computational intelligence in modelling aquatic processes...

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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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Hydroinformatics

modelling, information and communication technology, computer sciences

applied to problems of aquatic environment

with the purpose ofproper management


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


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


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


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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5

6

7

8

9

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


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


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

05

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


Pipe replacement options tested during optimization process

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NSGAX optimization software (Barreto & Solomatine): search for optimal pipe replacement option


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


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


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


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


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


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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50

100

150

200

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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?


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

2

4

6

8

200

400

600

100

200

300

400

500

600

700

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

2

2.5

3

3.5

4

4.5

5

Precipitation (t-1)

Pre

cipi

tatio

n

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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

)


Optimal model structure using recursive filter for baseflow separation

300 350 400 450 500 550 6000

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1400

Time(days)

Dis

char

ge

Baseflow Bagmati (Nepal)

Total flowBaseflow

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400

600

800

1000

1200

1400

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

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0

500

1000

1500

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2500

220 222 224 226 228 230 232 234 236 238 240

MMGMTarget

Bagmati catchment


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


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


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


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


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


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


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


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


Study area: Brue catchment, UK

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0 2000 4000 6000 8000 10000 12000 14000 16000

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


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

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40D

isch

arge

(m3 /s

)

Time (hours)

C1C2C3C4C5


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)

0 1000 2000 3000 4000 5000 6000 7000 8000-20

020

Time (hours)Res

idua

ls (m

3 /s)

C1C2C3C4C5

PIs by instance baseObservedPIs by regression

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char

ge (m

3 /s)

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dict

ion

boun

ds (m

3 /s)

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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


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?


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


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 == −∑


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

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[email protected]

computational intelligence in modelling aquatic processes...

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