Current Trends in Environmental Modelling with · PDF file · 2006-11-14Current...

2nd Summer School in Computational Biology

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Current Trends in Environmental Modelling with

Uncertainty

Jiří Hřebíček, Michal HejčMasaryk University, Brno, Czech Republic


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Content

• Introduction• Modified uncertainty analysis• Uncertainty analysis with Maple • Case study: Air pollution by the transport in

Czech Republic• Case study: Assessment of waste management

indicators in South Moravia region• Conclusions


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Introduction

• The important characteristic feature of environmental modelling is the complexity and uncertainty of its mathematicalrepresentation (uncertainty of formula).

• Imprecision of its input data is another characteristic feature, where it is not possible to omit influences of primary monitoring (e.g. gaps of data, errors of measuring facilities, human factor, etc).


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Uncertainty

Uncertainties in the scientific sense are the component of all aspects of the environmental modelling process.

They describe lack of knowledge about models, their parameters, constants, data, and beliefs.


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Sources of uncertainty

• the science underlying a model,• uncertainty in model parameters,• scientific constants and input data,• monitoring and observation errors,• and implementation uncertainty.


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Modified uncertainty analysis

The uncertainty analysis of environmental models and their algorithm implementation consists of following stages:

• characterization of input uncertainties,• uncertainty propagation,• characterization of model uncertainty,• characterization of the uncertainties in algorithm

predictions.


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Types of approaches

Proposed uncertainty analysis of environmental models issues from following approaches:– interval arithmetics,– fuzzy theory,– probabilistic analysis,– methodology of Checkland.


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Checkland SSM iterative approachSeven stages, forming a life cycle of mathematical model:

1. Finding out about the environmental problem situation. This is basic research into the problem area.

2. Expressing the situation through so-called “Rich Picture“.3. Selecting how to view the situation and producing root definitions.4. Building conceptual environmental models of what the system

must do for each root definitions. We have basic “Whats” from the root definitions. Now we begin to define “Hows”.

5. Comparison of the conceptual environmental models with the real world. We compare the results from steps 4 and 2.

6. Identify feasible and desirable changes. Are there ways of improving the situation?

7. Recommendations for taking action to improve the environmental problem situation.


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Appropriate ICT tools: CAS

Computer algebra based systems (CAS) involve:

• the direct symbolic and algebraic computation (SAC) of the governing equations of mathematical models and analytical/numerical solution of environmental problem and its visualization/presentation;

• and also the estimation of the sensitivity and uncertainty of model outputs with respect to model inputs.


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Appropriate ICT tools: CAS

The process of environmental modelling using CAS consists of the spiral cycle:

IDENTIFY – DEVELOP – IMPLEMENT –SOLVE – ANALYZE – MODIFY,

which shows the way how complex CAS automate all phases of environmental

modelling.


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IDENTIFY model components

SOLVE computationalmodel

MODIFY model for optimization

Environmental Problem

Extensive user communityMany journals, books, conferences, etc.

SAC user groupsSAC e-resources (problems & algorithms &

benchmark data)

Symbolic equationsSpecialized math functions

Matrix formulationAutomatic differentiation

OptimizationCurve fitting

Mathematical dictionaryEfficient user interface

Programming languagesConnectivity to SAC e-sources

Unique data structureInteractive debugger

Syntax checking utilitiesAdvanced help system

Analytical solutionHigh precision numerical solution

Fortran, C, Java, MS Visual basic code generation

Interactive graphicsSpecialized plots

2D & 3D animationsNatural equation display

“What if” analysisHTML, XML & MathML export

LaTeX, RFT, PDF export

Interactive worksheetsMultiple document interfaceReusable sub expressions

Maintains symbolic relationshipsTechnical document processingConnectivity to SAC e-sources

DEVELOP equations& algorithms

IMPLEMENT computational model

ANALYZE andpublish solutions


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The known SAC systems

Yacas,HartMath,The OpenXM project,Prologie,GiNaC,ArtLandia,Axiom,CoCoA,Derive,Algebra Domain Constructor,Fermat,GAP,GANITH,GRG,GRTensor,LiDIA,GNU DOE Maxima,

Magma,Maple,Mathematica,Mathomatic,MathSoft,MATLAB,MathTensor,Milo,MP,MuPAD,NTL,Pari,Reduce,Schur,Singular,SymbMath,TI-92 Calculator, and TI-92 Plus.


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Uncertainty Analysis with Maple

• Precise value of scientific constants (Scientific constants package)

• Implementation of interval arithmetic (Tolerances package).• Implementation of fuzzy theory (FuzzySet Package).• Implementation of probabilistic analysis (ScientificErrorAnalysis

package).• The Checkland's SSM is not automatically implemented in Maple.

Getting the online data and program codes (www.maplesoft.com).

The Sockets package of Maple allows getting data and program codes for the computation online from the web. In particular, itenables two independent Maple processes running on different computers on a network to communicate with one another


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Scientific constants implementation

• The Scientific Constants package of Maple provides access to the values of various constant physical quantities, for example, the velocity of light and the atomic weight of sodium. These values are required to solve equations in fields such as chemistry and physics.

• The Scientific Constants package also provides the units for each of the constant values, allowing for greater understanding of the equation as well as units matching for error checking of the solution.

• The quantities available in the Scientific Constants package are divided into two distinct categories.

• physical constants• properties of the chemical elements (and their isotopes)


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Scientific constants implementation


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Interval arithmetic implementation

The package Tolerance of Maple provides basic data types and operations for interval arithmetic as well as additional features for further interval computation.

It contains the type checking functions, all arithmetic functions including powers, trigonometric andhyperbolic ones, set operations on the interval, range operations for a given function, complex number support and some basic numeric methods as the Newton's method for finding a root of an uncertain function.


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Fuzzy theory implementation

The Fuzzy Sets toolbox of Maple allows constructing and working with fuzzy subsets of both the real line and of user-defined finite sets.

Its modules automatically generate fuzzy controllers from a collection of user-defined rules. This allows modelling, testing, and modifying fuzzy systems in the interactive Maple worksheet environment.


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Probabilistic Analysis Implementation

The ScientificErrorAnalysis package of Maple provides representation and construction of numerical quantities in Maple that has a central value and associated uncertainty or error, which is some measure of the degree of precision to which the quantity's value is known.

The associated uncertainty can be specified in absolute, relative, or units in the least digit form. In the returned object, the uncertainty is quantified in absolute form.


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Probabilistic Analysis Implementation


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Case Study: Air Pollution by the Transport

The emissions from transport in the Czech Republic has been analyzed with respect to uncertainties using ICT tools of Maple and SSM, where the implemented mathematical model of transport air emissions in Maple has issued from the well-known mathematical model COPERT III.


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COPERT III

We rearrange the model COPERT III in the following way:

• Treating the emission factors, fuel consumptions and transport powers as uncertain.

• Unifying the formulas for various pollutants.• Unifying the formulas for various transport

types.


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Checkland's SSM

• Of course, not all of these changes must be desirable, but using the Checkland's SSM iterative approach has allowed us to change the model afterwards, taking into account its bad properties which were not corresponded to the situation in the real world.

• Selected emission factors, which are based on measured values, used probabilistic approach, and further the direct dependence of the relationship of transport performances given in passenger kilometres or ton-kilometres were eliminated.


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Carbon dioxide (CO2) emissions (kg/inhabitant) generated in the Czech Republic by all types of transport

0

100

200

300

400

500

600

700

800

1995 2000 2001 2002 2003 2004

IndividualroadpassengerPublic roadpassenger

Road goods

Urban public

Rail - dieseltraction

Waterway

Air


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VOC emissions (kg/inhabitant) generated in the Czech Republic byall types of transport

0,00

1,00

2,00

3,00

4,00

5,00

6,00

1995 2000 2001 2002 2003 2004

Indiv idualroadpassengertransportPublic roadpassengertransport

Roadgoodstransport

Urbanpublictransport

Railtransport -dieseltractionWaterw aytransport


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Assessment of waste management indicators in South Moravia region

This case study presents the use of the modular approach, which divides the annually collected waste management data into particular modules.

Every module processes one portion of the data(portions may have intersections) and uses one method for uncertainty processing (interval arithmetic, probability approach, fuzzy approach or their combination).


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Modules’ dependencies

METADATA KNOWLEDGE

TRUSTWORTHINESS MODELS

RESULT CHECKFUNCTION

OPTIMUM


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Module models

The module model uses number of inhabitants in districtand evaluates the coefficients of production per capita in the district from its district primary collected data, uses archived coefficients of production per capita and year average production in the district d from the last year, etc.

The output of this model is estimated the annual regional municipal waste production and the annual regional municipal waste production from the last year.

∑d

pN dI

d1


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Modules Trustworthiness, Result

The module for Trustworthiness of the data estimates trustworthiness of primary collected data (it can be of the type fuzzy, interval or probability), and it takes into consideration primary data from past three years, models, knowledge and some parameters, which are suitable for optimization.

The module Result just computes the result from primary waste management data, its trustworthiness and the model.

Result takes all their optimization parameters and the output of this function is a value which will substitute the origin value in waste database, but with the confirmation of the government response officers in the case of sensitive data (e.g. demolitions waste, hazardous waste, etc.). Substituted values are mostly the same as in the primary data (more the data are uncertain means more substituted values). This function applied on all values produces the new database.

∑d

pN dI

d1


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Module Check function

The new database calculated by module Result is evaluated. During evaluation some evaluation criteria (again it can be used with some parameters) are used if possible. Evaluated database is checked by the check function (which is a function of the Result and some knowledge). In accordance with evaluation criteria the optimum database is founded in sufficient number of iterations, generally in cooperation with decision makers from the government of South Moravia.

∑d

pN dI

d1


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Conclusions

• The uncertainty handling is not further so problematic in environmental modelling using current ICT tools.

• Deeper knowledge of the mathematical model and the data together with uncertainty and sensitivity analysiscan show how much the input uncertainty influence the outcome of the model.


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Conclusions (cont.)

• Classification of the parameters and the data into clusters (where some of them are sufficient to be known roughly and some of them more accurately) can divide the problem of uncertainty into parts, solved by different approaches(interval arithmetic, fuzzy and probabilistic theory).


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Conclusions (cont.)

• The further opened question is the sense of using the SSM because now it seems that we need at least two iterations to get as good results as the original model.

• Such approach makes environmental modelling little bit slower, but the iteration guarantees that there are no useless formulas and keeps the model complexity at the lower bound corresponding to the results we have.


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Thank for your attention

Prof. Dr. Jiří Hřebíč[email protected]


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Acknowledgment

Supported by project MSM0021622412

INCHEMBIOL

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