve is the following one g t axt bxt c wes om a Gusan en mm 00 500 0 Figure 3 18 Organisation of the CSV file containing Gaussian measurements 3 4 2 6 Calibration process To see the distribution of the calibrated date select the measurement and click on the icon on the right hand side of the window See Figure 3 19 To come back on the scene panel you click a second time on it Now the dating is specified in the events scene and the distribution of the calibrated date may be seen in the event as can be seen in Figure 3 19 m HOU New Open Save Undo Redo New Date 14C Calibration process Distribution of calibrated date 0 0061 MAP 793 Mean 822 Std deviation 69 Q1 777 34 Q2 Median 823 08 Q3 868 64 HPD 95 1 688 902 86 7 920 963 8 3 Chronomodel 1 1 My project 1 chr Model MCMC Run Results Log Help STUDY PERIOD Apply 60 start s Mo T
63. rior distribution of cal ibrated dates P M t This method is adapted for calibrated measures namely radiocarbon measure ments or archeomagnetism measurements but not for typo chronological refer ences and when densities are multimodal e Metropolis Hastings algorithm using the parameter prior distribution P tia 0 This method is recommended when no calibration is needed namely for TL OSL gaussian measurements or typo chronological references e Metropolis Hastings algorithm using an adaptative Gaussian random walk This method is recommended when no calibration is needed or when there are stratigraphic constraints This method is adapted when the density to be ap proximated is unimodal 4 1 3 Drawings from the conditional posterior distribution of the variance of a calibrated measure 0 In this case only one method is implemented in ChronoModel the uniform skrink age as explained in Daniels 3 The full conditional density is unimodal hence the Metropolis Hastings algorithm can be implemented here The proposal density in volved is an adaptative Gaussian random walk 17 The variance of this proposal density is adapted during the process 4 1 4 Drawings from the full conditional posterior distribution of a bound If a bound is fixed no sampling is needed If a bound has a uniform distribution the full conditional distribution is also a uniform distribution 4 2 MCMC settings The Gibbs sampler
64. stribution Markov chain Monte Carlo MCMC is a general method based on drawing values of 0 from approximate distributions and then on correcting those draws to better approxi mate the target posterior distribution p 0 y The sampling is done sequentially with the distribution of the sampled draws depending on the past value drawn Indeed a Markov chain is a sequence of random variables 9 02 for which for any t the distribution of 9 given all previous s depend only on the recent value gt U 14 15 4 1 Choice of the MCMC algorithm convenient algorithm useful in many multidimensional problems is the Gibbs sam pler or conditional sampling 14 15 Let s say we want to approximate p 61 05 04 y The algorithm starts with a sample co 2M T 6 randomly selected The first step of the algorithm is to update the first value by sampling a candidate value of g knowing 9 9 from the full conditional distribution pO o 8 The next step is to find a candidate value gto knowing 99 9 eC using the full conditional distribution pos j 9 9 And so on Then the d step is to find a candidate value for 1 o Ri RP T of initial values 0 gU knowing g gto This process is then iteratively repeated d 1 d 1 Starting values In ChronoModel the initial value of each Markov chain is ran domly selected More details are given in the appendix In ChronoModel two main algor
65. t pus 1 or BED y s lt min 05 Eire Or min 072 max 8 41 1 gt 0 2 2 4 2 Prior information about the duration of the hiatus Prior information about the duration of the hiatus may be included in ChronoModel Hiatus may be fixed or known within a range of values e Fixed hiatus y Yfized e Hiatus known within a range of values y lin YMaxl Consequently all the events of phase P and phase P 4 1 have to verify the following constraint O n max 07 gt Y e min un Chapter 3 Use of ChronoModel 3 1 Installation From the website www chronomodel fr choose to download the software adapted to your computer e For MAC Click on the MAC Download button Then double click on the package to install the software Following the wizard window the software will be placed in the Applications directory Once done the logo of ChronoModel should be seen in he Applications directory e For Windows Click on the Windows Download button Then double click on the EXE to install the software 13 3 2 Creating a new project Opening a project After having launched ChronoModel the following interface appears See Figure 3 1 800 A Chronomodel Ready Figure 3 1 ChronoModel interface In order to create a new project click on v on the left hand side of the window This action will open a new window asking you to name this new project and to save it in a chosen directory
66. t s include prior information about the accession dates of Tutankamun and Horemheb We include the two fixed bounds as detailled in section 6 1 3 Here two modellings are possible using ChronoModel Figure 6 23 represents the first modelling in which bounds constrain events Figure 6 24 represents the second modelling in which bounds are included in separated phases using one phase for each bound and stratigraphic constraints are placed between phases However these two modellings give similar results The marginal posterior densities of all parameters are presented in Figure 6 25 The mean duration of the bouquets phase is 12 years associated with a credibility interval of 0 33 that is smaller than the one estimated without bounds bouquet1 A 2 LA L i bouquet2 i in A L M I EN bouquet2 bouquet1 Figure 6 23 First modelling 1 of Bouquets 1 and 2 including a phase and bounds bouquet1 SacA15966 bouquet2 A SacA18758 5acA15969 SacA15967 SacA18761 h SacA18759 SacA15970 LA SacA15968 SacA18762 A bouquet2 bouquet1 SacA18760 SacA15971 j SacA18763 SacA15972 SacA18764 Figure 6 24 Second modelling of Bouquets 1 and 2 including a phase and bounds Figure 6 25 Marginal posterior densities related to the modellisation of Bouquets 1 and 2 including a phase and bounds The densities of the minimum and the maximum are drawn in red the density of Bouquet 1
67. traint ready to be deleted PTS Do you really want to delete this 3 constraint m Figure 3 24 Confirmation box to delete a constraint 3 4 6 Using the grid By default the events scene is white A grid may be added using the icon situated on the left hand side of the window 3 4 7 Using the overview The overview may be useful when many elements are created in the event s scene To e watch the events scene from an overview use the icon MEME placed on the left hand side of the window 3 4 8 Exporting the image of the events scene LR Save Image Icon and save it either in PNG format or in Scalable Vector Graphics SVG format In both cases you will need to name the image and to choose the directory where to save the You may also export the image of the events scene by clicking on the image see Figure 3 25 If you choose the PNG format you will be ask for the image size factor and the number of dots per inch Save model image as savea oo A Tags ICE sp mi amp E Bureau Q FAVORITES x E Dropbox h Applications m Documents T l chargements DEVICES Disque distant Image png M Hide extension New Folder Figure 3 25 Dialog box used to name the image image and to choose the directory where to save the 3 9 Creating phases and constraints between phases In ChronoModel the events scene and the
68. udy is to date Sennefer s burial using ChronoModel Samples were extracted from different short lived plants leaves twigs etc on each bouquet in order to ensure the consistency of the dates All samples were radiocarbon dated All the datasets and chr files are available on the website of chronomodel at this address http chronomodel fr Quiles bouquets zip 6 1 1 Bouquet 1 Let s say we want to estimate the calendar date of the cut of bouquet 1 6 samples were extracted from this bouquet and radiocarbon dated Bouquet1 CSV contains all the datations related to Bouquet 1 The modelling of the event the cut of bouquet 1 by ChronoModel is represented by Figure 6 1 Each radiocarbon measurement is calibrated using IntCal09 curve No reservoir offset is taken into account The study period is chosen to start at 2 000 and end at 0 using a step of 1 year For the MCMC method the method used to generate new values at each step of the Gibbs sampler are the following ones these are the default settings for the event Bouquet 1 rejection sampling using a double exponential proposal for the distributions of the datations Metropolis Hastings algorithm using the posterior distribution of calibrated dates We start with 1 000 iterations in the Burn in period 1 000 iterations in each of the 100 maximum batches in the Adapt period and 10 000 iterations in the Acquire period using thinning intervals of 1 Only one chain is produced
69. xed bounds or bounds having a uniform distribution with a small period 8 years lead to similar results Numerical values are presented in Table 6 1 bouquet2 SacA15969 Douquet1 SacA15966 SacAl18761 SacA18758 Sac415970 SacA15967 Sac 18762 SacAl18759 SacA15971 SacAl596E 5acA18763 SacA18760 5acA15972 5acA18764 Figure 6 17 Modelling of Bouquets 1 and 2 without bounds bouquet 1 SacA15966 SacA18758 SacA15967 SacA18759 SacA15968 SacA18760 bouquet2 SacA15969 SacA18 761 5acA15970 SacA18762 5acA15971 5acA18763 5acA15972 SacA187654 Figure 6 18 Modelling of Bouquets 1 and 2 including bounds Event bouquet1 0 017 Tr ETEETETETITTUEOTYITTTTTY 00 1loo i 1460 1450 1440 1430 1420 1410 1400 1390 1380 1370 1360 1350 1340 1330 1320 1310 1300 1290 1280 1270 1260 1250 1240 1230 1220 1210 1200 1190 11 amp Event bouquet2 JJ M 0 013 0 pec it ee verry 1460 1450 1440 1430 1420 1410 1400 1390 1380 1370 1360 1350 1340 1330 1320 1310 1300 1290 1280 1270 1260 1250 1240 1230 1220 1210 1200 1190 ile Figure 6 19 Marginal posterior densities related to the modellisation of Bouquets 1 and 2 without bounds Event accession date Horemheb Event accession date Horemheb 14

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