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[cellml-discussion] Representing stochastic models in CellML


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  • From: r.britten at auckland.ac.nz (Randall Britten)
  • Subject: [cellml-discussion] Representing stochastic models in CellML
  • Date: Wed, 14 May 2008 12:10:01 +1200
Hi

Perhaps Skype would be better, since many locations would allow Skype
participation, but there are only a few locations that allow participation
via BestGRID.

Regards,
Randall

> -----Original Message-----
> From: cellml-discussion-bounces at cellml.org [mailto:cellml-discussion-
> bounces at cellml.org] On Behalf Of Andrew Miller
> Sent: Wednesday, 14 May 2008 11:58 a.m.
> To: Kevin Burrage
> Cc: For those interested in contributing to the development of CellML.
> Subject: Re: [cellml-discussion] Representing stochastic models in
> CellML
>
> Kevin Burrage wrote:
> > Andrew
> >
> > I have lots of ideas on this - but email not a good way of doing it.
> > Am in oxford at the moment but could skype you if you were interested.
>
> Hi Kevin,
>
> Sorry for the delay in getting back to you about this.
>
> There seems to be a lot of interest amongst members of the CellML
> community in Auckland about hearing your views on this; one suggestion
> I
> heard was to use the BestGRID rather than Skype so more people could be
> involved at this end (although I am not sure how convenient it would be
> for you to access the BestGRID facilities in Oxford so the timezones
> would work out).
>
> When would be the most convenient for you? Given that there are a
> number
> of people who have indicated a desire to be involved in these
> discussion, and also a few that are interested in sitting in to listen,
> I would suggest making it at earliest next week, to ensure everyone can
> get their schedules to fit, and to ensure that software is working
> ahead
> of time. Since you are, I presume, in a GMT+1 timezone, and we are in
> GMT+12, the best time would probably be early in the morning for you /
> night time for us, e.g. 7-8 AM Oxford / 6-7 PM NZ, 8-9 AM Oxford / 7-8
> PM NZ , or 9-10 AM Oxford / 8-9 PM NZ.
>
> Best regards,
> Andrew
>
> >
> > BW
> >
> > Kevin
> > -------------------------------------------------
> > Kevin Burrage
> > Professor of Computational Systems Biology, COMLAB, University of
> Oxford
> > and Professor of Computational Mathematics, IMB, University of
> Queensland
> > -------------------------------------------------
> >
> >
> >
> > On Wed, 23 Apr 2008 17:31:26 +1200
> > Andrew Miller <ak.miller at auckland.ac.nz> wrote:
> > Hi all,
> >
> > I have spent some time looking into how we might be able
> > to represent stochastic models in CellML. This is
> > something that would be useful to ensure we have properly
> > contemplated for CellML 1.2. I have pasted in the notes I
> > wrote on this below.
> >
> > Please let me know if you have any suggestions, comments,
> > or criticisms of the below document. At some point, this
> > will obviously need to be transformed into a more robust
> > proposal, but for now, I just want to make sure we keep
> > the option open to use the CellML 1.2 core to represent
> > stochastic models.
> >
> > Best regards,
> > Andrew
> >
> > -----
> >
> > Overall goal:
> > Describe a framework which can be used in CellML 1.2 to
> > represent a
> > range of
> > different systems which require stochastic differential
> > equations to
> > describe.
> >
> > Constraints:
> > Do not want to describe the procedure for solving the
> > model in core
> > CellML,
> > only the underlying mathematical / statistical model.
> > Want to express the model in such a way that the
> > procedure is computable
> > from the model.
> > Want there to be only one interpretation of the model.
> > Want the representation to be abstract enough that it is
> > meaningful for a
> > number of different fields, and not just chemical
> > equations in a well
> > stirred vessel.
> > Want the representation to work naturally when mixed
> > with systems of
> > ordinary
> > differential equations.
> >
> > Use cases:
> > Chemical reactions under the Chemical Master Equation
> > model of Gillespie:
> > We need to split these into separate species. This is
> > a Poisson process,
> > so there are simple ways to represent it.
> >
> > It is more efficient to represent models using Weiner
> > processes when
> > this
> > there are large enough numbers of molecules to justify
> > this but ODEs are
> > not being used.
> >
> > However, Poisson and Weiner processes are both Levy
> > Processes, that is,
> > they have stationary independent increments, are zero
> > at time zero, and
> > are cadlag. This is not necessarily a good thing
> > because some things we
> > want to model might have memory of past events or a
> > time dependence.
> >
> > How we can represent this in CellML:
> > For the continuous case, integral equations for the
> > increment in terms of
> > built in processes like Weiner and Poisson processes (I
> > don't believe
> > there
> > is a clean way to represent increments in MathML).
> >
> > Implementations will need to identify these and work out
> > the
> > distribution of
> > the time until the next event (good implementations
> > might be able to
> > perform
> > symbolic algebra to work this out, but most
> > implementations would probably
> > just recognise common cases like Poisson distributions
> > with arbitrary
> > parameters, and deal with expressions involving a Weiner
> > process by
> > sampling
> > from the increment distribution in each time step),
> > which could be put
> > into a
> > slightly generalised Gibson-Bruck type of framework
> > where we store the
> > time of
> > the next event.
> >
> > None of this helps for non stationary independent
> > processes, however.
> >
> > How could this be related to the standards:
> > It has been proposed that CellML 1.2 have a core
> > specification which
> > describes
> > the basic way of representing the mathematical structure
> > in very general
> > terms, and secondary specifications be used to narrow
> > CellML down to
> > specific
> > subsets which can be implemented in their entirety by a
> > actual software
> > packages.
> >
> > Core CellML 1.2 should be general enough to represent
> > concepts like
> > probability distributions (MathML allows new operators
> > to be defined,
> > so we
> > could create ones for our base types of stochastic
> > process). The ODE IV
> > secondary specification would not allow stochastic
> > models, while we would
> > have another alternative secondary specification which
> > extended the ODE IV
> > one to allow certain limited types of stochastic model
> > (limited to
> > types we
> > know how to solve).
> >
> > In terms of interaction with the typing system, in a
> > stochastic system we
> > have both reals and real-valued random variables. Once
> > we have one random
> > variable in our system, this will propagate to
> > everything else which
> > is affected by it, so most of the model will technically
> > be a random
> > variable.
> > However, we want to be able to easily mix stochastic
> > models with
> > existing ODE
> > IV models to create hybrid models, so we don't really
> > want the required
> > datatype to change just to connect up a random variable
> > to a non-random
> > variable. For this reason, I don't think it is
> > worthwhile to consider a
> > random variable of a certain type a different datatype,
> > and instead we
> > would
> > require tools to deduce this information if they require
> > it.
> >
>
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