MTV 14: Mixed, Patched, and Meta-Models

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Module 14 from Creative Modeling for Tech Vision covers models of mixed form, patching models together, and meta-models with the emphasis on simplified quantitative models (little models).

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MTV 14: Mixed, Patched, and Meta-Models

  1. 1. Creative Modeling for Technology Visionaries Qualitative & Simplified Quantitative Modeling for Product Creation Module 14: Mixed, Patched & Meta-Models David E. Goldberg University of Illinois at Urbana-Champaign Urbana, Illinois 61801 deg@uiuc.edu
  2. 2. Making Little Modeling Fully Usable Can 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. Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 2
  3. 3. This Module 2 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. Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 3
  4. 4. The Problem of Mixed Models Many of our models with explicit solution have a particular form: Some mixed cases, f and g unrelated. 2 examples: search & hierarchy. Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 4
  5. 5. 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. Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 5
  6. 6. Optimizing Transactions Imagine Transaction costs C Success probability p Marginal 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. Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 6
  7. 7. Trials vs. Search Success 10 profit value cost 8 Profit, Value, or Cost of Hire 6 4 2 0 -2 0 10 20 30 40 50 60 70 80 90 100 Annual Frequency of Hires Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 7
  8. 8. Turning Point Solution Set Costs = Value. V[1 – (1 – p )n] = Cn. Explicit differential solution easier. 10 profit value TP not bad 8 cost approximation, Profit, Value, or Cost of Hire 6 however. 4 2 0 -2 0 10 20 30 40 50 60 70 80 90 100 Annual Frequency of Hires Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 8
  9. 9. 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 = T 1 k + T2 d Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 9
  10. 10. 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 Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 10
  11. 11. 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: k(lnk)2 Compute the average T1/T2. Given the organization size m, compute the SOC k’. Compute the mean error |k- k’| m Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 11
  12. 12. Can Generalize Topology Mean error Can generalize to arbitrary exponents. Both linear and quadratic are nearly a as predictive. Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 12
  13. 13. 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? Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 13
  14. 14. How to Integrate LMs Integration 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. Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 14
  15. 15. 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 Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 15
  16. 16. Example from GAs: Sweet Spot Mixing, drift, and cross-competitive boundaries intersect. Experiments match shape. Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 16
  17. 17. Types of Patch Transitions 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. Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 17
  18. 18. A Patchquilt Approach Organization size: m Model1: Model2: Larger value governs (bigger better conflict resolution). Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 18
  19. 19. 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. Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 19
  20. 20. 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. Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 20
  21. 21. Interpretation versus Ignorance #1 Suppose ignorance falls off inversely with length: Cε/ℓ. Cε C= + C   Suppose cost of interpretation goes up as the number of terms: Cℓℓ. Form is an old friend. Cε * = C Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 21
  22. 22. Interpretation versus Ignorance #2 Suppose ignorance falls off geometrically with model length: ε = (1 − α )  Suppose cost of interpretation goes up as the number of terms. Form is essential same as search model. C = Cε (1 − α ) + C   Solution: 1 C where * = − ln r = − ln 1 − α r rCε Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 22
  23. 23. 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? Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 23
  24. 24. Sample Cost vs. Lowered Uncertainty Error goes down inversely as σ square root of number of samples, ε = 1/ 2 m. m Assume cost increase is linear. Total cost: Cε σ C = Cs m + 1/ 2 m Optimum: 2/3  Cε σ  m* =   2C    s  Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 24
  25. 25. 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. Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 25
  26. 26. The Bottom, Bottom Line Science 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. Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 26

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