07_12_19 Multi Objective Calibration (Mahyar)

Multi-Objective Evolutionary Optimization; Concept and Application to Calibration of

Rainfall-Runoff Model

Mahyar Shafii

December 2007

Table of ContentsTable of Contents

IntroductionIntroduction Classical Methods to Solve Multi-Objective Optimization Classical Methods to Solve Multi-Objective Optimization

ProblemsProblems Evolutionary Algorithm (EA) TerminologyEvolutionary Algorithm (EA) Terminology Multi-Objective Evolutionary Algorithms (MOEAs)Multi-Objective Evolutionary Algorithms (MOEAs) MOEA Application to Calibration of Conceptual Rainfall-Runoff MOEA Application to Calibration of Conceptual Rainfall-Runoff

ModelsModels Literature ReviewLiterature Review Concluding RemarksConcluding Remarks

Development of an Improved NSGA-IIDevelopment of an Improved NSGA-II

IntroductionIntroduction

Multi-Objective OptimizationMulti-Objective Optimization

Optimal solution in single-objective optimization is clearly defined.Optimal solution in single-objective optimization is clearly defined. In multi-objective optimization there is rather a set of alternative In multi-objective optimization there is rather a set of alternative

trade-offs, generally known as “trade-offs, generally known as “Pareto-OptimalPareto-Optimal” solutions. ” solutions.

Real-World ProblemReal-World Problem

Several incommensurable and

often competing objectives


Multi-Objective OptimizationMulti-Objective Optimization Basic Concept and TerminologyBasic Concept and Terminology

Find a vector of solutions:Find a vector of solutions:

mm inequality constraints: inequality constraints:

pp equality constraints: equality constraints:

kk objective functions objective functions

so that,so that,

Tnxxxx ],...,,[ **

2*

1*

mixgi ,...,2,10)(

pixhi ,...,2,10)(

Tk xfxfxfxf )](),...,(),([)( 21

)()(( * xfxf iiFx


Pareto Optimum conceptPareto Optimum concept

Vilfredo Pareto (1896)Vilfredo Pareto (1896)

XX** ЄЄ F FX X ЄЄ F F

or, there is at least one i or, there is at least one i ЄЄ I so that I so that

)()()( * IiXfXf ii

)()( * XfXf ii

f 2

f 1

Feasible Region

None of solutions in Pareto None of solutions in Pareto optimal set can be identified as optimal set can be identified as

better than the others unless better than the others unless preference information is preference information is

included (e.g. a ranking of the included (e.g. a ranking of the objectives).objectives).

Traditional ApproachesTraditional Approaches

Aggregating the objectives into a single and Aggregating the objectives into a single and parameterized objective function and performing several parameterized objective function and performing several runs with different parameter settings to achieve a set of runs with different parameter settings to achieve a set of solutions approximating the Pareto-optimal set.solutions approximating the Pareto-optimal set.

Weighting Method (Cohon, 1978)Weighting Method (Cohon, 1978)

Constraint Method (Cohon, 1978)Constraint Method (Cohon, 1978)

Goal Programming (Steuer, 1986)Goal Programming (Steuer, 1986)

Minimax Approach (Koski, 1984)Minimax Approach (Koski, 1984)

Traditional ApproachesTraditional Approaches

Difficulties with classical methods:Difficulties with classical methods:

Being sensitive to the shape of the Pareto-optimal front (e.g. Being sensitive to the shape of the Pareto-optimal front (e.g. weighting method).weighting method).

Need for problem knowledge which may not be available.Need for problem knowledge which may not be available.

Restrictions on their use in some application areas (Deb, 1999).Restrictions on their use in some application areas (Deb, 1999).

Need to several optimization runs to achieve the best parameter Need to several optimization runs to achieve the best parameter setting to obtain an approximation of the Pareto-optimal set.setting to obtain an approximation of the Pareto-optimal set.

Evolutionary Algorithms (EAs)Evolutionary Algorithms (EAs)

The term evolutionary algorithm (EA) stands for a class The term evolutionary algorithm (EA) stands for a class of stochastic optimization methods that simulate the of stochastic optimization methods that simulate the process of natural evolution. They are meta-heuristics process of natural evolution. They are meta-heuristics that attempt to apply the principles of neo-Darwinian that attempt to apply the principles of neo-Darwinian evolution to the creation of artificial intelligence evolution to the creation of artificial intelligence (machine learning) and to optimization.(machine learning) and to optimization.

Origins of EAs: Firstly proposed in the late 1950s Origins of EAs: Firstly proposed in the late 1950s leading to development of several EAs since the 1970s, leading to development of several EAs since the 1970s, mainly (Bäck, Hammel, and Schwefel 1997)mainly (Bäck, Hammel, and Schwefel 1997) Genetic Algorithms (GA)Genetic Algorithms (GA) Evolutionary Programming (EP) Evolutionary Programming (EP) Evolution Strategies (ES)Evolution Strategies (ES)


Basic Principles of EABasic Principles of EA Genotype versus PhenotypeGenotype versus Phenotype

Genotype is underlying genetic coding (Genes in GA)Genotype is underlying genetic coding (Genes in GA) Phenotype is expression of that coding forming a possible Phenotype is expression of that coding forming a possible

solution (Chromosome in GA)solution (Chromosome in GA) Mapping between G-space & P-spaceMapping between G-space & P-space SelectionSelection

giving a chance to each solution to reproduce a certain number of giving a chance to each solution to reproduce a certain number of times, dependent on their quality or so-called fitness values. times, dependent on their quality or so-called fitness values.

VariationVariation Imitating natural capability of creating ”new” living beings by Imitating natural capability of creating ”new” living beings by

means of recombination and mutation. means of recombination and mutation.


ATGCC

AGTCAGCACC

TGTCC Recombined offspring

ATGCCGCACCTGTCCAGTCA Parent chromosomes

• Recombination involves swapping sections of two individuals’ characteristics.

Note: This is not what occurs in nature

ATGCC

AGTCAGCACC

TGTCCA

T Random mutations in genetic composition

• Mutation results in a random change in one or more of an individual’s characteristics.


Recombination and produce offspring

Finish

Yes

Generating a set of individuals (Population)

Termination Criteria

Parent selection

No

Multi-Objective Evolutionary Multi-Objective Evolutionary Algorithms (MOEAs)Algorithms (MOEAs) Evolutionary algorithms do better than other Evolutionary algorithms do better than other

blind search strategies in multi-objective blind search strategies in multi-objective optimization (Fonseca and Fleming (1995); optimization (Fonseca and Fleming (1995); Valenzuela-Rend´on and Uresti-Charre Valenzuela-Rend´on and Uresti-Charre (1997)).(1997)).

At first, they were applied by At first, they were applied by functions functions aggregation.aggregation.

More recently, MOEAs were designed to More recently, MOEAs were designed to search decision spaces for the optimal tradeoffs search decision spaces for the optimal tradeoffs among a vector of objectives (Coello Coello, among a vector of objectives (Coello Coello, 2002). 2002).

Multi-Objective Evolutionary Multi-Objective Evolutionary Algorithms (MOEAs)Algorithms (MOEAs) Some representatives of MOEAs in operational Some representatives of MOEAs in operational

research through past years:research through past years:

a)a) Non-Dominated Sorting genetic Algorithm (NSGA), Srinivas Non-Dominated Sorting genetic Algorithm (NSGA), Srinivas et Deb, 1995.et Deb, 1995.

b)b) NSGA-II, Deb et al., 2002.NSGA-II, Deb et al., 2002.

c)c) Strength Pareto Evolutionary Algorithm (SPEA), Zitzler and Strength Pareto Evolutionary Algorithm (SPEA), Zitzler and Thiele, 1999.Thiele, 1999.

d)d) SPEA2, Zitzler et al., 2001.SPEA2, Zitzler et al., 2001.

e)e) Epsilon-NSGAII, Kollat and Reed, 2005.Epsilon-NSGAII, Kollat and Reed, 2005.

f)f) Multi-objective Shuffled Complex Evolution Metropolis Multi-objective Shuffled Complex Evolution Metropolis Algorithm (MOSCEM-UA), Vrugt et al., 2003.Algorithm (MOSCEM-UA), Vrugt et al., 2003.

MOEA Applications in Calibration of MOEA Applications in Calibration of Conceptual Rainfall-Runoff ModelsConceptual Rainfall-Runoff Models Conceptual rainfall-runoff (CRR) modelsConceptual rainfall-runoff (CRR) models

Calibration of RR models is a process in which Calibration of RR models is a process in which parameter adjustment is made so as to match parameter adjustment is made so as to match (as closely as possible) the dynamic behavior (as closely as possible) the dynamic behavior of the RR model to the observed behavior of of the RR model to the observed behavior of the catchment. the catchment.

MOEA Applications in Calibration of MOEA Applications in Calibration of Conceptual Rainfall-Runoff ModelsConceptual Rainfall-Runoff Models

PurelyRandom

Techniques

Local and GlobalSearch Algorithms

Evolutionary-PopulationBased Approaches Lumped Modeling Distributed Modeling

Literature ReviewLiterature Review

Some calibration results reveal that moving from a Some calibration results reveal that moving from a lumped model structure to a semi-distributed model lumped model structure to a semi-distributed model structure improves the simulation results (Ajami et al., structure improves the simulation results (Ajami et al., 2004).2004).

Single Objective Calibration Scheme

Multi-Objective Calibration Scheme

Gupta et al. (1998) have discussed the advantages of a Gupta et al. (1998) have discussed the advantages of a multiple-objective representation of the model multiple-objective representation of the model calibration problem and this scheme has been shown to calibration problem and this scheme has been shown to be applicable and desirable.be applicable and desirable.


Lumped Hydrological Modeling:Lumped Hydrological Modeling: Development an algorithmDevelopment an algorithm

Yapo et al. (1998): MOCOM-UAYapo et al. (1998): MOCOM-UA Cheng et al. (2002): Fuzzy Global Optimization and GACheng et al. (2002): Fuzzy Global Optimization and GA Khu et al. (2005): NSGA-II and kNNKhu et al. (2005): NSGA-II and kNN

Proposing General FrameworkProposing General Framework Wagener (2001)Wagener (2001)

Multi-Objective nature with aggregated functionMulti-Objective nature with aggregated function Madsen (2000)Madsen (2000) Seibert (2000)Seibert (2000)

Comparison between single and multiple objective formulationsComparison between single and multiple objective formulations Seibert (2000)Seibert (2000) Chahinian and Moussa (2007)Chahinian and Moussa (2007)

Working on objective functionsWorking on objective functions Yu and Yang (2000): Fuzzy Multi-Objective Function (FMOF)Yu and Yang (2000): Fuzzy Multi-Objective Function (FMOF)


Distributed Hydrological Modeling:Distributed Hydrological Modeling:

Proposing General FrameworkProposing General Framework Madsen (2003): Aggregated Objective FunctionMadsen (2003): Aggregated Objective Function

Development an algorithmDevelopment an algorithm Cheng et al. (2006): Following previous work by the same authors.Cheng et al. (2006): Following previous work by the same authors. Bekele and Nicklow (2007): NSGA-II for calibration of SWATBekele and Nicklow (2007): NSGA-II for calibration of SWAT

Comparison between single and multiple objective formulationsComparison between single and multiple objective formulations Schoups et al. (2005): Subsurface modelingSchoups et al. (2005): Subsurface modeling Parajka et al. (2007)Parajka et al. (2007)

Sensitivity and uncertainty analysisSensitivity and uncertainty analysis Muleta and Nicklow (2005): SWAT, but in a single-objective schemeMuleta and Nicklow (2005): SWAT, but in a single-objective scheme Parajka et al. (2007)Parajka et al. (2007)

Although a majority of prior studies have focused on CRR applications, there are an increasing number of recent studies focusing on developing multi-objective calibration strategies for distributed hydrological models such as:


Remarks and recommendationsRemarks and recommendations Modification to the study of Madsen (2003) in order to Modification to the study of Madsen (2003) in order to

develop an improved framework for calibration of RR develop an improved framework for calibration of RR process in distributed hydrological models process in distributed hydrological models considering:considering:

Application of MOEAs and resolving the problem Application of MOEAs and resolving the problem in a multi-objective formulation instead of function in a multi-objective formulation instead of function aggregation technique to obtain Pareto optimum.aggregation technique to obtain Pareto optimum.

Constraining input parameters.Constraining input parameters.


Remarks and recommendationsRemarks and recommendations Proper study on application of hybrid EAProper study on application of hybrid EA

Developing a framework to establish a criterion to choose a Developing a framework to establish a criterion to choose a solution among Pareto-optimum solutions and state as the solution among Pareto-optimum solutions and state as the final solution of the problemfinal solution of the problem

As the main weakness of MOEAs is that they require a As the main weakness of MOEAs is that they require a large number of function evaluations through consumption large number of function evaluations through consumption of a great deal of time, it would be promising to direct of a great deal of time, it would be promising to direct efforts towards application of meta-modeling to reduce the efforts towards application of meta-modeling to reduce the number of simulations. number of simulations.

Development of Improved NSGA-IIDevelopment of Improved NSGA-II

NSGA-II, Deb et al. (2002)NSGA-II, Deb et al. (2002)


InnovationsInnovations Application of Heuristic Genetic Operators Application of Heuristic Genetic Operators

(Crossover and Mutation)(Crossover and Mutation) Heuristic parent-centric recombination (PCX) operatorHeuristic parent-centric recombination (PCX) operator Adaptation by Fuzzy Logic Controller (FLC) Adaptation by Fuzzy Logic Controller (FLC)

Mathematical Test ProblemsMathematical Test Problems ZDT1, ZDT2, ZDT3, ZDT4, ZDT6, (Deb et al., 2002)ZDT1, ZDT2, ZDT3, ZDT4, ZDT6, (Deb et al., 2002)


Metrics of PerformanceMetrics of Performance Diversity Metrics Convergence MetricsDiversity Metrics Convergence Metrics


Results and ConclusionsResults and Conclusions

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ZDT1 ZDT2 ZDT3 ZDT4 ZDT6

Mathematical Test Functions

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Initial NSGA-II

Improved NSGA-II

Diversity MetricsDiversity Metrics


Results and ConclusionsResults and Conclusions

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ZDT1 ZDT2 ZDT3 ZDT4 ZDT6

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Initial NSGA-II

Improved NSGA-II

Convergence MetricsConvergence Metrics

Thanks for your kind attentionThanks for your kind attention

07_12_19 Multi Objective Calibration (Mahyar)

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