The multiobjective evolutionary annealing-simplex method and its application in calibrating hydrological models

Abstract

Optimisation problems related to water resources are, by nature, multiobjective, even if traditionally handled as single-objective. Current advances in hydrological modelling employ multiobjective approaches to treat the well-known problem of "equifinality", thus assessing the uncertainties related to the parameter estimation procedure. However, computer tools generating Pareto-optimal solutions are still particularly time-consuming, especially in real-world applications, with many parameters and many fitting criteria. Moreover, the mathematical concept of Pareto optimality often leads to solutions that are far away from an acceptable compromise between the conflicting objectives. Most multiobjective optimisation tools are adaptations of evolutionary algorithms. In multiobjective evolutionary optimisation two are the major goals: (a) guiding the search towards the Pareto-optimal front, and (b) generating a well-distributed set of nondominated solutions. Both are achieved through the fitness evaluation and selection procedures; using the fundamental principle of dominance, scalar fitness values are assigned to individuals, then evolved by employing the typical genetic operators (crossover, mutation). The multiobjective evolutionary annealing-simplex (MEAS) method is an innovative scheme, also comprising an evaluation phase and an evolution phase. The evaluation aims to assign a performance measure to each member of the population, which requires the comparison of all individuals against each other and against all criteria. A fitness strategy inspired from the strength-Pareto approach of Zitlzer and Thiele (IEEE Trans. Evol. Comp., 3(4), 1999), in addition to an extension of the definition of dominance, provides a large variety of discrete performance values. The population is guided towards a promising sub-region of the Pareto front (not the entire front), that contains representative trade-offs, among which the best-compromise may easily detected. The generation of solutions with extreme performance, i.e. too good against some criteria, too bad for the rest ones, is prohibited, by means of penalty functions. In this manner, the discrete fitness space is transformed to a continuous space, which may be explored through global search techniques. The latter (i.e., the evolution phase) is implemented through a set of combined deterministic and stochastic transition rules, most of them based on a simplex-evolving pattern. During evolution, the degree of randomness is controlled through an adaptive annealing cooling schedule, which automatically regulates the "temperature" of the system. The MEAS method was tested on a variety of benchmark functions taken from the literature, as well as on some challenging hydrological applications, formerly handled through weighted objective functions. The analysis indicate that the proposed algorithm locates good trade-offs among the conflicting objectives simultaneously being much more efficient if compared to other, well-established multiobjective evolutionary schemes.

Full text

European Geosciences Union (EGU) - General Assembly
Vienna, Austria, 25 - 29 April 2005
Session HS1: Hydroinformatics
The multiobjective evolutionary
annealing-simplex method
and its application in calibrating
hydrological models
Andreas Efstratiadis and Demetris Koutsoyiannis
Department of Water Resources, School of Civil Engineering,
National Technical University, Athens, Greece

Efstratiadis and Koutsoyiannis, The MEAS method and its application in calibrating hydrological models 2
Hydrological modelling and multiobjective
parameter estimation: The motivation
¾Complex (semi- or fully-distributed) models generate multiple output variables
at various sites →need for faithful reproduction of all model responses,
that are representative of the watershed behaviour
¾Due to the large number of parameters and their highly nonlinear interactions,
alternative sets with similarly good performance may be detected (the
“equifinality” problem) →need for establishment of “behavioural” (i.e.,
realistic, reliable and stable) parameter sets
¾Models are too weak against data and structural errors →need to assess
the sensitivity of parameters and the model predictive uncertainty
¾Multiple error measures, when aggregated to a single objective function,
formulate response surfaces that are strongly related to the aggregation
scheme →need to distinguish the optimisation criteria, to avoid scaling
problems and to investigate possible contradictory interactions
¾Automatic calibration methods, involving too extended, high-dimensional and
non-convex search spaces, are easily trapped by local optima or other
peculiarities →need for reducing the parameter boundaries, to assist the
searching procedure

Efstratiadis and Koutsoyiannis, The MEAS method and its application in calibrating hydrological models 3
Multiobjective optimisation: The story so far
¾“Philosophical” foundation (1880-1900): the concept of Pareto-Edgeworth
optimum, applied in sociology and welfare economics
¾Mathematical foundation (1950-1960): formulation of the vector maximum
problem by Kuhn and Tucker and first engineering applications
¾Plain aggregating approaches (1970): a priori definition of the best
compromise decision set, through the formulation of utility functions based on
weighting coefficients, articulation of preferences, goal-vectors, etc.
¾Population-based non-Pareto approaches (1980): formulation of sub-sets,
each one evaluated according to different criterion (by switching objectives),
and next shuffled and evolved through crossover and mutation (VEGA)
¾Dominance-based evolutionary approaches (1990): use of ranking
procedures, based on the principle of Pareto optimality, and techniques to
maintain diversity through fitness sharing, to generate representative trade-
offs among conflicting objectives (MOGA, NSGA, NPGA)
¾Modern approaches: revision of multiobjective evolutionary schemes, with
emphasis on efficiency, using faster ranking techniques, clustering methods
and elitism mechanisms (SPEA, SPEA-II, NSGA-II, PAES, MOMGA, etc.)

Efstratiadis and Koutsoyiannis, The MEAS method and its application in calibrating hydrological models 4
Multiobjective evolutionary algorithms:
General principles
1. According to the principle of dominance, a rank measure riis assigned to
each individual or group of individuals, where the best (lower) value
corresponds to non-dominated points, thus guiding the search towards the
Pareto front; a variety of rank values protects from high selection pressure.
2. A density measure σiis assigned to individuals, using sharing functions or
nearest neighbour techniques, to maintain diversity within population, thus
favouring the generation of well-distributed sets.
3. The selection process is implemented applying typical mechanisms (e.g.,
roulette, tournament), on the basis of dummy fitness of the form φi= φ(ri, σi).
4. The evolution process is implemented using the typical genetic operators.
In multiobjective evolutionary
search, due to the use of the
concept of dominance in fitness
evaluation, a discrete response
surface is created, which is
reformed at each generation.
A well-distributed
set, representative
of the Pareto front
f2
f1

Efstratiadis and Koutsoyiannis, The MEAS method and its application in calibrating hydrological models 5
Applying multiobjective evolutionary algorithms
for model calibration: Some drawbacks
¾Search is computationally demanding, especially in the case of complex
models with many parameters.
¾There is too little experience regarding problems with more than two criteria.
¾Fitting criteria are conflicting only in case of ill-posed structures or data.
¾The concept of dominance is not necessarily consistent with the concept of
“equifinality”; hence multiobjective search may result to non-behavioural,
albeit Pareto optimal, parameter sets, providing extreme performance, i.e. too
good against some criteria, too bad against the rest ones.
¾A best-compromise parameter set is required for operational purposes.
f1
f2
“Smooth” non-
dominated front; all
points correspond
to “behavioural”
parameter sets
f1
f2
Only a small part
of the Pareto front
corresponds to
“behavioural”
parameter sets

Efstratiadis and Koutsoyiannis, The MEAS method and its application in calibrating hydrological models 6
The multiobjective evolutionary annealing-
simplex (MEAS) method
Phase 1: Evaluation
A performance measure (fitness) is assigned, consisting of:
•a rank measure, based on a strength-Pareto scheme, which both ensures
convergence to the real Pareto front and diversity preservation;
•an indifference measure for further discrimination of indifferent solutions in
case of multiple (more than two) objectives;
•a feasibility measure, for guiding search toward a desirable region of the
Pareto front, thus providing acceptable trade-offs among conflicting objectives.
Phase 2: Evolution
Evolution is implemented according to transition rules that are based on a
simplex-annealing approach, where:
•a downhill simplex pattern, combining both deterministic and stochastic
transition rules, is employed for offspring generation;
•an adaptive annealing cooling schedule is used to control the degree of
randomness during evolution.

Efstratiadis and Koutsoyiannis, The MEAS method and its application in calibrating hydrological models 7
The MEAS method: Fitness assignment through
a strength-Pareto approach
0 (2)
0 (3)
0 (3)
0 (2)
2 (1) 13 (0)
3 (1)
6 (1)
5 (0)
Strength =
number of
dominated
individuals
Rank = sum
strength of all
dominators
Non-dominated
individuals have
zero rank
¾The concept is based on the SPEA
and SPEA-II methods (Zitzler and
Thiele, 1999; Zitzler et al., 2002).
¾For each individual, both
dominating and dominated points
are taken into account.
¾Formulates a integral response
surface that changes whenever a
new individual is generated.
¾Provides a large variety of rank
values (larger than any other known
ranking algorithm), as well as a sort
of “niching” mechanism, to preserve
population diversity.
¾A non-integral term is added to
fitness, to penalise individuals
excelling in fewer criteria than other
indifferent ones, with identical rank

Efstratiadis and Koutsoyiannis, The MEAS method and its application in calibrating hydrological models 8
The MEAS method: Restricting the feasible
objective space
¾Based on a concept inspired from the goal-programming method.
¾Requires the specification of a constraint vector ε= (ε1, …, εm) denoting the
boundaries of a desirable (“feasible”) region of the objective space.
¾Ensures a better insight on the most promising parts of the Pareto front,
where the best-compromise parameter set is suspected to be sited.
f2
f1
ε1
Constraint
vector ε
Non-feasible,
albeit optimal
sub-front
Computational steps
1. The maximum fitness value is
computed, i.e. Φ= max φ(i).
2. Each individual iis checked whether it
lies within the feasible space; if xij > εj
for the jth criterion, a square distance
penalty ∆εij = (xi–εi)2is added to φ(i).
3. All infeasible individuals are further
penalised by adding Φ; hence, they
become worse than any other feasible
individual, either dominated or not.
ε2
Optimal
sub-front
Feasible
objective
space

Efstratiadis and Koutsoyiannis, The MEAS method and its application in calibrating hydrological models 9
The MEAS method: A selection procedure
based on a simulated annealing strategy
f(i) = φ(i) + p(i) + rT
Deterministic component, y(i)Stochastic component, s(i)
Penalty measure
Dominance
term, φ(i)
Feasibility
term, p(i)
Unit random
number, r
Current system’s
temperature, T
Favours the survival
of feasible and non-
dominated solutions
Provides flexibility,
to escape from local
optima and handle
peculiarities of non-
convex spaces

Efstratiadis and Koutsoyiannis, The MEAS method and its application in calibrating hydrological models 10
The MEAS method: Evolving population
1. According to an elitism concept, the population is divided to non-dominated
(φ< 1) and dominated (φ> 1) individuals.
2. The system temperature is regulated in order to not exceed Tmax = ξ∆y,
where ξ≥1 parameter of the annealing cooling schedule and ∆ythe
difference between the best and worst fitness of current population.
3. From the entire population n+ 1 points are picked up, thus forming a
simplex in the n-dimensional search space; at least one simplex vertex is
selected from the dominated set, given that the latter is not empty.
4. The “weakest” individual wis detected by means of maximisation of f.
5. A crossover scheme is employed on the basis of a downhill simplex pattern;
if a better point x΄(“offspring”) is located, it replaces wand the temperature is
reduced by λ, where λ< 1 parameter of the annealing cooling schedule.
6. If recombination fails (i.e., any better solution cannot be found), the offspring
is generated via a random perturbation (mutation) of w, i.e. x΄= w+ ∆x.
For an earlier, single-objective implementation of the evolutionary annealing-simplex
method see: Efstratiadis and Koutsoyiannis (2002), Rozos et al. (2004)

Efstratiadis and Koutsoyiannis, The MEAS method and its application in calibrating hydrological models 11
The MEAS method: Simplex configurations
Outside contraction
Inside
contraction
Multiple expansion
(one-dimensional
minimisation)
Centroid Reflection
Weakest
vertex, wOffspring, x΄Offspring
Trial 3
(offspring)
Offspring
Trial 1
(reflection)
Trial 2

Efstratiadis and Koutsoyiannis, The MEAS method and its application in calibrating hydrological models 12
Performance assessment of MEAS method:
Test function SCH-2
0
2000
4000
6000
8000
10000
12000
-50 0 50 100
Initial set
0.0
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
-1.0 -0.5 0.0 0.5 1.0
Final set Pareto front
¾Taken from Schaffer
(1984)
¾Single control
variable, in the range
[-100, 100]
¾Extended feasible
objective space
¾Disconnected Pareto
set (1 ≤x≤2 and 4 ≤
x≤5)
¾Disconnected and
convex Pareto front
¾Population size = 100
¾Convergence to a non-dominated
set after 9366 function evaluations

Efstratiadis and Koutsoyiannis, The MEAS method and its application in calibrating hydrological models 13
Performance assessment of MEAS method:
Test function KUR
-15
-10
-5
0
5
10
15
20
25
-20 -15 -10 -5
Initial set After 25000 evaluations Final set
¾Taken from Kursawe
(1991)
¾3 control variables, in
the range [-5, 5]
¾Non-convex Pareto front
¾Population size = 100
¾Convergence to a non-
dominated set after
37563 function
evaluations

Efstratiadis and Koutsoyiannis, The MEAS method and its application in calibrating hydrological models 14
Performance assessment of MEAS method:
Test function POL
0
5
10
15
20
25
30
35
40
0 102030405060
Initial set Final set
¾Taken from Poloni
(1997)
¾Two control variables, in
the range [-π, π]
¾Non-convex and
disconnected Pareto
front
¾Population size = 100
¾Convergence to a non-
dominated set after 2218
function evaluations

Efstratiadis and Koutsoyiannis, The MEAS method and its application in calibrating hydrological models 15
Performance assessment of MEAS method:
Test function ZDT-2
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
0.0 0.2 0.4 0.6 0.8 1.0
Initial set Final set
Pareto front Non-dominated set
¾Taken from Zitzler et al.
(2000)
¾30 control variables, in
the range [0, 1]
¾Pareto set: 0 ≤x1≤1
and xi= 0, for i= 2,.., 30
¾Non-convex Pareto front
¾Population size = 100
¾Convergence to a
locally non-dominated
set after 16080 function
evaluations
¾Final set obtained after
25000 function
evaluations

Efstratiadis and Koutsoyiannis, The MEAS method and its application in calibrating hydrological models 16
Performance assessment of MEAS method:
Test function ZDT-3
-2.0
-1.0
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
0.0 0.2 0.4 0.6 0.8 1.0
Initial set Final set F-boundary
¾Taken from Zitzler et al.
(2000)
¾10 control variables, in
the range [0, 1]
¾Disconnected Pareto
set: 0 ≤x1≤1 and xi=
0, for i= 2,.., 10
¾Convex and
disconnected Pareto
front
¾Population size = 100
¾Convergence to a non-
dominated set after
12944 function
evaluations

Efstratiadis and Koutsoyiannis, The MEAS method and its application in calibrating hydrological models 17
Performance assessment of MEAS method:
Test function ZDT-6
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
0.30.50.70.9
Initial set Non-dominated set
Pareto front Final set
¾Taken from Zitzler et al.
(2000)
¾10 control variables, in
the range [0, 1]
¾Pareto set: 0 ≤x1≤1
and xi= 0, for i= 2,.., 10
¾Non-convex and non-
uniformly distributed
Pareto front
¾Population size = 100
¾Final set, with
satisfactory spread of
non-dominated points,
found after 150000
function evaluations

Efstratiadis and Koutsoyiannis, The MEAS method and its application in calibrating hydrological models 18
Multiobjective calibration of a complex
hydrological model: Study area
¾Watershed area ~2000 km2, with highly non-linear
interactions between surface and groundwater
processes and man-made interventions.
¾Main modelling issues:
a semi-distributed schematisation of the
hydrographic network;
a conceptualisation of surface processes,
based on spatial elements with homogenous
characteristics (hydrological response units,
HRU) and fitting to each one a soil moisture
accounting model of six parameters;
a multi-cell groundwater scheme, with two
parameters assigned to each cell;
a water management model, estimating the
optimal system fluxes (flows, abstractions).
¾Model components: 5 sub-basins, 6 HRU, 35
groundwater cells

Efstratiadis and Koutsoyiannis, The MEAS method and its application in calibrating hydrological models 19
Multiobjective calibration of a complex
hydrological model: Main assumptions
¾Observed series: daily discharge measurements at the basin outlet (Karditsa
tunnel), sparse (1-2 per month) discharge measurements at six main karstic
springs, contributing more than 50% of total runoff
¾Control period: October 1984-September 1990 (calibration period), October
1990-September 1994 (validation period)
¾Calibration criteria: determination coefficients of monthly discharge series at
the basin outlet and the main spring sites (number of objectives = 7)
¾Control variables: soil moisture capacity (K) and recession rate for
percolation (µ), assigned to each HRU, conductivity (C) of each virtual cell
that represents spring dynamics (search space dimension = 18)
¾Feasible search space: 0 < Ki< 1000 (in mm), 0 < µi< 1 (dimensionless),
0.000001 < Ci< 0.5 (in m/s)
¾Algorithmic inputs: sample size = 50, maximum function evaluations = 5000
¾Other model parameters: obtained through an earlier single-objective
optimisation scenario, based on a weighted objective function and handled by
combining automatic and manual calibration methods (Rozos et al., 2004)

Efstratiadis and Koutsoyiannis, The MEAS method and its application in calibrating hydrological models 20
Multiobjective calibration of a complex hydrological
model: Characteristic trade-offs
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
0.20 0.25 0.30 0.35 0.40 0.45 0.50
Error for Mavroneri springs runoff
Error for Melas spring runoff
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.20 0.25 0.30 0.35
Error for Lilea springs runoff
Error for Mavroneri
springs runoff
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
0.12 0.13 0.14 0.15 0.16 0.17
Error for Karditsa runoff
Error for Melas spring runoff
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
0.20 0.25 0.30 0.35
Error for Lilea springs runoff
Error for Melas springs
runoff
Trade-offs represent: (a) modelling errors due to the complexity of processes
(negative correlation of some spring hydrographs with precipitation); and (b)
data errors, due to the construction of control series based on few observations

Efstratiadis and Koutsoyiannis, The MEAS method and its application in calibrating hydrological models 21
Multiobjective calibration of a complex hydrological
model: Restricting the objective space
-3.50
-3.00
-2.50
-2.00
-1.50
-1.00
-0.50
0.00
0.50
1.00
Determ ination coefficient
Minimum
Maximum
Optimal
-3.00
-2.50
-2.00
-1.50
-1.00
-0.50
0.00
0.50
1.00
Determ ination coefficient
Minimum
Maximum
Optimal
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
Karditsa
Lilea
Mavroneri
Ag.
Paraskevi
Erkyna
Melanas
Polygyra
Determ ination coefficient
Minimum
Maximum
Optimal
-0.50
-0.25
0.00
0.25
0.50
0.75
1.00
Karditsa
Lilea
Mavroneri
Erkyna
Melanas
Determination coefficient
Minimum
Maximum
Optimal
Unbounded objective space, calibration Unbounded objective space, validation
Bounded objective space, calibration Bounded objective space, validation

Efstratiadis and Koutsoyiannis, The MEAS method and its application in calibrating hydrological models 22
Concluding remarks
¾Despite the impressive progress of last years regarding the development of
evolutionary multiobjective optimisation techniques, limited experience
exist on operational applications of hydrological interest, and most of them
restricted to two-dimensional objective spaces.
¾When fitting hydrological models on numerous observed responses,
irregular Pareto fronts are formed due to structural and data errors.
¾In case of complex, ill-posed hydrological models with many parameters, a
multiobjective calibration approach is necessary to:
reduce uncertainties regarding the parameter estimation procedure;
investigate acceptable trade-offs between optimisation criteria;
guide the search towards promising areas of both the objective and the
parameter space.
¾The MEAS algorithm is an innovative scheme, suitable for challenging
hydrological calibration problems, which combines: (a) a fitness evaluation
procedure based on a strength-Pareto approach and a feasibility concept,
(b) an evolving pattern based on the downhill simplex method, and (c) a
simulated annealing strategy, to control randomness during evolution.

Efstratiadis and Koutsoyiannis, The MEAS method and its application in calibrating hydrological models 23
References
¾Efstratiadis, A., and D. Koutsoyiannis, An evolutionary annealing-simplex algorithm for global
optimisation of water resource systems, in Proceedings of the Fifth International Conference on
Hydroinformatics, Cardiff, UK, 1423-1428, IWA, 2002.
¾Kursawe, F., A variant of evolution strategies for vector optimization, in Parallel Problem Solving from
Nature, H. P. Schwefel and R. Manner (editors), 193-197, Springer-Verlag, Berlin, 1991.
¾Poloni, C., Hybrid GA for multiobjective aerodynamic shape optimization, in Genetic Algorithms in
Engineering and Computer Science, G. Winter, J. Periaux, M. Galan, and P. Cuesta (editors), New
York, Wiley, 397-414, 1997.
¾Rozos, E., A. Efstratiadis, I. Nalbantis, and D. Koutsoyiannis, Calibration of a semi-distributed model
for conjunctive simulation of surface and groundwater flows, Hydrological Sciences Journal, 49(5),
819-842, 2004.
¾Shaffer, J. D., Multiple objective optimization with vector evaluated genetic algorithms, in Proceedings
of an International Conference on Genetic Algorithms and their Applications, J. J. Grefenstette
(editor), 93-100, Pittsburgh, 1985.
¾Zitzler, E., K. and L. Thiele, Multiobjective evolutionary algorithms: A comparative case study and the
strength pareto approach, IEEE Transactions on Evolutionary Computation, 3(4), 257-271, 1999.
¾Zitzler, E., K. K. Deb, and L. Thiele, Comparison of multiobjective evolutionary algorithms: Empirical
results, Evolutionary Computation, 8(2), 173-195, 2000.
¾Zitzler, E., M. Laumanns, and L. Thiele, SPEA 2: Improving the strength Pareto evolutionary algorithm
for multiobjective optimization, in Evolutionary Methods for Design, Optimization and Control, K.
Giannakoglou, D. Tsahalis, J. Periaux, K. Papailiou, and T. Fogarty (editors), 19-26, Barcelona, 2002.

Efstratiadis and Koutsoyiannis, The MEAS method and its application in calibrating hydrological models 24
This presentation is available on-line at:
http://itia.ntua.gr/e/docinfo/644
Poster presentation of the hydrological model:
Friday, 29 April 2005, 17:30 - 19:00, area Z028
Contact info:
andreas@itia.ntua.gr
dk@itia.ntua.gr