1. Yap Von Bing NUS Statistics
Empirical modelling
Yap Von Bing
NUS Statistics

2. aims
Experience sharing
What is empirical modeling?
A possible programme
Summary

3. outline
A classification of modelling activities
Teaching programme

4. Reality (HOW?) Models (of WHAT?) Classification Abstract object
Empirical (backward)
Hypothetical (forward)

5. Some examples
Measured leaf areas vs time
Probabilistic visualization of the Sierpinski gasket
Differential equations for an epidemic
Probabilistic estimation of area of circle
Motion of a tossed stone
Growth of a fixed deposit with compound interest
Optimisation of parking spaces

6. task
Classify the examples
Did you teach any of these or other examples in class? Lessons? Challenges?

7. Authentic data
Quadratic: ng_body
Sinusoidal: temperature
Exponential: radioactivity

8. Programme (level 1)
State aim: given real data, make prediction
Choose functional form
Sketch a curve on data plot
Notice imperfect agreement
Discuss source of discrepancy

9. Free fall data
A steel ball is released from a height.
A machine is used to measure the time for it to drop 0.25, 0.50, 0.75 and 1.00 metres.
Time (s)
0.00
0.22
0.33
0.39
0.45
Distance (m)
0.25
0.50
0.75
1.00

11. Level 2: Goodness-of-fit
Intuition first: clearly contrasting curves
Technique second: confirms intuition, then technique “takes over”
Qualitative before quantitative

12. Beware of (Almost) perfection

13. Level 3
Define criterion: least square
Demonstrate with activities

14. Exact curve-fitting (I)
Is there a quadratic curve that passes through these points?
(1,3), (2,9), (3,19)
(1,3), (2,9), (3,19), (4,33)
(1,3), (2,9), (3,1)

15. Exact curve-fitting (II)
Is there a unique quadratic curve that passes through (0,0) and (2,2)?
Is there an exponential curve that passes through
(1,1) and (2,2)?
(1,1) and (3,1)?

17. Level 4: random error
Drop a stone from height. Distance travelled is
y(t) = 4.9t2 + e
e is a measurement error.
Statisticians often assume e is a random number of mean 0 and variance s2.

18. Suppose e ~ N(0, s2), where s = 0.04, as suggested by data.
Computer Simulation
Suppose e ~ N(0, s2), where s = 0.04, as suggested by data. t
0.00
0.22
0.33
0.39
0.45
1st e
0.01
-0.03
0.06
0.05
y = 4.9t2 + e
0.21
0.51
0.81
1.04
2nd
0.02
-0.05
0.04
0.03
030
0.48
0.79
1.03

20. Programme summary
Level
Activities 1
Clarify aim, plot data, visual interpretation 2
Goodness-of-fit: qualitative 3
Goodness-of-fit: quantitative
Exact curve-fitting 4
Quantifying random errors
Statistical models

21. Many modelling issues are not mathematical.
Overall summary
Many modelling issues are not mathematical.
Quality of description and prediction is hard to assess.
The mathematics tends to be advanced: multivariable calculus (least square), random variables (measurement errors).
Computer simulation and graphing help build ideas.

22. Modeling tasks should not be graded like routine mathematical tasks.
Suggestion
Modeling tasks should not be graded like routine mathematical tasks.
Grades preferable to marks.
The big idea of “modeling” (at least empirical modeling) is beyond mathematics.