Yap Von Bing NUS Statistics Empirical modelling Yap Von Bing NUS Statistics

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

outline A classification of modelling activities Teaching programme

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

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

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

Authentic data Quadratic: https://en.wikipedia.org/wiki/Equations_for_a_falli ng_body Sinusoidal: temperature Exponential: radioactivity

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

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

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

Beware of (Almost) perfection https://hpcaspersa.wikispaces.com/Ch2_CaspertS http://www.schoolphysics.co.uk/age14-16/Nuclear%20physics/text/Half_life/index.html

Level 3 Define criterion: least square Demonstrate with activities

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)

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)?

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.

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

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

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.

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.