Fig 1.
Flow diagram of the main processes captured with SPOTPY.
Multiple cycle black arrows indicate the possibility of parallelization of the iterating algorithms. The black box returning the simulation and evaluation data can be filled with any model.
Table 1.
Settings of the algorithms used in the case studies.
Fig 2.
Three-dimensional surface plot of the Rosenbrock function.
Colors from red (bad) to violet (optimal) represent the corresponding objective function (RMSE) for a parameter setting of x and y.
Fig 3.
Trace plot of the two dimensional Rosenbrock function.
The trace is shown as a blue line and the global optimum of the function as a broken red line. The x-axes show the number of iterations, while the y-axes show the value of the parameters x and y from -10 to 10.
Fig 4.
Three-dimensional surface plot of the Griewank function.
Colors from red (bad) to violet (optimal) represent the corresponding objective function (RMSE) for a parameter setting of x and y.
Fig 5.
Surface plot of the Griewank function.
Background colours showed from orange (bad response) to yellow (optimal response). Black dots show the sampled 5,000 parameter combinations. The x-axis shows the range of parameter x and the y-axis of parameter y.
Fig 6.
Three-dimensional surface plot of the Ackley function.
Colors from red (bad) to violet (optimal) represent the corresponding objective function (RMSE) for a parameter setting of x and y.
Fig 7.
Objective function traces of the Ackley function.
Setup with 2, 3, 5, 10, 20, 30 and 50 domains from the vector x of the Ackley function. All algorithms sampled 15,000 parameter combinations. The shown objective function on the y-axis is the root mean squared error (RMSE). The x-axis shows the number of iterations.
Fig 8.
Best CMF runs for simulating soil moisture.
Found with 10,000 iterations of the different algorithms realized with SPOTPY. The resulting different curves are very similar and overlap most of the time.
Fig 9.
Prior distribution (blue line) of input parameters of CMF.
Posterior distribution (green line) as the best 10% of the samples, plotted only for the Bayesian approaches. The optimal parameter setting is marked with a vertical red line.
Fig 10.
Comparison of measured and observed CO2 emission simulated with LDNDC (top panels).
Best model runs were derived with four different objective functions using a Latin Hypercube sampling approach (n = 50,000 model urns). The objective function BIAS is shown in red, r2 in green, RMSE in light blue and AI in dark blue. Observed values are shown as black dots. Middle panels depict classified residual error counts of simulated CO2 emissions for each model. The dashed black lines in the correlation plots of observed versus simulated CO2 emissions (bottom panels) show the theoretical optimal fit.
Table 2.
Capabilities of the different algorithms implemented in SPOTPY.