Produce children (crossover and mutation)

Replace unfit individuals

, Continue until one Individual meets success criteria! E.g., achieving an r2 > 0.6 measured vs. calculated data

Figure 13 Conceptual diagram for the design of evolutionary algorithms. Modified from Morrall D (2006) Ecological application of genetic algorithms. In: Recknagel F (ed.) Ecological Informatics, 2nd edn., pp. 69-83. New York: Springer.

distinctively designed for assembling the explicit model represented as multivariate functions or rule sets. Therefore EAs serve as powerful tools for knowledge discovery as well.

The hybrid evolutionary algorithms (HEAs) have been ad hoc designed as flexible tool for inducing predictive multivariate functions and rule sets from ecological time-series data. The conceptual framework of the application of HEA to rule discovery in limnological time-series data is represented in Figure 14. It indicates that similar to supervised ANN, the training of HEA aims at the optimal approximation of the calculated output Yc to the observed (desired) output Yo. However, by contrast,

HEA iteratively adjusts the rule structure and parameter values rather than input weights in order to minimize the error (Yo — Yc).

The detailed algorithm for the rule discovery and parameter optimization by HEA is shown in Figure 15. HEA uses genetic programming (GP) to generate and optimize the structure of rule sets and a genetic algorithm (GA) to optimize the parameters of a rule set. GP is an extension of GA in which the genetic population consists of computer programs of varying sizes and shapes. In standard GP, computer programs can be represented as parse trees, where a branch node represents an element from a function set (arithmetic operators, logic operators, elementary functions of at least one argument), and a leaf node represents an element from a terminal set (variables, constants, and functions of no arguments). These symbolic programs are subsequently evaluated by means of 'fitness cases'. Fitter programs are selected for recombination to create the next generation by using genetic operators, such as crossover and mutation. This step is iterated for consecutive generations until the termination criterion of the run has been satisfied. A general GA is used to optimize the random parameters in the rule set.

Figures 16 and 17 illustrate the structure, input sensitivity, and ¿-fold cross-validation of a rule-based agent for 7-day-ahead forecasting of Microcystis biomass developed by HEA.

The rule in Figure 16a is the result of using 42 years of merged limnological data of the South African lakes Hartbeespoort, Roodeplaat, and Rietvlei for the

Input x

Initial population

Next generation

Input x

Initial population

Next generation

Observed output yOBS

Algal abundance

Figure 14 Conceptual framework of the application of HEA for rule discovery in limnological time-series data.

Observed output yOBS

Algal abundance

Figure 14 Conceptual framework of the application of HEA for rule discovery in limnological time-series data.

training of HEA. The sensitivity analysis in Figure 16b indicates that both water temperature and Secchi depth are key driving variables for low biovolumes of Microcystis of up to 14 cm3 m~3 reflected by the THEN branch of the rule as well as for high biovolumes of up to 350 cm3 m~3 reflected by the ELSE branch of the rule. As a result of ¿-fold cross-validation, the parameters p1 and p2 have been evolved to water temperature functions which provide the agent an extra mechanism for adaptation to lake-specific seasonal conditions.

The ¿-fold cross-validation of the rule-based agent for Microcystis achieved r values of 0.31 for Lake Hartbeespoort, 0.34 for Lake Roodeplaat, and 0.75 for Lake Rietvlei (Figure 17).

Successful applications of EA have been demonstrated for cross-sectional data of fish populations as well as macroinvertebrate communities in streams, and for time-series data of plankton communities in lakes and rivers, and biological wastewater treatment.

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