Singleobjection function cannot describe the characteristics of the complicated hydrologic system. Consequently, it stands to reason that multiobjective functions are needed for calibration of hydrologic model. The multiobjective algorithms based on the theory of nondominate are employed to solve this multiobjective optimal problem. In this paper, a novel multiobjective optimization method based on differential evolution with adaptive Cauchy mutation and Chaos searching (MODECMCS) is proposed to optimize the daily streamflow forecasting model. Besides, to enhance the diversity performance of Pareto solutions, a more precise crowd distance assigner is presented in this paper. Furthermore, the traditional generalized spread metric (SP) is sensitive with the size of Pareto set. A novel diversity performance metric, which is independent of Pareto set size, is put forward in this research. The efficacy of the new algorithm MODECMCS is compared with the nondominated sorting genetic algorithm II (NSGAII) on a daily streamflow forecasting model based on support vector machine (SVM). The results verify that the performance of MODECMCS is superior to the NSGAII for automatic calibration of hydrologic model.
Some research has been reported that streamflow processes are affected by several known factors, such as rainfall and soil moisture, and many potential factors. Consequently, the streamflow processes always result in being nonlinear and timevarying. And the differences between the characteristics of high flow processes and low flow processes are very significant most of time. Therefore, it tends to be very difficult to predict the streamflow processes.
Firstly, selecting and constructing a proper model are the great and first important step of the research. Statistics forecast models [
As the model is established, the next vital work is to make calibration of the hydrologic model. Calibration of hydrologic model is to find the optimal parameters which could make the model match the characteristic of the hydrology as good as possible. Usually, the hydrological models are calibrated under the framework of singleobjective paradigm. And the major work is focused on choosing the singleobjective function and the optimization strategy to optimize that measure. However, several research reports reveal that singleobjective functions often cannot properly describe all characteristics of the hydrological models [
In this paper, a new multiobjective optimization method named multiobjective optimization method based on differential evolution with adaptive Cauchy mutation and Chaos searching (MODECMCS) is proposed to optimize a threeparameter support vector machine model for daily streamflow forecasting. And the efficacy of the new algorithm MODECMCS is compared with the NSGAII algorithm on a threeparameter SVM based daily streamflow forecasting model.
This paper is organized as follows: Section
In this section, we will give a simple introduction about the NSGAII, and after that the details of the proposed multiobjective algorithm will be described.
NSGAII, which is an improvement over the NSGA, is first proposed by Deb et al. [
Flowchart of NSGAII.
Several main features of the algorithm NSGAII can be summarized as follows:
NSGAII is significantly more efficient than the original version of the algorithm. And NSGAII can reduce the computational complexity into
NSGAII employs a CrowdedComparison Operator to maintain the diversity of the nondominated Pareto solutions. Each nondominated solution has a new attributecrowd value, which is calculated through the CrowdedComparison Operator. And each nondominated solution is ranked by the crowd value. The nondominated solution with higher crowd value is more likely to be preserved to the next population. The diversity of the NSGAII is superior to the original version of the algorithm.
NSGAII does not have the parametershare parameter, which is designed in the original version of the algorithm and can hardly set a proper value. This will make the NSGAII be more flexible and easier to use.
The proposed multiobjective algorithm MODECMCS is based on the DE algorithm; thus the DE algorithm will be introduced first, and then the details about MODECMCS are presented.
DE is firstly proposed by Storn and Price [
The mutant vectors
Thereafter, the discrete recombination is employed to create vectors
DE adopts a greedy selection method, and then the next generation can be gained by
Details of the proposed multiobjective algorithm MODECMCS are discussed in this section. The MODECMCS mainly contains 7 operators as follows.
Crowd distance assignment.
For each nondominated individual
Begin
If1
Else1
Begin
If2
Else2
Begin
add
If3 the length of
Begin
Calculate the crowd distance of individuals in
Delete the individuals with the minimum crowd distance;
End if3
End else2
End else1
End for
For each pair of selected candidate individual
Begin
If
Else if
Else if
End for
If the length of
Begin
Sort the individuals of
Calculate the crowd distance of individuals in
Select the first
End if
If the value is below the preset threshold
The procedure of the Chaos searching operator in this paper is as Algorithm
For each individual
Begin
Set
Generate a
While (
Begin
Calculate
Map the
Calculate the new individual generated from the current individual according to:
the Chaos searches in a small scope;
If
Begin
Replace
Delete the individuals dominated by
Break while;
End if
Else
Begin
Replace
End else
End while
End for
With the above 7 operators, the flowchart of the proposed multiobjective algorithm MODECMCS can be expressed as Figure
The flowchart of MODECMCS.
In this paper, two common indexes are employed. Otherwise, a novel index for evaluating diversity of the solutions is proposed. The details of the three indexes are as follows.
GD is used to estimate the convergence of the algorithm. It measures how far the obtained set of nondominated solutions is from the real Pareto frontier. The definition of GD is as
The generalized spread metric SP measures the diversity of the obtained set of solutions [
In practical applications, the size of the obtained set of Pareto solutions may be different using different algorithms (in this research, NSGAII generates 100 Pareto optimal solutions while MODECMCS generates 30). At this situation, taking SP as the diversity evaluation indicator is unsuitable and inequitable here, as the SP value is very sensitive with the length of sample. The explanation is as follows.
Suppose
As
To overcome this problem, a novel index SPC is proposed in this paper. We convert the problem into calculating some indicator independent of sample length.
First, a series of distance DIS is constructed base on the definition of
Second, a series REF with the length
Then, the index SPC is defined as calculating the correlation coefficient between DIS and REF; the calculated equation is as
The value of SPC can be used to measure the diversity of generated Pareto optimal solutions, and the larger the SPC is, the better the diversity of the set of Pareto optimal solutions is.
In this part, the MODECMCS algorithm is employed to solve 5 benchmark problems. And the performance is compared with other reported results.
ZDT1, ZDT2, ZDT3, ZDT4, and ZDT6, are adopted, and the features of these test problems are given in Table
Benchmark test problems.
Problem 

Variable bounds  Objective functions  Optimal solutions  Comments 

ZDT1  30 



Convex 


ZDT2  30 



Nonconvex 


ZDT3  30 



Convex, discontinuous 


ZDT4  10 



Nonconvex 


ZDT6  10 



Nonconvex, nonuniform 
To make equal comparison with the results in [
The metrics GD and SP are selected to measure the performance of algorithms. The mean and variance of the values of the two metrics are calculated by MODECMCS on 30 individual tests, and the two metrics are shown in Table
Statistics of results on metrics GD and SP.
Index  Problems  NSGAII  MODECMCS  

Average  Std  Average  Std  
GD  ZDT1  0.033482  0.004750 

0.000055 
ZDT2  0.072391  0.031689 

0.000081  
ZDT3  0.114500  0.007940 

0.000073  
ZDT4  0.513053  0.118460 

0.000010  
ZDT6  0.296564  0.013135 

0.000011  


SP  ZDT1  0.390307  0.001876 

0.010209 
ZDT2  0.430776  0.004721 

0.007355  
ZDT3  0.738540  0.019706 

0.009318  
ZDT4  0.702612  0.064648 

0.013693  
ZDT6  0.668025  0.009923 

0.008142 
In this section, the performance of MODECMCS is compared with NSGAII on calibration of a threeparameter SVM based daily streamflow forecasting model. In this case study, we are especially concerned with the ability of finding the approximate nondominated frontier.
Figure
Location of the study area (Yangtze River Basin).
The daily streamflow forecasting model used in this research is a threeparameter SVM model, which is similar with the SVM model in [
The parameters of the daily streamflow forecasting SVM model are optimized by NSGAII and the proposed MODECMCS algorithm with two pairs of objective functions: mean squared logarithmic error (MLSE) versus mean squared derivative error (MSDE) and mean fourthpower error (M4E) versus mean squared derivative error (MSDE) [
The MSLE function is more suitable for low flows due to the logarithmic transformation while the M4E is considered as an indicator of goodnessoffit to high flows as larger deviations are given more contributions. The MSDE can be taken as an indicator of the fit of the shape of hydrograph. As the MSDE function does not take into account bias between observed and simulated value, so it cannot be used as the only objective function for calibration of hydrologic models; it should be used in combination with MSLE or M4E.
For NSGAII, the control parameters are preset as follows: the size of the population is set to 100, the crossover rate is set to 0.8, the mutation rate is set to 0.02, and the maximum number of function evaluations is set to 100000. And for MODECMCS, the following values of control parameters are selected: the size of the population is set to 100, the size of the archive set is set to 30 (to lower computation load), the threshold
The prediction results of the threeparameter SVM model are shown in Figures
Pareto plots of the prediction results when the objective functions are MSLE and MSDE.
Pareto plots of the prediction results when the objective functions are MSDE and M4E.
The prediction uncertainty ranges associated with the Pareto optimal solutions generated by MODECMCS algorithm when the objective functions are MSLE and MSDE.
The prediction uncertainty ranges associated with the Pareto optimal solutions generated by MODECMCS algorithm when the objective functions are MSLE and MSDE.
From Figures
Convergence performance of two algorithms.
Algorithm  GD (MSLE versus MSDE)  GD 

NSGAII  5.21 
2.96 
MODECMCS 


From Table
However, in multiobjective optimization, we usually have two goals. The first one is to generate an approximation Pareto optimal set which are as close as possible to the optimal frontier, and the second one is to maintain the diversity of the solutions. We have to emphasize that the second goal is as important as the first one. This means that generating a better set of Pareto optimal solutions is not enough. Thus, to enhance the diversity of the generated approximation Pareto optimal set, the improved distance assignment operator, which has been discussed in Section
The SPC values obtained from NSGAII and the proposed MODECMCS are given in Table
Comparisons of SPC between NSGAII and MODECMCS.
Algorithm  SPC (MSLE versus MSDE)  SPC 

NSGAII  0.996562  0.993955 
MODECMCS 


From the above analysis, it can be noted that the performance of the proposed MODECMCS in this paper is better than the NSGAII in terms of searching better Pareto optimal solutions and maintaining the diversity of the generated solutions.
Furthermore, the multiobjective calibration can help the modelers have a better understanding of the hydrology models. Figures
From Figures
This paper presents a novel multiobjective algorithm named MODECMCS for automatic calibration of hydrology model. Considering the drawback of the DE algorithm of being easily trapped in local minimum, the adaptive Cauchy mutation operator is introduced. And to generate better Pareto optimal solutions (which are closer to the true Pareto optimal frontier), we employ the local Chaos searching operator. Forward, the MODECMCS uses a more precise crowd distance assignment than that of the NSGAII. Besides, as the traditional generalized spread metric SP is sensitive with the size of Pareto set, a novel diversity performance metric SPC, which is independent of Pareto set size, is proposed in this research.
The proposed MODECMCS algorithm is firstly employed to solve 5 benchmark problems. The results present that the MODECMCS is more powerful than NSGAII in both providing an approximation of the true Pareto frontier and maintaining the diversity of Pareto set. Furthermore, the performance of the proposed MODECMCS algorithm for generating approximate Pareto optimal solutions is compared with the NSGAII algorithm for calibration of a threeparameter SVM based daily streamflow forecasting model. And two pairs of objective functions, which are MSLE versus MSDE and MSDE versus M4E, are considered. The results of the case study also prove that the performance of the proposed MODECMCS is better than the NSGAII in terms of both searching better Pareto optimal solutions and maintaining the diversity of the generated solutions. Moreover, the multiobjective calibration of hydrology models can help us understand the models better and then put forward improving proposal.
The authors declare no competing interests.
Yi Liu and Jun Guo conceived and designed the experiments; Huaiwei Sun and Yueran Wang performed the experiments; Yueran Wang and Wei Zhang developed the model code and performed the simulations; Yi Liu and Jun Guo prepared the paper with contributions from all coauthors.
The authors appreciate the support from the National Natural Science Foundation Project of China (NSFC) (no. 51509095), the CRSRI Open Research Program (Program SN:CKWV2014220/KY), and the State Key Program of State Grid Hunan Electric Power Company of China (no. 5216A514003Z).