58
P.A. Whigham . G.B. Fogel
predict the concentration of chlorophyll-a (Whigham and Recknagel 1999) and
allows the equations to express relationships between variables as a function of
past values, average values and nonlinear mathematical functions such as
exponential, logarithms and power functions. The expressive nature of the
equations allows more detailed exploration of the patterns then could be achieved
using a statistical approach, since the constraints of independence and linearity are
not required.
4.4.2
Optimisation of Difference Equations
There have been many developments of differential and difference equations to
predict ecological response. Often there are difficulties in tuning the parameters of
these equations; since the equations respond in a nonlinear fashion and so simple
hill climbing search algorithms (i.e. dynamic programming) do not perform
adequately. Since these problems can be framed as an optimization of the
parameters of the difference equation, evolutionary algorithms are a suitable
approach. One such example has been to use a GA to tune the parameters of a
difference equation, with the parameters constrained within known physical
limits. Each candidate solution (population member) represented a vector of the
parameter values, and the fitness function was a measure of how weIl the
difference equation predicted the measured data describing the freshwater system
(Whigham and Recknagel 1999). Constraining the parameter values and using
independent training and test data sets the equation parameters were evolved to
produce far greater accuracy and generalization ability (the RMSE for the unseen
test period of 1986 and 1993 was originally 91.46 and reduced to 46.75, as shown
in Figure 4.3). This gave the evolved difference equation accuracy that was
comparable to neural network and GP applications for the same data set.
Extensions to this work investigated evolving components of the difference
equation to derive new terms (such as the grazing term) to substitute in the
equation. This allowed an exploration of other forms of representation for this
term, and concluded by demonstrating that the grazing term is likely to not be a
linear function of chlorophyIl-a concentration (Whigham and Recknagel 2000).
This work was also extended to demonstrate that a complete differential equation
could be evolved, however as the degrees of freedom increased, the possibility of
exploiting other relationships in the data, and therefore not producing a physically
based solution, became more likely.
P.A. Whigham . G.B. Fogel
predict the concentration of chlorophyll-a (Whigham and Recknagel 1999) and
allows the equations to express relationships between variables as a function of
past values, average values and nonlinear mathematical functions such as
exponential, logarithms and power functions. The expressive nature of the
equations allows more detailed exploration of the patterns then could be achieved
using a statistical approach, since the constraints of independence and linearity are
not required.
4.4.2
Optimisation of Difference Equations
There have been many developments of differential and difference equations to
predict ecological response. Often there are difficulties in tuning the parameters of
these equations; since the equations respond in a nonlinear fashion and so simple
hill climbing search algorithms (i.e. dynamic programming) do not perform
adequately. Since these problems can be framed as an optimization of the
parameters of the difference equation, evolutionary algorithms are a suitable
approach. One such example has been to use a GA to tune the parameters of a
difference equation, with the parameters constrained within known physical
limits. Each candidate solution (population member) represented a vector of the
parameter values, and the fitness function was a measure of how weIl the
difference equation predicted the measured data describing the freshwater system
(Whigham and Recknagel 1999). Constraining the parameter values and using
independent training and test data sets the equation parameters were evolved to
produce far greater accuracy and generalization ability (the RMSE for the unseen
test period of 1986 and 1993 was originally 91.46 and reduced to 46.75, as shown
in Figure 4.3). This gave the evolved difference equation accuracy that was
comparable to neural network and GP applications for the same data set.
Extensions to this work investigated evolving components of the difference
equation to derive new terms (such as the grazing term) to substitute in the
equation. This allowed an exploration of other forms of representation for this
term, and concluded by demonstrating that the grazing term is likely to not be a
linear function of chlorophyIl-a concentration (Whigham and Recknagel 2000).
This work was also extended to demonstrate that a complete differential equation
could be evolved, however as the degrees of freedom increased, the possibility of
exploiting other relationships in the data, and therefore not producing a physically
based solution, became more likely.