MTV 14: Mixed, Patched, and Meta-Models

Creative Modeling forTechnology VisionariesQualitative & Simplified Quantitative Modeling for Product Creation

Module 14: Mixed, Patched & Meta-Models

David E. GoldbergUniversity of Illinois at Urbana-ChampaignUrbana, Illinois 61801

[email protected]

Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved.2

Making Little Modeling Fully UsableCan sometimes squeeze models with mixed terms.

Usually need to combine or patch multiple models.

Can move beyond models of phenomena and create meta-models.

Will have the basic tools for any LMing task.


This Module2 mixed models:

Search or transaction costs.

Hierarchical organization.

Integrating models with patchquilts: Can we integrate two or models?

Models of models:Can we go quantitative about modeling itself?

Consider various tradeoffs of models themselves.

Short wrapup of course.


The Problem of Mixed ModelsMany of our models with explicit solution have a particular form:

Some mixed cases, f and g unrelated.2 examples: search & hierarchy.


Searching for Things of Value at Cost

Searching for something of value.

Each trial has a cost C.

Each trial is successful with probability p.

When found object realizes value V.

Could be a job (transaction costs), an employee, or other thing of value.

Optimize number of trials by maximizing expected profit.


Optimizing TransactionsImagine

Transaction costs CSuccess probability pMarginal value of successful transaction V

Profit P of nth transaction:P = V[1 – (1 – p )n] – Cn

Maximizing profit yields:n* = [ln V/C + ln r] / r,

where q = 1 – p, r = -ln q Example: p = 0.1, V = 10, C = 0.1, n* = 22.4.


Trials vs. Search Success

0 10 20 30 40 50 60 70 80 90 100-2

0

2

4

6

8

10

Annual Frequency of Hires

Pro

fit,

Val

ue,

or C

ost

of H

ire

profitvalue cost


Turning Point Solution

Set Costs = Value.V[1 – (1 – p )n] = Cn.

Explicit differential solution easier.

TP not badapproximation,however.

0 10 20 30 40 50 60 70 80 90 100-2

0

2

4

6

8

10

Annual Frequency of Hires

Pro

fit,

Val

ue,

or C

ost

of H

ire

profitvalue cost


Imagine a Hierarchy

Bosses communicate down the hierarchy to the bosses at the next level sequentially.

Within teams there is communication with specified topology.

What team size k minimizes the total communication m.

T = T1k + T2d


Optimal Span of Control

o m = total organization size.

o k = team size.

o d = organization depth.

o Minimize total coordination communication in organization.

o

o

o


Results on Texas School Data

Does this work in real organizations?678 Texas schools from 1994 to 1997, total 2712 data points (m,k) [Moore & Bohte, 2000, 2003].Measuring the quality of models:

Compute the average T1/T2.

Given the organization size m, compute the SOC k’.

Compute the mean error |k-k’| m

k(lnk)

2


Can Generalize Topology

Can generalize to arbitrary exponents.

Both linear and quadratic are nearly as predictive.

aM

ean

erro

r


A Simple Approach to Complexity

Complex systems complex, because they have multiple facets.

Cannot expect simple model to model then all.

Different tradeoffs govern in different parts of “phase space.”

Can we build multiple models and integrate them some how?


How to Integrate LMsIntegration methods:

Asymptotic methods of applied math & physics.Probabilistic blending.Patchquilt integration.

First two add complexity of their own.Let’s KISS (keep it simple stupid).Consider patchquilt integration.


Example from Fluids: Pipeflow

Laminar flows: model is analytical.

Turbulent flow model is partially analytical, partially a curve fit.

Patchquilt is discontinuous (region of instability) yet principled.

Moody diagram a patchquilt


Example from GAs: Sweet Spot

Mixing, drift, and cross-competitive boundaries intersect.

Experiments match shape.


Types of PatchTransitions occur at critical values of a dimensionless parameter.

Usually determined experimentally.Rapid qualitative change in system.

Intersecting inequalities:Boundaries of region determine feasibility.Example: Sweet Spot.

Intersecting equalities:Requires conflict management scheme.Often bigger or smaller is better.


A Patchquilt Approach

Organization size: m

Model1:

Model2:

Larger value governs (bigger better conflict resolution).


Meta-Models: Models of Models

Can we characterize models mathematically?

Would like models that are optimum in some sense.

Will consider two cases:Theoretical modeling: Error vs. cost of interpretation.

Empirical modeling: Reliability vs. cost of sampling.


Interpretation versus Ignorance

Little models are easily interpreted, but sometimes they are not as accurate as others.

This is the fundamental tradeoff of modeling.

Considered in information theory as minimum description length.

Let’s do bounding little models of little models.


Interpretation versus Ignorance #1Suppose ignorance falls off inversely with length: Cε/ℓ.

Suppose cost of interpretation goes up as the number of terms: Cℓℓ.

Form is an old friend.

CCC

C

C*


Interpretation versus Ignorance #2Suppose ignorance falls off geometrically with model length:

Suppose cost of interpretation goes up as the number of terms.

Form is essential same as search model.

Solution: where

1

CCC 1

rC

C

r ln

1* 1lnr


Consider Typical Lab Setting

Can sample more and improve accuracy of mean measurement.

But each sample has a cost.

Stat classes teach more samples better, but what if we can quantify cost of error and cost of sample.

Wouldn’t we want to minimize total costs?


Sample Cost vs. Lowered Uncertainty

Error goes down inversely as square root of number of samples, m.

Assume cost increase islinear.

Total cost:

Optimum:

2/1mCmCC s

2/1m

3/2

2*

sC

Cm


Bottom Line

Can sometimes handle mixed little models.

Oftentimes can integrate LMs in patches.

LMs of LMs give important insights.

What about larger lessons of course about modeling generally.


The Bottom, Bottom LineScience is relentless in search of better knowledge: Models are end.Engineering is relentless in search of better designs: Models are instrumental to that end.Need for category creators shifts emphasis to qual theory & LMs for first quant.Engineers have always written on napkins & scratched out bounding calcs on back of envelopes.Real world of engineering thought is dressed up in scientific garb until unrecognizable.Doing so is disservice & misleading.Road to tech vision is paved with bricks of effective creative modeling.

MTV 14: Mixed, Patched, and Meta-Models

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