A Core Course on Modeling

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A Core Course on Modeling

Contents • What is a Formal Model?

• A Practical Route to Formal Models

• Example 1: The Detergent Problem

• Example 2: The Chimney Sweepers Problem

• Example 3: The Peanut Butter Problem

•The relation wizard

•The function selector• Summary

• References to lecture notes + book

• References to quiz-questions and homework assignments (lecture notes)

Week 4-The Function of Functions

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What is a Formal Model? What is the meaning of + ?

resistors: Rtot = R1 + R2

springs: Ctot = 1/(1/C1+1/C2)

resistors: Rtot = 1/(1/R1 + 1/R2)

springs: Ctot = C1 + C2

the intuition of ‘addition’, ‘accumulation’:


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What is a Formal Model? What is the meaning of + ?


Conclusion: there is always a need for interpretation going from relations in ‘real world’ to mathematical relations

There is no simple, generic way to infer mathematical relations from relations in rela world

We need a (heuristic) process to do so.

the intuition of ‘addition’, ‘accumulation’:

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A Practical Route to Formal Models Heuristics to arrive at formal expressions:

• meaningful names

• chain of dependencies

• todo-list

• dimensional analysis

• wisdom of the crowds

• two models is better than one model

• when is a model good enough?

• Baron von Münchhausen

In a conceptual model, properties are always part of a concept (‘myCar.wheel.diameter’). In a formal model, properties may be ‘just’ quantities (‘myCarWheelDiameter) with names that may be meaningfully abbreviated (‘mCWhD’).

General scheme: start with the quantitiy needed for your purpose, and try to express this in other quantities

•in the simplest possible form

•with as few as possible assumptions

•such that the quantity is written as function of the other quantities

•continue with the arguments of the function

Everytime when introducing a new quantity, add it to the todo list. When a quantity is expressed into in something known (= a value or another expression), take it off the todo list. When the todo list is empty, model’s first version is ready.

When seeking a mathematical expression,

•if possible, use dimensional considerations to find needed expression …

•… but at least verify expression with dimensional analysis

•… even if this may require inventing units and dimensions.

Often values need guessing. If you can involve a number of people, let them guess independently. This gives (a) a more accurate estimate if there are no systematic errors and (b) an idea of the variation.

Otherwise: try to relate the unknown values to values you (and your friends) may know.

Sometimes, there are two or more routes to (part) of your model. Implement them all, and compare the results. The spread in results is a clue to the reliability of the achievable outcome.

A model is never complete and fully accurate. But, given its purpose, it can be complete and accurate enough. (See chapter 6.) Regularly check if your purpose is met.

Once you have your first version running, you can:

•use it to find out which of the uncertainties of your inputs are the most dominant try to get these more accurate if necessary

•find out which modifications could be worthwhile


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The Detergent Problem

“What is the total amount of detergent

annually dumped in the

Environment in the Netherlnds?


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amAnDetDmp =

nrAnWshs * detPWsh

relations dimensions assumptions todo[kg / year] =

[wash / year] * [kg / wash]

Washing laundry is the only way detergent gets into the environment

amAnDetDmpnrAnWshsdetPWshnrAnWshsPFamnrFamIHnrPIHnrPPFam

nrAnWshs =

nrAnWshsPFam * nrFamIH

[wash / year] =

[wash / (fam *year)] * [fam]

No institutional laundry washing, only families

nrFamIH =

nrPIH / nrPPFam

[fam] =

[people] / [people / fam]

Everybody belongs to exactly one family: families are disjoint

nrPIH = 17 0.5 million [people] common knowledge

nrPPFam = 2.2 0.2 [people/fam] public domain

nrAnWshsPFam = 100 20 [wash / year] wisdom of the crowds

Det:detergent; An: annual; Wsh: wash; Fam: family; P: people; am: amount; Dmp: dump

detPWsh = 0.17 0.03 [kg / wash] wisdom of the crowds

todo list is empty model is ready



http://www.keesvanoverveld.com/Accel/accel.htm?script=laundry.txt

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This type of model is a ‘thumbnail calculation’:

• OK for quick order of magnitude estimations

• Uses straightforward substitutions, only based on dimension analysis

• Work with intervals to get an impression of the variation of the answers (143 67 M; correct value according to various sites such as http://wiki.watmooi.nl/pages/Wassen_en_onderhoud is 150 M kg)

• Purpose: pub quizzes, trivial pursuit, …

a.k.a. ‘sledgehammer estimation’



http://wiki.watmooi.nl/pages/Wassen_en_onderhoud



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About accuracymodel: outcome = function of inputs y=f(x1,x2,…)

y=f/x1 x1 + f/x2 x2 + …

So:|y|=|f/x1| |x1| + |f/x2| |x2| + …

This is a very pessimistic upperbound: all quantities need to conspire to give worst case deviation.

In chapter 6, we will get a more realistic estimate.



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The Chimney Sweepers Problem

“How many chimney sweepers

work in Amsterdam?”


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nrChSwIA =

nrChIA * nrSwPCh

relations dimensions assumptions todo[Sw / A] =

[Ch / A] * [SW / Ch]

Amsterdam ch.-sweepers sweep Amsterdam chimneys only

nrChSwIAnrChIAnrSwPChnrChPFamnrFamIAnrPIAnrPPFam

nrChIA =

nrChPFam * nrFamIA

[Ch / A] =

[Ch / Fam] * [Fam / A]

ch.-sweepers sweep only chimneys on family houses

nrFamIA =

nrPIA / nrPPFam

[Fam / A] =

[P / A] / [P / Fam]

nrPPFam is the same everywhere (does not depend on ‘Amsterdam’)

nrPIA = 790000 [P] common knowledge

nrPPFam = 2.2 0.2 [P/Fam] public domain

nrChPFam (= 1/nrFamPCh) =0.10.02 [Ch/Fam]

wisdom of the crowds

Sw=sweeper; Ch=chimney;A=Amsterdam;Fam=Family;P=people;Se=Service;



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nrSwPCh =

nrSwPSe * nrSePCh

relations dimensions assumptions todo[Sw / Ch] =

[Sw*year/Se] * [Se/(Ch*year)]

Introduce time to associate sweeper’s ca-pacity to chimney’s need

nrChSwIAnrChIAnrSwPChnrChPFamnrFamIAnrPIAnrPPFamnrSwPSenrSePChtimeP1SetimeP1Sw

nrSwPSe =

timeP1Se / timeP1Sw

[Sw * year / Se] =

[hour / Se] / [hour / (Sw*year)]

assume average times (i.e., no season influences etc.)

timeP1Sw = 1200100 hour / Sw * year)

Sw=sweeper; Ch=chimney;A=Amsterdam;Fam=Family;P=people;Se=Service;

timeP1Se = 20.25 hour / Se wisdom of the crowds

work year = 1600 hours

nrSePCh = 1 Se /( Ch * year) insurance requirement

todo list is empty model is ready

but what does it mean ? NOTHING, since we formulated no purpose

The Chimney Sweepers Problem Notice: two different units, both with dimension time. To verify that units are consistent, substitute back into expressions for nrSwPSe and nrSwPCh: check that all units multiply and divide to produce the correct final result.

In this case: nrSwPCh=timeP1Se * nrSePCh / timeP1Sw has unit ‘sweeper / chimney’ – which is correct.


To find an expression for nrSwPCh, ask: ‘what links the nr of sweepers to the nr chimneys?’. Answer: sweepers service chimneys. ‘How many services’ (1) relates to the capacity (=available resource) of a sweeper, and (2) to the need of a chimney (=needed resource). Here, ‘resource’=time. The capacity of a sweeper is expressed in the time he works ([Sw*year]); The ‘need’ of a chimney is therefore expressed ([Ch*year]).

http://www.keesvanoverveld.com/Accel/accel.htm?script=chimneySweepers1.txt

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but what does it mean ? NOTHING, since we formulated no purpose

What purposes could we think of:

are there at least 300 Chimney Sweepers so that we can begin a professional journal?

so: we only need to know if NrChSwIA > 300

are there less than 50 Chimney Sweepers so that we can have next year’s ChSw convention meeting in the Restaurant ‘the Swinging Sweeper?’

so: we only need to know if NrChSwIA <50

are there about as many Chimney Sweepers as there are Sewer Cleaners so that we can form efficient ‘Chimney and Sewage Control and Service Units’?

so: we only need to know if NrChSwIA is between 50 and 60…

… each purpose poses different challenges / allows different approximations in our model.



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To assess credibility of a model:

•confront with actual measurements

•confront with outcome of a second, independent model

• how much soot and ashes are disposed of by the municipal Ash & Soot Depot?

• how often do you see a chimney sweeper at work (wisdom of the crowds)?

• how much money do people in A. spend annually in cleaning their chimneys?

check out a model for the later case:

(notice: this model is not completely independent from the previous; it uses some common quantities)

The Chimney Sweepers Problem doing experiments: beyond the scope

of modeling


http://www.keesvanoverveld.com/Accel/accel.htm?script=chimneySweepers2.txt

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Thusfar, all mathematical relations were obtained via dimension analysis.

Dimensions are more generic than just SI dimensions/units.

Dimensional analysis often works for models of a particular form:

y = x1n1 * x2

n2 * x3n3 … = i xi

ni,

where (integer) ni can be both larger and smaller than 0.

How to go about when other forms are needed?



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The Peanut Butter Problem

“How to get rich

by selling a new brand

of

peanut butter?”

PB


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income =

pricePerItem * nrSoldItems

relations dimensions assumptions todo

[Euro / year] =

[Euro/myPB] * [myPB/year]

no discount with larger quantities per purchase

profitincomeexpensespricePerItemnrSoldItemsnrSoldTotalmarketShare

nrSoldItems =

nrSoldTotal * marketShare

[myPB / year] =

[allPB / year] * [myPB/allPB]

my PB will not increase the total market

pricePerItem = …

nrSoldTotal = … [allPB/year] from a neutral marketing bureau

marketShare = …

problems:

• I have a choice for pricePerItem

• marketShare depends on pricePerItem



profit = income - expenses [Euro / year] = [Euro/year] no taxes, no inflation

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problems:



Approach 1: glass box (glass jar …;-):

pricePerItem

mar

ketS

hare

0

1

??



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problems:




pricePerItem

mar

ketS

hare

0

1

What mechanism determines marketShare(pricePerItem)?• monotonically decreasing• between 0 and 1• asymptote: marketShare0 if pricePerItem • asymptote: marketShare1 if pricePerItem - • what sort of mathematical dependency ???


If something gets more expensive, the chance people will buy it decreases

A market share cannot be <0; it could be >1 but only if it creates additional request

If something gets sufficiently expensive, nobody will buy itIf something gets sufficiently cheap, every potentially interested customer will buy (or get!) it

It would need a glass box model of customers’ brains to derive this dependency from first principles


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problems:




pricePerItem

mar

ketS

hare

0

1

cheapestcompetitor

most expensivecompetitor

1st guess: straight lines

• advantage: simple

• disadvantage: not smooth

• uncertain: does this represent the actual behavior?



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problems:




pricePerItem

mar

ketS

hare

0

1

cheapestcompetitor


2nd, 3rd guess: arctan, logistic, …?

• advantage: smooth

• disadvantage: more parameters?

• what values for the parameters?

• uncertain if this follows the actual behaviour

The Peanut Butter Problem Depending on what you are going to DO with the math (e.g., optimisation), smoothness can be important


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problems:




pricePerItem

mar

ketS

hare

0

1

cheapestcompetitor




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problems:



Approach 2: black box: use panel of test subjects; ask them if they would buy your PB for price X

pricePerItem

mar

ketS

hare

0

1 do a smooth curve fit (e.g., splines) that satisfies 0 marketShare 1



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problems:



Approach 2: black box: use panel of test subjects; ask them if they would buy your PB for price X

pricePerItem

mar

ketS

hare

0

1 do a smooth curve fit (e.g., splines) that satisfies 0 marketShare 1

but, anyway:

let us assume we have some function:

then the model predicts the income

as function of the pricePerItem.



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Revisit the peanut butter example:

income=pricePerItem * nrSoldItems

nrSoldItems=nrSoldTotal * marketShare

marketShare=f(pricePerItem)

for convenience, introduce abbreviations:



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Revisit the peanut butter example:

inc =pPI * nSI (inc=income; pPI=pricePerItem; nSI=nrSoldItems)

nSI = nST * mSh (nST=nrSoldTotal; mSh=marketShare)

mSh=f(pPI)

So: inc = pPI * nST * mSh(pPI)

This is naive: first, realize that nST = i nSTi, i ranges over customers.

So: inc = pPI * mSh(pPI) * i nST(i)

Improved model: inc = pPI * i nST(i) * mSh(pPI, i)

The Peanut Butter Problem This is irrelevant if all customers would buy equal amounts of peanutbutter. But: there are mega-consumers and mini-consumers !

Realize that the decision to choose MY peanutbutter is taken by a customer. Suppose that the majority of mega customers would decide against my peanutbutter – then the original model with mSh(pPI) instead of mSh(pPI,i) would give an overestimate misleading.

So: think about which arguments a function should depend on!


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To go from conceptual model to formal model:

while your purpose is not satisfied:

•start with quantity you need for the purpose•put this on the to-do list•while the todo list is not empty:

•take a quantity from the todo list•think: what does it depend on?

•if depends on nothing substitute constant value (perhaps with uncertainty bounds)•else give an expression for it

•if possible, use dimensional analysis•propose suitable mathematical expression•think about assumptions•in any case, verify dimensions

•add newly introduced quantities to the todo list•todo list is empty: evaluate your model•check if purpose is satisfied; if not, refine your model

• ready



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To go from conceptual model to formal model:

while your purpose is not satisfied:

•start with quantity you need for the purpose•put this on the to-do list•while the todo list is not empty:

•take a quantity from the todo list•think: what does it depend on?

•if depends on nothing substitute constant value (perhaps with uncertainty bounds)•else give an expression for it

•if possible, use dimensional analysis•propose suitable mathematical expression•think about assumptions•in any case, verify dimensions

•add newly introduced quantities to the todo list•todo list is empty: evaluate your model•check if purpose is satisfied; if not, refine your model

• ready

splinearctan

asymptot

efunction selectorfunction selector

relation wizardrelation wizard

logistic

function

optim

isatio

n The Peanut Butter Problem


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• Conceptual model formal model : not in a formally provable correct way;

• Appropriate naming

• Structure

• Chain of dependencies: the formal model as a directed acyclic graph;

•What mechanism?

•What quantities drive this mechanism?

•What is the qualitative behavior of the mechanism?

•What is the mathematical expression to describe this mechanism?

• To-do-list : all intermediate quantities are found and elaborated in turn;

• Formation of mathematical expressions:

•dimensional analysis mathematical expressions, e.g in the case of proportionality

•the Relation Wizard can help finding appropriate fragments of mathematics;

•the Function Selector can help finding an appropriate expression for a desired behavior;

•wisdom of the crowds can help improve the accuracy of guessed values;

Summary


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A Core Course on Modeling

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