540
ANDREAS OSCHLIES
can explore large regions of the parameter space and reduce the chance
of getting trapped in a local minimum early on. An example is the
Markov Chain Monte Carlo method [Harmon and Challenor, 1997], that
also addresses estimating the posterior error distribution of the optimal
parameter values. Construction of other stochastic minimisation methods was guided by attempts to understand the emergence of apparently
optimal structures in nature. One of these is the concept of simulated
annealing that was used by Matear [1995] to optimise parameters of a
suite of ecosystem models to observations at Station P in the subpolar
North Pacific. The simulated annealing technique is analogous to the
thermodynamics that describe cooling and cristallisation of liquids. It
consists of an iterative random selection of state vectors within a slowly
narrowing probability distribution around the “best” parameter vector
of the previous iteration. The width of this probability distribution decreases with increasing iteration number as does the probability of new
parameter vectors being accepted with a higher cost function value than
the “best” one of the previous iteration. A finite probability of uphill
steps in the cost-function landscape is required to escape local minima.
The probability is formulated in terms of the Boltzman factor that describes energy fluctuations in statistical mechanics, where the probability
of transitions to more energetic states increases with temperature. If the
temperature is decreased slowly enough to avoid getting trapped in local
minima, this can eventually lead to a liquid cristallising into a perfect
lattice that represents the state with lowest possible potential energy.
Replacing potential energy by the cost function value, this behaviour is
mimicked by the minimisation algorithm with a “temperature” parameter in the Boltzman factor decreasing with increasing iteration number.
For typical ecosystem-model applications, several tenthousand evaluations of the cost function are needed to obtain a robust result. Still,
convergence to the global minimum cannot generally be proven.
Hurtt and Armstrong [1996] employed simulated annealing to minimise the model-data misfit for a new, implicitly size-structured ecosystem model at Bermuda and later [Hurtt and Armstrong, 1999] extended
this approach to test simultaneous optimisation of a similar model at
the distinct sites of the Bermuda Atlantic Time series Study (BATS,
32 ◦ N, 64 ◦ W) and Ocean Weather Ship India (59 ◦ N, 19 ◦ W). While they
found that different ecological processes had to be considered at the two
locations to achieve a reasonable fit, Schartau and Oschlies [2003a,b]
ported that simultaneous optimisation of a NPZD model at three sites
(BATS, OWS India, and the site of the North Atlantic Bloom Experiment at 47 ◦ N, 20 ◦ W) worked almost as well (or as badly!) as separate
optimisations at the individual locations. Their optimisation method
re
ANDREAS OSCHLIES
can explore large regions of the parameter space and reduce the chance
of getting trapped in a local minimum early on. An example is the
Markov Chain Monte Carlo method [Harmon and Challenor, 1997], that
also addresses estimating the posterior error distribution of the optimal
parameter values. Construction of other stochastic minimisation methods was guided by attempts to understand the emergence of apparently
optimal structures in nature. One of these is the concept of simulated
annealing that was used by Matear [1995] to optimise parameters of a
suite of ecosystem models to observations at Station P in the subpolar
North Pacific. The simulated annealing technique is analogous to the
thermodynamics that describe cooling and cristallisation of liquids. It
consists of an iterative random selection of state vectors within a slowly
narrowing probability distribution around the “best” parameter vector
of the previous iteration. The width of this probability distribution decreases with increasing iteration number as does the probability of new
parameter vectors being accepted with a higher cost function value than
the “best” one of the previous iteration. A finite probability of uphill
steps in the cost-function landscape is required to escape local minima.
The probability is formulated in terms of the Boltzman factor that describes energy fluctuations in statistical mechanics, where the probability
of transitions to more energetic states increases with temperature. If the
temperature is decreased slowly enough to avoid getting trapped in local
minima, this can eventually lead to a liquid cristallising into a perfect
lattice that represents the state with lowest possible potential energy.
Replacing potential energy by the cost function value, this behaviour is
mimicked by the minimisation algorithm with a “temperature” parameter in the Boltzman factor decreasing with increasing iteration number.
For typical ecosystem-model applications, several tenthousand evaluations of the cost function are needed to obtain a robust result. Still,
convergence to the global minimum cannot generally be proven.
Hurtt and Armstrong [1996] employed simulated annealing to minimise the model-data misfit for a new, implicitly size-structured ecosystem model at Bermuda and later [Hurtt and Armstrong, 1999] extended
this approach to test simultaneous optimisation of a similar model at
the distinct sites of the Bermuda Atlantic Time series Study (BATS,
32 ◦ N, 64 ◦ W) and Ocean Weather Ship India (59 ◦ N, 19 ◦ W). While they
found that different ecological processes had to be considered at the two
locations to achieve a reasonable fit, Schartau and Oschlies [2003a,b]
ported that simultaneous optimisation of a NPZD model at three sites
(BATS, OWS India, and the site of the North Atlantic Bloom Experiment at 47 ◦ N, 20 ◦ W) worked almost as well (or as badly!) as separate
optimisations at the individual locations. Their optimisation method
re