So, you want to learn about modeling ...

Please ask questions & volunteer answers- for my benefit as well as yours.

Yes …I believe there’s a question there in

the back.”

Please ask questions & volunteer answers- for my benefit as well as yours.

Yes …I believe there’s a question there in

the back.”

Scientific descriptions: tedious names, numbers & statistics.Artistic descriptions: prettier & work OK.

describe & predict.

Science is valued for useful predictions about the future:IF (do this) THEN (that will happen), ELSE …

Based on understanding processes: ‘laws acting around us’

Science is like a blabbermouth who ruins a movie

by telling you how it ends!Well I say there are some

things we don't want to know! Important things!

Ned Flanders In "Lisa the Skeptic" The Simpsons

Science is a process for learning about natureScience is a process for learning about nature in which competing ideas about how the world works are measured against observations (Feynman 1965, 1985).

http://plato.stanford.edu/entries/models-science/Models in ScienceModels in ScienceModels are of central importance in many scientific contexts. The centrality of models such as the billiard ball model of a gas, the Bohr model of the atom, the MIT bag model of the nucleon, the Gaussian-chain model of a polymer, the Lorenz model of the atmosphere, the Lotka-Volterra model of predator-prey interaction, the double helix model of DNA, agent-based and evolutionary models in the social sciences, or general equilibrium models of markets in their respective domains are cases in point. Scientists spend a great deal of time building, testing, comparing and revising models, and much journal space is dedicated to introducing, applying and interpreting these valuable tools. In short, models are one of the principle instruments of modern sciencemodels are one of the principle instruments of modern science.

Level Models Themes

Primary•A model is different from the real thing but helps us to learn about the real thing. •Simple analogies help to explain how things work.

Elementary

•Changes in models should reflect changes in the real thing. •Models come in many types - maps, geometric figures, number sequences, analogies, graphs, sketches, diagrams. •Models are useful tools for understanding a process or natural phenomenon.

Middle Grades

•Models are used to think about processes that happen too fast or slow, or are too large or too small to view directly. •Different models can represent the same real object. •Technology is useful in constructing models.

Secondary•Mathematical models help to understand relationships. •Useful models predict behavior of objects and events successfully. •Applying technology enhances models and understanding.

Vehicles for Understanding and Doing Science: Models

See: http://plato.stanford.edu/entries/models-science/

“I have deeply regretted that I did not proceed far enough at least to understand something of the great principles of mathematics; for men thus endowed seem to have an extra sense.”

- The Autobiography of Charles Darwin (pg. 8)

The good Christian should beware of mathematicians and all those who make empty prophecies. The danger already exists that mathematicians have made a covenant with the devil to darken the spirit and confine man in the bonds of Hell. - St. Augustine

Charles Elton, in reference to work by Lotka - “Like most mathematicians, he takes the hopeful biologist to the edge of a pond, points out that a good swim will help his work, and then pushes him in and leaves him to drown."

You can’t pull anything out of the hat that you didn’t put in to begin with.

Chapter 1IntroductionThere is a story that during his stay at the court of Catherine II of Russia, the great Swiss mathematician Euler got into an argument about the existence of God. To defeat the Voltairians in the battle of wits, the great mathematician called for a blackboard, on which he wrote:

(x + y)2 = x2 + 2xy+ y2

Therefore God exists.

Unable to dispute the relevance of the equation (which they did not understand), and unwilling to confess their ignorance, the literati were forced to accept his argument!Much of the vigour of behavioural ecology stems from the use of mathematical models. But, as this little story illustrates, there is a certain mystique surrounding mathematics and mathematical modelling. ...

The widespread use of analogies in cognition, including scientific reasoning, has been well documented ... people use analogies to solve problems.

Paul Thagard Philosophy Department University of Waterloo

The {standard} model of analogical transfer ... First, someone attempts to solve a target problem, and then remembers a similar source problem for which a solution is known. Then the target problem is solved by adapting the solution to the source problem

The serendipity model ... a target problem is recalled and solved using a source accidentally encountered after initial solutions fail. Darwin had long wondered about how biological evolution occurs, and only found a solution when he read Malthus's ideas about ... the struggle for existence. {Ah-ha! Epiphany!}

Theoretical analogies are important in the development of explanatory hypotheses. Physics ... comparison of sound with water waves and of light waves with sound waves. Biology ... Darwin's analogy between natural and artificial selection.Animal models provide generated theoretical analogies that are at least suggestive about the causes of diseases in humans.

Metaphors and Analogies in Scientific Thinking

Holyoak and Thagard (1995) identify three constraints that must be satisfied by a good analogy: 1. Similarity: The source and the target must share some common properties. Both minds and computers process and store information; ... solve problems. 2. Structure: ... there should be an overall correspondence in structure. Here the mind/computer analogy becomes more slippery. 3. Purpose: The creation of analogies is guided by the problem-solver's goals. Analogies are not fixed forever - they can be modified.

A good analogy seems to accurately or completely represent critical relationships and concepts while less important attributes are considered irrelevant. 

Sir William Thomson (later: Lord Kelvin): "I never satisfy myself until I can make a mechanical model of a thing. If I can make a mechanical model, I can understand it. …

An analogy is comparing one thing that we understand to something that we do not understand to gain insight into the unknown. Therefore, analogies are used to explain or clarify.

The Blind Men and the Elephant

John Godfrey Saxe's ( 1816-1887) version of the famous Indian legendhttp://www.noogenesis.com/pineapple/blind_men_elephant.html

It was six men of IndostanTo learning much inclined,

Who went to see the Elephant(Though all of them were blind),

That each by observationMight satisfy his mind.

The First approached the Elephant,And happening to fall

Against his broad and sturdy side,At once began to bawl:

"God bless me! but the ElephantIs very like a wall!"

...

And so these men of IndostanDisputed loud and long,Each in his own opinion

Exceeding stiff and strong,Though each was partly in the right,

And all were in the wrong!

The challenge is to make useful analogiesthat do more good than harm,

& not to get too hung up on which one is the correct or “true” one.

The theoretical mathematical modelis defined, often implicitly as“if … then …” w/ the machineryof math churning out the implications “then” from the assumptions “if.”Math is “truth preserving” ifdone properly (“internal validity”)

The natural system is something of interest,w/ things we can measure,out there in the “real world”

The theoretical hypothesisis a tentative assertion

that the model correspondsto aspects of the natural system

(“external validity”)

If the premise “If …” turns out to not be valid

in particular cases,the theoretical hyp of

external validity is rejected,but the “model: If … then …”

isn’t really wrong;it just isn’t very useful

because it doesn’t specifythe “else …” part.

You can’t pull anything out of the hat that you didn’t put in to begin with.

RealityReality is that which, when you stop believing in it, doesn't go away. Philip K. Dick, Do Androids Dream of Electric Sheep? US science fiction author

(1928 - 1982)

IfIf [God exists] thenthen [he will save me]NOTNOT [saved] implies NOTNOT [he exists].

Is the logic is internally validinternally valid? Does the conclusion follows properly from the premise?Is the premise externally validexternally valid?

Any system can be represented by a block diagram consisting of an input, an output, and a law (or function f (•) that determines (maps) input to output: out = f(in).

in law out

law

in f(x(t)) out

predictiona) empirical: measure y & x only, curve fit (Black Box) and forecast.b) mechanistic: measure structural components-parameters: ex: y = a∙x + b

x(t)

t

y(t)

t

?

x(t)

t

in f(x(t)) out

controla) via inputb) via mechanism {later: inputs that change state of sys.}

y(t)

t

??

x(t) f(x(t)) y(t)

law

x(t)

t

in out ?

y(t)

t

system identificationtesting hypotheses about mechanism

If models are tools, what are they useful for? (from Doucet &Sloep pg. 296)

x(t)

t

in f(x(t)) out

controla) via inputb) via mechanism

y(t)

t

??

Mechanisms include behavioral & other choices of inputs

like season, habitat & social partners, etc.( “ecological theaters” )

Evolution by natural selectionsorts among mechanisms(“The evolutionary play” )on basis of performance

in particular “ecological theaters.”

Empirical vs Mechanistic Models?

In hierarchical systemslike the biosphere,the distinction betweenempirical phenomenaand causal mechanismis “frame sensitive”

In reductionist approach,every mechanism can be treated as a phenomenonneeding an explanationone level down.Your unmeasured, statistical parameters become my dependent variables.

Ecosystem responses to global climate change: Moving beyond color mapping. Schmitz et al. 2003.BIOSCIENCE 53 (12):199-1205..

In practice, there is somemechanism implicit in allempirical models & there are empirical parametersin all mechanistic models.

Levins, R. 1966. The strategy of model building in population biology. American Scientist 54 : 421-431.

Orzack, S.H. & Sober, E. 1993. A critical assessment of Levins's The Strategy of Model Building in Population Biology (1966). The Quarterly Review of Biology 68:533-546.

Levins, R. 1993. A response to Orzack and Sober: formal analysis and the fluidity of science. The Quarterly Review of Biology 68:547-555.

Mathematical models in population biology can be evaluated in terms of 3 general characteristics (Levins 1966): precisionprecision, generalitygenerality, and realismrealism. Any model will represent a compromise in these characteristics.

It is of course desirable to work with manageable models which maximize generality, realism, and precision toward the overlapping but not identical goals of understanding, predicting, and modifying nature. But this cannot be done. Therefore, several alternative strategies have evolved:1. Sacrifice generality to realism and precision. This is the approach of Bolling, (e.g., 1959), of many fishery biologists, arm of Watt (1956). These workers can reduce the parameters to the" relevant to the shortterm behavior of their organism, make fairly accurate measurements, solve numerically on the computer, and end with precise testable predictions applicable to these particular situations.2. Sacrifice realism to generality and precision. Kerner (1957), Leigh (1965), and most physicists who enter population biology work in this tradition which involves setting up quite general equations work which precise results may be obtained. Their equations am clearly unrealistic. For instance, they use the Volterra predator‑prey system which omit time lags, physiological states, and the effect of a species population density on its own rate of increase, But them workers hope that their model is analogous to assumptions of frictionless systems or perfect gases They expect that many of the unrealistic assumptions will cancel each other, that small deviations from realism result in small deviations in the conclusions, and that, in any case the way in which nature departs from theory will suggest where further complications will be useful. Starting with precision they hope to increase realism.3. Sacrifice precision to realism and generality. This approach is favored by MacArthur (1965) and myself. Since we are really concerned in the long run with qualitative rather than quantitative results (which are only important in testing hypotheses) we can resort to very flexible models often graphical, which generally assume that functions are increasing or decreasing, convex or concave, greater or less than some value, instead of specifying the mathematical form of an equation. This means that the predictions we can make am also expressed as inequalities