How Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence


 Jocelin Lang
 2 years ago
 Views:
Transcription
1 1 st Internatonal Symposum on Imprecse Probabltes and Ther Applcatons, Ghent, Belgum, 29 June 2 July 1999 How Sets of Coherent Probabltes May Serve as Models for Degrees of Incoherence Mar J. Schervsh Teddy Sedenfeld Carnege Mellon Unversty Joseph B.Kadane Abstract We ntroduce two ndces for the degree of ncoherence n a set of lower and upper prevsons: maxmzng the rate of loss the ncoherent boomaer experences n a Dutch Boo, or maxmzng the rate of proft the gambler acheves who maes Dutch Boo aganst the ncoherent boomaer. We report how effcent boomang s acheved aganst these two ndces n the case of ncoherent prevsons for events on a fnte partton, and for ncoherent prevsons that nclude also a smple random varable. We relate the epsloncontamnaton model to effcent boomang n the case of the rate of proft. Keywords. Dutch Boo, coherence, εcontamnaton model 1 Introducton It s a famlar remar that defnett s Dutch Boo argument provdes a smple dchotomy between coherent and ncoherent prevsons. For our presentaton here, consder the followng verson of hs argument, whch we present as a twoperson, zerosum game between a Booe, who s the subject of the argument, and a Gambler, who s the opponent. Let X be a (bounded) random varable defned on some space S of possbltes. The Booe s requred to offer hs/her prevson p(x) on the condton that the Gambler may then choose a real quantty α X,p(X) resultng n a payoff to the Booe of α X,p(X) [X  p(x) ] wth the opposte payoff to the Gambler a zerosum game. The Booe s prevsons for a set of random varables are ncoherent f there s a (fnte) selecton of nonzero α s by the Gambler that results, by summng, n a (unformly) negatve payoff to the Booe and a (unformly) postve payoff to the Gambler. The Booe s prevsons are coherent, otherwse. Ths leads to defnett s Dutch Boo Theorem The Booe s prevsons are coherent f and only f they are the expectatons of a (fntely addtve) probablty dstrbuton. defnett extends hs analyss to nclude assessments of condtonal prevsons, gven an event F, through calledoff wagers usng the ndcator for F, χ F, of the form α X,p(X),F χ F [X  p(x) ] Moreover, when the random varables X are restrcted to ndcator functons for events, E, the Booe s prevsons are coherent f and only f they are the condtonal probabltes of a sngle (fntely addtve) probablty. In ths case, the magntude α E,p(E),F s the stae for each wager, and the sgn of α E,p(E),F, postve or negatve, determnes whether the Booe bets respectvely, on or aganst E, calledoff f F fals to occur. It s a famlar concern, apprecated by many at ths conference, that defnett s crteron of coherence requres that the Booe posts a sngle prevson, or calledoff prevson gven F, for each X. For bettng on events, ths amounts to statng hs/her far (calledoff) odds : odds that the Gambler may use regardless the sgn of the coeffcent α. In response to ths concern, the game has been relaxed to permt what C.A.B.Smth [4]
2 called lower and upper pgnc odds. That s, n the case of ndcator varables, the Booe may post one prevson p a lower probablty used wth postve α for wagerng on E, and another prevson q an upper probablty used wth negatve α for wagerng aganst E. In effect, the Booe asserts that at odds of p : 1 p or less he/she wll bet on E, whereas at odds of q : 1 q or greater he/she wll bet aganst E. defnett s Dutch Boo theorem generalzes n ths settng to assert, roughly, that the Booe s lower and upper prevsons are coherent f and only f they are, respectvely, the lower and upper expectatons of a convex set of (fntely addtve) probablty dstrbuton. (See [3] for a precse statement of ths result.) Ths generalzaton, however, retans the ntal dchotomy: the Booe s prevsons are coherent or else the Gambler can mae a Dutch Boo an acheve a sure return. 2 Degrees of Incoherence In [2], we ntroduce two ndces of ncoherence: a rate of loss for the Booe and a rate of proft for the Gambler. These ndex the amount of the Gambler s suregan aganst ether of two escrow accounts, accounts that reflect the porton of the total stae each player contrbutes. The rate of loss ndexes the Gambler s guaranteed sure gan (.e., the mnmum of the Booe s assured loss) aganst the proporton of the total stae contrbuted by the Booe. The rate of proft ndexes the Gambler s guaranteed sure gan aganst hs/her own contrbuton to the total stae. In what follows, we focus on the second of these two ndces: the rate of proft acheved by the Gambler. Of course, there are more than these two ways of formalzng degrees of ncoherence. Nau [1] gves a flexble framewor that ncorporates our rate of loss as a specal case, for example. 2.1 Incoherence for events n a partton Let {A: = 1,...n} be a partton of the sureevent by n nonempty events, wth n > 1. Let 0 = p = q =1 be the Booe s lower and upper prevsons for the A ( = 1,..., n). Let s + = Σ q and let s = Σ p, so that the Booe s ncoherent f ether s + < 1 or 1 < s . Theorem 2 (from [2]): (1) If s + < 1 then the rate of guaranteed proft equals (1 s + )/s + and s acheved when the Gambler sets all the α = 1/ s +. (2) If s  > 1, then the Gambler maxmzes the mnmum rate of proft by choosng the staes accordng to the followng rule: Let * be the frst such that n p < 1+ ( 1) p = n + 1 n wth * = n f ths equalty always fals. Then the Gambler sets α all equal and postve for > n *+1, and sets α = 0 for all < n*. The Fgure below llustrates ths result for the case wth n = 3 atoms, a ternary partton. The set of coherent defnettprevsons s represented by the tranglular hyperplane: the smplex wth extreme values {(1,0,0), (0,1,0), (0,0,1)}. The set of ncoherent lower probabltes, where s  > 1, les above t. The selected hyperplane n the fgure s comprsed of lower probabltes wth s  = 1.5. For those lower prevsons n the whteregon, outsde the projecton of the coherent smplex, the Gambler maxmzes hs/her rate of proft (whch equals 3/7) by gnorng the Booe s prevson on A3, and achevng boo by havng the Booe bet on each of A1 and A 2, at equal staes. That would be the case f the Booe s lower prevsons were (.6,.7,.2). If, however, the Booe s prevsons were nsde the projecton of the coherent smplex, e.g., (.5,.5,.5), then the Gambler s rate of proft s only 1/3, acheved wth equal staes on each of the three atoms. 2.2 Incoherence wth prevsons for a smple random varable Next, consder the addton of a sngle random varable defned by a (fnte) partton, {A: = 1,...n}, as n the subsecton above. Let X be a (smple) random varable defned on these n events. For the next result, we assume that the the Booe gves prevsons p = p(a () ), ordered to be ncreasng n p, whch are sngly coherent, 0 < p < 1. Also, the Booe gves a prevson for X.. For smplcty, we state the followng result for the case s < 1. Defne these seven quanttes, s = Σ p µ = Σ x p and δ =  µ. p n, 1 s+, = = n + 1 s = = v ( ) = p X x p (1 s, ) x1 v = 1 n ( ) = (1 s xn + + +, ) = n + 1 p x p p X
3 Two atom strategy regon (1,0,1) (0,0,1) (1,1,1) (0,1,1) (0,0,0) (1,0,0) (.6,.7,.4) (.6,.7,.3 (.6,.7,.2) (1,1,0) (0,1,0) Theorem 6 (of [2]) The Gambler acheves the maxmum guaranteed rate of proft, as follows: 1) If δ < (1s)x 1, let * be the smallest value of such that v  () < 0. Then set α X = 1, set α = x x * for < * and set α = 0 for > *. 2) If δ > (1s)x n, let * be the smallest value of such that v + () < 0. Then set α X = 1, set α = x + x n* for > n* and α = 0 for < n *. 3) If (1s)x 1 < δ < (1s)x n, then set α X = 0 and apply the prevous theorem,.e., gnore the Booe s prevson for X but, nstead, use solely the ncoherence among the p. A Corollary to ths Theorem s nterestng and ntellgble on ts own. Havng already gven the (possbly ncoherent) prevsons p, and now oblged to provde the addtonal prevson, the Booe can as how to avod ncreasng the rate of proft that the Gambler may acheve. Corollary The Gambler s rate of proft after learnng the Booe s prevson does not ncrease f and only f satsfes: µ + (1s) x 1 < < µ + (1s) x n. That s, the corollary dentfes the Booe s mnmax strateges for augmentng the prevsons p for the events A, wth a sngle new prevson for X. Ths corollary apples to calledoff bettng as a specal case:
4 Consder the ternary partton and random varable X whose values are gven n the second row of the followng table. Thus, a 1 a 2 a α[1p(x)] αp(x) are the three correspondng payoffs to the Booe assocated wth the wager α[x  p(x)]. Then, e.g., wth s < 1, havng already announced the prevsons p ( = 1, 2, 3), the Booe s mnmax strateges for restranng the Gambler s rateofproft satsfes: p 1 + p 2 < < p 1 + p s. It s nterestng to note that choosng the pseudobayes condtonal value = p 2 /( p 2 + p 3 ) always satsfes these nequaltes. In other words, the ncoherent Booe can tae advantage of the fact that the pseudobayes solutonn s mnmax. You don t have to be coherent to le Bayes solutons! Of course, f s = 1, so that the Booe s coherent, the sole mnmax soluton s just the Bayes soluton. 3 Epsloncontamnaton and the rate of guaranteed proft The Gambler s decsons n the frst of the two Theorems, n secton 2.1 above, can be explaned wth an εcontamnaton model, through the Bayesan dual to the mnmax strateges for ths case. For the Gambler to accept wagers when the Booe offers upper probabltes, the Gambler must fnd these wagers acceptable as lower probabltes n a ratonal decson. Smlarly, for the Gambler to accept wagers when the Booe offers lower probabltes, the Gambler must fnd these wagers acceptable as upper probabltes n a ratonal decson. Gven a fxed probablty dstrbuton, p*, an ε contamnaton model of probabltes s a set of probabltes Mp* = {(1ε)p* + εq: 0 < ε < 1}, wth q an arbtrary probablty. Equvalently for fnte algebras, an εcontamnated model s gven by specfyng a coherent set of lower probabltes for the atoms of the algebra. When the Booe s ncoherent wth upper probabltes, s + < 1, these may be the coherent lower probabltes for the Gambler usng an εcontamnaton model. In fact, the Gambler maxmzes hs/her expected rate of proft accordng to ths (convex) set by wagerng as ndcated n (1) of the Theorem. When the Booe s ncoherent wth lower probabltes, s  > 1, t s not always the case that these can be the coherent upper probabltes for an ε contamnaton model. Precsely when the Booe s lower probabltes fall wthn the projecton of the coherent smplex, when they fall wthn the trangular regon llustrated n the Fgure, then the Gambler may use these as the coherent upper probabltes from an e contamnaton model. Otherwse, the Gambler fts the largest εcontamnaton model that s allowed by the Booe s offers. Expressed n other words, the strateges reported by the Theorem are those whch gve the Gambler a postve expected value for each component wager used to mae the Dutch Boo, and these relate to an εcontamnaton model, as just explaned. The same analyss apples to the second of the two Theorems, n secton 2.2 above. Ths case nvolves the Gambler s rate of guaranteed proft when the Booe s prevsons nclude a set of bets on a fnte partton and a prevson for one (smple) random varable defned on that partton. The nequaltes of Theorem 6 correspond, n precsely the same way, to the upper and lower expectatons from an εcontamnaton model, based on the Booe s ncoherent upper prevsons,.e., when s < 1. The conference presentaton ncludes, also, results for the parallel case when s >1. Then Gambler s maxmn strateges for securng an effcent Dutch Boo, reflect the added complcaton of truncaton of the ε contamnaton model, just as n the correspondng case (s > 1) for the Theorem of secton Concluson Ths presentaton ntroduces the use of a convex set of coherent probabltes, the εcontamnaton model, as the Bayes dual solutons to a Gambler s maxmn strateges for what we call the guaranteed rate of proft n mang effcent Dutch Boo aganst an ncoherent Booe. The two cases dscussed here nclude (1) ncoherent upper and lower prevsons for events n a fnte partton, and (2) a context where the Booe ncludes a prevson for a smple random varable defned on ths same partton.
5 Ongong wor (to be reported at the conference) specfces the correspondng convex set of probabltes that are dual to the Gambler s maxmn strateges for maxmzng the Booe s guaranteed rate of loss n each of these two cases. These sets nvolve fxng both upper and lower probablty bounds on the atoms of the fnte algebra, rather than merely fxng the lower probabltes, as s done n an εcontamnaton model. References [1] Nau, R.F. (1989), Decson analyss wth ndetermnate or ncoherent probabltes. Ann. Oper. Res [2] Schervsh, M.J., Sedenfeld, T., and Kadane, J.B. (1998), Two Measures of Incoherence: How Not to Gamble If You Must, T.R. #660 Dept. of Statstcs, Carnege Mellon Unv., Pgh. PA (A Postscrpt fle s avalable at: [3] Sedenfeld, T., Schervsh, M.J., and Kadane, J.B. (1990), Decsons wthout orderng. In Actng and Reflectng (W.Seg, ed.) Kluwer Academc Publshers, Dorddrecht. [4] Smth, C.A.B. (1961), Consstency n Statstcal Inference and Decson, J.R.S.S. B, 23, 125.
NONCONSTANT SUM REDANDBLACK GAMES WITH BETDEPENDENT WIN PROBABILITY FUNCTION LAURA PONTIGGIA, University of the Sciences in Philadelphia
To appear n Journal o Appled Probablty June 2007 OCOSTAT SUM REDADBLACK GAMES WITH BETDEPEDET WI PROBABILITY FUCTIO LAURA POTIGGIA, Unversty o the Scences n Phladelpha Abstract In ths paper we nvestgate
More informationPSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 12
14 The Chsquared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed
More informationLuby s Alg. for Maximal Independent Sets using Pairwise Independence
Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent
More informationWhat is Candidate Sampling
What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble
More informationNPAR TESTS. OneSample ChiSquare Test. Cell Specification. Observed Frequencies 1O i 6. Expected Frequencies 1EXP i 6
PAR TESTS If a WEIGHT varable s specfed, t s used to replcate a case as many tmes as ndcated by the weght value rounded to the nearest nteger. If the workspace requrements are exceeded and samplng has
More informationThe OC Curve of Attribute Acceptance Plans
The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4
More informationCommunication Networks II Contents
8 / 1  Communcaton Networs II (Görg)  www.comnets.unbremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP
More informationbenefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).
REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or
More informationTHE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek
HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo
More informationOPTIMAL INVESTMENT POLICIES FOR THE HORSE RACE MODEL. Thomas S. Ferguson and C. Zachary Gilstein UCLA and Bell Communications May 1985, revised 2004
OPTIMAL INVESTMENT POLICIES FOR THE HORSE RACE MODEL Thomas S. Ferguson and C. Zachary Glsten UCLA and Bell Communcatons May 985, revsed 2004 Abstract. Optmal nvestment polces for maxmzng the expected
More informationMultivariate EWMA Control Chart
Multvarate EWMA Control Chart Summary The Multvarate EWMA Control Chart procedure creates control charts for two or more numerc varables. Examnng the varables n a multvarate sense s extremely mportant
More information1 Example 1: Axisaligned rectangles
COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton
More informationII. PROBABILITY OF AN EVENT
II. PROBABILITY OF AN EVENT As ndcated above, probablty s a quantfcaton, or a mathematcal model, of a random experment. Ths quantfcaton s a measure of the lkelhood that a gven event wll occur when the
More informationPLANAR GRAPHS. Plane graph (or embedded graph) A graph that is drawn on the plane without edge crossing, is called a Plane graph
PLANAR GRAPHS Basc defntons Isomorphc graphs Two graphs G(V,E) and G2(V2,E2) are somorphc f there s a onetoone correspondence F of ther vertces such that the followng holds:  u,v V, uv E, => F(u)F(v)
More informationPERRON FROBENIUS THEOREM
PERRON FROBENIUS THEOREM R. CLARK ROBINSON Defnton. A n n matrx M wth real entres m, s called a stochastc matrx provded () all the entres m satsfy 0 m, () each of the columns sum to one, m = for all, ()
More informationInstitute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic
Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange
More informationAn Alternative Way to Measure Private Equity Performance
An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate
More informationRecurrence. 1 Definitions and main statements
Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.
More informationHow Much to Bet on Video Poker
How Much to Bet on Vdeo Poker Trstan Barnett A queston that arses whenever a gae s favorable to the player s how uch to wager on each event? Whle conservatve play (or nu bet nzes large fluctuatons, t lacks
More informationLogistic Regression. Lecture 4: More classifiers and classes. Logistic regression. Adaboost. Optimization. Multiple class classification
Lecture 4: More classfers and classes C4B Machne Learnng Hlary 20 A. Zsserman Logstc regresson Loss functons revsted Adaboost Loss functons revsted Optmzaton Multple class classfcaton Logstc Regresson
More information1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)
6.3 /  Communcaton Networks II (Görg) SS20  www.comnets.unbremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes
More informationSolution of Algebraic and Transcendental Equations
CHAPTER Soluton of Algerac and Transcendental Equatons. INTRODUCTION One of the most common prolem encountered n engneerng analyss s that gven a functon f (, fnd the values of for whch f ( = 0. The soluton
More information1.1 The University may award Higher Doctorate degrees as specified from timetotime in UPR AS11 1.
HIGHER DOCTORATE DEGREES SUMMARY OF PRINCIPAL CHANGES General changes None Secton 3.2 Refer to text (Amendments to verson 03.0, UPR AS02 are shown n talcs.) 1 INTRODUCTION 1.1 The Unversty may award Hgher
More informationModule 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..
More informationExtending Probabilistic Dynamic Epistemic Logic
Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008 Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σalgebra: a set
More informationA Probabilistic Theory of Coherence
A Probablstc Theory of Coherence BRANDEN FITELSON. The Coherence Measure C Let E be a set of n propostons E,..., E n. We seek a probablstc measure C(E) of the degree of coherence of E. Intutvely, we want
More informationTHE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES
The goal: to measure (determne) an unknown quantty x (the value of a RV X) Realsaton: n results: y 1, y 2,..., y j,..., y n, (the measured values of Y 1, Y 2,..., Y j,..., Y n ) every result s encumbered
More informationHYPOTHESIS TESTING OF PARAMETERS FOR ORDINARY LINEAR CIRCULAR REGRESSION
HYPOTHESIS TESTING OF PARAMETERS FOR ORDINARY LINEAR CIRCULAR REGRESSION Abdul Ghapor Hussn Centre for Foundaton Studes n Scence Unversty of Malaya 563 KUALA LUMPUR Emal: ghapor@umedumy Abstract Ths paper
More informationU.C. Berkeley CS270: Algorithms Lecture 4 Professor Vazirani and Professor Rao Jan 27,2011 Lecturer: Umesh Vazirani Last revised February 10, 2012
U.C. Berkeley CS270: Algorthms Lecture 4 Professor Vazran and Professor Rao Jan 27,2011 Lecturer: Umesh Vazran Last revsed February 10, 2012 Lecture 4 1 The multplcatve weghts update method The multplcatve
More informationEfficient Project Portfolio as a tool for Enterprise Risk Management
Effcent Proect Portfolo as a tool for Enterprse Rsk Management Valentn O. Nkonov Ural State Techncal Unversty Growth Traectory Consultng Company January 5, 27 Effcent Proect Portfolo as a tool for Enterprse
More information2.4 Bivariate distributions
page 28 2.4 Bvarate dstrbutons 2.4.1 Defntons Let X and Y be dscrete r.v.s defned on the same probablty space (S, F, P). Instead of treatng them separately, t s often necessary to thnk of them actng together
More informationQuantization Effects in Digital Filters
Quantzaton Effects n Dgtal Flters Dstrbuton of Truncaton Errors In two's complement representaton an exact number would have nfntely many bts (n general). When we lmt the number of bts to some fnte value
More information1 Approximation Algorithms
CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons
More informationPowerofTwo Policies for Single Warehouse MultiRetailer Inventory Systems with Order Frequency Discounts
Powerofwo Polces for Sngle Warehouse MultRetaler Inventory Systems wth Order Frequency Dscounts José A. Ventura Pennsylvana State Unversty (USA) Yale. Herer echnon Israel Insttute of echnology (Israel)
More informationA Computer Technique for Solving LP Problems with Bounded Variables
Dhaka Unv. J. Sc. 60(2): 163168, 2012 (July) A Computer Technque for Solvng LP Problems wth Bounded Varables S. M. Atqur Rahman Chowdhury * and Sanwar Uddn Ahmad Department of Mathematcs; Unversty of
More informationI. SCOPE, APPLICABILITY AND PARAMETERS Scope
D Executve Board Annex 9 Page A/R ethodologcal Tool alculaton of the number of sample plots for measurements wthn A/R D project actvtes (Verson 0) I. SOPE, PIABIITY AD PARAETERS Scope. Ths tool s applcable
More information+ + +   This circuit than can be reduced to a planar circuit
MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to
More informationEE201 Circuit Theory I 2015 Spring. Dr. Yılmaz KALKAN
EE201 Crcut Theory I 2015 Sprng Dr. Yılmaz KALKAN 1. Basc Concepts (Chapter 1 of Nlsson  3 Hrs.) Introducton, Current and Voltage, Power and Energy 2. Basc Laws (Chapter 2&3 of Nlsson  6 Hrs.) Voltage
More informationSupport Vector Machines
Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.
More informationIMPROVEMENT OF CONVERGENCE CONDITION OF THE SQUAREROOT INTERVAL METHOD FOR MULTIPLE ZEROS 1
Nov Sad J. Math. Vol. 36, No. 2, 2006, 009 IMPROVEMENT OF CONVERGENCE CONDITION OF THE SQUAREROOT INTERVAL METHOD FOR MULTIPLE ZEROS Modrag S. Petkovć 2, Dušan M. Mloševć 3 Abstract. A new theorem concerned
More informationThe Development of Web Log Mining Based on ImproveKMeans Clustering Analysis
The Development of Web Log Mnng Based on ImproveKMeans Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.
More informationMoment of a force about a point and about an axis
3. STATICS O RIGID BODIES In the precedng chapter t was assumed that each of the bodes consdered could be treated as a sngle partcle. Such a vew, however, s not always possble, and a body, n general, should
More informationComment on Rotten Kids, Purity, and Perfection
Comment Comment on Rotten Kds, Purty, and Perfecton PerreAndré Chappor Unversty of Chcago Iván Wernng Unversty of Chcago and Unversdad Torcuato d Tella After readng Cornes and Slva (999), one gets the
More informationSIX WAYS TO SOLVE A SIMPLE PROBLEM: FITTING A STRAIGHT LINE TO MEASUREMENT DATA
SIX WAYS TO SOLVE A SIMPLE PROBLEM: FITTING A STRAIGHT LINE TO MEASUREMENT DATA E. LAGENDIJK Department of Appled Physcs, Delft Unversty of Technology Lorentzweg 1, 68 CJ, The Netherlands Emal: e.lagendjk@tnw.tudelft.nl
More informationSolutions to the exam in SF2862, June 2009
Solutons to the exam n SF86, June 009 Exercse 1. Ths s a determnstc perodcrevew nventory model. Let n = the number of consdered wees,.e. n = 4 n ths exercse, and r = the demand at wee,.e. r 1 = r = r
More information9.1 The Cumulative Sum Control Chart
Learnng Objectves 9.1 The Cumulatve Sum Control Chart 9.1.1 Basc Prncples: Cusum Control Chart for Montorng the Process Mean If s the target for the process mean, then the cumulatve sum control chart s
More informationErrorPropagation.nb 1. Error Propagation
ErrorPropagaton.nb Error Propagaton Suppose that we make observatons of a quantty x that s subject to random fluctuatons or measurement errors. Our best estmate of the true value for ths quantty s then
More informationCHAPTER 14 MORE ABOUT REGRESSION
CHAPTER 14 MORE ABOUT REGRESSION We learned n Chapter 5 that often a straght lne descrbes the pattern of a relatonshp between two quanttatve varables. For nstance, n Example 5.1 we explored the relatonshp
More informationPassive Filters. References: Barbow (pp 265275), Hayes & Horowitz (pp 3260), Rizzoni (Chap. 6)
Passve Flters eferences: Barbow (pp 6575), Hayes & Horowtz (pp 360), zzon (Chap. 6) Frequencyselectve or flter crcuts pass to the output only those nput sgnals that are n a desred range of frequences (called
More informationNasdaq Iceland Bond Indices 01 April 2015
Nasdaq Iceland Bond Indces 01 Aprl 2015 Fxed duraton Indces Introducton Nasdaq Iceland (the Exchange) began calculatng ts current bond ndces n the begnnng of 2005. They were a response to recent changes
More informationANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING
ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING Matthew J. Lberatore, Department of Management and Operatons, Vllanova Unversty, Vllanova, PA 19085, 6105194390,
More informationA Fault Tree Analysis Strategy Using Binary Decision Diagrams.
A Fault Tree Analyss Strategy Usng Bnary Decson Dagrams. Karen A. Reay and John D. Andrews Loughborough Unversty, Loughborough, Lecestershre, LE 3TU. Abstract The use of Bnary Decson Dagrams (BDDs) n fault
More informationx f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60
BIVARIATE DISTRIBUTIONS Let be a varable that assumes the values { 1,,..., n }. Then, a functon that epresses the relatve frequenc of these values s called a unvarate frequenc functon. It must be true
More informationMAPP. MERIS level 3 cloud and water vapour products. Issue: 1. Revision: 0. Date: 9.12.1998. Function Name Organisation Signature Date
Ttel: Project: Doc. No.: MERIS level 3 cloud and water vapour products MAPP MAPPATBDClWVL3 Issue: 1 Revson: 0 Date: 9.12.1998 Functon Name Organsaton Sgnature Date Author: Bennartz FUB Preusker FUB Schüller
More informationJ. Parallel Distrib. Comput.
J. Parallel Dstrb. Comput. 71 (2011) 62 76 Contents lsts avalable at ScenceDrect J. Parallel Dstrb. Comput. journal homepage: www.elsever.com/locate/jpdc Optmzng server placement n dstrbuted systems n
More informationImplied (risk neutral) probabilities, betting odds and prediction markets
Impled (rsk neutral) probabltes, bettng odds and predcton markets Fabrzo Caccafesta (Unversty of Rome "Tor Vergata") ABSTRACT  We show that the well known euvalence between the "fundamental theorem of
More informationwhere the coordinates are related to those in the old frame as follows.
Chapter 2  Cartesan Vectors and Tensors: Ther Algebra Defnton of a vector Examples of vectors Scalar multplcaton Addton of vectors coplanar vectors Unt vectors A bass of noncoplanar vectors Scalar product
More informationCausal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting
Causal, Explanatory Forecastng Assumes causeandeffect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of
More informationInequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.
Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.
More informationv a 1 b 1 i, a 2 b 2 i,..., a n b n i.
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are
More informationSection C2: BJT Structure and Operational Modes
Secton 2: JT Structure and Operatonal Modes Recall that the semconductor dode s smply a pn juncton. Dependng on how the juncton s based, current may easly flow between the dode termnals (forward bas, v
More informationGibbs Free Energy and Chemical Equilibrium (or how to predict chemical reactions without doing experiments)
Gbbs Free Energy and Chemcal Equlbrum (or how to predct chemcal reactons wthout dong experments) OCN 623 Chemcal Oceanography Readng: Frst half of Chapter 3, Snoeynk and Jenkns (1980) Introducton We want
More informationPortfolio Loss Distribution
Portfolo Loss Dstrbuton Rsky assets n loan ortfolo hghly llqud assets holdtomaturty n the bank s balance sheet Outstandngs The orton of the bank asset that has already been extended to borrowers. Commtment
More informationFormula of Total Probability, Bayes Rule, and Applications
1 Formula of Total Probablty, Bayes Rule, and Applcatons Recall that for any event A, the par of events A and A has an ntersecton that s empty, whereas the unon A A represents the total populaton of nterest.
More informationOn the Optimal Control of a Cascade of HydroElectric Power Stations
On the Optmal Control of a Cascade of HydroElectrc Power Statons M.C.M. Guedes a, A.F. Rbero a, G.V. Smrnov b and S. Vlela c a Department of Mathematcs, School of Scences, Unversty of Porto, Portugal;
More informationLossless Data Compression
Lossless Data Compresson Lecture : Unquely Decodable and Instantaneous Codes Sam Rowes September 5, 005 Let s focus on the lossless data compresson problem for now, and not worry about nosy channel codng
More information"Research Note" APPLICATION OF CHARGE SIMULATION METHOD TO ELECTRIC FIELD CALCULATION IN THE POWER CABLES *
Iranan Journal of Scence & Technology, Transacton B, Engneerng, ol. 30, No. B6, 789794 rnted n The Islamc Republc of Iran, 006 Shraz Unversty "Research Note" ALICATION OF CHARGE SIMULATION METHOD TO ELECTRIC
More informationApproximation algorithms for allocation problems: Improving the factor of 1 1/e
Approxmaton algorthms for allocaton problems: Improvng the factor of 1 1/e Urel Fege Mcrosoft Research Redmond, WA 98052 urfege@mcrosoft.com Jan Vondrák Prnceton Unversty Prnceton, NJ 08540 jvondrak@gmal.com
More informationIntroduction: Analysis of Electronic Circuits
/30/008 ntroducton / ntroducton: Analyss of Electronc Crcuts Readng Assgnment: KVL and KCL text from EECS Just lke EECS, the majorty of problems (hw and exam) n EECS 3 wll be crcut analyss problems. Thus,
More informationWe are now ready to answer the question: What are the possible cardinalities for finite fields?
Chapter 3 Fnte felds We have seen, n the prevous chapters, some examples of fnte felds. For example, the resdue class rng Z/pZ (when p s a prme) forms a feld wth p elements whch may be dentfed wth the
More informationA Lyapunov Optimization Approach to Repeated Stochastic Games
PROC. ALLERTON CONFERENCE ON COMMUNICATION, CONTROL, AND COMPUTING, OCT. 2013 1 A Lyapunov Optmzaton Approach to Repeated Stochastc Games Mchael J. Neely Unversty of Southern Calforna http://wwwbcf.usc.edu/
More informationThe Analysis of Outliers in Statistical Data
THALES Project No. xxxx The Analyss of Outlers n Statstcal Data Research Team Chrysses Caron, Assocate Professor (P.I.) Vaslk Karot, Doctoral canddate Polychrons Economou, Chrstna Perrakou, Postgraduate
More informationFeature selection for intrusion detection. Slobodan Petrović NISlab, Gjøvik University College
Feature selecton for ntruson detecton Slobodan Petrovć NISlab, Gjøvk Unversty College Contents The feature selecton problem Intruson detecton Traffc features relevant for IDS The CFS measure The mrmr measure
More informationOn Competitive Nonlinear Pricing
On Compettve Nonlnear Prcng Andrea Attar Thomas Marott Franços Salané February 27, 2013 Abstract A buyer of a dvsble good faces several dentcal sellers. The buyer s preferences are her prvate nformaton,
More informationSolution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt.
Chapter 9 Revew problems 9.1 Interest rate measurement Example 9.1. Fund A accumulates at a smple nterest rate of 10%. Fund B accumulates at a smple dscount rate of 5%. Fnd the pont n tme at whch the forces
More informationNonparametric Estimation of Asymmetric First Price Auctions: A Simplified Approach
Nonparametrc Estmaton of Asymmetrc Frst Prce Auctons: A Smplfed Approach Bn Zhang, Kemal Guler Intellgent Enterprse Technologes Laboratory HP Laboratores Palo Alto HPL200286(R.) November 23, 2004 frst
More informationEnabling P2P Oneview Multiparty Video Conferencing
Enablng P2P Onevew Multparty Vdeo Conferencng Yongxang Zhao, Yong Lu, Changja Chen, and JanYn Zhang Abstract MultParty Vdeo Conferencng (MPVC) facltates realtme group nteracton between users. Whle P2P
More informationWeek 6 Market Failure due to Externalities
Week 6 Market Falure due to Externaltes 1. Externaltes n externalty exsts when the acton of one agent unavodably affects the welfare of another agent. The affected agent may be a consumer, gvng rse to
More informationOptimal Bidding Strategies for Generation Companies in a DayAhead Electricity Market with Risk Management Taken into Account
Amercan J. of Engneerng and Appled Scences (): 86, 009 ISSN 94700 009 Scence Publcatons Optmal Bddng Strateges for Generaton Companes n a DayAhead Electrcty Market wth Rsk Management Taken nto Account
More informationThe Choice of Direct Dealing or Electronic Brokerage in Foreign Exchange Trading
The Choce of Drect Dealng or Electronc Brokerage n Foregn Exchange Tradng Mchael Melvn & Ln Wen Arzona State Unversty Introducton Electronc Brokerage n Foregn Exchange Start from a base of zero n 1992
More informationProduction. 2. Y is closed A set is closed if it contains its boundary. We need this for the solution existence in the profit maximization problem.
Producer Theory Producton ASSUMPTION 2.1 Propertes of the Producton Set The producton set Y satsfes the followng propertes 1. Y s nonempty If Y s empty, we have nothng to talk about 2. Y s closed A set
More informationLoop Parallelization
  Loop Parallelzaton C52 Complaton steps: nested loops operatng on arrays, sequentell executon of teraton space DECLARE B[..,..+] FOR I :=.. FOR J :=.. I B[I,J] := B[I,J]+B[I,J] ED FOR ED FOR analyze
More informationDamage detection in composite laminates using cointap method
Damage detecton n composte lamnates usng contap method S.J. Km Korea Aerospace Research Insttute, 45 EoeunDong, YouseongGu, 35333 Daejeon, Republc of Korea yaeln@kar.re.kr 45 The contap test has the
More informationProject Networks With MixedTime Constraints
Project Networs Wth MxedTme Constrants L Caccetta and B Wattananon Western Australan Centre of Excellence n Industral Optmsaton (WACEIO) Curtn Unversty of Technology GPO Box U1987 Perth Western Australa
More informationGeneralizing the degree sequence problem
Mddlebury College March 2009 Arzona State Unversty Dscrete Mathematcs Semnar The degree sequence problem Problem: Gven an nteger sequence d = (d 1,...,d n ) determne f there exsts a graph G wth d as ts
More informationGraph Theory and Cayley s Formula
Graph Theory and Cayley s Formula Chad Casarotto August 10, 2006 Contents 1 Introducton 1 2 Bascs and Defntons 1 Cayley s Formula 4 4 Prüfer Encodng A Forest of Trees 7 1 Introducton In ths paper, I wll
More informationL10: Linear discriminants analysis
L0: Lnear dscrmnants analyss Lnear dscrmnant analyss, two classes Lnear dscrmnant analyss, C classes LDA vs. PCA Lmtatons of LDA Varants of LDA Other dmensonalty reducton methods CSCE 666 Pattern Analyss
More informationComplex Number Representation in RCBNS Form for Arithmetic Operations and Conversion of the Result into Standard Binary Form
Complex Number epresentaton n CBNS Form for Arthmetc Operatons and Converson of the esult nto Standard Bnary Form Hatm Zan and. G. Deshmukh Florda Insttute of Technology rgd@ee.ft.edu ABSTACT Ths paper
More informationA GameTheoretic Approach for Minimizing Security Risks in the InternetofThings
A GameTheoretc Approach for Mnmzng Securty Rsks n the InternetofThngs George Rontds, Emmanoul Panaouss, Aron Laszka, Tasos Daguklas, Pasquale Malacara, and Tansu Alpcan Hellenc Open Unversty, Greece
More informationEXAMPLE PROBLEMS SOLVED USING THE SHARP EL733A CALCULATOR
EXAMPLE PROBLEMS SOLVED USING THE SHARP EL733A CALCULATOR 8S CHAPTER 8 EXAMPLES EXAMPLE 8.4A THE INVESTMENT NEEDED TO REACH A PARTICULAR FUTURE VALUE What amount must you nvest now at 4% compoune monthly
More informationLecture 3: Force of Interest, Real Interest Rate, Annuity
Lecture 3: Force of Interest, Real Interest Rate, Annuty Goals: Study contnuous compoundng and force of nterest Dscuss real nterest rate Learn annutymmedate, and ts present value Study annutydue, and
More informationRECOGNIZING DIFFERENT TYPES OF STOCHASTIC PROCESSES
RECOGNIZING DIFFERENT TYPES OF STOCHASTIC PROCESSES JONG U. KIM AND LASZLO B. KISH Department of Electrcal and Computer Engneerng, Texas A&M Unversty, College Staton, TX 778418, USA Receved (receved date)
More informationAn InterestOriented Network Evolution Mechanism for Online Communities
An InterestOrented Network Evoluton Mechansm for Onlne Communtes Cahong Sun and Xaopng Yang School of Informaton, Renmn Unversty of Chna, Bejng 100872, P.R. Chna {chsun,yang}@ruc.edu.cn Abstract. Onlne
More informationTHE TITANIC SHIPWRECK: WHO WAS
THE TITANIC SHIPWRECK: WHO WAS MOST LIKELY TO SURVIVE? A STATISTICAL ANALYSIS Ths paper examnes the probablty of survvng the Ttanc shpwreck usng lmted dependent varable regresson analyss. Ths appled analyss
More informationTrafficlight a stress test for life insurance provisions
MEMORANDUM Date 006097 Authors Bengt von Bahr, Göran Ronge Traffclght a stress test for lfe nsurance provsons Fnansnspetonen P.O. Box 6750 SE113 85 Stocholm [Sveavägen 167] Tel +46 8 787 80 00 Fax
More informationDETERMINATION THERMODYNAMIC PROPERTIES OF WATER AND STEAM
DETERMINATION THERMODYNAMIC PROPERTIES OF WATER AND STEAM S. Sngr, J. Spal Afflaton Abstract Ths work presents functons of thermodynamc parameters developed n MATLAB. There are functons to determne: saturaton
More informationNordea G10 Alpha Carry Index
Nordea G10 Alpha Carry Index Index Rules v1.1 Verson as of 10/10/2013 1 (6) Page 1 Index Descrpton The G10 Alpha Carry Index, the Index, follows the development of a rule based strategy whch nvests and
More informationThe covariance is the two variable analog to the variance. The formula for the covariance between two variables is
Regresson Lectures So far we have talked only about statstcs that descrbe one varable. What we are gong to be dscussng for much of the remander of the course s relatonshps between two or more varables.
More informationVENTILATION MEASUREMENTS COMBINED WITH POLLUTANT CONCENTRATION MEASUREMENTS DISCRIMINATES BETWEEN HIGH EMISSION RATES AND INSUFFICIENT VENTILATION
VENTILTION MESREMENTS OMINED WITH OLLTNT ONENTRTION MESREMENTS DISRIMINTES ETWEEN HIGH EMISSION RTES ND INSFFIIENT VENTILTION Mkael orlng 1, Hans Stymne 2, Magnus Mattsson 2, and laes lomqvst 2 1 Department
More informationCalculation of Sampling Weights
Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a twostage stratfed cluster desgn. 1 The frst stage conssted of a sample
More information