Introduction to dynamical system modellingszh/teaching/matlabmodeling/Lecture2... · 2013-01-10 · Introduction to dynamical system modelling Dynamical systems What is a system?

Introduction to dynamical system modelling


Shan He

School for Computational ScienceUniversity of Birmingham

Module 06-23836: Computational Modelling with MATLAB


Outline

Outline of Topics

Dynamical systems

Concepts about modelling

How to model?

How to use models for scientific research


Dynamical systems

What is a system?

A System: consisting of interconnected components, built orevolved with a desired purpose.Examples:

I Toilet tank.

I Car engine.

I Brain.

I Bird flock.

I Climatesystem.


Dynamical systems

What is a system?


I Toilet tank.

I Car engine.

I Brain.

I Bird flock.

I Climatesystem.


Dynamical systems

What is a system?


I Toilet tank.

I Car engine.

I Brain.

I Bird flock.

I Climatesystem.


Dynamical systems

What is a system?


I Toilet tank.

I Car engine.

I Brain.

I Bird flock.

I Climatesystem.


Dynamical systems

What is a system?


I Toilet tank.

I Car engine.

I Brain.

I Bird flock.

I Climatesystem.


Dynamical systems

What is a dynamical system?

I Static vs. Dynamical.

I Static system: current inputs =⇒ outputs.

Static systemInputs xt Outputs y

I Dynamical system: history + current inputs =⇒ outputs.

Dynamical system

Memory

Inputs xt Outputs y


Dynamical systems






Dynamical system

Memory

Inputs xt Outputs y


Dynamical systems






Dynamical system

Memory

Inputs xt Outputs y


Dynamical systems


I Outputs depend on the present and past values of the inputs.

I Changes over time.

I Sometimes called dynamic systems or sequential systems.

I Mathematically described with differential or differenceequations.


Dynamical systems







Dynamical systems







Dynamical systems







Dynamical systems

Which one is a dynamical system?


I Toilet tank.

I Car engine.

I Brian.

I Bird flock.

I Climate system.

I X

I X

I X

I X

I X


Dynamical systems



I Toilet tank.

I Car engine.

I Brian.

I Bird flock.

I Climate system.

I X

I X

I X

I X

I X


Dynamical systems



I Toilet tank.

I Car engine.

I Brian.

I Bird flock.

I Climate system.

I X

I X

I X

I X

I X


Dynamical systems



I Toilet tank.

I Car engine.

I Brian.

I Bird flock.

I Climate system.

I X

I X

I X

I X

I X


Dynamical systems



I Toilet tank.

I Car engine.

I Brian.

I Bird flock.

I Climate system.

I X

I X

I X

I X

I X


Dynamical systems



I Toilet tank.

I Car engine.

I Brian.

I Bird flock.

I Climate system.

I X

I X

I X

I X

I X


Dynamical systems

Biological systems


I Biological systems change over time.

I Biological systems are dynamical systems.

I Biological systems are complex: emergence.

I Reductionism might not work well

I System point of view on biological systems is new - SystemsBiology.


Dynamical systems

Biological systems








Dynamical systems

Biological systems








Dynamical systems

Biological systems








Dynamical systems

Biological systems









What are models?

When we say models, we usually mean computational/mathmeticalmodels.

I Model: a description of a system usingmathematical/computational concepts and language.

I Mathematical model: a set of variables and a set ofequations that establish relationships between the variables.

I Computational model: a computer program thatimplements computational techniques, e.g., rules, automata,petri nets or artificial neural networks to describe a system.

I Can be hybrid: Mathematical models + Computationalmodels.



What are models?








What are models?








What are models?








Why use models?

We use model to describe phenomena and understand phenomena.More specifically:

I Predict: Make testable prediction

I Explain: Reveal underlying mechanisms or rule out particularexplanations

I Discover: Propose new questions

I Guide: Data collection or experiments design



Why use models?








Why use models?








Why use models?








Example: Climate models

By building climate models, we can:

I Predict: How temperature changes in the next few days?

I Explain: How the global atmosphere is operating?

I Discover: How the carbon dioxide affects our climate?

I Guide: Collect climate data.


























How to model?

Modelling methods

I Equation based methods:

I Differential equation: Ordinary Differential Equation, Partialdifferential equation.

I Statistical methods: linear regression, multilevel model andStructural equation model.

I Game theoretic methods.

I Agent-based methods.

I Other methods: data driven methods.


How to model?

Modelling methods

I Equation based methods:I Differential equation: Ordinary Differential Equation, Partial

differential equation.

I Statistical methods: linear regression, multilevel model andStructural equation model.





How to model?

Modelling methods


differential equation.I Statistical methods: linear regression, multilevel model and

Structural equation model.





How to model?

Modelling methods



Structural equation model.I Game theoretic methods.




How to model?

Modelling methods







How to model?

Modelling methods







How to model?

Modelling principles

I Question: How to build the best model to model a system?

I No answer, even no a single definition to ‘the best model’.I One good attempt is Minimum Description Length principle,

or Occam’s razor.

I “The best model is the one that is the smallest.”

I Einstein’s razor’ (Better!):

I The model should be made as simple as possible, but nosimpler.


How to model?



I No answer, even no a single definition to ‘the best model’.

I One good attempt is Minimum Description Length principle,or Occam’s razor.





How to model?




or Occam’s razor.





How to model?




or Occam’s razor.I “The best model is the one that is the smallest.”




How to model?








How to model?





I Einstein’s razor’ (Better!):I The model should be made as simple as possible, but no

simpler.


How to model?

How to put into practice?

I found this paper very useful: 5 Simple principles of modelling byProfessor Mike Pidd. Here are the 5 principles:

I Model simple, think complicated.

I Be parsimonious, start small and add.

I Divide and conquer, avoid mega models.

I Don’t fall in love with data.

I Model building may feel like muddling through.


How to model?









How to model?









How to model?









How to model?









How to model?

Modelling process

SimulationDefinitionIdentification

ValidationAnalysis Satisfied?

No

Yes


How to model?

Modelling process: Identification

I Identify the system to model (What)

I Gene regulatory networks.I Cellular signalling networks.I Prey-predator interaction.

I Or more importantly, identify the problems to answer (Why).I For example, if we model prey-predator interaction, we wish to

answer:

I How predators affect prey populations, and vice-versa?I Whether a prey-predator system is stable or collapse?


How to model?


I Identify the system to model (What)I Gene regulatory networks.

I Cellular signalling networks.I Prey-predator interaction.


answer:



How to model?


I Identify the system to model (What)I Gene regulatory networks.I Cellular signalling networks.

I Prey-predator interaction.


answer:



How to model?


I Identify the system to model (What)I Gene regulatory networks.I Cellular signalling networks.I Prey-predator interaction.


answer:



How to model?



I Or more importantly, identify the problems to answer (Why).

I For example, if we model prey-predator interaction, we wish toanswer:



How to model?




answer:



How to model?




answer:I How predators affect prey populations, and vice-versa?

I Whether a prey-predator system is stable or collapse?


How to model?




answer:I How predators affect prey populations, and vice-versa?I Whether a prey-predator system is stable or collapse?


How to model?

Modelling process: Definition



No

Yes


How to model?


The most important step. General suggestions are:I Draw a detailed picture:

I Defines all components.I Defines the relations between components.

I Define the parameters and variables to be used in the model.I Check literature:

I Any descriptions about the components?I Any relations reported?I What parameters and variables used in similar studies?

I You might need to simplify your assumptions.


How to model?



I Defines all components.

I Defines the relations between components.





How to model?








How to model?




I Define the parameters and variables to be used in the model.

I Check literature:




How to model?








How to model?





I Any descriptions about the components?

I Any relations reported?I What parameters and variables used in similar studies?



How to model?





I Any descriptions about the components?I Any relations reported?

I What parameters and variables used in similar studies?



How to model?








How to model?








How to model?

Modelling process: Simulation



No

Yes


How to model?

Modelling process: Simulation

This simply means, in our module, to execute your MATLABprogram ;)


How to model?

Modelling process: Validation



No

Yes


How to model?


I This simply means compare the outputs from your MATLABprogram with experimental data.

I Decision time: Do the results match the experimental data?

I Answer: Yes =⇒ AnalysisI Answer: No =⇒ Definition. We need to check:

I Simple typos?I Oversimplified assumptions?I Missing components?I Incorrect parameters?


How to model?







How to model?




I Answer: Yes =⇒ Analysis

I Answer: No =⇒ Definition. We need to check:



How to model?







How to model?





I Simple typos?

I Oversimplified assumptions?I Missing components?I Incorrect parameters?


How to model?





I Simple typos?I Oversimplified assumptions?

I Missing components?I Incorrect parameters?


How to model?





I Simple typos?I Oversimplified assumptions?I Missing components?

I Incorrect parameters?


How to model?







How to model?

Modelling process: Analysis



No

Yes


How to model?


I By analysing the results, we aim to answer:

I What do the results imply or suggested?I What is new and what did not know?I What prediction can we make?

I If the analysis proposes new questions, we need to go back toidentification.


How to model?


I By analysing the results, we aim to answer:I What do the results imply or suggested?

I What is new and what did not know?I What prediction can we make?



How to model?


I By analysing the results, we aim to answer:I What do the results imply or suggested?I What is new and what did not know?

I What prediction can we make?



How to model?


I By analysing the results, we aim to answer:I What do the results imply or suggested?I What is new and what did not know?I What prediction can we make?



How to model?


I By analysing the results, we aim to answer:I What do the results imply or suggested?I What is new and what did not know?I What prediction can we make?




Modelling process and scientific method

We should always put modelling into the context of scientificmethod:

1. Make general observations of phenomena

2. Formulate a hypothesis that is consistent with yourobservations.

3. Use the hypothesis to make predictions.

4. Develop experiments to test your hypothesis and modify thehypothesis in the light of your results.

5. Repeat steps 3&4 until there are no discrepancies betweenhypothesis and phenomena.









































No

Yes

Formulate hypothesis

Develop

experiments

Phenomena

Step 1

Step 2

Step 3

Step 4



Too abstract?

After learning MATLAB next week, in week 4, I will follow theprocess to model predator-prey intreactions using OrdinaryDifferential Equations.

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Introduction to dynamical system modellingszh/teaching/matlabmodeling/Lecture2... · 2013-01-10 · Introduction to dynamical system modelling Dynamical systems What is a system?