CHAPTER 10 Hybridisation of process-based ecosystem …CHAPTER 10 Hybridisation of process-based ecosystem models with ... Figure 1: Flowchart of the evolutionary algorithm for optimising

CHAPTER 10

Hybridisation of process-based ecosystem models with evolutionary algorithms: multi-objective optimisation of process and parameter representations of the lake simulation library SALMO-OO

H. Cao & F. RecknagelSchool of Earth and Environmental Sciences, University of Adelaide, Adelaide, Australia.

1 Introduction

Over the past three decades, numerous lake ecosystem models incorporating phyto- and zoo-plankton population dynamics as well as nutrient cycles have been developed and published [1–8], which were represented by complex ordinary differential equations (ODE). Those models usually consist of numerous state and driving variables, process equations as well as parameters, which are rigidly networked by the ODE. The process-based lake simulation library SALMO-OO [9–12] is an attempt to overcome rigidity of ecosystem models by hierarchically modularis-ing complex ODE and replacing rigid process representations by optional arrays of alternative process representations implemented by object-oriented programming [13].

The present study demonstrates by means of SALMO-OO that an object-oriented implemented simulation library also gains fl exibility with regard to hybridisation with evolutionary algorithms (EA).

SALMO-OO as documented by Recknagel and Cao (Chapter 9) is based on more than 100 process equations and constant parameters.

Even though the processes are causally represented in some cases, the knowledge is limited and the representations are shallow and vague. Therefore, fi rst SALMO-OO was embodied with EA to redefi ne or extend vague process representations in its complex ODE. EA utilised merged data of the same category lakes to discover process functions being generic for the lake category refl ected by the merged data.

The numerous constant parameters estimated from experimental and literature data signifi -cantly determine process dynamics and simulation results [14]. Therefore, secondly, EA were applied to merged data of the same category lakes to optimise and adapt crucial model parame-ters to distinct conditions of specifi c lake categories. Whilst traditional optimisation techniques allowed fi netuning constant parameter values within their observed variance [15, 16], the use of

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170 Handbook of Ecological Modelling and Informatics

EA allows representing seasonal and trophic relationships of parameter ranges by nonlinear tem-perature and phosphate functions being generic for the lake category refl ected by the merged data [17]. Moreover, EA facilitate multi-objective optimisation, e.g. of same metabolic parameters, of competing diatoms, green and blue-green algae at the same time.

2 Evolutionary algorithm for the optimisation of process representations and parameters

In recent years, EA proved to be powerful tools for inducing both predictive and explanatory models for a variety of problems [18, 19]. They gain their highly adaptive capability from simu-lating principles of natural selection and genetic variation. They use suitable coding to represent possible solutions to a problem and guide the search for the ‘optimum solution’ by means of genetic operators and the principle of ‘survival of the fi ttest’. EA can be divided into the follow-ing four categories: genetic algorithms, evolutionary strategy, evolutionary programming and genetic programming (GP) [20–22]. In the context of the present study, we use basically the GP approach.

The fl owchart for optimising process representations and parameters in SALMO-OO by means of EA is shown in Fig. 1 and is explained in the following sections.

Figure 1: Flowchart of the evolutionary algorithm for optimising process representations and parameters in SALMO-OO.


Hybridisation of Process-Based Ecosystem Models 171

2.1 Encoding

We extend the representation of a single tree in standard GP to a vector of binary trees denoted as (T1, T2, …, TK) where Ti corresponds to the specifi c process/parameter model and K is the total number of processes/parameters to be optimised as function models. For example, a representation of a parameter by a function of the form

(sin(T)/P * 8.2) + (ln(P/T) − T) (1)

can be represented as a binary tree illustrated in Fig. 2. Besides this, the maximum depth of per tree is restricted by a constant D, which refl ects the complexity of a model.

2.2 Fitness evaluation

Given an individual Pi in the population of process or parameter representations denoted as (T1, T2, …, TK), fi rst, we replace K processes or parameters with corresponding functions T1 ~ TK and get their calculated values based on input data of the independent variables. Second, we integrate those values into the whole ODE of SALMO-OO and calculate all output variables such as phosphate P, nitrate N, total algae A, diatoms A1, green algae A2, blue-green algae A3, zooplankton Z, detritus D and dissolved oxygen O. Depending on the availability of measured data for the output variables, our objective is to simultaneously minimise the errors between the measured and calculated data, which requires multi-objective optimisation. Depending on the availability of measured data for phosphate, total algae or algal groups and zooplankton the total error TE is either defi ned as

2 2 2

1

ˆˆ ˆ(P P ) (A A ) (Z Z )m

i i i i i ii

m=

⎡ ⎤− + − + −⎣ ⎦∑ (2)

or

2 2 2 2 2

1

ˆ ˆ ˆˆ ˆ(P P ) (A1 A1 ) (A2 A2 ) (A3 A3 ) (Z Z ),

m

i i i i i i i i i ii

m=

⎡ ⎤− + − + − + − + −⎣ ⎦∑ (3)

Figure 2: Example (1) of the representation for a single parameter by a function in GP.



where m is the total day number, Pi, Ai (A1i, A2i, A3i), Zi and Pi , ( Ai , A1i , A2i ), A3i Zi are the measured and calculated values of P, A (A1, A2, A3), Z on the ith day, respectively.

The experiments for optimising process representations by means of EA were applied to data of one lake where TE was used as fi tness function.

The experiments for optimising parameters were applied separately to merged data of lakes belonging to a certain lake category. By searching for the best functional representation of param-eters for lakes of the same category, we separately calculated the total errors by eqns (2) and (3), and used their sum in eqn (4) as fi tness value:

1

Fitness TE ,L

ii=

= ∑ (4)

where L is the number of lakes considered in one category.

2.3 Genetic operators

As each individual is represented as a vector of trees, there are two levels of crossover, namely vector-level crossover and tree-level crossover.

Figure 3 illustrates the procedure of vector-level crossover. Given two parents Parent1 and Parent2, the vector-level crossover is performed by randomly selecting one tree from them each time until the total number of process or parameter representations K is reached.

By contrast, the tree-level crossover is performed between each pair of Ti(1) and Ti(2). A node within each tree is randomly selected as a crossover point. After swapping the subtrees rooted at the crossover points, two new trees are produced. Then either of them is used as offspring tree based on the condition that its maximum depth does not exceed D. Figure 4 illustrates an exam-ple for tree-level crossover between Parent1 and Parent2.

Similarly, the tree-level mutation begins by randomly selecting a tree from the parent and also a node within the tree as the mutation point, replacing the subtree rooted at the mutation point with a randomly generated new subtree, thus producing an offspring tree. Figure 5 illustrates an example of tree-level mutation.

3 Optimisation of process representations in SALMO-OO by means of EA

3.1 Experimental settings and measures

The processes equations of SALMO-OO determine mass balances of the state variables and refl ect a typical pelagic food web. The following processes were exemplarily chosen for optimising

Figure 3: Illustrations of vector-level crossover of process or parameter representations.



process representations by means of EA: the temperature term of phytoplankton respiration RAT, the ingestion rate in the dark depending on phytoplankton and zooplankton biomass GDB, the temperature term of zooplankton respiration RZT and the zooplankton mortality ZMO. In addition, a new process called the nutrient competition of phytoplankton ACO was evolved in a fi nal experiment. Figure 6 illustrates the experimental design for optimising process repre-sentations by using EA. The algorithm consists of four consecutive steps. After fi nishing one step, the previous process equation in SALMO-OO is replaced by the actually best-performing function, and the algorithm proceeds with the next step. For the processes RAT and ACO,

Figure 4: Illustrations of tree-level crossover of process or parameter representations.

Notes:Parent1: cos(P)*31.6/T−ln(|P|)Parent2: sin(T)+exp(2.34-T/P)Offspring1:

cos(P)*31.6+exp(2.34-T/P)sin(T)/T−ln(|P|)

Offspring2:

Figure 5: Illustration of tree-level mutation of process or parameter representations.

Notes:Parent: (sin(T)/P*8.2)+(ln(P/T)−T)Offspring: sin(T)/P*8.2)+(ln(T*T)



the so-described procedure is carried out simultaneously for the algal groups diatoms A1, green algae A2 and blue-green algae A3. Process equations and defi nitions of SALMO-OO are provided in Table 1.

3.1.1 SALMO-OO by using EAThe procedure for optimising process representations by EA was tested by annual data of 1978 of the dimictic eutrophic Lake Bautzen, Germany. Fifty independent runs were conducted for each step. All experiments were performed on a Hydra supercomputer (IBM eServer 1350 Linux) with a peak speed of 1.2 TFlops by using C programming language. Parameter settings of EA were as follows: MAXGEN = 50, N = 100, D = 4, function set = {+, −, ∗, /, sin, cos, exp, ln}.

To measure the goodness of model fi tting, the root mean squared error (RMSE) for the state variable X (i.e., P, A, A1, A2, A3, Z) was calculated as follows:

2

1

ˆ(X X )RMSE( ) ,

m

i iiX

m=

−=

∑ (5)

where m is the total day number, Xi and ˆiX are the measured value and calculated value of X on

ith day, respectively.

Figure 6: Experimental design for optimising process representations of SALMO-OO by using EA.



3.2 Results and discussion

Table 2 summarises the optimised process representations discovered by the EA approach as well as the corresponding RMSE and r2 values for the state variables P, A, Z before and after opti-misation based on data for Lake Bautzen of 1978. In steps 1 and 4, three separate functions for each of the three algal groups were obtained for both processes RAT and ACO in SALMO-OO. Another three optimised functions for the processes GDB, RZT and ZMO resulted from steps 2 and 3. The RMSE and r2 values in Table 2 indicate that the model validity for the three state variables phosphate, total algae and zooplankton gradually improved from step 1 to step 4. For example, the r2 for phosphate changed from 0.17 before the optimisation to 0.67 after step 4 of the optimisation. The r2 for total algae improved from 0.004 to 0.87, and for zooplankton from 0.65 to 0.81.

Figure 7 illustrates the changing process dynamics before optimising process representations in Table 1 and after optimising process representations in Table 2. As can be seen in Fig. 7, the process dynamics of RXT [2, 3] and GDB have changed signifi cantly both in magnitude and overall trends. In contrast, both RZT and ZMO remained rather similar with only slightly differ-ent magnitudes. As for the new process ACO, the three algae groups appeared to have different seasonal dynamics for the nutrient competition. The distinctive negative values of ACO [2] indicated a higher impact of the nutrient competition on green algae compared with diatoms and blue-green algae. This fi nding seems to correspond with the often rather opportunistic than competitive behaviour of green algae.

Figure 8 illustrates the simulation results of SALMO and SALMO-EA step by step for phos-phate, total algae and zooplankton on Lake Bautzen. These graphs clearly show how the simula-tion results for the three outputs are improved step by step. In the fi rst step, by SALMO-EA, the timing of total algae in SALMO is moved ahead and coincides with the observed data very well and thus, the simulated total algae values are improved signifi cantly. The simulation result of zooplankton is also improved in this step by decreasing the overestimation in SALMO. On the

Table 1: Equations and defi nitions for the processes of SALMO-OO to be optimised by EA (1: diatoms, 2: green algae, 3: blue-green algae).



contrary, the simulated phosphate values get a little worse than SALMO. However, in the follow-ing three steps, the results for phosphate are getting better and better. In step 4, the simulated curve of phosphate in SALMO-EA matches the observed data so well that almost all the mea-sured data points closely surround it. As for total algae and zooplankton, by evolving the corre-sponding processes step by step, the simulated values for some measured data points get closer to the observed values and more and more points drop on the simulation curves until the best results are achieved in step 4. Overall, the above experimental results have demonstrated that EA are effi cient tools to either optimise representations of existing processes or introduce representa-tions of new processes to complex ODE such as SALMO-OO. Results show that the simulation results for phosphate, total algae and zooplankton can be signifi cantly improved.

4 Case study of parameter optimisation in SALMO-OO

4.1 Experimental settings and measures

SALMO-OO contains 128 constant parameters that are either estimated from fi eldwork and laboratory experiments or adopted from the literature. Typically, the most sensitive parameters in phytoplankton population models are optimised to give more realistic and accurate simulation results. These parameters include the maximum photosynthesis under optimal conditions (PHO-MAX), the optimal algal respiration (RATOPT) and the maximum grazing rate by zooplankton (GMAX). EA can be used to determine temperature and phosphate functions for PHOMAX and RATOPT by building different functions simultaneously for diatoms, green algae and blue-green algae and also for GMAX to build a generic temperature and phosphate function for each lake category. For case studies, we choose two typical lake categories: I – warm-monomictic and hypertrophic; II – dimictic and mesotrophic. To carry out the experiments, the following four

Table 2: Optimised representations for selected processes of SALMO-OO discovered by EA.



example lakes were selected: Hartbeespoort (South Africa, 2003–2004) and Klipvoor (South Africa, 2003–2004) (I); Saidenbach (Germany, 1975) and Weida (Germany, 1984) (II). The gen-eral characteristics of these four lakes are described in the literature [11]. The constant values and value ranges of the parameters to optimise for these lakes in SALMO-OO are listed in Table 3. For each category, we use the merged data of two example lakes to build generic parameter function models by using EA and conduct 20 independent runs for each lake category. The experimental running environment, the parameter settings of EA and the measures are the same as Section 3.1.

Figure 7: Step-wise changes of process dynamics of SALMO-OO before and after process optimisation based on data of Lake Bautzen in 1978.



4.2 Results and discussion

Table 4 lists the optimised parameter representations for the two lake categories ‘warm-monomictic hypertrophic’ and ‘dimictic mesotrophic’ resulting from 20 runs of the EA. It also includes the RMSE and r2 values of the state variables phosphate P, total algae A, diatoms A1, green algae A2, blue-green algae A3 and zooplankton Z before and after parameter optimisation. The EA have induced functions for PHOMAX[1-3], RATOPT[1-3] and GMAX depending on phosphate (P) and water temperature (T), which are specifi c for each lake category. The EA induced the func-tions by combining the arithmetic operators (+, –, *, /) with elementary functions (sin, cos, exp, ln). The comparison of RMSE and r2 values for the six state variables indicates that quantitative validity of SALMO-OO has signifi cantly improved between before and after optimising param-eter representations of SALMO-OO by EA.

Figure 8: Step-wise changes of simulation results for phosphate, total algae, zooplankton of SALMO-OO before and after process optimisation based on data of Lake Bautzen in 1978; measured data from Benndorf and Recknagel [4].



Figure 9 illustrates the representations of the parameters PHOMAX[1-3], RATOPT[1-3] and GMAX as discovered for ‘warm-monomictic hypertrophic’ lakes, and Fig. 10 illustrates the parameter representations for ‘dimictic mesotrophic’ lakes as documented in Table 4. It is impor-tant to note that the optimisation of all parameter representations by EA was restricted by the parameter ranges as defi ned in Table 3. Results in Figs 9 and 10 show in general distinct season-alities of PHOMAX[1-3] and RATOPT[1-3], whilst GMAX stays largely constant during the years, except for Lake Klipvoor. This may indicate that GMAX is not as sensitive as the other parameters to specifi c lake conditions or seasons. The results also indicate that green algae prefer much higher PHOMAX values than diatoms and blue-green algae.

Figure 11 illustrates the validation results for the state variables phosphate P, total algae A and zooplankton Z of SALMO-OO two warm-monomictic hypertrophic lakes, and Fig. 12 for two dimictic mesotrophic lakes before and after parameter optimisation by EA. With regard to total algae A, the parameter optimisation by EA achieves a substantial improvement for most of the lakes except for Lake Klipvoor. The improved results for total algae were achieved in particular for the Lakes Saidenbach and Weida by a more realistic simulation of seasonal peaks and troughs corresponding much better with the observed biomass dynamics of phytoplankton. The results for total algae for Lake Hartbeespoort (Fig. 11) improved as well after parameter optimisation by EA that were largely achieved by fi netuning the magnitudes during spring (September-October) and summer (December-January). However, the optimisation of the above-mentioned parameters by EA had little effect on improving the simulation result of zooplankton of all four lakes.

Figure 13 illustrates the validation results for the state variables diatoms A1, green-algae A2 and blue-green algae A3 of SALMO-OO two warm-monomictic hypertrophic lakes, and Fig. 14 for two dimictic mesotrophic lakes before and after parameter optimisation by EA. Due to lim-ited data, the validation can be made only for the simulation results of Lakes Hartbeespoort, Klipvoor and Weida. Whilst SALMO-OO overestimates most observed points of three algal groups in Lake Hartbeespoort, the parameter optimisation by EA produces much more realistic values, in particular, during spring (September-October) and summer (December_January). The parameter optimisation improves the spring peak (September) of diatoms as well as the overall simulation for blue-green algae in Lake Klipvoor. However, before and after parameter optimi-sation SALMO-OO largely overestimates four observed peaks of green algae. For Lake Weida,

Table 3: Values and ranges of parameters of SALMO-OO to be optimised by EA (1: diatoms, 2: green algae, 3: blue-green algae).

Parameters Defi nition Constant value Value range

PHOMAX [1-3] Maximum photosynthesis under optimal conditions (day–1)

PHOMAX[1] = 2.37PHOMAX[2] = 3.3PHOMAX[3] = 2.37

PHOMAX[1]: [1, 3]PHOMAX[2]: [2, 4]PHOMAX[3]: [1, 3]

RATOPT [1-3] Optimum algal respiration (day–1)

Hartbeespoort 0.08 [0.03, 0.5]Klipvoor 0.1Saidenbach 0.25Weida 0.25

GMAX Maximum grazing rate by zoo-plankton (day–1)

Hartbeespoort 0.45 [0.3, 1.3]Klipvoor 0.45Saidenbach 1.3Weida 0.65



the parameter optimisation by EA achieves a satisfactory simulation result for diatoms, which not only matches the timing and magnitude of observed data very well, but also reaches the early summer (May) peak exactly (Fig. 14). As for green algae, simulation results improve after param-eter optimisation by changing the timing of seasonal dynamics.

Table 4: Optimised representations of the parameters PHOMAX[1-3], RATOPT[1-3] and GMAX of SALMO-OO for two lake categories performed by EA after 20 runs.



In summary, the experimental results for optimising representations of key parameters in SALMO-OO by means of EA demonstrated that the validation results were slightly improved for phosphate, but distinctively improved for total algae and three algal groups (diatoms, green algae and blue-green algae). However, little improvements have been achieved for zooplankton.

Figure 9: Illustration of representations of the parameters PHOMAX[1-3], RATOPT[1-3] and GMAX for warm-monomictic hypertrophic lakes as shown in Table 4.

Figure 10: Illustration of representations of the parameters PHOMAX[1-3], RATOPT[1-3] and GMAX for dimictic mesotrophic lakes as shown in Table 4.



Figure 11: Validation results for the state variables phosphate P, total algae A and zooplankton Z of SALMO-OO before and after the parameter optimisation by EA for two warm-monomictic hypertrophic lakes; measured data from Van Ginkel et al. [23].

Figure 12: Validation results for the state variables phosphate P, total algae A and zooplankton Z of SALMO-OO before and after parameter optimisation by EA for two dimictic mesotrophic lakes; measured data for Saidenbach from Benndorf and Recknagel [4]; measured data for Weida from Saenger (personal communication).



Figure 13: Validation results for the state variables diatoms A1, green-algae A2 and blue-green al-gae A3 of SALMO-OO before and after parameter optimisation by EA for two warm-monomictic hypertrophic lakes; measured data for Hartbeespoort and Klipvoor from Van Ginkel et al.[23].

Figure 14: Validation results for the state variables diatoms, green-algae and blue-green algae of SALMO-OO before and after parameter optimisation by EA for two dimictic mesotrophic lakes; measured data for Saidenbach from Benndorf and Recknagel[4]; measured data for Weida from Saenger (personal communication).



5 Conclusions and future work

The hybridisation of complex ODE, such as the lake simulation library SALMO-OO with EA, designed for the optimisation of either process representations or parameter representations, pro-vides a great potential to improve the validity of highly complex process-based models. The proposed hybrid model has demonstrated signifi cant improvements of the model validity for the lake categories ‘warm-monomictic hypertrophic’ and ‘dimictic mesotrophic’ after the optimisa-tion of either process or parameter representations.

Future research will focus on the combined optimisation of process and parameter represen-tations and its validation for additional lake categories such as ‘dimictic eutrophic’ and ‘warm-monomictic mesotrophic’. It will also aim at the causal interpretation of induced process and parameter representations to improve understanding of the ecology of different lake categories.

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CHAPTER 10 Hybridisation of process-based ecosystem …CHAPTER 10 Hybridisation of process-based ecosystem models with ... Figure 1: Flowchart of the evolutionary algorithm for optimising