Modelling elements and methods
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Modelling elements and methods. Building a simulation model. Step 1 ) Get acquinted with the system Step 2 ) Define the dynamic problem Step 3)Construct a conceptual modell Step 4 ) Define the causal loops or casual relationships

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Modelling elements and methods

Modelling elements and methods


Modelling elements and methods

Building a simulation model

Step 1)Get acquinted with the system

Step 2)Define the dynamic problem

Step 3)Construct a conceptual modell

Step 4)Define the causal loops or casual relationships

Step 5)Express the relationships mathematically in equations

Step 6)Get values of parameters

Step 7)Implement the mathematical relationships in a computer program

Step 8)Run the model

Step 9)Judge if the results are reasonable by comparing to callibration data or ”common sence”

Step 10) Sensitivity analysis

Step 11)Repeat 3-10 to improve the model and parameter estimates

Step 12) Validate the model by independent data

Step 13) Apply the model


Modelling elements and methods

Step 1)Get acquinted with the system

Step 2)Define the dynamic problem

Fundamental dynamic patterns


Modelling elements and methods

Exponential growth

Po = 5

r = 0.1


Modelling elements and methods

Exponential decay

Po = 100

r = -0.1


Modelling elements and methods

S-shaped growth

Overshot


Modelling elements and methods

Oscillation

Disturbance

Equilibrium


Modelling elements and methods

Equilibrium

?


Modelling elements and methods

Chaos

Chaos theory

describes complex motion and the dynamics of sensitive systems.

Chaotic systems are mathematically deterministic but nearly impossible to predict: Unpredicted courses of events

Discovered by Edward Lorenz in 1961 (Butterfly Effect):Weather forecast based on previous calculations and giving not equal precision data.

Chaos theory

http://www.mathjmendl.org/chaos/


Modelling elements and methods

Step 3) Construct a conceptual modell

-stock and flow diagram

  • Identify stocks and flows

  • Connect the stocks by flow

  • Identify other elements (variables, parameters or constants) that affect the flows

  • Connect all elements with arrows for the direction of dependency

  • For each element that has incoming arrows there has to be an equation

  • Units for stocks and flows have to be consistent, eg indivuals – individuals per year or g m-2 – g m-2 s-1

Stock

Other elements

Flow

Connector for direction of dependency


Modelling elements and methods

Step 3) Construct a conceptual modell

Example bird population

Potential birth rate

Births

Birth rate

Carrying capacity

Bird population

Death rate

Deaths


Modelling elements and methods

Step 4) Define the causal loops or casual relationships

  • Causal Loop Diagram (CLD):

  • A simplified conseptual model where all elements of the model are connected with arrows for dependency

  • A simplified understanding of a complex problem

  • A common language to convey the understanding

  • A way of explaining cause and effect relationships

  • Explanation of underlying feedback systems

  • A help for understanding the overall system behaviour


Modelling elements and methods

Conceptual modelling: causal loops diagrams (CLD)

Negative feedback:

-

+

x

y

or

x

y

+

-

The higher value of x

– the higher value of y

The higher value of y

– the lower value of x

The higher value of x

– the lower value of y

The higher value of y

– the higher value of x

Positive feedback:

+

-

x

y

x

y

+

-

The higher value of x

– the higher value of y

The higher value of y

– the higher value of x

The lower value of x – the lower value of y

The lower value of y – the lower value of x

An odd number if negative dependcies in a loop means negative feedback


Modelling elements and methods

Example bird population

Potential birth rate

+

Births

+

Birth rate

+

(+)

+

(-)

-

+

Carrying capacity

Bird population

(-)

+

-

Death rate

+

Deaths


Modelling elements and methods

Some problems in the conceptual modelling phase:

  • What is relevant for the model? Sort out essentials

  • At what level do we simulate: micro or macro level

  • Static and dynamic factors ?

  • What are the boundaries of the system?

  • Time horizon ?


Modelling elements and methods

Step 5) Express the relationships mathematically in equations

Determine what type of model you will make

- functional or mechanistic

Use ”standard” equations if possible

Analyse relationships with a curve fitting tool


Modelling elements and methods

Dynamic models: many functions are differential equations

Biological systems: most of the differential equations can not be solved to analytical functions

Thanks to computers: numerical approximations

But: Numerical method is an approximation to the true solution


Modelling elements and methods

Dynamic models

change of state variables with time

Xt : status of X at time t

X/ t : rate of change

Xt+1=Xt + X

Continuous model in time: dt is infinitive small

Discrete model: t is a period of time


Modelling elements and methods

Numeric and analytical solutions

Nt+1= Nt+ N

N = r Nt

r is the net growth rate


Modelling elements and methods

Example bird population – numerical solution

Potential birth rate (rpot)

Births (B)

Birth rate (rb)

Carrying capacity (CC)

Bird population (P)

Death rate (rd)

Deaths (D)

For each element that has incoming arrows there has to be an equation


Modelling elements and methods

Step 6)Get values of parameters

Parameter estimation from

Physical laws

Physical based experiments/observations

General description of ecological processes

Best guess of an expert

Your own intuition

From

literature

Level of trust


Modelling elements and methods

Step 7) Implement the mathematical relationships in a computer program

Now we have a solution on paper:

Next step is writing the model to a computer program

Computer modelling – programming

’Telling’ a computer what to do

Compiling = ’translating’

Debugging = finding/correcting errors in the code

What you need:

Discipline and attention to detail

Good memory

Abstract thinking

With a good conceptual model and some general structure it is rather ”easy”


Modelling elements and methods

Programming and computer implementation

Computer ’languages’ : ’telling a computer what to do’:

Basic

Fortran (1950), Fortran IV (1966), Fortran77, Fortran90,

Visual Fortran (Formula translation)

Pascal, Delphi

C, C++

Java, J, J++

Phyton

Matlab

Stella

Simulink

SIMILE

Excel

SQL


Modelling elements and methods

  • Computer implementation

  • = all steps neccessary to translate a mathematical description of a model into a computer program and should work in a useful way

  • is rather time consuming and thus ’expensive’

  • should produce a flexible program: easy to adapt

  • should produce a ’user-friendly’ program: both for user of the model as user of the source code


Modelling elements and methods

  • General structure for a dynamical model

  • Get input parameter values

  • Get start values of stocks and other state variables

  • Loop in which the timestep is increased by one for each cycle

    • Read driving variables and apply the equation of the processes to get the flows of the model

    • Update the stocks for time+1

  • End of loop


Modelling elements and methods

Step 8) Run the model


Modelling elements and methods

Step 9) Judge if the results are reasonable by comparing to callibration data or ”common sence”

Reasons for ”poor” results

Bugs in the computer implementation

Wrong understanding of the dynamical problem

Using an application outside the model´s development conditions

Normal need for parameter callibration


Modelling elements and methods

Step 10)Sensitivity analysis

Varying parameters and/or variables independently

This may highlight the weakness in the model and indicate which parameters or variables need much attention and high accuracy

  • A well known technique: Monte Carlo simulations

  • A random value is selected for each of the tasks/parameters,

    • based on a range (pseudo random)

  • - The model is applied repeately, each time with another random value

  • - A typical Monte Carlo simulation calculates the model hundreds or

  • thousands of times.

  • Book: 3.4.2 – 3.5.3


    Modelling elements and methods

    If needed go back to improve the model

    Step 1)Get acquinted with the system

    Step 2)Define the dynamic problem

    Step 3)Construct a conceptual modell

    Step 4)Define the causal loops or casual relationships

    Step 5)Express the relationships mathematically in equations

    Step 6)Get values of parameters

    Step 7)Implement the mathematical relationships in a computer program

    Step 8)Run the model

    Step 9)Judge if the results are reasonable by comparing to callibration data or ”common sence”

    Step 10) Sensitivity analysis

    Step 11) Repeat 3-10 to improve the model and parameter estimates

    Step 12) Validate the model by independent data

    Step 13) Apply the model to new situations


    Modelling elements and methods

    Step 12) Validate the model by independent data

    - to assure that the model is correct

    Simulation models are simplifications of the real world. If you leave out (unimportant) factors and only describe the system by capturing the important factors, you have to prove that the model is still usefull

    Verification: concerned with building the model right

    Validation: concerned with building the right model.

    ’Validation is the determination as to whether model behavior departs from real system behavior sufficiently farto jeopardize model objectives’


    Modelling elements and methods

    Validation:

    Compare modelled and measured values by ’goodness-of-fit’

    • Try to use standard statistical tests !

    • Comparing qualitive similarity is often used, but be careful!


    Modelling elements and methods

    Accuracy of measurements

    Coincidence: Difference between validation and model data

    Associotion: Similarity in trends between validation and model data

    High coincidence

    and association

    Low coincidence

    high association

    High coincidence

    low association


    Modelling elements and methods

    Measures of coincidence

    Student’s t test

    Observations (O), Modelled value (P)

    Book section 3.3.1


    Modelling elements and methods

    Measures of association

    Regression analysis

    Correlation coefficient, r

    F-statistics

    Plotting of residuals

    Book section 3.3.1


    Modelling elements and methods

    Accuracy is often used as the complement of error:

    95% accuracy implies 5% error

    But accuracy refers also often to the fidelity (= trohet) with which the model represents the processes and relationsships


    Modelling elements and methods

    • Sometimes it is not possible to validate a model

    • Then there are other options:

    • perform a sensitivity analysis

    • compare with other validated models or compartment of other models


    Modelling elements and methods

    Step 13) Apply the model

    to new situations

    Interpolation and extrapolation in time and space

    Test of new policies, methods etc

    Pack the model in a way that is suitable for the end user

    -Who is the end user

    -What will the model do?

    -How can the application guard against input error

    -How can the application guard agains misinterpretation of the results

    -What documentation is needed


    Modelling elements and methods

    Steps in modelling

    Book: chapter 2

    • Problem formulation

      • Conceptual model construction

      • System boundaries

      • CLD

      • Variables, parameters and settings

      • Reference behavior

    • Model construction

      • From conceptional model to quantitative model

      • Parameterization/Verification

      • Sensitivity and robustness testing

      • Model validation

    • Model use

      • Scenario analysis

      • Backcasting and forecasting

      • Application/Use..


    Modelling elements and methods

    Exercise in conceptual model/causual loop construction from 2011’s exam

    Think about a house and its heating system. Assume a simple dynamical model that consists of these variables:         - Temperature inside the house         - Outdoor temperature         - Target temperature set at the thermostats of the radiators         - Energy content of the house         - Heat production from the radiators         - Heat loss to the surrounding Put these variables together in a conceptual model diagram and a causal loop diagram and explain what the diagrams tell you.


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