tadd info on MCMC walkers - cosmo - front and backend for Markov-Chain Monte Carlo inversion of cosmogenic nuclide concentrations
(HTM) git clone git://src.adamsgaard.dk/cosmo
(DIR) Log
(DIR) Files
(DIR) Refs
(DIR) README
(DIR) LICENSE
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(DIR) commit 92b401241f5faf7348a8517d6bd5d82bad18500e
(DIR) parent 34d863dbc2986f2eff0ac8ff390333c91e8fa532
(HTM) Author: Anders Damsgaard <anders.damsgaard@geo.au.dk>
Date: Fri, 27 Nov 2015 16:23:12 +0100
add info on MCMC walkers
Diffstat:
M pages/methods.html | 122 +++++++++++++++++++------------
1 file changed, 76 insertions(+), 46 deletions(-)
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(DIR) diff --git a/pages/methods.html b/pages/methods.html
t@@ -16,8 +16,8 @@
Constraining the landscape history and past erosion rates in
previously glaciated terrains is, however, notoriously
difficult because it involves a large number of unknowns.
- This webpage uses an approach based on the Markov Chain
- Monte Carlo (MCMC) technique. The model framework currently
+ This tool uses an approach based on the Markov Chain Monte
+ Carlo (MCMC) technique. The model framework currently
incorporates any combination of the following terrestrial
cosmogenic nuclides (TCNs) <sup>10</sup>Be, <sup>26</sup>Al,
<sup>14</sup>C, and <sup>21</sup>Ne in order to constrain a
t@@ -68,74 +68,104 @@
of the model parameters.
</p>
- <p>Given a single value of model parameters
+ <p>When model parameters
(ε<sub>int</sub>, ε<sub>gla</sub>,
<i>t</i><sub>degla</sub>,
- δ<sup>18</sup>O<sub>threshold</sub>), the TCN
- concentration after the duration of e.g. the entire
- Quaternary period in a sample can be computed. This
- <i>forward model</i> describes a history of exhumation and
- TCN production in a sample volume as it experiences the
- variable physical environment of the Pleistocene.
+ δ<sup>18</sup>O<sub>threshold</sub>) are varied within
+ specified limits, they can be thought of as orthogonal axes
+ creating a coordinate system in higher-order space. Every
+ position in this model space is associated with a certain
+ set of model parameter values.
</p>
- <p>When model parameters
+ <p>Given a single value of model parameters
(ε<sub>int</sub>, ε<sub>gla</sub>,
<i>t</i><sub>degla</sub>,
- δ<sup>18</sup>O<sub>threshold</sub>) are allowed to
- vary within specified limits, they can be thought of as
- orthogonal axes creating a coordinate system in higher-order
- space. Every position in this model space is associated with
- a certain set of model parameter values.
- </p>
+ δ<sup>18</sup>O<sub>threshold</sub>) within the
+ specified limits, the TCN concentration after the duration
+ of e.g. the entire Quaternary period in a sample can be
+ computed. This <i>forward model</i> describes a possible
+ history of exhumation and TCN production in a sample volume
+ as it experiences the variable physical environment of the
+ Pleistocene.</p>
+
+ </div>
+
+ <div id="twostage" class="subsection scrollspy">
+ <h4 class="header blue-text light">
+ Two-stage glacial-interglacial forward model</h4>
+ <p>The forward model builds on the assumption of a
+ "two-stage uniformitarianism", meaning that the processes
+ that operated during the Holocene also operated during
+ earlier interglacials with comparable intensity. Likewise,
+ the erosion rate during the past glacial periods is assumed
+ to be comparable.</p>
+
+ <p>The model approach assumes that glacial periods were
+ characterized by 100% shielding and no exposure, which would
+ require more than 10 m of ice thickness for production due
+ to spallation (>50 m for muons). Interglacial periods are
+ assumed to have been characterized by 100% exposure and zero
+ shielding.</p>
</div>
<div id="mcmcwalker" class="subsection scrollspy">
<h4 class="header blue-text light">
What is a MCMC walker?</h4>
<p>
- A MCMC walker is a numerical entity which sequentially
- explores the model parameter space in order to obtain the
- best result between a forward-model and an observational
- dataset. During each iteration
+ A MCMC walker is in this context a numerical entity which
+ sequentially explores the model parameter space in order to
+ obtain the closest match between the forward model and the
+ observational dataset of TCNs. During each iteration
the walker takes its current position in model space, plugs
- the parameter value into the forward-model, and
+ the parameter value into the forward model, and
evaluates if the output result matches the observational
record better or worse than the output at its previous
position in model space. If the new results better matches
- the observed dataset, it continues walking along the same
- path in model space with a small random perturbation.
+ the observed dataset, it continues walking along
+ approximately in the same direction in model space.
</p>
<p>
Starting at a random place inside the model space, a burn-in
phase of 1000 iterations is first used to make a crude
- search of the entire model space.
- The burn-in phase is followed by a similar but more detailed
- and local search of the model space, based on the best-fit
- model parameters from the burn-in phase. The weighted
- least-squared misfit to observed TCN concentrations is used
- to evaluate the likelyhood for the combinations of
- model parameter values.
+ search of the entire model space. The burn-in phase is
+ followed by a similar but more detailed and local search of
+ the model space, based on the best-fit model parameters from
+ the burn-in phase. The weighted least-squared misfit to
+ observed TCN concentrations is used to evaluate the
+ likelyhood for the combinations of model parameter values.
+ The MCMC walker continues exploring the model space until it
+ is sufficiently satisfied with the best model parameter
+ estimate it has found.
</p>
- </div>
- <div id="twostage" class="section scrollspy">
- <h3 class="header blue-text">
- Two-stage glacial-interglacial model</h3>
- <p>The model concept builds on the assumption of a
- "two-stage uniformitarianism", meaning that the processes
- that operated during the Holocene also operated during
- earlier interglacials with comparable intensity. Likewise,
- the erosion rate during the past glacial periods is assumed
- to be comparable.</p>
+ <p>
+ For a given observational data set more than one set of
+ model parameters may produce forward models which
+ sufficiently satisfy the MCMC walker as solution
+ approximations. In this case the solution is
+ <i>non-unique</i>. Even worse, a single MCMC walker may find
+ an area in model space which seemingly is in good
+ correspondence with the observational data set, but is
+ missing a much better set of model parameters since they are
+ located somewhere entirely different in the model space. In
+ order to mitigate these issues, MCMC inversions are often
+ performed using several MCMC walkers. The starting point of
+ each MCMC walker is chosen at random, resulting in unique
+ walks through the model space. If a single walker is caught
+ in an area of non-ideal solutions, chances are that the
+ other walkers will find the area of better model parameters.
+ </p>
- <p>The model approach assumes that glacial periods were
- characterized by 100% shielding and no exposure, which would
- require more than 10 m of ice thickness for production due
- to spallation (>50 m for muons). Interglacial periods are
- assumed to have been characterized by 100% exposure and zero
- shielding.</p>
+ <p>
+ The computational time depends on the number of MCMC
+ walkers. When casually trying out the calculator we
+ recommend using low numbers of MCMC walkers (1 to 2) in
+ order to obtain fast results and reduce load on the server.
+ When attempting to produce high-quality reliable results,
+ the number of walkers should be increased (3 to 4).
+ </p>
</div>