computational intelligence in modelling aquatic processes...
TRANSCRIPT
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Computational intelligence in modelling aquatic processes,
optimizing models, and representing uncertainty
Dr. Dimitri P. SolomatineProfessor of Hydroinformatics
D.P. Solomatine. Computational intelligence 2
UNESCO-IHE: Hydroinformatics Core
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D.P. Solomatine. Computational intelligence 3
Hydroinformatics
modelling, information and communication technology, computer sciences
applied to problems of aquatic environment
with the purpose ofproper management
D.P. Solomatine. Computational intelligence 4
Hydroinformatics system: flow of information
Earth observation, monitoring
Numerical Weather Prediction Models
Data modelling, integration with hydrologic and hydraulic models
Access to modellingresults
Data Models Knowledge Decisions
Decision support
Map of flood probability
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Tools and application areas in Hydroinformatics
Methods and toolssimulation modellingcomputational intelligenceinformation and knowledge systemssoftware integration technologiessystems engineering, optimizationdecision support systems
Application areas: surface and ground water resources coastal systemsurban systems environmental systems
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Main modelling paradigms
Physically-based model (also called process, simulation, knowledge-based, numerical) is based on the understanding of the underlying processes in the system
examples: river models based on main principles of water motion, expressed in differential equations, solved using finite-difference approximations
Data-driven model is based on the recorded values of variables characterising the system. They need much less knowledge about physical behaviour
examples: statistical regression model linking input and output
Agent-based model consists of dynamically interacting (competing) computational codes (agents)
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Examples of using CI in modelling aquatic processes
1. Optimization of models’ parameters
2. Optimization of water assets
3. Data-driven modelling
4. Optimisation of models’ structure
5. Predictive models of models’ uncertainty
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1. CI in optimization of models’parameters
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Random search in model calibration
there is no analytical expression for model error E(P), so we cannot use efficient gradient-based search typically it is a multi-extremum problem random search methods
Input X
Physical system
Modely = f (X, P)
Measured output yMES
Model output yMOD
Model error E(yMES-yMOD, P)
Error small enough?
Stop
YesNo, update model parameters P
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Adaptive cluster covering (ACCO) algorithmSolomatine (1995, 1999)
Main principles:reductionclusteringadaptationperiodic randomization
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Optimization tool GLOBE in calibration of a rainfall-runoff model
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Global optimization tool GLOBE:the following algorithms are implemented
GA with a one-point crossover, and with a choice between the real-valued or binary 15-bit coding, various random bit mutation, between the tournament and fitness rank selection, and between elitist and non-elitist versionsCRS2 (controlled random search, by Price 1983)CRS4 (modification by Ali & Storey 1994)Multis - multistart algorithm (based on Powell-Brent direct minimization) M-Simplex - multistart algorithm (based on downhill simplex method of Nelder and Mead)adaptive cluster covering (ACCO)adaptive cluster covering with local search (ACCOL)adaptive cluster descent (ACD);adaptive cluster descent with local search (ACDL)
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Performance of various GO algorithms in calibration of a hydrological model
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6
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8
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ThousandsFunction evaluations
Func
tion
valu
eCRS2
GA
ACCOL/3LC
Multis
M-Simplex
CRS4
ACDL/3LC
SIRT rainfall-runoff model (8-var.)
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2. Optimization of water assets
Using EPANET models and single- and multi-objective optimization
of water distribution systems
Abebe, A.J., and Solomatine D. (1998). Application of global optimization to the design of pipe networks. Proc. Int. Conf. on Hydroinformatics, Balkema, Rotterdam.L. Alfonso, A. Jonoski, D.P. Solomatine (2009). Multi-objective optimization of operational responses for contaminant flushing in water distribution networks. ASCE J. Water Res. Planning & Management.
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Rehabilitation of the water distribution network using EPANET model
Test example: Hanoi Network (fragment)Model: EPANET
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Model-based optimization
Penalty CostBased on
Nodal HeadViolationCOST2
Extract NodalHeads from
OutPut
Run NetworkSimulation
Model
Update InputFile of
SimulatorCalculate
Actual Cost ofthe Network
COST1
CalculateTotal Cost
COST1+COST2
CorrespondingPipe Diameters
from CommercialPipe Database
EPAUPD
NET.INP NET.RPT
G L O B E
START
STOP
Optimal Solution
Obtained?
File ContainingPotential Solutions
(Pipe Indices)
File With TotalCost of Network(Response File)
E P A C O S T
EPANET
Optimization tool
Hydraulic model
Optimal solutions
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Multi-objective optimisation of operational responses for contaminant flushing in WDN
Test case study: fragment of the WDN in Villavicencio, Columbia
Contam
H1
H2
V2
V3
V4
V5
V7
Trace J9
0.00
0.00
0.00
50.00
percent
Trace J9
0.00
0.00
0.00
50.00
percent
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Solutions
One of the solutions: close valve V7, open hydrant H2
Contam
H1
H2
V2
V3
V4
V5
V7
Trace J9
0.00
0.00
0.00
50.00
percent
Trace J9
0.00
0.00
0.00
50.00
percent
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Multi-objective optimization: finding the Pareto layer of "good solutions"
The "best" solution is to be selected by a decision maker
Pareto front for optimisation problem Sector 11 Villavicencio
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1015202530354045
0 5 10 15 20 25 30 35 40 45 50
Number of movements
Affe
cted
junc
tions
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Urban drainage networks optimizationRehabilitation of drainage
networks using hydrodynamic models and multi-objective
optimization
W. Barreto Cordero, R.K. Price, D.P. Solomatine, Z. Vojinovic. Approaches to multi-objective multi-tier optimization in urban drainage planning. Proc. 7th Intern. Conf. on Hydroinformatics, Nice, Research Publishing, 2006. W. Barreto Cordero, Z. Vojinovic, R. Price, D.P. Solomatine. A Multi-objective Evolutionary Approach for Rehabilitation of Urban Drainage Systems. J Water Res. Planning & Mang., 2009.
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Hydrodynamic modelling
MOUSE or SWMM models can be used to model flows
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Pipe replacement options tested during optimization process
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NSGAX optimization software (Barreto & Solomatine): search for optimal pipe replacement option
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Results of optimization: set of solutions
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2
Floo
d da
mag
e du
e to
ove
rflo
ws
Costs of implementing rehabilitation option
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2. Data-driven modelling
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Data-Driven Modelling
Uses (numerical) data (time series) describing some physical processEstablishes functions that link variables
outputs = F (inputs)Uses statistics, machine learning, computational intelligence to build FCan be used for forecastingValuable when physical processes are unknownAlso useful to emulate complex physically-based models (surrogate models)
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Data-Driven Modelling: care is needed
Difficulties with extrapolation (working outside the variables’ range)
A solution: exhaustive data collection, optimal construction of the calibration set
Care needed if the time series is not stationaryA solution: to build several models responsible for different regimes
Need to ensure that the relevant physical variables are included
A solution: use correlation and average mutual information analysis
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Complementary Role for Physically Based and Data-Driven Modelling
Replication of complex physically based models using data driven modelsData assimilation (updating state variables)Residual error modelling - separate modelling of errors using data driven models
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Data-driven rainfall-runoff models: Case study Sieve (Italy)
mountaneous catchment in Southern Europearea of 822 sq. km
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SIEVE: visualization of data
variables for building a decision tree model were selected on the basis of cross-correlation analysis and average mutual information:
inputs: rainfalls REt, REt-1, REt-2, REt-3, flows Qt, Qt-1
outputs: flows Qt+1 or Qt+3
FLOW1: effective rainfall and discharge data
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0 500 1000 1500 2000 2500
Time [hrs]
Discharge [m3/s]
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Discharge [m3/s]Eff.rainfall [mm]
Effective rainfall [mm]
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Using data-driven methods in rainfall-runoff modelling
Available data:rainfalls Rtrunoffs (flows) Qt
Inputs: lagged rainfalls Rt Rt-1 … Rt-LOutput to predict: Qt+T
Model: Qt+T = F (Rt Rt-1 … Rt-L … Qt Qt-1 Qt-A … Qtup Qt-1
up …)(past rainfall) (autocorrelation) (routing)
Questions: how to find the appropriate lags? (lags embody the physical properties of the catchment)how to build non-linear regression function F ?
QQtt
QQttupup
RRtt
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Choosing the “relevant” input variables
Input parameters are determined by maximizing the Average mutual information
Techniques used:MLP, RBF networks, SVM regression, M5 model trees
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( , )( , ) log
( ) ( )i j
AB i jAB AB i j
a b A i b j
P a bI P a b
P a P b⎡ ⎤
= ⎢ ⎥⎢ ⎥⎣ ⎦
∑
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Neural Machine: predicting flows
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ANN verification RMSE=11.353NRMSE=0.234COE=0.9452
MT verificationRMSE=12.548NRMSE=0.258COE=0.9331
SIEVE: Predicting Q(t+3) three hours ahead
Prediction of Qt+3 : Verification performance
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0 20 40 60 80 100 120 140 160 180t [hrs]
Q [m
3 /s]
ObservedModelled (ANN)Modelled (MT)
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Computational intelligence in generating inundation maps, Yellow River Commission
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4. Optimization of model structures
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Modular Models
Questions: Should ML algorithms in Module 1, 2, 3 discover “hidden”processes?Or Shall we ask domain experts (humans) to specify such processes?
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Modular models: Methods of data splitting
Grouping (clustering) data using machine learning methods:k-means (A)Fuzzy c-meansSelf-organising maps
Applying hydrological knowledge for flow separation:Tracer-based methodsThreshold-based flow separationConstant-slope method for baseflow separation (B)Recursive filter for baseflow separation (C)
Objectives for this work:we compare highlightedmethodsBuild optimal modular models
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Modular models using clustering
Modular Models are built for each cluster of data0
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Precipitation (mm/hr)
Fore
cast
Dis
char
ge (m
³/s)
-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.5
0
0.5
1
1.5
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Precipitation (t-1)
Pre
cipi
tatio
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K-means cluster (Bagmati training data set)
P (current precipitation)Q (current discharge)
Qt+
1(f
orec
ast d
isch
arge
)
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Optimal model structure using recursive filter for baseflow separation
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Time(days)
Dis
char
ge
Baseflow Bagmati (Nepal)
Total flowBaseflow
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Time(days)
Dis
char
ge
Baseflow Bagmati (Nepal)
Total flowBaseflow
Parameter a=0.01 (Recession coefficient) Parameter a= 0.99 (Recession coefficient )
( ) ( )max 1 max
max
1 11
k kk
BFI ab a BFI Qb
aBFI−− + −
=−
Parameter BFImax=0.5 (Chapman and Maxwell 1996)
Ekhardt 2005
Model optimization by GeneticAlgorithms(GA)
Baseflowfilter
Model 1
Model 2
0fb max ,BFI a
CalculateError(RMSE)
Measured Flow
tQ1tQ +
G. Corzo and D.P. Solomatine (2007). Knowledge-based modularization and global optimization of ANN models in hydrologic forecasting. Neural Networks, 20, 528–536. .
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Performance of the Modular Model using recursive filter vs Single (global) model
-500
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2500
220 222 224 226 228 230 232 234 236 238 240
MMGMTarget
Bagmati catchment
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Combination of process and data-driven models in hydrological forecasting (Meuse)
ANN
HBV
HBV
HBV HBV
HBVHBV
HBV
HBV Model
Sub-Basin 1
Sub-Basin 4
Sub-Basin 5
Sub-Basin 2
Sub-Basin 3
Weather Forecast N
Weather Forecasts 1
Data from gauges
SpecialisedANN
Adaptive ensemble
Delft-FEWS
Sobek
Spatial combination
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5. Predictive models of models’ uncertainty
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Sources of uncertainty in modelling
Inputs Model parameters Calibration data
Model
X(t) Q(t)p
y = M(x, s, θ) + εs + εθ + εx + εy
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Let’s try to encapsulate uncertainty in a CI model
1. UNNEC = machine learning model of the past residual errors of the optimal process model is built
2. MLUE = machine learning model of the process model’s parametric uncertainty (Monte Carlo simulation results) is built
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Model uncertainty predictor = model of the (hydrological) model errors
UNEEC:
UNcertainty Estimation based on local Errors and Clustering
D.L. Shrestha, D.P. Solomatine (2006). Machine learning approaches for estimation of prediction interval for the model output. Neural Networks, 19(2), 225-235.
D.P. Solomatine, D.L. Shrestha. A novel method to estimate model uncertainty using neural networks and other machine learning methods. Water Resour. Res.
UNESCO-IHE European Commission
machine learning model of the past errors of the optimal process model is built
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UNEEC: assumptions, constraints
Assumptions
Model error is an indicator of the model uncertainty
Model error depends on the hydrologic conditions and can be predicted
Model errors are similar for similar hydrological conditions
Constraints
Model structure and parameters are fixed
Need to re-train the error model with the changes in the catchment characteristics (e.g. land use change)
Data hungry, more data are needed for reliable results
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UNEEC: features
Features
The method relies on the concept of optimality instead of equifinality
Estimates the integral uncertainty without attempt to disaggregate contribution given by the different sources of uncertainty
It can be used with any kind of model
No assumptions about the model error properties are made
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Error (Qt-Qt’)
Flow Qt-1
Rainfall Rt-2
past records (examples in the space of inputs)
Outputiμ∑
iN
iμ∑
=1
iN
iμ∑α
=12/
iN
iμ∑α−
=1)2/1(
Prediction interval
Error distribution in cluster
Idea 1: local modelling of errors
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Idea 2: Use fuzzy clustering of examples to generate training data sets
New record. The trained f L and f U models will estimate the prediction interval
Error limits(or prediction intervals)
Flow Qt-1
Rainfall Rt-2
past records (examples in the space of inputs)
Output
Lclus
Nclus
clusexampleclus
Lexample PICPI ∑
=
=1
,μ
•μclus,,example is the membership grade of the example to cluster clus
Train regression (ANN) models:
PIL = fL (X)PIU = fU (X)
Non-linear regression (ANN or M5 model
tree)
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Using instance-based learning
Error limits(or prediction intervals)
Flow Qt-1
Rainfall Rt-2
past records (examples in the space of inputs)
New record
Output
Lclus
Nclus
clusexampleclus
Lexample PICPI ∑
=
=1
,μ
•μclus,,example is the membership grade of the example to cluster clus
Instance based learning
∗∗
∗The distance function is computed to estimate fuzzy weight
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Clustering (finding groups of data in the space characterisinghydro-meteo condition): K-means clustering, fuzzy C-means clustering
UNEEC details. Step 1: clustering
Obj. function⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
∑ ∑== =
c
j
N
iji
mjim DVUJVU
1 1
2,,),(),min( μ
Constraint ic
jji ∀=∑
=,1
1,μ
Distance22
, Ajiji vxD −=
Degree ofFuzzification
1≥m
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iLj ePIC =
UNEEC details. Step 2: Determining Prediction Interval (PI) for each cluster
∑ ji,μ
∑=
N
iji
1,μ
∑=
N
iji
1,2/ μα
∑−=
N
iji
1,)2/1( μα
iUj ePIC =
∑<∑==
N
ijiji
i
ki
1,,
12/: μαμ
∑−<∑==
N
ijiji
i
ki
1,,
1)2/1(: μαμ
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UNEEC details. Step 3, 4, 5: Building and using the model
Lj
c
jji
Li PICPI ∑=
=1, μ U
jc
jji
Ui PICPI ∑=
=1, μ
Step 3: Generation of Step 3: Generation of Prediction intervals for Prediction intervals for
each exampleeach example
)X( uL
uL fPI = )X(U
uU fPI =
Step 4: Building the Step 4: Building the uncertainty Modeluncertainty Model
)X( vL
uL fPI = )X( v
Uu
U fPI =Step 5: Using the Step 5: Using the uncertainty Modeluncertainty Model
Uii
Ui PIyPL += ˆL
iiLi PIyPL += ˆ
Model Outputs with Model Outputs with uncertainty boundsuncertainty bounds
Inde
pend
ent C
ompu
tatio
n
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UNEEC methodology
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Study area: Brue catchment, UK
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Time (houry) (1994/6/24 05:00 - 1996/05/31 13:00)
Disc
harg
e (m
3 /s)
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Rain
fall
(mm
/hou
r)
Calibration data (8760 points):
Validation data (8217 points):
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Conceptual Hydrological model HBV
LZ
UZ
SM
RF
R
PERC
EA
Q=Q0+Q1Q1
Transformfunction
SP
Q0
SF
CFLUX
IN
SF – SnowRF – RainEA – EvapotranspirationSP – Snow coverIN – InfiltrationR – RechargeSM – Soil moistureCFLUX – Capillary transportUZ – Storage in upper reservoirPERC – PercolationLZ – Storage in lower reservoirQo – Fast runoff componentQ1 – Slow runoff componentQ – Total runoff
LZ
UZ
SM
RFRF
RR
PERCPERC
EAEA
Q=Q0+Q1Q1Q1
Transformfunction
SP
Q0Q0
SFSF
CFLUXCFLUX
ININ
SF – SnowRF – RainEA – EvapotranspirationSP – Snow coverIN – InfiltrationR – RechargeSM – Soil moistureCFLUX – Capillary transportUZ – Storage in upper reservoirPERC – PercolationLZ – Storage in lower reservoirQo – Fast runoff componentQ1 – Slow runoff componentQ – Total runoff
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Experiments setup
HBV model was calibrated using the global optimization algorithmACCO (Solomatine et al., 1999)The simulation errors were computed for the given model structure and given parameter setsInput variables for uncertainty model were selected based on theanalysis of average mutual information
PItL , PIt
UV , UQtOutput
REt-8, REt-9, REt-10, Qt-1, Qt-2, Qt-3
REt-8, REt-9, REt-
10, Qt-1, Qt-2, Qt-3
Rt , EtInput
Variables used for uncertaintyprocessor
Variables usedfor clustering
HBV modelVariables
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Clustering result example
0 1000 2000 3000 4000 5000 6000 7000 80000
5
10
15
20
25
30
35
40D
isch
arge
(m3 /s
)
Time (hours)
C1C2C3C4C5
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Estimation of uncertainty bounds
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Dis
char
ge (m
3 /s)
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Pre
dict
ion
boun
ds (m
3 /s)
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020
Time (hours)Res
idua
ls (m
3 /s)
C1C2C3C4C5
PIs by instance baseObservedPIs by regression
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Dis
char
ge (m
3 /s)
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Pre
dict
ion
boun
ds (m
3 /s)
4700 4750 4800 4850 4900 4950 5000-2
0
2
Time (hours)Res
idua
ls (m
3 /s)
C1C2C3C4C5
PIs by instance baseObservedPIs by regression
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2. MLUE methodMachine Learning in Uncertainty Estimation
machine learning model of the process model’s Monte Carlo simulation results is built
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Monte Carlo simulation of parametric uncertaintyy = M(x, s, θ) + εs + εθ + εx + εy
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Monte Carlo simulation of parametric uncertainty
Consider the model M calculating y (e.g., discharge)y(t) = M (X(t), p)
where X(t) = vector of inputs (precipitation, temperature etc) known for t = 1,…, Tp = vector of parameters (soil properties, roughness, etc)
Monte Carlo approach:sample N parameter vectors pirun the model for each of them yi(t) = M (X(t), pi) and generate N outputs (leave some of them if GLUE used)assess distribution of Qi(t) for each time moment t (or its parameters - mean, variance, prediction intervals, quantiles)
The problem:How to assess the parametric uncertainty of the model M for t = T+1 when new input data X(t+1) is fed?
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Issues with MC
Issues with re-running MC for new inputs:1) convergence of the Monte Carlo simulation is very slow (O(N^-0.5)) so larger number of runs needed to establish a reliable estimate of uncertainties2) number of simulation increases exponentially with the dimension of the parameter vector ((O(n^d)) to cover the entire parameter domain
Idea:encapsulate the results of MC simulation in a machine learning model
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MLUE Methodology (1)
Consider the sources of the uncertainty analysis to be conducted within the framework of Monte Carlo simulationExecute the MC simulations to generate the data
yi(t) = M (X(t), pi)
Estimate the uncertainty measures of the MC realizations, e.g., mean, variance, prediction intervals, quantiles
In this study, we use two quantiles (say, 5% and 95%), forming the prediction interval PI
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MLUE Methodology (2)
Analyze the dependency of the uncertainty measures (quantiles) on the input and state variables of the hydrological model
we used Correlation and Average mutual information analysis
Select the input variables for machine learning model based on the dependency analysisTrain the machine learning model U to predict the uncertainty measures of MC realizations PI = U (X)Validate machine learning model U by estimating the uncertainty measures with the “new” input data
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Validation
Measuring predictive capability of uncertainty model U (measures the accuracy of uncertainty models in approximating the quantiles of the model outputs generated by MC simulations)
Coefficient of correlation (r) and root mean squared error (RMSE)Measuring the statistics of the uncertainty estimation (i.e. goodness of the model U as uncertainty estimator)
Prediction interval coverage probability (PICP) and mean prediction interval (MPI) (Shrestha & Solomatine 2006, 2008)
Visualizing such as scatter and time plot of the prediction intervals obtained from the MC simulation and their predicted values
1
1
1, with = 0, otherwise
n
tL Ut t t
PICP Cn
PL y PLC
==
⎧ ≤ ≤⎪⎨⎪⎩
∑
1
1 ( )n
U Lt t
tMPI PL PL
n == −∑
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Conclusions
CI plays an important role in solving various problems related to aquatic environmentMore collaboration between the “water people” and “CI people” needed – to ensure that the right problems are solved and the best methods used
Challenges:to promote the idea of optimization in the water communityto integrate CI methods into the existing decision support frameworks which are based on process models