itimprove MCMC description - cosmo - front and backend for Markov-Chain Monte Carlo inversion of cosmogenic nuclide concentrations Err mx1.adamsgaard.dk 70 hgit clone git://src.adamsgaard.dk/cosmo URL:git://src.adamsgaard.dk/cosmo mx1.adamsgaard.dk 70 1Log /src/log.gph mx1.adamsgaard.dk 70 1Files /src/files.gph mx1.adamsgaard.dk 70 1Refs /src/refs.gph mx1.adamsgaard.dk 70 1README /src/file/README.md.gph mx1.adamsgaard.dk 70 1LICENSE /src/file/LICENSE.gph mx1.adamsgaard.dk 70 i--- Err mx1.adamsgaard.dk 70 1commit 34d863dbc2986f2eff0ac8ff390333c91e8fa532 /src/commit/34d863dbc2986f2eff0ac8ff390333c91e8fa532.gph mx1.adamsgaard.dk 70 1parent 6bbc2f1eca168add56b0057cd8692abdd08448bb /src/commit/6bbc2f1eca168add56b0057cd8692abdd08448bb.gph mx1.adamsgaard.dk 70 hAuthor: Anders Damsgaard URL:mailto:anders.damsgaard@geo.au.dk mx1.adamsgaard.dk 70 iDate: Fri, 27 Nov 2015 16:07:08 +0100 Err mx1.adamsgaard.dk 70 i Err mx1.adamsgaard.dk 70 iimprove MCMC description Err mx1.adamsgaard.dk 70 i Err mx1.adamsgaard.dk 70 iDiffstat: Err mx1.adamsgaard.dk 70 i M pages/methods.html | 53 ++++++++++++++++++++++++++----- Err mx1.adamsgaard.dk 70 i Err mx1.adamsgaard.dk 70 i1 file changed, 45 insertions(+), 8 deletions(-) Err mx1.adamsgaard.dk 70 i--- Err mx1.adamsgaard.dk 70 1diff --git a/pages/methods.html b/pages/methods.html /src/file/pages/methods.html.gph mx1.adamsgaard.dk 70 it@@ -65,7 +65,28 @@ Err mx1.adamsgaard.dk 70 i (δ18Othreshold) is specified Err mx1.adamsgaard.dk 70 i with uniform probability across the linear Err mx1.adamsgaard.dk 70 i parameter interval. The user specifies the bounds Err mx1.adamsgaard.dk 70 i- of the model parameters, which define the model space. Err mx1.adamsgaard.dk 70 i+ of the model parameters. Err mx1.adamsgaard.dk 70 i+

Err mx1.adamsgaard.dk 70 i+ Err mx1.adamsgaard.dk 70 i+

Given a single value of model parameters Err mx1.adamsgaard.dk 70 i+ (εint, εgla, Err mx1.adamsgaard.dk 70 i+ tdegla, Err mx1.adamsgaard.dk 70 i+ δ18Othreshold), the TCN Err mx1.adamsgaard.dk 70 i+ concentration after the duration of e.g. the entire Err mx1.adamsgaard.dk 70 i+ Quaternary period in a sample can be computed. This Err mx1.adamsgaard.dk 70 i+ forward model describes a history of exhumation and Err mx1.adamsgaard.dk 70 i+ TCN production in a sample volume as it experiences the Err mx1.adamsgaard.dk 70 i+ variable physical environment of the Pleistocene. Err mx1.adamsgaard.dk 70 i+

Err mx1.adamsgaard.dk 70 i+ Err mx1.adamsgaard.dk 70 i+

When model parameters Err mx1.adamsgaard.dk 70 i+ (εint, εgla, Err mx1.adamsgaard.dk 70 i+ tdegla, Err mx1.adamsgaard.dk 70 i+ δ18Othreshold) are allowed to Err mx1.adamsgaard.dk 70 i+ vary within specified limits, they can be thought of as Err mx1.adamsgaard.dk 70 i+ orthogonal axes creating a coordinate system in higher-order Err mx1.adamsgaard.dk 70 i+ space. Every position in this model space is associated with Err mx1.adamsgaard.dk 70 i+ a certain set of model parameter values. Err mx1.adamsgaard.dk 70 i

Err mx1.adamsgaard.dk 70 i Err mx1.adamsgaard.dk 70 i Err mx1.adamsgaard.dk 70 it@@ -73,13 +94,29 @@ Err mx1.adamsgaard.dk 70 i

Err mx1.adamsgaard.dk 70 i What is a MCMC walker?

Err mx1.adamsgaard.dk 70 i

Err mx1.adamsgaard.dk 70 i- forward responses are computed based on an initial set of Err mx1.adamsgaard.dk 70 i- model parameters that is proposed using the Err mx1.adamsgaard.dk 70 i- Metropolis-Hastings technique. A burn-in phase of 1000 Err mx1.adamsgaard.dk 70 i- iterations is first used to make a crude initial search of Err mx1.adamsgaard.dk 70 i- the model space. This step is followed by a more detailed Err mx1.adamsgaard.dk 70 i- and local search of the model space based on the best-fit Err mx1.adamsgaard.dk 70 i- model parameters from the burn-in phase. Err mx1.adamsgaard.dk 70 i+ A MCMC walker is a numerical entity which sequentially Err mx1.adamsgaard.dk 70 i+ explores the model parameter space in order to obtain the Err mx1.adamsgaard.dk 70 i+ best result between a forward-model and an observational Err mx1.adamsgaard.dk 70 i+ dataset. During each iteration Err mx1.adamsgaard.dk 70 i+ the walker takes its current position in model space, plugs Err mx1.adamsgaard.dk 70 i+ the parameter value into the forward-model, and Err mx1.adamsgaard.dk 70 i+ evaluates if the output result matches the observational Err mx1.adamsgaard.dk 70 i+ record better or worse than the output at its previous Err mx1.adamsgaard.dk 70 i+ position in model space. If the new results better matches Err mx1.adamsgaard.dk 70 i+ the observed dataset, it continues walking along the same Err mx1.adamsgaard.dk 70 i+ path in model space with a small random perturbation. Err mx1.adamsgaard.dk 70 i+

Err mx1.adamsgaard.dk 70 i+ Err mx1.adamsgaard.dk 70 i+

Err mx1.adamsgaard.dk 70 i+ Starting at a random place inside the model space, a burn-in Err mx1.adamsgaard.dk 70 i+ phase of 1000 iterations is first used to make a crude Err mx1.adamsgaard.dk 70 i+ search of the entire model space. Err mx1.adamsgaard.dk 70 i+ The burn-in phase is followed by a similar but more detailed Err mx1.adamsgaard.dk 70 i+ and local search of the model space, based on the best-fit Err mx1.adamsgaard.dk 70 i+ model parameters from the burn-in phase. The weighted Err mx1.adamsgaard.dk 70 i+ least-squared misfit to observed TCN concentrations is used Err mx1.adamsgaard.dk 70 i+ to evaluate the likelyhood for the combinations of Err mx1.adamsgaard.dk 70 i+ model parameter values. Err mx1.adamsgaard.dk 70 i

Err mx1.adamsgaard.dk 70 i Err mx1.adamsgaard.dk 70 i Err mx1.adamsgaard.dk 70 .