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

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

Fundamental dynamic patterns

Step 1) Get acquinted with the

system

Step 2) Define the dynamic

problem

0

10

20

30

40

50

60

0 5 10 15 20 25 30

rPdtdP / rtePtP 0)(

Po = 5

r = 0.1

Exponential growth

0

20

40

60

80

100

120

0 5 10 15 20 25 30

Po = 100

r = -0.1

Exponential decay

0

10

20

30

40

50

60

0 10 20 30 40 50 60

S-formad tillväxt

Carrying capacity

S-shaped growth

Overshot

𝑟=𝑟𝑝𝑜𝑡𝐶𝐶−𝑃𝐶𝐶

0

5

10

15

20

25

0 50 100 150

Oscillerande tillväxt

Störning

Jämvikt (equilibrium)

Oscillation

Disturbance

Equilibrium

Equilibrium

?

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/

Chaos

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

StockFlow Other elements

Connector for direction

of dependency

Step 3) Construct a conceptual

modellExample bird population

Bird population

Births

Carrying capacity

DeathsDeath rate

Birth rate

Potential birth rate

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


Conceptual modelling: causal loops diagrams (CLD)

x y

+

The lower value of x – the lower value of yThe lower value of y – the lower value of x

Negative feedback:

-or x y

+

-

x y+

+

x y-

-

The higher value of x– the higher value of yThe higher value of y– the lower value of x

The higher value of x– the lower value of yThe higher value of y– the higher value of x

The higher value of x– the higher value of yThe higher value of y– the higher value of x

Positive feedback:

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

Example bird population

Carrying capacity

Death rate

Birth rate

Potential birth rate

Bird population

Deaths

Births

(-)

-

-+

+

++

+

+

+

(-)(+)

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 ?


Determine what type of model you will make- functional or mechanistic

Use ”standard” equations if possible

Analyse relationships with a curve fitting tool

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

Dynamic models change of state variables with time

Xt : status of X at time tX/ t : rate of changeXt+1=Xt + X

Continuous model in time: dt is infinitive small

Discrete model: t is a period of time

Numeric and analytical solutions

rteNN 0

r is the net growth rate

Nt+1= Nt + N

N = r Nt

0

50

100

150

200

250

300

350

400

0 5 10 15 20 25 30

Population

Analytical

N0 = 100r = 5%Δ = 1

Example bird population – numerical solution

Bird population (P)

Births (B)

Carrying capacity (CC)

Deaths (D)Death rate (rd)

Birth rate (rb)

Potential birth rate (rpot)

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

𝑟𝑏=𝑟 𝑝𝑜𝑡𝐶𝐶−𝑃𝐶𝐶

𝐵=𝑟𝑏𝑃

𝑃 𝑡+1=𝑃+𝐵−𝐷

𝐷=𝑟 𝑑𝑃


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 o

f trust

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 detailGood memoryAbstract thinking

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

Step 7) Implement the mathematical relationships in a computer

program

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

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

Visual Fortran (Formula translation)Pascal, DelphiC, C++Java, J, J++Phyton

Matlab

StellaSimulinkSIMILE

ExcelSQL

Programming and computer implementation

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

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


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

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 orthousands of times.

Book: 3.4.2 – 3.5.3


If needed go back to improve the

modelStep 1) Get acquinted with the system

Step 2) Define the dynamic problem

Step 3) Construct a conceptual modell




Step 7) Implement the mathematical relationships in a computer program


Step 9) Judge if the results are reasonable by comparing to callibration data

or ”common sence”


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

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 far to jeopardize model objectives’

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!

Accuracy of measurements

Coincidence: Difference between validation and model data

Associotion: Similarity in trends between validation and model data

High coincidenceand association

1:1 line 1:1 line

High coincidencelow association

1:1 line

Low coincidencehigh association

Measures of coincidence

Book section 3.3.1

𝑅𝑜𝑜𝑡𝑚𝑒𝑎𝑛𝑠𝑞𝑢𝑎𝑟𝑒𝑑𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛=√∑𝑖=1𝑛

(𝑂𝑖−𝑃 𝑖)2

𝑛

Student’s t test

𝑅𝑜𝑜𝑡𝑚𝑒𝑎𝑛𝑠𝑞𝑢𝑎𝑟𝑒𝑑𝑒𝑟𝑟𝑜𝑟 (%)=100𝑂 √∑𝑖=1

𝑛

(𝑂 𝑖− 𝑃 𝑖)2

𝑛

𝐵𝑖𝑎𝑠 :𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒𝑒𝑟𝑟𝑜𝑟 (%)=100𝑂

∑𝑖=1

𝑛

(𝑂𝑖− 𝑃 𝑖)

𝑛

Observations (O), Modelled value (P)

Measures of association

Regression analysis

Correlation coefficient, r

F-statistics

Plotting of residuals

Book section 3.3.1

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

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

to new situationsInterpolation and extrapolation in time and spaceTest 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

Step 13) Apply the model

Problem formulationConceptual model constructionSystem boundariesCLDVariables, parameters and settingsReference behavior

- Model constructionFrom conceptional model to quantitative modelParameterization/VerificationSensitivity and robustness testingModel validation

- Model useScenario analysisBackcasting and forecastingApplication/Use..

Steps in modelling Book: chapter 2

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.

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

Documents

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