Introduction Case 1: Out of Eden Walk 13. How We Compare: Technical Details Case 2: Climate . . . • How the number of comments r ( t ) depends on time t ? Case 3: Software . . . Placing Bio-Weapon . . . – exponential model: r ( t ) ≈ r 0 ( t ) = A · exp( − α · t ); Meteorological Sensors – power law model: r ( t ) ≈ r 0 ( t ) = A · t − α . UAVs Patrolling the . . . • To check which model is more adequate, we use the Optimal Placement Tests chi-square criterion Feedback for Students � ( r ( t ) − r 0 ( t )) 2 χ 2 def Home Page = . r 0 ( t ) Title Page t • To estimate A and α , we use both Least Squares ◭◭ ◮◮ � i → min and robust ( ℓ 1 ) estimation � e 2 | e i | → min. ◭ ◮ i i • Result: the power law is more adequate: Page 14 of 106 – for exponential model H 0 , p ≪ 0 . 05, so H 0 is re- Go Back jected; Full Screen – for power law model H 0 , p ≫ 0 . 05, so H 0 is not Close rejected. Quit
Introduction Case 1: Out of Eden Walk 14. Comparison Results Case 2: Climate . . . Case 3: Software . . . χ 2 χ 2 χ 2 Dispatch Title N c p p p p, 1 p e p p, 1 e Placing Bio-Weapon . . . Let’s Walk 271 30.6 30.0 31,360 0 . 33 0 . 37 0.00 Meteorological Sensors Sole Brothers 61 22.1 22.8 83 0 . 76 0 . 74 0.00 UAVs Patrolling the . . . The Glorious Optimal Placement Tests Boneyard 59 16.3 18.6 262 0.96 0.91 0.00 Feedback for Students The Self-Love Home Page Boat 67 63.1 60.0 124 0.00 0.00 0.00 Title Page Go Slowly–Work ◭◭ ◮◮ Slowly 91 33.0 31.5 821 0 . 24 0 . 29 0.00 The Camel and ◭ ◮ the Gyrocopter 52 28.4 24.6 72 0 . 45 0 . 65 0.00 Page 15 of 106 Lines in Sand 69 21.4 18.3 89 0.81 0 . 92 0.00 Go Back Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 15. Out of Eden Walk: Conclusions Case 2: Climate . . . • To improve teaching and learning, it is important to Case 3: Software . . . know how knowledge propagates. Placing Bio-Weapon . . . Meteorological Sensors • Traditional models of knowledge propagation are sim- UAVs Patrolling the . . . ilar to differential-equations-based models in physics. Optimal Placement Tests • Recently, an alternative fractal-motivated power-law Feedback for Students model of knowledge propagation was proposed. Home Page • We compare this model with the traditional model on Title Page the example of the Out of Eden Walk project. ◭◭ ◮◮ • It turns out that for the related data, the power law is ◭ ◮ indeed a more adequate description. Page 16 of 106 • This shows that the fractal-motivated power law is a Go Back more adequate description of knowledge propagation. Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 16. Analytical Techniques for Knowledge Use, on Case 2: Climate . . . the Example of Climate Variability Case 3: Software . . . • Global warming is a statistically confirmed long-term Placing Bio-Weapon . . . phenomenon. Meteorological Sensors UAVs Patrolling the . . . • Somewhat surprisingly, its most visible consequence is: Optimal Placement Tests – not the warming itself but Feedback for Students – the increased climate variability. Home Page • In this talk, we explain why increased climate variabil- Title Page ity is more visible than the global warming itself. ◭◭ ◮◮ • In this explanation, use general system theory ideas. ◭ ◮ Page 17 of 106 Go Back Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 17. Formulation of the Problem Case 2: Climate . . . • Global warming usually means statistically significant Case 3: Software . . . long-term increase in the average temperature. Placing Bio-Weapon . . . Meteorological Sensors • Researchers have analyzed the expected future conse- UAVs Patrolling the . . . quences of global warming: Optimal Placement Tests – increase in temperature, Feedback for Students – melting of glaciers, Home Page – raising sea level, etc. Title Page • A natural hypothesis was that at present, we would see ◭◭ ◮◮ the same effects, but at a smaller magnitude. ◭ ◮ • This turned out not to be the case. Page 18 of 106 • Some places do have the warmest summers and the Go Back warmest winters in record. Full Screen • However, other places have the coldest summers and the coldest winters on record. Close Quit
Introduction Case 1: Out of Eden Walk 18. Formulation of the Problem (cont-d) Case 2: Climate . . . • What we actually observe is unusually high deviations Case 3: Software . . . from the average. Placing Bio-Weapon . . . Meteorological Sensors • This phenomenon is called increased climate variabil- UAVs Patrolling the . . . ity . Optimal Placement Tests • A natural question is: why is increased climate vari- Feedback for Students ability more visible than global warming? Home Page • A usual answer is that the increased climate variability Title Page is what computer models predict. ◭◭ ◮◮ • However, the existing models of climate change are still ◭ ◮ very crude. Page 19 of 106 • None of these models explains why temperature in- Go Back crease has slowed down in the last two decades. Full Screen • It is therefore desirable to provide more reliable expla- nations. Close Quit
Introduction Case 1: Out of Eden Walk 19. A Simplified System-Theory Model Case 2: Climate . . . • Let us consider the simplest model, in which the state Case 3: Software . . . of the Earth is described by a single parameter x . Placing Bio-Weapon . . . Meteorological Sensors • In our case, x can be an average Earth temperature or UAVs Patrolling the . . . the temperature at a certain location. Optimal Placement Tests • We want to describe how x changes with time. Feedback for Students • In the first approximation, dx Home Page dt = u ( t ), where u ( t ) are Title Page external forces. ◭◭ ◮◮ • We know that, on average, these forces lead to a global warming, i.e., to the increase of x ( t ). ◭ ◮ Page 20 of 106 • Thus, the average value u 0 of u ( t ) is positive. Go Back def • We assume that the random deviations r ( t ) = u ( t ) − u 0 Full Screen are i.i.d., with some standard deviation σ 0 . Close Quit
Introduction Case 1: Out of Eden Walk 20. Conclusions Case 2: Climate . . . • By solving this equation, we get an analytical model, Case 3: Software . . . in which: Placing Bio-Weapon . . . Meteorological Sensors – the systematic part x s ( t ) corresponds to global warm- UAVs Patrolling the . . . ing, while Optimal Placement Tests – the random part x r ( t ) corresponds to climate vari- Feedback for Students ability. Home Page • It turns out that on the initial stages of this process: Title Page – climate variability effects are indeed much larger ◭◭ ◮◮ – than the effects of global warming. ◭ ◮ • This is exactly what we currently observe. Page 21 of 106 • Similar conclusions can be made if we consider more Go Back complex multi-parametric models. Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 21. Analytical Techniques for Knowledge Use, on Case 2: Climate . . . the Example of Software Migration Case 3: Software . . . • In many aspects of our daily life, we rely on computer Placing Bio-Weapon . . . systems: Meteorological Sensors UAVs Patrolling the . . . – computer systems record and maintain the student grades, Optimal Placement Tests Feedback for Students – computer systems handle our salaries, Home Page – computer systems record and maintain our medical records, Title Page – computer systems take care of records about the ◭◭ ◮◮ city streets, ◭ ◮ – computer systems regulate where the planes fly, etc. Page 22 of 106 • Most of these systems have been successfully used for Go Back years and decades. Full Screen • Every user wants to have a computer system that, once implemented, can effectively run for a long time. Close Quit
Introduction Case 1: Out of Eden Walk 22. Need for Software Migration/Modernization Case 2: Climate . . . • Computer systems operate in a certain environment; Case 3: Software . . . they are designed: Placing Bio-Weapon . . . Meteorological Sensors – for a certain computer hardware – e.g., with sup- UAVs Patrolling the . . . port for words of certain length, Optimal Placement Tests – for a certain operating system, programming lan- Feedback for Students guage, interface, etc. Home Page • Eventually, the computer hardware is replaced by a Title Page new one. ◭◭ ◮◮ • While all the efforts are made to make the new hard- ◭ ◮ ware compatible with the old code, there are limits. Page 23 of 106 • As a result, after some time, not all the features of the old system are supported. Go Back • In such situations, it is necessary to adjust the legacy Full Screen software so that it will work on a new system. Close Quit
Introduction Case 1: Out of Eden Walk 23. Software Migration and Modernization Is Dif- Case 2: Climate . . . ficult Case 3: Software . . . • At first glance, software migration and modernization Placing Bio-Weapon . . . sounds like a reasonably simple task: Meteorological Sensors UAVs Patrolling the . . . – the main intellectual challenge of software design is Optimal Placement Tests usually when we have to invent new techniques; Feedback for Students – in software migration and modernization, these tech- Home Page niques have already been invented. Title Page • Migration would be easy if every single operation from ◭◭ ◮◮ the legacy code was clearly explained and justified. ◭ ◮ • The actual software is far from this ideal. Page 24 of 106 • In search for efficiency, many “tricks” are added by programmers that take into account specific hardware. Go Back • When the hardware changes, these tricks can slow the Full Screen system down instead of making it run more efficiently. Close Quit
Introduction Case 1: Out of Eden Walk 24. How Migration Is Usually Done Case 2: Climate . . . • When a user runs a legacy code on a new system, the Case 3: Software . . . compiler produces thousands of error messages. Placing Bio-Weapon . . . Meteorological Sensors • Usually, a software developer corrects these errors one UAVs Patrolling the . . . by one. Optimal Placement Tests • This is a very slow and very expensive process: Feedback for Students – correcting each error can take hours, and Home Page – the resulting salary expenses can run to millions of Title Page dollars. ◭◭ ◮◮ • There exist tools that try to automate this process by ◭ ◮ speeding up the correction of each individual error. Page 25 of 106 • These tools speed up the required time by a factor of Go Back even ten. Full Screen • However, still thousands of errors have to be handled individually. Close Quit
Introduction Case 1: Out of Eden Walk 25. Resulting Problem: Need to Speed up Migra- Case 2: Climate . . . tion and Modernization Case 3: Software . . . • Migration and modernization of legacy software is a Placing Bio-Weapon . . . ubiquitous problem. Meteorological Sensors UAVs Patrolling the . . . • It is thus desirable to come up with ways to speed up Optimal Placement Tests this process. Feedback for Students • In this dissertation: Home Page – we propose such an idea, and Title Page – we show how expert knowledge can help in imple- ◭◭ ◮◮ menting this idea. ◭ ◮ Page 26 of 106 Go Back Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 26. Our Main Idea Case 2: Climate . . . • Modern compilers do not simply indicate an error. Case 3: Software . . . Placing Bio-Weapon . . . • They usually provide a reasonably understandable de- Meteorological Sensors scription of the type of an error; for example: UAVs Patrolling the . . . – it may be that a program is dividing by zero, Optimal Placement Tests – it may be that an array index is out of bound. Feedback for Students • Some of these types of error appear in numerous places Home Page in the software. Title Page • Our experience shows that in many such places, these ◭◭ ◮◮ errors are caused by the same problem in the code. ◭ ◮ • So, instead of trying to “rack our brains” over each Page 27 of 106 individual error, a better idea is Go Back – to look at all the errors of the given type, and Full Screen – come up with a solution that would automatically eliminate the vast majority of these errors. Close Quit
Introduction Case 1: Out of Eden Walk 27. Need for Analytical Models Case 2: Climate . . . • This idea saves time only if we have enough errors of Case 3: Software . . . a given type. Placing Bio-Weapon . . . Meteorological Sensors • We thus need to predict how many errors of different UAVs Patrolling the . . . type we will encounter. Optimal Placement Tests • There are currently no well-justified software models Feedback for Students that can predict these numbers. Home Page • What we do have is many system developers who have Title Page an experience in migrating and modernizing software. ◭◭ ◮◮ • It is therefore desirable to utilize their experience. ◭ ◮ • So, we need to build an analytical model based on ex- Page 28 of 106 pert knowledge. Go Back Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 28. Expert Knowledge about Software Migration Case 2: Climate . . . • A reasonable idea is to start with n 1 errors of the most Case 3: Software . . . frequent type. Placing Bio-Weapon . . . Meteorological Sensors • Then, we should concentrate on n 2 errors of the second UAVs Patrolling the . . . most frequent type, etc. Optimal Placement Tests • So, we want to know the numbers n 1 , n 2 , . . . , for which Feedback for Students Home Page n 1 ≥ n 2 ≥ . . . ≥ n k − 1 ≥ n k ≥ n k +1 ≥ . . . Title Page • We know that for every k , n k +1 is somewhat smaller ◭◭ ◮◮ than n k . ◭ ◮ • Similarly, n k +2 is more noticeably smaller than n k , etc. Page 29 of 106 • After formalizing the n k < n k +1 rule, we get n k +1 = Go Back f ( n k ) . Full Screen • Which function f ( n ) should we choose? Close Quit
Introduction Case 1: Out of Eden Walk 29. Which Function f ( n ) Should We Choose? Case 2: Climate . . . • A migrated software package usually consists of two Case 3: Software . . . (or more) parts. Placing Bio-Weapon . . . Meteorological Sensors • We can estimate n k +1 in two different ways: UAVs Patrolling the . . . – We can use n k = n (1) k + n (2) to predict k Optimal Placement Tests n k +1 ≈ f ( n k ) = f ( n (1) k + n (2) Feedback for Students k ) . Home Page – Or, we can use n (1) to predict n (1) k +1 , n (2) to predict k k Title Page n (2) k +1 , and add them: n k +1 ≈ f ( n (1) k ) + f ( n (2) k ) . ◭◭ ◮◮ • It is reasonable to require that these estimates coincide: ◭ ◮ f ( n (1) k + n (2) k ) = f ( n (1) k ) + f ( n (2) k ) . Page 30 of 106 • So, f ( a + b ) = f ( a ) + f ( b ) for all a and b , thus f ( a ) = Go Back f (1) + . . . + f (1) ( a times), and f ( a ) = f (1) · a . Full Screen • Thus, n k +1 = c · n k , i.e., n k +1 /n k = const. Close Quit
Introduction Case 1: Out of Eden Walk 30. Empirical Data: Values n k for Migrating a Case 2: Climate . . . Health-Related C Package from 32 to 64 Bits Case 3: Software . . . Here, n ab is stored in the a-th column (marked ax) and Placing Bio-Weapon . . . b-th row (marked xb). Meteorological Sensors UAVs Patrolling the . . . 0x 1x 2x 3x 4x 5x 6x 7x Optimal Placement Tests x0 – 308 95 47 13 5 2 1 Feedback for Students x1 7682 301 91 38 13 4 2 1 Home Page x2 4757 266 85 34 12 4 2 1 Title Page x3 3574 261 81 34 12 4 2 1 x4 2473 241 76 30 11 3 2 1 ◭◭ ◮◮ x5 2157 240 69 24 9 3 2 1 ◭ ◮ x6 956 236 58 21 8 3 2 1 Page 31 of 106 x7 769 171 57 19 8 3 1 1 x8 565 156 50 17 8 2 1 1 Go Back x9 436 98 47 17 6 2 1 – Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 31. Empirical Data: Values n k for Migrating a Case 2: Climate . . . Health-Related C Package from 32 to 64 Bits Case 3: Software . . . Here, n ab is stored in the a-th column (marked ax) and Placing Bio-Weapon . . . b-th row (marked xb); e.g., n 23 = 81. Meteorological Sensors UAVs Patrolling the . . . 0x 1x 2x 3x 4x 5x 6x 7x Optimal Placement Tests x0 – 308 95 47 13 5 2 1 Feedback for Students x1 7682 301 91 38 13 4 2 1 Home Page x2 4757 266 85 34 12 4 2 1 Title Page x3 3574 261 81 34 12 4 2 1 x4 2473 241 76 30 11 3 2 1 ◭◭ ◮◮ x5 2157 240 69 24 9 3 2 1 ◭ ◮ x6 956 236 58 21 8 3 2 1 Page 32 of 106 x7 769 171 57 19 8 3 1 1 x8 565 156 50 17 8 2 1 1 Go Back x9 436 98 47 17 6 2 1 – Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 32. How Accurate is This Estimate? Case 2: Climate . . . • One can easily see that for k ≤ 9, we indeed have Case 3: Software . . . n k +1 ≈ c · n k , with c ≈ 0 . 65-0.75. Placing Bio-Weapon . . . Meteorological Sensors • Thus, the above simple rule described the most fre- UAVs Patrolling the . . . quent errors reasonably accurately. Optimal Placement Tests • However, starting with k = 10, the ratio n k +1 /n k be- Feedback for Students comes much closer to 1. Home Page • Thus, the one-rule estimate is no longer a good esti- Title Page mate. ◭◭ ◮◮ • A natural idea is this to use two rules: ◭ ◮ – in addition to the rule that n k +1 is somewhat smaller Page 33 of 106 than n k , Go Back – let us also use the rule that n k +2 is more noticeably Full Screen smaller than n k . Close Quit
Introduction Case 1: Out of Eden Walk 33. Two-Rules Approach: Results Case 2: Climate . . . • Similar arguments lead to the following analytical model Case 3: Software . . . Placing Bio-Weapon . . . n k = A 1 · exp( − b 1 · k ) + A 2 · exp( − b 2 · k ) . Meteorological Sensors UAVs Patrolling the . . . • This double-exponential model indeed describes the above data reasonably accurately: Optimal Placement Tests Feedback for Students – for k ≤ 9, the data is a good fit with an an expo- Home Page nential model for which ρ = n k +1 /n k ≈ 0 . 65-0.75; Title Page – for k ≥ 10, the data is a good fit with another exponential model, for which ρ 10 ≈ 2-3. ◭◭ ◮◮ ◭ ◮ Page 34 of 106 Go Back Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 34. Practical Consequences Case 2: Climate . . . • For small k , the dependence n k rapidly decreases with k . Case 3: Software . . . Placing Bio-Weapon . . . • So, the values n k corresponding to small k constitute Meteorological Sensors the vast majority of all the errors. UAVs Patrolling the . . . • In the above example, 85 percent of errors are of the Optimal Placement Tests first 10 types; thus: Feedback for Students Home Page – once we learn to repair errors of these types, – the remaining number of un-corrected errors de- Title Page creases by a factor of seven. ◭◭ ◮◮ • This observation has indeed led to a significant speed- ◭ ◮ up of software migration and modernization. Page 35 of 106 Go Back Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 35. Software Migration: Conclusion Case 2: Climate . . . • In many practical situations, we need to migrate legacy Case 3: Software . . . software to a new hardware and system environment. Placing Bio-Weapon . . . Meteorological Sensors • If we run the software package in the new environment, UAVs Patrolling the . . . we get thousands of difficult-to-correct errors. Optimal Placement Tests • As a result, software migration is very time-consuming. Feedback for Students • A reasonable way to speed up this process is to take Home Page into account that: Title Page – errors can be naturally classified into categories, ◭◭ ◮◮ – often all the errors of the same category can be ◭ ◮ corrected by a single correction. Page 36 of 106 • Coming up with such a joint correction is also some- Go Back what time-consuming. Full Screen • The corresponding additional time pays off only if we have sufficiently many errors of this category. Close Quit
Introduction Case 1: Out of Eden Walk 36. Software Migration: Conclusion (cont-d) Case 2: Climate . . . • Coming up with a joint correction is time-consuming. Case 3: Software . . . Placing Bio-Weapon . . . • This additional time pays off only if we have sufficiently Meteorological Sensors many errors of this category. UAVs Patrolling the . . . • So, it is desirable to be able to estimate the number of Optimal Placement Tests errors n k of different categories k . Feedback for Students Home Page • We show that expert knowledge leads to a double- exponential model in good accordance w/observations. Title Page ◭◭ ◮◮ ◭ ◮ Page 37 of 106 Go Back Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 37. Other Results from the Dissertation: Case 2: Climate . . . A Brief Overview Case 3: Software . . . • Problem: optimal sensor placement under uncertainty. Placing Bio-Weapon . . . Meteorological Sensors • First situation: placing bio-weapon detectors. UAVs Patrolling the . . . • Objective: minimize the expected risk. Optimal Placement Tests • Second situation: patrolling the border with UAVs. Feedback for Students Home Page • Objective: minimize the probability of undetected smug- Title Page gling. ◭◭ ◮◮ • Third situation: placing meteorological sensors. ◭ ◮ • Objective: maximize accuracy with which we know all Page 38 of 106 relevant quantities. Go Back • Solution: we describe analytical expressions for opti- mal sensor placement for all three situations. Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 38. Other Results from the Dissertation (cont-d) Case 2: Climate . . . • Problem: optimal placement tests. Case 3: Software . . . Placing Bio-Weapon . . . • Situation: inability to solve problems causes discom- Meteorological Sensors fort which hinders ability to solve further problems. UAVs Patrolling the . . . • Objective: minimize this discomfort. Optimal Placement Tests • Solution: we come up with an analytical expression for Feedback for Students Home Page the optimal placement testing. Title Page • Problem: optimal selection of class-related feedback for students. ◭◭ ◮◮ • Empirical fact: immediate feedback makes learning, on ◭ ◮ average, twice faster. Page 39 of 106 • Solution: an analytical expression derived from first Go Back principles explains the empirical fact. Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 39. Acknowledgments Case 2: Climate . . . • My deep gratitude to my committee: Drs. V. Kreinovich, Case 3: Software . . . D. Pennington, C. Tweedie, S. Starks, & O. Kosheleva. Placing Bio-Weapon . . . Meteorological Sensors • I also want to thank all the faculty, staff, and students UAVs Patrolling the . . . of the Computational Science program. Optimal Placement Tests • My special thanks to Drs. A. Gates and M.-Y. Leung, Feedback for Students to J. Maxey and L. Valera, and to C. Aguilar-Davis. Home Page • Last but not the least, my thanks and my love to my Title Page family: ◭◭ ◮◮ – to my sons Sam and Joseph, ◭ ◮ – to my parents Hortensia and Antonio, Page 40 of 106 – to my sister Martha, and Go Back – to my brothers Antonio, Victor, and Javier. Full Screen Close Quit
Introduction Case 1: Out of Eden Walk Appendix 1: Global Warming Case 2: Climate . . . vs. Climate Variability Case 3: Software . . . Placing Bio-Weapon . . . Meteorological Sensors UAVs Patrolling the . . . Optimal Placement Tests Feedback for Students Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 41 of 106 Go Back Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 40. Towards the Second Approximation Case 2: Climate . . . • Most natural systems are resistant to change: other- Case 3: Software . . . wise, they would not have survived. Placing Bio-Weapon . . . def Meteorological Sensors • So, when y = x − x 0 � = 0, a force brings y back to 0: dy UAVs Patrolling the . . . dt = f ( y ); f ( y ) < 0 for y > 0, f ( y ) > 0 for y < 0. Optimal Placement Tests • Since the system is stable, y is small, so we keep only Feedback for Students linear terms in the Taylor expansion of f ( y ): Home Page f ( y ) = − k · y, so dy Title Page dt = − k · y + u 0 + r ( t ) . ◭◭ ◮◮ • Since this equation is linear, its solution can be repre- ◭ ◮ sented as y ( t ) = y s ( t ) + y r ( t ), where Page 42 of 106 dy s dy r dt = − k · y s + u 0 ; dt = − k · y r + r ( t ) . Go Back • Here, y s ( t ) is the systematic change (global warming). Full Screen • y r ( t ) is the random change (climate variability). Close Quit
Introduction Case 1: Out of Eden Walk 41. An Empirical Fact That Needs to Be Explained Case 2: Climate . . . • At present, the climate variability becomes more visi- Case 3: Software . . . ble than the global warming itself. Placing Bio-Weapon . . . Meteorological Sensors • In other words, the ratio y r ( t ) /y s ( t ) is much higher UAVs Patrolling the . . . than it will be in the future. Optimal Placement Tests • The change in y is determined by two factors: Feedback for Students – the external force u ( t ) and Home Page – the parameter k that describes how resistant is our Title Page system to this force. ◭◭ ◮◮ • Some part of global warming may be caused by the ◭ ◮ variations in Solar radiation. Page 43 of 106 • Climatologists agree that global warming is mostly caused Go Back by greenhouse effect etc., which lowers resistance k . Full Screen • What causes numerous debates is which proportion of the global warming is caused by human activities. Close Quit
Introduction Case 1: Out of Eden Walk 42. An Empirical Fact to Be Explained (cont-d) Case 2: Climate . . . • Since decrease in k is the main effect, in the 1st ap- Case 3: Software . . . proximation, we consider only this effect. Placing Bio-Weapon . . . Meteorological Sensors • In this case, we need to explain why the ratio y r ( t ) /y s ( t ) UAVs Patrolling the . . . is higher now when k is higher. Optimal Placement Tests • To gauge how far the random variable y r ( t ) deviates Feedback for Students from 0, we can use its standard deviation σ ( t ). Home Page • So, we fix values u 0 and σ 0 , st. dev. of r ( t ). Title Page • For each k , we form the solutions y s ( t ) and y r ( t ) cor- ◭◭ ◮◮ responding to y s (0) = 0 and y r (0) = 0. ◭ ◮ • We then estimate the standard deviation σ ( t ) of y r ( t ). Page 44 of 106 • We want to prove that, when k decreases, the ratio Go Back σ ( t ) /y s ( t ) also decreases. Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 43. Estimating the Systematic Deviation y s ( t ) Case 2: Climate . . . • We need to solve the equation dy s Case 3: Software . . . dt = − k · y s + u 0 . Placing Bio-Weapon . . . • If we move all the terms containing y s ( t ) to the left- Meteorological Sensors hand side, we get dy s ( t ) UAVs Patrolling the . . . + k · y s ( t ) = u 0 . dt Optimal Placement Tests def • For an auxiliary variable z ( t ) = y s ( t ) · exp( k · t ), we get Feedback for Students Home Page dz ( t ) = dy s ( t ) · exp( k · t ) + y s ( t ) · exp( k · t ) · k = Title Page dt dt � dy s ( t ) � ◭◭ ◮◮ exp( k · t ) · + k · y s ( t ) . dt ◭ ◮ • Thus, dz ( t ) = u 0 · exp( k · t ) , so z ( t ) = u 0 · exp( k · t ) − 1 Page 45 of 106 , dt k Go Back and y s ( t ) = exp( − k · t ) · z ( t ) = u 0 · 1 − exp( − k · t ) Full Screen . k Close Quit
Introduction Case 1: Out of Eden Walk 44. Estimating the Random Component y r ( t ) Case 2: Climate . . . • For the random component, we similarly get Case 3: Software . . . � t Placing Bio-Weapon . . . y r ( t ) = exp( − k · t ) · r ( s ) · exp( k · s ) ds, so Meteorological Sensors 0 UAVs Patrolling the . . . � t � t y r ( t ) 2 = exp( − 2 k · t ) · Optimal Placement Tests ds dv r ( s ) · r ( v ) · exp( k · s ) · exp( k · v ) , 0 0 Feedback for Students and σ 2 ( t ) = E [ y r ( t ) 2 ] = Home Page � t � t Title Page exp( − 2 k · t ) · ds dv E [ r ( s ) · r ( v )] · exp( k · s ) · exp( k · v ) . 0 0 ◭◭ ◮◮ • Here, E [ r ( s ) · r ( v )] = E [ r ( s )] · E [ r ( v )] = 0 and E [ r 2 ( s )] = ◭ ◮ σ 2 0 , so � t Page 46 of 106 σ 2 ( t ) = E [ y r ( t ) 2 ] = exp( − 2 k · t ) · ds σ 2 0 · exp( k · s ) · exp( k · s ) . Go Back 0 Full Screen 0 · 1 − exp( − 2 k · t ) • Thus, σ 2 ( t ) = σ 2 . 2 k Close Quit
Introduction Case 1: Out of Eden Walk 45. Analyzing the Ratio σ ( t ) /y s ( t ) Case 2: Climate . . . 0 · 1 − exp( − 2 k · t ) , y s ( t ) = u 0 · 1 − exp( − k · t ) Case 3: Software . . . • σ 2 ( t ) = σ 2 . 2 k k Placing Bio-Weapon . . . = σ 2 ( t ) s ( t ) = σ 2 · (1 − exp( − 2 k · t )) · k 2 Meteorological Sensors def 0 • Thus, S ( t ) 2 k · (1 − exp( − k · t )) 2 . u 2 y 2 UAVs Patrolling the . . . 0 Optimal Placement Tests • Here, 1 − exp( − 2 k · t ) = (1 − exp( − k · t )) · (1+exp( − k · t )) , so S ( t ) = σ 2 Feedback for Students · (1 + exp( − k · t )) · k 0 2 · (1 − exp( − k · t )) . Home Page u 2 0 Title Page • When the k is large, exp( − k · t ) ≈ 0, and S ( t ) ≈ σ 2 · k 0 2 . u 2 ◭◭ ◮◮ 0 • This ratio clearly decreases when k decreases. ◭ ◮ Page 47 of 106 • So, when the Earth’s resistance k will decrease, the ratio σ ( t ) /y s ( t ) will also decrease. Go Back • Thus, we will start observing mainly the direct effects Full Screen of global warming – unless we do something about it. Close Quit
Introduction Case 1: Out of Eden Walk 46. Discussion Case 2: Climate . . . • We made a simplifying assumption that the climate Case 3: Software . . . system is determined by a single parameter x (or y ). Placing Bio-Weapon . . . Meteorological Sensors • A more realistic model is when the climate system is UAVs Patrolling the . . . determined by several parameters y 1 , . . . , y n . Optimal Placement Tests • In this case, in the linear approximation, the dynamics Feedback for Students is described by a system of linear ODEs Home Page n � dy i dt = − a ij · y j ( t ) + u i ( t ) . Title Page j =1 ◭◭ ◮◮ • In the generic case, all eigenvalues λ k of the matrix a ij ◭ ◮ are different. Page 48 of 106 • In this case, a ij can be diagonalized by using the linear Go Back combinations z k ( t ) corresponding to eigenvectors: Full Screen dz k dt = − λ k · z k ( t ) + � u k ( t ) . Close Quit
Introduction Case 1: Out of Eden Walk 47. Discussion (cont-d) Case 2: Climate . . . • Reminder: we have a system of equations Case 3: Software . . . Placing Bio-Weapon . . . dz k dt = − λ k · z k ( t ) + � u k ( t ) . Meteorological Sensors UAVs Patrolling the . . . • For each of these equations, we can arrive at the same Optimal Placement Tests conclusion: Feedback for Students – the current ratio of the random to systematic effects Home Page is much higher Title Page – than it will be in the future. ◭◭ ◮◮ • So, our explanations holds in this more realistic model ◭ ◮ as well. Page 49 of 106 Go Back Full Screen Close Quit
Introduction Case 1: Out of Eden Walk Appendix 2. Two-Rules Approach Case 2: Climate . . . to Software Migration Case 3: Software . . . Placing Bio-Weapon . . . Meteorological Sensors UAVs Patrolling the . . . Optimal Placement Tests Feedback for Students Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 50 of 106 Go Back Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 48. Two-Rules Approach Case 2: Climate . . . • Once we know n k and n k +1 , we need to estimate n k +2 = Case 3: Software . . . f ( n k , n k +1 ) . Placing Bio-Weapon . . . Meteorological Sensors • When the software package consists of two parts, we UAVs Patrolling the . . . can estimate n k +2 in two different ways: Optimal Placement Tests – We can use the overall numbers n k = n (1) k + n (2) and k Feedback for Students n k +1 = n (1) k +1 + n (2) k +1 and predict Home Page n k +2 ≈ f ( n k , n k +1 ) = f ( n (1) k + n (2) k , n (1) k +1 + n (2) k +1 ) . Title Page – Alternatively, we can predict the values n (1) k +2 and ◭◭ ◮◮ n (2) k +2 , and add up these predictions: ◭ ◮ n k +2 ≈ f ( n (1) k , n (1) k +1 ) + f ( n (2) k , n (2) k +1 ) . Page 51 of 106 • It is reasonable to require that these two approaches Go Back lead to the same estimate, i.e., that we have Full Screen f ( n (1) k + n (2) k , n (1) k +1 + n (2) k +1 ) = f ( n (1) k , n (1) k +1 )+ f ( n (2) k , n (2) k +1 ) . Close Quit
Introduction Case 1: Out of Eden Walk 49. Two-Rules Approach (cont-d) Case 2: Climate . . . • Reminder: for all a ≥ a ′ and b ≥ b ′ , we have Case 3: Software . . . f ( a + b, a ′ + b ′ ) = f ( a, a ′ ) + f ( b, b ′ ) . Placing Bio-Weapon . . . Meteorological Sensors • One can show that this leads to n k +2 = c · n k + c ′ · n k +1 UAVs Patrolling the . . . for some c and c ′ , and thus, to Optimal Placement Tests n k = A 1 · exp( − b 1 · k ) + A 2 · exp( b 2 · k ) . Feedback for Students Home Page • In general, b i are complex numbers – leading to oscil- Title Page lating sinusoidal terms. ◭◭ ◮◮ • We want n k ≥ n k +1 , so there are no oscillations, both b i are real. ◭ ◮ Page 52 of 106 • Without losing generality, we can assume that b 1 < b 2 . Go Back • If A 1 > A 2 , then the first term always dominates. Full Screen • But we already know that an exponential function is not a good description of n k . Close Quit
Introduction Case 1: Out of Eden Walk 50. Two-Rules Model Fits the Data Case 2: Climate . . . • Thus, to fit the empirical data, we must use models Case 3: Software . . . with A 1 < A 2 . In this case: Placing Bio-Weapon . . . Meteorological Sensors – for small k , the second – faster-decreasing – term UAVs Patrolling the . . . dominates: n k ≈ A 2 · exp( − b 2 · k ); Optimal Placement Tests – for larger k , the first – slower-decreasing – term Feedback for Students dominates: n k ≈ A 1 · exp( − b 1 · k ). Home Page • This double-exponential model indeed describes the Title Page above data reasonably accurately: ◭◭ ◮◮ – for k ≤ 9, the data is a good fit with an an expo- ◭ ◮ nential model for which ρ = n k +1 /n k ≈ 0 . 65-0.75; Page 53 of 106 – for k ≥ 10, the data is a good fit with another exponential model, for which ρ 10 ≈ 2-3. Go Back Full Screen Close Quit
Introduction Case 1: Out of Eden Walk Appendix 3. First Case of Sensor Case 2: Climate . . . Placement: Placing Bio-Weapon Case 3: Software . . . Detectors Placing Bio-Weapon . . . Meteorological Sensors UAVs Patrolling the . . . Optimal Placement Tests Feedback for Students Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 54 of 106 Go Back Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 51. Sensor Placement: Case Study Case 2: Climate . . . • Biological weapons are difficult and expensive to de- Case 3: Software . . . tect. Placing Bio-Weapon . . . Meteorological Sensors • Within a limited budget, we can afford a limited num- UAVs Patrolling the . . . ber of bio-weapon detector stations. Optimal Placement Tests • It is therefore important to find the optimal locations Feedback for Students for such stations. Home Page • A natural idea is to place more detectors in the areas Title Page with more population. ◭◭ ◮◮ • However, such a commonsense analysis does not tell us ◭ ◮ how many detectors to place where. Page 55 of 106 • To decide on the exact detector placement, we must Go Back formulate the problem in precise terms. Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 52. Towards Precise Formulation of the Problem Case 2: Climate . . . • The adversary’s objective is to kill as many people as Case 3: Software . . . possible. Placing Bio-Weapon . . . Meteorological Sensors • Let ρ ( x ) be a population density in the vicinity of the UAVs Patrolling the . . . location x . Optimal Placement Tests • Let N be the number of detectors that we can afford Feedback for Students to place in the given territory. Home Page • Let d 0 be the distance at which a station can detect an Title Page outbreak of a disease. ◭◭ ◮◮ • Often, d 0 = 0 – we can only detect a disease when the ◭ ◮ sources of this disease reach the detecting station. Page 56 of 106 • We want to find ρ d ( x ) – the density of detector place- Go Back ment. � Full Screen • We know that ρ d ( x ) dx = N. Close Quit
Introduction Case 1: Out of Eden Walk 53. Optimal Placement of Sensors Case 2: Climate . . . • We want to place the sensors in an area in such a way Case 3: Software . . . that Placing Bio-Weapon . . . Meteorological Sensors – the largest distance D to a sensor UAVs Patrolling the . . . – is as small as possible. Optimal Placement Tests • It is known that the smallest such number is provided Feedback for Students by an equilateral triangle grid: Home Page Title Page h ✛ ✲ ◭◭ ◮◮ r r r r ❆ ✁ ❆ ✁ ❆ ✁ ❆ ❆ ❆ ❑ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ❆ ◭ ◮ ❆ h ❆ ✁ ❆ ✁ ❆ ✁ ❆ ❆ ❆❆ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ❯ ❆ Page 57 of 106 ❆ ✁ ❆ ✁ ❆ ✁ r r r ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ Go Back ✁ ❆ ✁ ❆ ✁ ❆ ✁ ✁ ❆ ✁ ❆ ✁ ❆ ✁ Full Screen ✁ ❆ ✁ ❆ ✁ ❆ ✁ r r r r Close Quit
Introduction Case 1: Out of Eden Walk For the equilateral triangle placement, points which are Case 2: Climate . . . closest to a given detector forms a hexagonal area: h Case 3: Software . . . ✛ ✲ Placing Bio-Weapon . . . r r r r ❆ ✁ ❆ ✁ ❆ ✁ ❆ ❆ ❆ ❑ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ❆ ✟ ❍ ✟ ❍ Meteorological Sensors ❆ h ❆ ✁ ✟ ❆ ✁ ❍ ❆ ✁ ❆ ✟ ❍ ❆ ❆ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ❆ ❆ ❯ UAVs Patrolling the . . . ❆ ✁ ❆ ✁ ❆ ✁ r r r ✁ ❆ ✁ ❆ ✁ ❆ Optimal Placement Tests ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❍ ✟ ❍ ✟ ✁ ❆ ✁ ❍ ✟ ❆ ✁ ❆ ✁ ✟ ❍ Feedback for Students ✁ ❆ ✁ ❆ ✁ ❆ ✁ Home Page ✁ ❆ ✁ ❆ ✁ ❆ ✁ r r r r Title Page This hexagonal area consists of 6 equilateral triangles: ◭◭ ◮◮ h ✛ ✲ ◭ ◮ r r r r ❆ ✁ ❆ ✁ ❆ ✁ ❆ ❑ ❆ ❆ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✟ ❆ ❍ ✟ ❍ ❆ h ❆ ✁ ✟ ❆ ✁ ❍ ❆ ✁ ❆ ✟ ❍ Page 58 of 106 ❆ ❆❆ ❩ ✚ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ❩ ✚✚ ❯ ❆ ❆ ✁ ❩ ❆ ✁ ❆ ✁ Go Back ✚ ❩❩ r r r ✁ ❆ ✁ ❆ ✁ ❆ ✚ ✚ ❩ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ✟ ❍ ❍ ✟ Full Screen ✁ ❆ ✁ ❍ ✟ ❆ ✁ ❆ ✁ ❍ ✟ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ✁ ❆ ✁ ❆ ✁ ❆ ✁ Close r r r r Quit
Introduction Case 1: Out of Eden Walk 54. Optimal Placement of Sensors (cont-d) Case 2: Climate . . . • In each △ , the height h/ 2 is related to the side s by Case 3: Software . . . √ √ the formula h 3 3 Placing Bio-Weapon . . . 2 = s · cos(60 ◦ ) = s · 2 , hence s = h · 3 . Meteorological Sensors • Thus, the area A t of each triangle is equal to UAVs Patrolling the . . . √ √ Optimal Placement Tests A t = 1 2 · s · h 2 = 1 3 · 1 3 3 2 · h 2 = 12 · h 2 . 2 · Feedback for Students Home Page • So, the area A s of the whole set is equal to 6 times the √ Title Page 3 2 · h 2 . triangle area: A s = 6 · A t = ◭◭ ◮◮ • In a region of area A , there are A · ρ d ( x ) sensors, they ◭ ◮ cover area ( A · ρ d ( x )) · A s . √ Page 59 of 106 3 2 · h 2 • The condition A = ( A · ρ d ( x )) · A s = ( A · ρ d ( x )) · Go Back � 2 c 0 Full Screen def √ implies that h = � , with c 0 = 3 . ρ d ( x ) Close Quit
Introduction Case 1: Out of Eden Walk 55. Estimating the Effect of Sensor Placement Case 2: Climate . . . • The adversary places the bio-weapon at a location which Case 3: Software . . . is the farthest away from the detectors. Placing Bio-Weapon . . . Meteorological Sensors • This way, it will take the longest time to be detected. UAVs Patrolling the . . . • For the grid placement, this location is at one of the Optimal Placement Tests vertices of the hexagonal zone. Feedback for Students • At these vertices, the distance from each neighboring Home Page √ 3 Title Page detector is equal to s = h · 3 . ◭◭ ◮◮ c 0 c 1 • By know that h = � , so s = � , with ◭ ◮ ρ d ( x ) ρ d ( x ) √ √ √ Page 60 of 106 4 3 3 · 2 c 1 = 3 · c 0 = . 3 Go Back • Once the bio-weapon is placed, it starts spreading until Full Screen it reaches the distance d 0 from the detector. Close Quit
Introduction Case 1: Out of Eden Walk 56. Effect of Sensor Placement (cont-d) Case 2: Climate . . . c 1 Case 3: Software . . . • The bio-weapon is placed at a distance s = � ρ d ( x ) Placing Bio-Weapon . . . from the nearest sensor. Meteorological Sensors • Once the bio-weapon is placed, it starts spreading until UAVs Patrolling the . . . it reaches the distance d 0 from the detector. Optimal Placement Tests Feedback for Students • In other words, it spreads for the distance s − d 0 . Home Page • During this spread, the disease covers the circle of ra- Title Page dius s − d 0 and area π · ( s − d 0 ) 2 . ◭◭ ◮◮ • The number of affected people n ( x ) is equal to: ◭ ◮ � � 2 c 1 n ( x ) = π · ( s − d 0 ) 2 · ρ ( x ) = π · Page 61 of 106 � − d 0 · ρ ( x ) . ρ d ( x ) Go Back Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 57. Precise Formulation of the Problem Case 2: Climate . . . • For each location x , the number of affected people n ( x ) Case 3: Software . . . is equal to: Placing Bio-Weapon . . . � � 2 Meteorological Sensors c 1 n ( x ) = π · � − d 0 · ρ ( x ) . UAVs Patrolling the . . . ρ d ( x ) Optimal Placement Tests • The adversary will select a location x for which this Feedback for Students number n ( x ) is the largest possible: Home Page � � 2 Title Page c 1 π · . n = max � − d 0 · ρ ( x ) ◭◭ ◮◮ x ρ d ( x ) ◭ ◮ • Resulting problem: Page 62 of 106 – given population density ρ ( x ), detection distance d 0 , Go Back and number of sensors N , Full Screen – find a function ρ d ( x ) that minimizes the above ex- � pression n under the constraint ρ d ( x ) dx = N . Close Quit
Introduction Case 1: Out of Eden Walk 58. Main Lemma Case 2: Climate . . . • Reminder: we want to minimize the worst-case damage Case 3: Software . . . n = max n ( x ). Placing Bio-Weapon . . . x Meteorological Sensors • Lemma: for the optimal sensor selection, n ( x ) = const. UAVs Patrolling the . . . • Proof by contradiction: let n ( x ) < n for some x ; then: Optimal Placement Tests Feedback for Students – we can slightly increase the detector density at the Home Page locations where n ( x ) = n , Title Page – at the expense of slightly decreasing the location density at locations where n ( x ) < n ; ◭◭ ◮◮ – as a result, the overall maximum n = max n ( x ) will ◭ ◮ x decrease; Page 63 of 106 – but we assumed that n is the smallest possible. Go Back • Thus: n ( x ) = const; let us denote this constant by n 0 . Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 59. Towards the Solution of the Problem Case 2: Climate . . . • We have proved that n ( x ) = const = n 0 , i.e., that Case 3: Software . . . � � 2 Placing Bio-Weapon . . . c 1 Meteorological Sensors n 0 = π · � − d 0 · ρ ( x ) . ρ d ( x ) UAVs Patrolling the . . . Optimal Placement Tests • Straightforward algebraic transformations lead to: Feedback for Students √ ρ d ( x ) = 2 · 3 1 Home Page · � 2 . � 9 Title Page c 2 d 0 + � ρ ( x ) ◭◭ ◮◮ ◭ ◮ • The value c 2 must be determined from the equation � Page 64 of 106 ρ d ( x ) dx = N. Go Back Full Screen • Thus, we arrive at the following solution. Close Quit
Introduction Case 1: Out of Eden Walk 60. Solution Case 2: Climate . . . • General case: the optimal detector location is charac- Case 3: Software . . . terized by the detector density Placing Bio-Weapon . . . √ Meteorological Sensors ρ d ( x ) = 2 · 3 1 · � 2 . � UAVs Patrolling the . . . 9 c 2 d 0 + � Optimal Placement Tests ρ ( x ) Feedback for Students Home Page • Here the parameter c 2 must be determined from the √ � 2 · 3 1 Title Page equation · � 2 dx = N. � 9 c 2 ◭◭ ◮◮ d 0 + � ρ ( x ) ◭ ◮ • Case of d 0 = 0 : in this case, the formula for ρ d ( x ) takes Page 65 of 106 a simplified form ρ d ( x ) = C · ρ ( x ) for some constant C . Go Back • In this case, from the constraint, we get: Full Screen ρ d ( x ) = N · ρ ( x ) , where N p is the total population . Close N p Quit
Introduction Case 1: Out of Eden Walk 61. Towards More Relevant Objective Functions Case 2: Climate . . . • We assumed that the adversary wants to maximize the Case 3: Software . . . � number ρ ( x ) dx of people affected by the bio-weapon. Placing Bio-Weapon . . . Meteorological Sensors • The actual adversary’s objective function may differ UAVs Patrolling the . . . from this simplified objective function. Optimal Placement Tests • For example, the adversary may take into account that Feedback for Students different locations have different publicity potential. Home Page • In this case, the adversary maximizes the weighted Title Page � def value A � ρ ( x ) dx , where � ρ ( x ) = w ( x ) · ρ ( x ). ◭◭ ◮◮ • Here, w ( x ) is the importance of the location x . ◭ ◮ • From the math. viewpoint, the problem is the same – Page 66 of 106 w/“effective population density” � ρ ( x ) instead of ρ ( x ). Go Back • Thus, if we know w ( x ), we can find the optimal detec- Full Screen tor density ρ d ( x ) from the above formulas. Close Quit
Introduction Case 1: Out of Eden Walk Appendix 4. Second Case of Case 2: Climate . . . Optimal Sensor Placement: Case 3: Software . . . Placing Meteorological Sensors Placing Bio-Weapon . . . Meteorological Sensors UAVs Patrolling the . . . Optimal Placement Tests Feedback for Students Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 67 of 106 Go Back Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 62. How Temperatures etc. Change from One Spa- Case 2: Climate . . . tial Location to Another: A Model Case 3: Software . . . • Each environmental characteristic q changes from one Placing Bio-Weapon . . . spatial location to another. Meteorological Sensors UAVs Patrolling the . . . • A large part of this change is unpredictable (i.e., ran- dom). Optimal Placement Tests Feedback for Students • A reasonable value to describe the random component Home Page of the difference q ( x ) − q ( x ′ ) is the variance Title Page def V ( x, x ′ ) = E [(( q ( x ) − E [ q ( x )]) − ( q ( x ′ ) − E [ q ( x ′ )])) 2 ] . ◭◭ ◮◮ • Comment: we assume that averages are equal. ◭ ◮ • Locally, processes should not change much with shift Page 68 of 106 x → x + s : V ( x + s, x ′ + s ) = V ( x, x ′ ). Go Back • For s = − x ′ , we get V ( x, x ′ ) = C ( x − x ′ ) for Full Screen def C ( x ) = V ( x, 0) . Close Quit
Introduction Case 1: Out of Eden Walk 63. A Model (cont-d) Case 2: Climate . . . • In general, the further away the points x and x ′ , the Case 3: Software . . . larger the difference C ( x − x ′ ). Placing Bio-Weapon . . . Meteorological Sensors • In the isotropic case, C ( x − x ′ ) depends only on the distance D = | x − x ′ | 2 = ( x 1 − x ′ 1 ) 2 + ( x 2 − x ′ UAVs Patrolling the . . . 2 ) 2 . Optimal Placement Tests • It is reasonable to consider a scale-invariant depen- Feedback for Students dence C ( x ) = A · D α . Home Page • In practice, we may have more changes in one direction Title Page and less change in another direction. ◭◭ ◮◮ • E.g., 1 km in x is approximately the same change as 2 ◭ ◮ km in y . Page 69 of 106 • The change can also be mostly in some other direction, not just x - and y -directions. Go Back Full Screen • Thus, in general, in appropriate coordinates ( u, v ), we have C = A · D α for D = ( u − u ′ ) 2 + ( v − v ′ ) 2 . Close Quit
Introduction Case 1: Out of Eden Walk 64. Model: Final Formulas Case 2: Climate . . . • In general, C = A · D α , for D = ( u − u ′ ) 2 + ( v − v ′ ) 2 in Case 3: Software . . . appropriate coordinates ( u, v ). Placing Bio-Weapon . . . Meteorological Sensors • In the original coordinates x 1 and x 2 , we get: UAVs Patrolling the . . . C ( x − x ′ ) = A · D α , where Optimal Placement Tests 2 2 � � Feedback for Students g ij · ( x i − x ′ i ) · ( x j − x ′ D = j ) = Home Page i =1 j =1 Title Page g 11 · ( x 1 − x ′ 1 ) 2 +2 g 12 · ( x 1 − x ′ 1 ) · ( x 2 − x ′ 2 )+ g 22 · ( x 2 − x ′ 2 ) 2 . ◭◭ ◮◮ • From the computational viewpoint, we can include A into g ij if we replace g ij with A 1 /α · g ij , then ◭ ◮ Page 70 of 106 C ( x − x ′ ) = � 2 ) 2 � α Go Back 1 ) 2 + 2 g 12 · ( x 1 − x ′ g 11 · ( x 1 − x ′ 1 ) · ( x 2 − x ′ 2 ) + g 22 · ( x 2 − x ′ Full Screen • We can use these formulas to find the optimal sensor Close locations. Quit
Introduction Case 1: Out of Eden Walk Appendix 5. Third Case of Case 2: Climate . . . Optimal Sensor Placement: Case 3: Software . . . Patrolling the Border with UAVs Placing Bio-Weapon . . . Meteorological Sensors UAVs Patrolling the . . . Optimal Placement Tests Feedback for Students Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 71 of 106 Go Back Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 65. Optimal Use of Mobile Sensors: Case Study Case 2: Climate . . . • Remote areas of international borders are used by the Case 3: Software . . . adversaries: to smuggle drugs, to bring in weapons. Placing Bio-Weapon . . . Meteorological Sensors • It is therefore desirable to patrol the border, to mini- UAVs Patrolling the . . . mize such actions. Optimal Placement Tests • It is not possible to effectively man every single seg- Feedback for Students ment of the border. Home Page • It is therefore necessary to rely on other types of surveil- Title Page lance. ◭◭ ◮◮ • Unmanned Aerial Vehicles (UAVs): ◭ ◮ – from every location along the border, they provide Page 72 of 106 an overview of a large area, and Go Back – they can move fast, w/o being slowed down by clogged roads or rough terrain. Full Screen • Question: what is the optimal trajectory for these UAVs? Close Quit
Introduction Case 1: Out of Eden Walk 66. How to Describe Possible UAV Patrolling Strate- Case 2: Climate . . . gies Case 3: Software . . . • Let us assume that the time between two consequent Placing Bio-Weapon . . . overflies is smaller the time needed to cross the border. Meteorological Sensors UAVs Patrolling the . . . • Ideally, such a UAV can detect all adversaries. Optimal Placement Tests • In reality, a fast flying UAV can miss the adversary. Feedback for Students Home Page • We need to minimize the effect of this miss. Title Page • The faster the UAV goes, the less time it looks, the more probable that it will miss the adversary. ◭◭ ◮◮ • Thus, the velocity v ( x ) is very important. ◭ ◮ Page 73 of 106 • By a patrolling strategy, we will mean a f-n v ( x ) de- scribing how fast the UAV flies at different locations x . Go Back Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 67. Constraints on Possible Patrolling Strategies Case 2: Climate . . . 1) The time between two consequent overflies should be Case 3: Software . . . smaller the time T needed to cross the border: Placing Bio-Weapon . . . Meteorological Sensors – the time during which a UAV passes from the loca- tion x to the location x +∆ x is equal to ∆ t = ∆ x UAVs Patrolling the . . . v ( x ); Optimal Placement Tests – thus, the overall flight time is equal to the sum of Feedback for Students these times: � Home Page dx T = v ( x ) . Title Page ◭◭ ◮◮ 2) UAV has the largest possible velocity V , so we must have v ( x ) ≤ V for all x . ◭ ◮ 1 def Page 74 of 106 It is convenient to use the value s ( x ) = v ( x ) called slow- Go Back ness , so � � � = 1 Full Screen def T = s ( x ) dx ; s ( x ) ≥ S . V Close Quit
Introduction Case 1: Out of Eden Walk 68. Simplification of the Constraints Case 2: Climate . . . • Since s ( x ) ≥ S , the value s ( x ) can be represented as Case 3: Software . . . def S + ∆ s ( x ), where ∆ s ( x ) = s ( x ) − S . Placing Bio-Weapon . . . Meteorological Sensors • The new unknown function satisfies the simpler con- UAVs Patrolling the . . . straint ∆ s ( x ) ≥ 0. Optimal Placement Tests • In terms of ∆ s ( x ), the requirement that the overall Feedback for Students � time be equal to T has a form T = S · L + ∆ s ( x ) dx . Home Page • This is equivalent to: Title Page � ◭◭ ◮◮ T 0 = ∆ s ( x ) dx, where: ◭ ◮ • L is the total length of the piece of the border that Page 75 of 106 we are defending, and Go Back def • T 0 = T − S · L . Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 69. Detection at Crossing Point x Case 2: Climate . . . • Let h be the width of the border zone from which an Case 3: Software . . . adversary (A) is visible. Placing Bio-Weapon . . . Meteorological Sensors • Then, the UAV can potentially detect A during the UAVs Patrolling the . . . time h/v ( x ) = h · s ( x ). Optimal Placement Tests • So, the UAV takes ( h · s ( x )) / ∆ t photos, where ∆ t is Feedback for Students the time per photo. Home Page • Let p 1 be the probability that one photo misses A. Title Page • It is reasonable to assume that different detection er- ◭◭ ◮◮ rors are independent. ◭ ◮ • Then, the probability p ( x ) that A is not detected is Page 76 of 106 p ( h · s ( x )) / ∆ t , i.e., p ( x ) = exp( − k · s ( x )) , where: 1 Go Back = 2 h def k ∆ t · | ln( p 1 ) | . Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 70. Strategy Selected by the Adversary Case 2: Climate . . . • Let w ( x ) denote the utility of the adversary succeeding Case 3: Software . . . in crossing the border at location x . Placing Bio-Weapon . . . Meteorological Sensors • Let us first assume that we know w ( x ) for every x . UAVs Patrolling the . . . • According to decision theory, the adversary will select Optimal Placement Tests a location x with the largest expected utility Feedback for Students u ( x ) = p ( x ) · w ( x ) = exp( − k · s ( x )) · w ( x ) . Home Page Title Page • Thus, for each slowness function s ( x ), the adversary’s gain G ( s ) is equal to ◭◭ ◮◮ G ( s ) = max u ( x ) = max [exp( − k · s ( x )) · w ( x )] . ◭ ◮ x x Page 77 of 106 • We need to select a strategy s ( x ) for which the gain Go Back G ( s ) is the smallest possible. Full Screen G ( s ) = max u ( x ) = max [exp( − k · s ( x )) · w ( x )] → min s ( x ) . x x Close Quit
Introduction Case 1: Out of Eden Walk 71. Towards an Optimal Strategy for Patrolling Case 2: Climate . . . the Border Case 3: Software . . . • Let x m be the location at which the utility u ( x ) = Placing Bio-Weapon . . . exp( − k · s ( x )) · w ( x ) attains its largest possible value. Meteorological Sensors UAVs Patrolling the . . . • If we have a point x 0 s.t. u ( x 0 ) < u ( x m ) and s ( x 0 ) > S : Optimal Placement Tests – we can slightly decrease the slowness s ( x 0 ) at the Feedback for Students vicinity of x 0 (i.e., go faster in this vicinity) and Home Page – use the resulting time to slow down (i.e., to go Title Page slower) at all locations x at which u ( x ) = u ( x m ). ◭◭ ◮◮ • As a result, we slightly decrease the value ◭ ◮ u ( x m ) = exp( − k · s ( x m )) · w ( x m ) . Page 78 of 106 • At x 0 , we still have u ( x 0 ) < u ( x m ). Go Back • So, the overall gain G ( s ) decreases. Full Screen • Thus, when the adversary’s gain is minimized, we get Close u ( x ) = u 0 = const whenever s ( x ) > S. Quit
Introduction Case 1: Out of Eden Walk 72. Towards an Optimal Strategy (cont-d) Case 2: Climate . . . • Reminder: for the optimal strategy, Case 3: Software . . . Placing Bio-Weapon . . . u ( x ) = w ( x ) · exp( − k · s ( x )) = u 0 whenever s ( x ) > S. Meteorological Sensors u 0 UAVs Patrolling the . . . • So, exp( − k · s ( x )) = w ( x ), hence Optimal Placement Tests s ( x ) = 1 k · (ln( w ( x )) − ln( u 0 )) and ∆ s ( x ) = 1 Feedback for Students k · ln( w ( x )) − ∆ 0 . Home Page Title Page = 1 def • Here, ∆ 0 k · ln( u 0 ) − S. ◭◭ ◮◮ • When s ( x ) gets to s ( x ) = S and ∆ s ( x ) = 0, we get ◭ ◮ ∆ s ( x ) = 0. Page 79 of 106 • Thus, we conclude that Go Back � 1 � Full Screen ∆ s ( x ) = max k · ln( w ( x )) − ∆ 0 , 0 . Close Quit
Introduction Case 1: Out of Eden Walk 73. An Optimal Strategy: Algorithm Case 2: Climate . . . • Reminder: for some ∆ 0 , the optimal strategy has the Case 3: Software . . . form Placing Bio-Weapon . . . � 1 � Meteorological Sensors ∆ s ( x ) = max k · ln( w ( x )) − ∆ 0 , 0 . UAVs Patrolling the . . . Optimal Placement Tests • How to find ∆ 0 : from the condition that Feedback for Students � Home Page ∆ s ( x ) dx = Title Page � 1 � � ◭◭ ◮◮ max k · ln( w ( x )) − ∆ 0 , 0 dx = T 0 . ◭ ◮ • Easy to check: the above integral monotonically de- Page 80 of 106 creases with ∆ 0 . Go Back • Conclusion: we can use bisection to find the appropri- Full Screen ate value ∆ 0 . Close Quit
Introduction Case 1: Out of Eden Walk Appendix 6. Designing Optimal Case 2: Climate . . . Placement Tests Case 3: Software . . . Placing Bio-Weapon . . . Meteorological Sensors UAVs Patrolling the . . . Optimal Placement Tests Feedback for Students Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 81 of 106 Go Back Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 74. Assessing the Initial Knowledge Level Case 2: Climate . . . • Computers enable us to provide individualized learn- Case 3: Software . . . ing, at a pace tailored to each student. Placing Bio-Weapon . . . Meteorological Sensors • In order to start the learning process, it is important to UAVs Patrolling the . . . find out the current level of the student’s knowledge. Optimal Placement Tests • Usually, such placement tests use a sequence of N prob- Feedback for Students lems of increasing complexity. Home Page • If a student is able to solve a problem, the system gen- Title Page erates a more complex one. ◭◭ ◮◮ • If a student cannot solve a problem, the system gener- ◭ ◮ ates an easier one, etc. Page 82 of 106 • Once we find the exact level of student’s knowledge, the actual learning starts. Go Back Full Screen • It is desirable to get to actual leaning as soon as pos- sible, i.e., to minimize the # of placement problems. Close Quit
Introduction Case 1: Out of Eden Walk 75. Bisection – Optimal Search Procedure Case 2: Climate . . . • At each stage, we have: Case 3: Software . . . Placing Bio-Weapon . . . – the largest level i at which a student can solve, & Meteorological Sensors – the smallest level j at which s/he cannot. UAVs Patrolling the . . . • Initially, i = 0 (trivial), j = N + 1 (very tough). Optimal Placement Tests • If j = i + 1, we found the student’s level of knowledge. Feedback for Students Home Page def • If j > i + 1, give a problem on level m = ( i + j ) / 2: Title Page – if the student solved it, increase i to m ; ◭◭ ◮◮ – else decrease j to m . ◭ ◮ • In both cases, the interval [ i, j ] is decreased by half. Page 83 of 106 • In s steps, we decrease the interval [0 , N + 1] to width ( N + 1) · 2 − s . Go Back Full Screen • In s = ⌈ log 2 ( N +1) ⌉ steps, we get the interval of width ≤ 1, so the problem is solved. Close Quit
Introduction Case 1: Out of Eden Walk 76. Need to Account for Discouragement Case 2: Climate . . . • Every time a student is unable to solve a problem, Case 3: Software . . . he/she gets discouraged. Placing Bio-Weapon . . . Meteorological Sensors • In bisection, a student whose level is 0 will get ≈ UAVs Patrolling the . . . log 2 ( N + 1) negative feedbacks. Optimal Placement Tests • For positive answers, the student simply gets tired. Feedback for Students Home Page • For negative answers, the student also gets stressed and frustrated. Title Page • If we count an effect of a positive answer as one, then ◭◭ ◮◮ the effect of a negative answer is w > 1. ◭ ◮ • The value w can be individually determined. Page 84 of 106 • We need a testing scheme that minimizes the worst- Go Back case overall effect. Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 77. Analysis of the Problem Case 2: Climate . . . • We have x = N + 1 possible levels of knowledge. Case 3: Software . . . Placing Bio-Weapon . . . • Let e ( x ) denote the smallest possible effect needed to Meteorological Sensors find out the student’s knowledge level. UAVs Patrolling the . . . • We ask a student to solve a problem of some level n . Optimal Placement Tests • If s/he solved it (effect = 1), we have x − n possible Feedback for Students levels n , . . . , N . Home Page • The effect of finding this level is e ( x − n ), so overall Title Page effect is 1 + e ( x − n ). ◭◭ ◮◮ • If s/he didn’t (effect w ), his/her level is between 0 and ◭ ◮ n , so we need effect e ( n ), with overall effect w + e ( n ). Page 85 of 106 • Overall worst-case effect is max(1+ e ( x − n ) , w + e ( n )). Go Back • In the optimal test, we select n for which this effect is Full Screen the smallest, so e ( x ) = min 1 ≤ n<x max(1+ e ( x − n ) , w + e ( n )) . Close Quit
Introduction Case 1: Out of Eden Walk 78. Resulting Algorithm Case 2: Climate . . . • For x = 1, i.e., for N = 0, we have e (1) = 0. Case 3: Software . . . Placing Bio-Weapon . . . • We know that e ( x ) = min 1 ≤ n<x max(1+ e ( x − n ) , w + e ( n )) . Meteorological Sensors • We can use this formula to sequentially compute the UAVs Patrolling the . . . values e (2), e (3), . . . , e ( N + 1). Optimal Placement Tests Feedback for Students • We also compute the corresponding minimizing values Home Page n (2), n (3), . . . , n ( N + 1). Title Page • Initially, i = 0 and j = N + 1. ◭◭ ◮◮ • At each iteration, we ask to solve a problem at level ◭ ◮ m = i + n ( j − i ): Page 86 of 106 – if the student succeeds, we replace i with m ; Go Back – else we replace j with m . Full Screen • We stop when j = i + 1; this means that the student’s level is i . Close Quit
Introduction Case 1: Out of Eden Walk 79. Example 1: N = 3 , w = 3 Case 2: Climate . . . • Here, e (1) = 0. Case 3: Software . . . Placing Bio-Weapon . . . • When x = 2, the only possible value for n is n = 1, so Meteorological Sensors e (2) = min 1 ≤ n< 2 { max { 1 + e (2 − n ) , 3 + e ( n ) }} = UAVs Patrolling the . . . Optimal Placement Tests max { 1 + e (1) , 3 + e (1) } = max { 1 , 3 } = 3 . Feedback for Students Home Page • Here, e (2) = 3, and n (2) = 1. Title Page • To find e (3), we must compare two different values n = 1 and n = 2: ◭◭ ◮◮ ◭ ◮ e (3) = min 1 ≤ n< 3 { max { 1 + e (3 − n )) , 3 + e ( n ) }} = Page 87 of 106 min { max { 1+ e (2) , 3+ e (1) } , max { 1+ e (1) , 3+ e (2) }} = Go Back min { max { 4 , 3 } , max { 1 , 6 }} = min { 4 , 6 } = 4 . Full Screen • Here, min is attained when n = 1, so n (3) = 1. Close Quit
Introduction Case 1: Out of Eden Walk 80. Example 1: N = 3 , w = 3 (cont-d) Case 2: Climate . . . • To find e (4), we must consider three possible values Case 3: Software . . . n = 1, n = 2, and n = 3, so Placing Bio-Weapon . . . Meteorological Sensors e (4) = min 1 ≤ n< 4 { max { 1 + e (4 − n ) , 3 + e ( n )) }} = UAVs Patrolling the . . . Optimal Placement Tests min { max { 1 + e (3) , 3 + e (1) } , max { 1 + e (2) , 3 + e (2) } , Feedback for Students max { 1 + e (1) , 3 + e (3) }} = Home Page min { max { 5 , 3 } , max { 4 , 6 } , max { 1 , 7 }} = Title Page min { 5 , 6 , 7 } = 5 . ◭◭ ◮◮ • Here, min is attained when n = 1, so n (4) = 1. ◭ ◮ Page 88 of 106 Go Back Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 81. Example 1: Resulting Procedure Case 2: Climate . . . • First, i = 0 and j = 4, so we ask a student to solve a Case 3: Software . . . problem at level i + n ( j − i ) = 0 + n (4) = 1. Placing Bio-Weapon . . . Meteorological Sensors • If the student fails level 1, his/her level is 0. UAVs Patrolling the . . . • If s/he succeeds at level 1, we set i = 1, and we assign Optimal Placement Tests a problem of level 1 + n (3) = 2. Feedback for Students Home Page • If the student fails level 2, his/her level is 1. Title Page • If s/he succeeds at level 2, we set i = 2, and we assign a problem of level 2 + n (3) = 3. ◭◭ ◮◮ • If the student fails level 3, his/her level is 2. ◭ ◮ Page 89 of 106 • If s/he succeeds at level 3, his/her level is 3. Go Back • We can see that this is the most cautious scheme, when each student has at most one negative experience. Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 82. Example 2: N = 3 and w = 1 . 5 Case 2: Climate . . . • We take e (1) = 0. Case 3: Software . . . Placing Bio-Weapon . . . • When x = 2, then Meteorological Sensors e (2) = min 1 ≤ n< 2 { max { 1 + e (2 − n ) , 3 + e ( n ) }} = UAVs Patrolling the . . . Optimal Placement Tests max { 1 + e (1) , 1 . 5 + e (1) } = max { 1 , 1 . 5 } = 1 . 5 . Feedback for Students Home Page • Here, e (2) = 1 . 5, and n (2) = 1. Title Page • To find e (3), we must compare two different values n = 1 and n = 2: ◭◭ ◮◮ ◭ ◮ e (3) = min 1 ≤ n< 3 { max { 1 + e (3 − n )) , 1 . 5 + e ( n ) }} = Page 90 of 106 min { max { 1+ e (2) , 1 . 5+ e (1) } , max { 1+ e (1) , 1 . 5+ e (2) }} = Go Back min { max { 2 . 5 , 1 . 5 } , max { 1 , 3 }} = min { 2 . 5 , 3 } = 2 . 5 . Full Screen • Here, min is attained when n = 1, so n (3) = 1. Close Quit
Introduction Case 1: Out of Eden Walk 83. Example 2: N = 3 and w = 1 . 5 (cont-d) Case 2: Climate . . . • To find e (4), we must consider three possible values Case 3: Software . . . n = 1, n = 2, and n = 3, so Placing Bio-Weapon . . . Meteorological Sensors e (4) = min 1 ≤ n< 4 { max { 1 + e (4 − n ) , 1 . 5 + e ( n )) }} = UAVs Patrolling the . . . Optimal Placement Tests min { max { 1+ e (3) , 1 . 5+ e (1) } , max { 1+ e (2) , 1 . 5+ e (2) } , Feedback for Students max { 1 + e (1) , 1 . 5 + e (3) }} = Home Page min { max { 3 . 5 , 1 . 5 } , max { 2 . 5 , 3 } , max { 1 , 4 }} = Title Page min { 3 . 5 , 3 , 4 } = 3 . ◭◭ ◮◮ • Here, min is attained when n = 2, so n (4) = 2. ◭ ◮ Page 91 of 106 Go Back Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 84. Example 2: Resulting Procedure Case 2: Climate . . . • First, i = 0 and j = 4, so we ask a student to solve a Case 3: Software . . . problem at level i + n ( j − i ) = 0 + n (4) = 2. Placing Bio-Weapon . . . Meteorological Sensors • If the student fails level 2, we set j = 2, and we assign UAVs Patrolling the . . . a problem of level 0 + n (2) = 1: Optimal Placement Tests – if the student fails level 1, his/her level is 0; Feedback for Students – if s/he succeeds at level 1, his/her level is 1. Home Page • If s/he succeeds at level 2, we set i = 2, and we assign Title Page a problem at level 2 + n (2) = 3: ◭◭ ◮◮ – if the student fails level 3, his/her level is 2; ◭ ◮ – if s/he succeeds at level 3, his/her level is 3. Page 92 of 106 • We can see that in this case, the optimal testing scheme Go Back is bisection. Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 85. A Faster Algorithm May Be Needed Case 2: Climate . . . • For each n from 1 to N , we need to compare n different Case 3: Software . . . values. Placing Bio-Weapon . . . Meteorological Sensors • So, the total number of computational steps is propor- UAVs Patrolling the . . . tional to 1 + 2 + . . . + N = O ( N 2 ). Optimal Placement Tests • When N is large, N 2 may be too large. Feedback for Students Home Page • In some applications, the computation of the optimal testing scheme may take too long. Title Page • For this case, we have developed a faster algorithm for ◭◭ ◮◮ producing a testing scheme. ◭ ◮ • The disadvantage of this algorithm is that it is only Page 93 of 106 asymptotically optimal. Go Back Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 86. A Faster Algorithm for Generating an Asymp- Case 2: Climate . . . totically Optimal Testing Scheme Case 3: Software . . . • First, we find the real number α ∈ [0 , 1] for which Placing Bio-Weapon . . . α + α w = 1. Meteorological Sensors UAVs Patrolling the . . . • This value α can be obtained, e.g., by applying bisec- tion to the equation α + α w = 1. Optimal Placement Tests Feedback for Students • At each iteration, once we know bounds i and j , we Home Page ask the student to solve a problem at the level Title Page m = ⌊ α · i + (1 − α ) · j ⌋ . ◭◭ ◮◮ • This algorithm is similar to bisection, expect that bi- ◭ ◮ section corresponds to α = 0 . 5. Page 94 of 106 • This makes sense, since for w = 1, the equation for α takes the form 2 α = 1, hence α = 0 . 5. Go Back • For w = 2, the solution to the equation α + α 2 = 1 is Full Screen √ 5 − 1 Close the well-known golden ratio α = ≈ 0 . 618. 2 Quit
Introduction Case 1: Out of Eden Walk Appendix 7. Towards Optimal Case 2: Climate . . . Feedback for Students Case 3: Software . . . Placing Bio-Weapon . . . Meteorological Sensors UAVs Patrolling the . . . Optimal Placement Tests Feedback for Students Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 95 of 106 Go Back Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 87. Towards Optimal Teaching Case 2: Climate . . . • One of the main objectives of a course – calculus, physics, Case 3: Software . . . etc. – is to help students understand its main concepts. Placing Bio-Weapon . . . Meteorological Sensors • Of course, it is also desirable that the students learn UAVs Patrolling the . . . the corresponding methods and algorithms. Optimal Placement Tests • However, understanding is the primary goal. Feedback for Students Home Page • If a student does not remember a formula by heart, she can look it up. Title Page • However: ◭◭ ◮◮ – if a student does not have a good understanding of ◭ ◮ what, for example, is a derivative, Page 96 of 106 – then even if this student remembers some formulas, Go Back he will not be able to decide which formula to apply. Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 88. How to Gauge Student Understanding Case 2: Climate . . . • To properly gauge student’s understanding, several dis- Case 3: Software . . . ciplines have developed concept inventories . Placing Bio-Weapon . . . Meteorological Sensors • These are sets of important basic concepts and ques- UAVs Patrolling the . . . tions testing the students’ understanding. Optimal Placement Tests • The first such Force Concept Inventory (FCI) was de- Feedback for Students veloped to gauge the students’ understanding of forces. Home Page • A student’s degree of understanding is measured by the Title Page percentage of the questions that are answered correctly. ◭◭ ◮◮ • The class’s degree of understanding is measured by av- ◭ ◮ eraging the students’ degrees. Page 97 of 106 • An ideal situation is when everyone has a perfect 100% Go Back understanding; in this case, the average score is 100%. Full Screen • In practice, the average score is smaller than 100%. Close Quit
Introduction Case 1: Out of Eden Walk 89. How to Compare Different Teaching Techniques Case 2: Climate . . . • We can measure the average score µ 0 before the class Case 3: Software . . . and the average score µ f after the class. Placing Bio-Weapon . . . Meteorological Sensors • Ideally, the whole difference 100 − µ 0 disappears, i.e., UAVs Patrolling the . . . the students’ score goes from µ 0 to µ f = 100. Optimal Placement Tests • In practice, of course, the students’ gain µ f − µ 0 is Feedback for Students somewhat smaller than the ideal gain 100 − µ 0 . Home Page • It is reasonable to measure the success of a teaching Title Page method by which portion of the ideal gain is covered: ◭◭ ◮◮ = µ f − µ 0 def g . ◭ ◮ 100 − µ 0 Page 98 of 106 Go Back Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 90. Empirical Results Case 2: Climate . . . • It turns out that the gain g does not depend on the Case 3: Software . . . initial level µ 0 , on the textbook used, or on the teacher. Placing Bio-Weapon . . . Meteorological Sensors • Only one factor determines the value g : the absence or UAVs Patrolling the . . . presence of immediate feedback. Optimal Placement Tests • In traditionally taught classes, Feedback for Students Home Page – where the students get their major feedback only after their first midterm exam, Title Page – the average gain is g ≈ 0 . 23. ◭◭ ◮◮ • For the classes with an immediate feedback, the aver- ◭ ◮ age gain is twice larger: g ≈ 0 . 48. Page 99 of 106 • In this talk, we provide a possible geometric explana- Go Back tion for this doubling of the learning rate. Full Screen Close Quit
Introduction Case 1: Out of Eden Walk 91. Why Geometry Case 2: Climate . . . • Learning means changing the state of a student. Case 3: Software . . . Placing Bio-Weapon . . . • At each moment of time, the state can be described by Meteorological Sensors the scores x 1 , . . . , x n on different tests. UAVs Patrolling the . . . • Each such state can be naturally represented as a point Optimal Placement Tests ( x 1 , . . . , x n ) in the n -dimensional space. Feedback for Students Home Page • In the starting state S , the student does not know the material. Title Page • The desired state D describes the situation when a ◭◭ ◮◮ student has the desired knowledge. ◭ ◮ • When a student learns, the student’s state of knowl- Page 100 of 106 edge changes continuously. Go Back • It forms a (continuous) trajectory γ which starts at the Full Screen starting state S and ends up at the desired state D . Close Quit
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