SCNC1111 Scientific Method and Reasoning

Part II a: Quantitative Reasoning: Mathematics

Lecture 4 Mathematical Modelling

2014 09 23 Dr. William M.Y. Cheung

Previously on SCNC1111

Observation

Hypothesis

Prediction

Experimental Test

Confirmation or Falsification

(Generalization via

induction)

(Deduction)

A good theory must not be ambiguous!

How to come up with a non-ambiguous

theory?

What do you mean by being

non-ambiguous?

Also Previously on SCNC1111 Mathematics help us to be concrete

and precise! It would be difficult to be less

ambiguous when you tell me My age is 18 year and 36 days 1 day. The balance of my bank account (S)

changes with time (t), following the equation = 5000 sin + 5000.

Being Non-ambiguous via Being Mathematical

So we should 1. express our hypothesis in terms of

equations and numerical parameters e.g. = 5000 sin + 5000

2. make numerical predictions e.g. The balance of Williams bank account is going to be $123.4 $0.1 on 24 September, 2014.

3. compare with numerical measurements from experiments e.g. go and check the exact balance of Williams bank account on 24 September, 2014

A Potential Problem Not everything in our lives comes with numbers and

equations e.g. marriage and divorce

What factors lead to divorce?

In what way do these factors influence relationship?

Can we make concrete predictions?

Lets make a Mathematical Model to

describe all these phenomena!

Models What comes to your mind when we mention model?

What properties do they share in common?

Fashion models?

Scale models?

Animal models for disease studies?

Features of Models In all the above instances, the model is an

idealized representation or example for thinking about a real-life phenomenon, such as Wearing clothes; Manufacturing aeroplanes, robots; Town planning and building construction; Human diseases.

Mathematical Modelling Mathematical modelling is the description

of an experimentally delineated phenomenon by means of mathematics, with a view to capturing the salient aspects of the phenomenon at hand. Hugo van den Berg. Mathematical Models of

Biological Systems, Oxford, 2011. Page 3 Not all aspects need to be captured!

E.g. if you are only concerned about how the population of cows changes with time,

then it is OK to assume all cows are spherical!

9

Source: openlibrary.org Source: openlibrary.org

How to Come Up with Mathematical Models

1. Stare hard at the problem: identify important aspects; make appropriate assumptions; decide on the key quantities our

model should describe; (equally importantly) forget about

non-essential factors. E.g. if you are wondering how water

flows in the Three Gorges project, you will not care about the water

molecules.

How about Marriages? Seminal work by Professor John Gottman

of the University of Washington in the 1980s and 1990s

His prediction of which couples would divorce within a 4-year period is 94% accurate.

His team stared hard at the problem: They interviewed lots of couples; The interviews were videotaped; How a couple interacts with each other was

then analyzed.

Important Aspects Important aspects

Some previous studies considered gender differences in communication styles cause divorces.

Gottman suggested how a couple influences each other is more important.

Whether one has a positive personality also plays a role.

Anger is not the most destructive emotion in marriage:

Happy couples also fight.

Key Quantity and Assumptions Key quantity

Happiness as a function of time, t! Happiness of husband, x(t) Happiness of wife, y(t) Negative value means you are unhappy!

Assumptions Everyone tends to his/her own intrinsic happiness value

when one is single. Call it 0 or 0. Its value is larger for a more positive person.

When in a relationship, ones happiness is increased or decreased based on the partners happiness.

Different person reacts differently.

Now What? 2. Translate your problem into a

mathematical one: describe how the key quantities

affect one another Draw a flow chart!

using mathematical tools that can help you to draw conclusions.

Flip through your mathematics textbook as there can be more than one approaches!

How to Handle Dice-throwing We are used to treating the results of

throwing a die as probabilistic:

a chance of 1/6 to get 1, 2, 3, 4, 5, or 6

However mechanics informs us that the result can be determined with certainty

if we are given all the necessary information, such as

the force we apply; the air current nearby; how the die bounces.

How to Handle Dice-throwing Too complicated!

And we wont get all the necessary information anyway.

Lets revert back to Probability.

So the approach adopted depends on what we would like to achieve! We need not be entirely correct,

since we ignore irrelevant aspects.

SCNC1111Scientific Method and ReasoningPart II a: Quantitative Reasoning: MathematicsPreviously on SCNC1111Also Previously on SCNC1111Being Non-ambiguous via Being MathematicalA Potential ProblemModelsFeatures of ModelsMathematical ModellingSlide Number 9How to Come Up with Mathematical ModelsHow about Marriages?Important AspectsKey Quantity and AssumptionsNow What?How to Handle Dice-throwingHow to Handle Dice-throwing