Extending Probabilistic Dynamic Epistemic Logic


 Amberly Malone
 3 years ago
 Views:
Transcription
1 Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008
2 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 of subsets of S contanng, whch s closed under complements and countable unons and ntersectons. 3 µ : A [0, 1] s a probablty measure, that s µ(s) = 1 and µ( ) = 0 If {A 1, A 2,...} s a countable set of parwse dsjont elements of A, then µ( j=1 A j) = j=1 µ(a j). (Countable addtvty) If A = P(S), then the probablty space s called dscrete, and the probablty functon µ can be vewed as mappng each element of S nto [0, 1].
3 How the σalgebra helps us The σalgebra lets us restrct the doman of the probablty measure. Restrctng the doman of the probablty measure lets us reflect uncertanty about the probablty of ndvdual elements of S. Probablstc Epstemc Logc offers a dfferent way of handlng uncertanty about probabltes; t lets us express uncertanty about probablty spaces.
4 Probablstc Epstemc Model Defnton Let Φ be a set of proposton letters, and I be a set of agents. A probablstc epstemc model s a tuple M = (X, { } I,, {P,x } I,x X ), where X s a fnte set (a subset of X 2 ) s an epstemc relaton for each agent I, that s x y f consders y possble from x s a functon assgnng to each proposton letter p the set of states where t s true. for each agent and state x, the probablty space P,x s defned as the tuple (S,x, A,x, µ,x ), where S,x X s the sample space (fnte because X s fnte) A,x s a σalgebra µ,x : A,x [0, 1] s a probablty measure over S,x
5 Addng probablty formulas Add to epstemc logc probablty formulas of the form P (ϕ) q for a ratonal number q. Evaluate the truth of such a formula at a ponted model (M, x) (where M s a probablstc epstemc model and x s a state n M) n the followng way. (M, x) P (ϕ) q ff µ,x ([[ϕ]] S,x ) q, where [[ϕ]] s the set of states n M where ϕ s true. Of course, ths defnton only works f [[ϕ]] S,x A,x. We can ensure ths by ether 1 requrng each A,x be large enough so that [[ϕ]] S,x A,x s guaranteed for all ϕ, 2 extendng the functon µ,x to all subsets. Inner and outer measures are two such extensons, and they need not obey all condtons of a measure.
6 Inner and Outer Probabltes Let (S, A, µ) The nner and outer measures are defned on any set Y P(S) by (outer measure) µ (Y ) = nf{µ(b) : Y B, B A} (nner measure) µ (Y ) = sup{µ(b) : B Y, B A} When the σalgebra A s fnte, these are equvalent to (outer measure) µ (Y ) = µ( {B : Y B, B A}) (nner measure) µ (Y ) = µ( {B : B Y, B A}) The nner and outer measures are related by µ (Y ) = 1 µ (Y ) Nether the nner nor the outer measure s n general a measure.
7 Observaton about fnte σalgebras Any fnte σalgebra A can be characterzed by an equvalence relaton. The equvalence relaton s such that the equvalence classes are the sets {A A : x A} for each x S. Some of these sets produced are dentcal for dfferent x s. Although S may be nfnte, the there wll only be fntely many equvalence classes. A σalgebra can be generated by the equvalence classes of any equvalence relaton.
8 Fagn, Halpern, and Tuttle example Suppose there are two agents and k. 1 k s frst gven a bt 0 or 1. k learns he has ths bt, s aware that k receved a bt, but does not know what bt k receved. 2 k flps a far con and looks at the result. sees k look at the result, but does not what the result s. 3 k performs acton s f the con agrees wth the bt (gven that heads agrees wth 1 and tals agrees wth 0), and performs acton d otherwse. Ths example s from R. Fagn & J. Halpern (1994) Reasonng about Knowledge and Probablty. Journal of the ACM 41:2, pp
9 Dscusson There are four possble sequences of events: (1, H), (1, T ), (0, H), (0, T ) (note that the acton s or d s determned from the frst two). Untl k performs the acton s or d, agent consders any of these four states possble. (1, H) (1, T ) (0, H) (0, T ) We ndcate s uncertanty between two states usng a double arrow between the two states. In partcular, an arrow from state x to state y ndcates that consders y possble f x s the actual state. (Before the bt s gven, k s epstemc relaton wll be the same).
10 Here s a possblty for s probablty spaces. The sample space enclosed n a box, and the σalgebra equvalence classes are enclosed n the dotted ovals. (1, H) (1, T ) (0, H) (0, T ) M 1 The sample space s the same as the set of states consders possble. Indvdual states cannot be measurable (otherwse 0 or 1 must be assgned a probablty).
11 Another possblty has a sample space contanng only the states wth the correct bt (but recall that consders all states possble and both sample spaces possble). (1, H) (1, T ) (0, H) (0, T ) M 2 Wthout assgnng probablty to the bt, can now assgn a probablty to the actons s and d.
12 Here s uncertan among 4 probablty spaces. (1, H) (1, T ) (0, H) (0, T ) M 3
13 Modelng a sequence of events It s suggested that each of these models may reasonably represent s probablty spaces at a certan stage n the sequence of events (but to make to make better sense of the transton, we add a lttle more n parentheses that was not n the orgnal statement of the example): M 1 wth the tme before the bt s gven to k (suppose does not yet know that k wll perform acton s or d). M 2 wth the tme after the bt s gven to k, (after k tells he wll do ether s or d dependng on the con toss,) but before the con s flpped. M 3 wth the tme after the con s tossed, (after k spontaneously offers a bet about what acton he wll take,) but before k performs hs acton.
14 acton model Defnton (Acton Model) An acton model (Σ, { }, {P,σ }, pre) s a probablstc epstemc model wth the valuaton functon replaced by a functon pre whch assgns to each σ Σ a class of ponted probablstc epstemc models. Each element σ Σ s called an acton type.
15 Update Product We defne a mechancal procedure called the update product for transformng one model nto another model gven an acton (represented by an acton model). We defne the update product n two stages. 1 The frst, called the unrestrcted product, s to take a product that does not make use of the pre functon. 2 The second, called the relatvzaton, s to restrct the unrestrcted product to a set of states characterzed by the pre functon.
16 fnte product measure Defnton (Fnte product space) The product space of probablty spaces (S 1, A 1, µ 1 ) and (S 2, A 2, µ 2 ) s (S 3, A 3, µ 3 ), where 1 S 3 = S 1 S 2 s the Cartesan product. 2 A 3 s the smallest σalgebra contanng {A B : A A 1, B A 2 } 3 The probablty measure s defned as µ 3 (A) = n µ 1 (B k )µ 2 (C k ) k=1 where B k A 1, C k A 2, and A = n =1 B k C k
17 unrestrcted product Defnton (unrestrcted product) The unrestrcted product between a probablstc epstemc model M and an acton model Σ s M U Σ wth the followng components: 1 X = X Σ 2 (x, σ) (z, τ) ff x z and σ τ 3 p = p Σ 4 We defne P,(x,σ) to be the fnte product space between P,x and P,σ
18 relatvzaton Defnton (relatvzaton) The relatvzaton of a probablstc epstemc model M to Y X s gven by M R Y wth the followng components: 1 X Y = Y 2 x Y z ff x z and x, z Y 3 p Y = p Y 4 For x Y, f µ,x (Y ) = 0, then defne P,x to be the trval probablty space on the sngleton x. Otherwse 1 S Y,x = S,x Y 2 A Y,x s the σalgebra generated by {A Y : A A,x } 3 The probablty measure s defned by µ Y,x (A) = µ,x (A Y ) µ,x (Y )
19 Establshng Addtvty Theorem Let (S, A, µ) be a probablty space, such that A s a fnte σalgebra. If A, B A and Y S, then µ (A B Y ) = µ (A Y ) + µ (B Y ) µ (A B Y ) = µ (A Y ) + µ (B Y ) The proof of the outer measure part rests on the fact that Ŷ = {C : Y C : C A} A, and that A B Y s a dsjont unon of Â Y and B Y. The proof of the nner measure part follows smlar reasonng.
20 The update product Defnton Update Product Gven a probablstc epstemc model M and an acton model Σ, let Y = {(x, σ) : (M, x) pre(σ)} Then the update product between M and Σ, wrtten M Σ s M Σ = (M U Σ) R Y
21 What acton models should be used? From M 1 to M 2, there are two events: 1 a semprvate announcement of the bt to k 2 a publc announcement that k plans to do ether acton s or d. From M 2 to M 3, there are two events: 1 a semprvate announcement to k of the result of the con toss 2 a publc announcement regardng k s bet offer We frst consder gong from M 1 to M 2 usng just one acton model, and smlarly from M 2 to M 3 wth just one acton model. We then consder gong from M 1 to M 2 usng a sequence of two acton models, and smlarly from M 2 to M 3.
22 Semprvate announcement The relatonal structure of a semprvate announcement s gven by, k σ τ, k and k s probablty spaces: σ τ
23 M 1 to M 2 (1H, σ) (1T, σ) (1H, τ) (1T, τ) (0H, σ) (0T, σ) (0H, τ) (0T, τ)
24 M 1 to M 2 (1H, σ) (1T, σ) (1H, τ) (1T, τ) (0H, σ) (0T, σ) (0H, τ) (0T, τ)
25 (1H, σ) (1T, σ) (0H, τ) (0T, τ)
26 M 2 to M 3 (1H, σ) (1T, σ) (1H, τ) (1T, τ) (0H, σ) (0T, σ) (0H, τ) (0T, τ)
27 M 2 to M 3 (1H, σ) (1T, σ) (1H, τ) (1T, τ) (0H, σ) (0T, σ) (0H, τ) (0T, τ)
28 (1H, σ) (1T, τ) (0H, σ) (0T, τ)
29 From M 1 to M 2 frst stage: semprvate announcement relatonal structure:, k σ τ, k s probablty space: σ τ k s probablty spaces: σ τ pre(σ) ncludes states wth 1, and pre(τ) ncludes states wth 0.
30 From M 1 to M 2 second stage: publc announcement relatonal structure:, k, k σ τ, k Ths s the publc announcement the precondton of σ or the precondton of τ as long as no state satsfes both precondtons. and k s probablty spaces: σ τ pre(σ) ncludes states wth 1, and pre(τ) ncludes states wth 0.
31 From M 2 to M 3 The semprvate and publc announcement acton models are the same n all components except for the precondton functon pre. Instead of 1, the precondton of σ s H Instead of 0, the precondton of τ s T.
32 M 1, k, k, k (1, H) (1, T ), k, k, k, k (0, H) (0, T ), k, k, k
33 M 1, k, k, k (1, H) (1, T ), k, k, k, k (0, H) (0, T ), k, k, k
34 M 1, k, k, k (1, H) (1, T ), k, k, k, k (0, H) (0, T ), k, k, k
35 M 1, k, k, k (1, H) (1, T ), k, k, k, k (0, H) (0, T ), k, k, k
36 after 1st semprvate announcement, k, k, k (1, H) (1, T ) (0, H) (0, T ), k, k, k
37 after 1st semprvate announcement, k, k, k (1, H) (1, T ) (0, H) (0, T ), k, k, k
38 after 1st semprvate announcement, k, k, k (1, H) (1, T ) (0, H) (0, T ), k, k, k
39 after 1st semprvate announcement, k, k, k (1, H) (1, T ) (0, H) (0, T ), k, k, k
40 after frst publc announcement (M 2 ), k, k, k (1, H) (1, T ) (0, H) (0, T ), k, k, k
41 after frst publc announcement (M 2 ), k, k, k (1, H) (1, T ) (0, H) (0, T ), k, k, k
42 after frst publc announcement (M 2 ), k, k, k (1, H) (1, T ) (0, H) (0, T ), k, k, k
43 after frst publc announcement (M 2 ), k, k, k (1, H) (1, T ) (0, H) (0, T ), k, k, k
44 after 2nd semprvate announcement, k, k (1, H) (1, T ) (0, H) (0, T ), k, k
45 after 2nd semprvate announcement, k, k (1, H) (1, T ) (0, H) (0, T ), k, k
46 after 2nd semprvate announcement, k, k (1, H) (1, T ) (0, H) (0, T ), k, k
47 after 2nd semprvate announcement, k, k (1, H) (1, T ) (0, H) (0, T ), k, k
48 M 3, k, k (1, H) (1, T ) (0, H) (0, T ), k, k
49 M 3, k, k (1, H) (1, T ) (0, H) (0, T ), k, k
50 M 3, k, k (1, H) (1, T ) (0, H) (0, T ), k, k
51 M 3, k, k (1, H) (1, T ) (0, H) (0, T ), k, k
52 Recordng the past Here are two ways: lst hstory or statc hstory: Defne a lst hstory to be a lst (S 0, S 1,..., S n ) of probablstc epstemc model, for whch for each j, S j+1 = S j A for some acton model A. We may want to fx some underlyng structure of A for techncal reasons. Defne a statc hstory to be an augmented probablstc epstemc model (X, { } I,, {P,x }, Y ), where Y s a bnary relaton over X, and a number of condtons are mposed n order to ensure that xyz can adopt the readng x follows from z and some acton. The goal s to ensure that the statc hstory s structurally equvalent to a lst hstory. When provng completeness, we have found t easer to use the statc hstores.
53 Need for nonmeasurable sets n completeness proof? The answer to ths s stll unknown. Here are some comments: Completeness for Probablstc Epstemc Logc (whch nvolves nonmeasurable sets) constructs a fltraton that has dscrete measures. A smlar fltraton can be constructed for a probablstc dynamc epstemc logc wth a pasttme operator, but the fltraton wll guarantee all condtons needed to reflect update products. I suggest usng truthpreservng model transformatons to convert the fltraton nto a true statc hstory. It s yet unknown whether we would beneft from beng able to transform a model nto one where probabltes are not dscrete.
Recurrence. 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 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 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 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 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 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 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 information8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by
6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng
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 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 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 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 informationImplementation of Deutsch's Algorithm Using Mathcad
Implementaton of Deutsch's Algorthm Usng Mathcad Frank Roux The followng s a Mathcad mplementaton of Davd Deutsch's quantum computer prototype as presented on pages  n "Machnes, Logc and Quantum Physcs"
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 informationDEFINING %COMPLETE IN MICROSOFT PROJECT
CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMISP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,
More informationn + d + q = 24 and.05n +.1d +.25q = 2 { n + d + q = 24 (3) n + 2d + 5q = 40 (2)
MATH 16T Exam 1 : Part I (InClass) Solutons 1. (0 pts) A pggy bank contans 4 cons, all of whch are nckels (5 ), dmes (10 ) or quarters (5 ). The pggy bank also contans a con of each denomnaton. The total
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 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 informationIn our example i = r/12 =.0825/12 At the end of the first month after your payment is received your amount in the account, the balance, is
Payout annutes: Start wth P dollars, e.g., P = 100, 000. Over a 30 year perod you receve equal payments of A dollars at the end of each month. The amount of money left n the account, the balance, earns
More informationHow Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence
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
More informationCombinatorial Agency of Threshold Functions
Combnatoral Agency of Threshold Functons Shal Jan Computer Scence Department Yale Unversty New Haven, CT 06520 shal.jan@yale.edu Davd C. Parkes School of Engneerng and Appled Scences Harvard Unversty Cambrdge,
More informationToday s class. Chapter 13. Sources of uncertainty. Decision making with uncertainty
Today s class Probablty theory Bayesan nference From the ont dstrbuton Usng ndependence/factorng From sources of evdence Chapter 13 1 2 Sources of uncertanty Uncertan nputs Mssng data Nosy data Uncertan
More informationPOLYSA: A Polynomial Algorithm for Nonbinary Constraint Satisfaction Problems with and
POLYSA: A Polynomal Algorthm for Nonbnary Constrant Satsfacton Problems wth and Mguel A. Saldo, Federco Barber Dpto. Sstemas Informátcos y Computacón Unversdad Poltécnca de Valenca, Camno de Vera s/n
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 informationNONCONSTANT 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 informationQUANTUM MECHANICS, BRAS AND KETS
PH575 SPRING QUANTUM MECHANICS, BRAS AND KETS The followng summares the man relatons and defntons from quantum mechancs that we wll be usng. State of a phscal sstem: The state of a phscal sstem s represented
More informationgreatest common divisor
4. GCD 1 The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no
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 informationDynamic OnlineAdvertising Auctions as Stochastic Scheduling
Dynamc OnlneAdvertsng Auctons as Stochastc Schedulng Isha Menache and Asuman Ozdaglar Massachusetts Insttute of Technology {sha,asuman}@mt.edu R. Srkant Unversty of Illnos at UrbanaChampagn rsrkant@llnos.edu
More informationAnswer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy
4.02 Quz Solutons Fall 2004 MultpleChoce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multplechoce questons. For each queston, only one of the answers s correct.
More informationINTERPRETING TRUE ARITHMETIC IN THE LOCAL STRUCTURE OF THE ENUMERATION DEGREES.
INTERPRETING TRUE ARITHMETIC IN THE LOCAL STRUCTURE OF THE ENUMERATION DEGREES. HRISTO GANCHEV AND MARIYA SOSKOVA 1. Introducton Degree theory studes mathematcal structures, whch arse from a formal noton
More informationREGULAR MULTILINEAR OPERATORS ON C(K) SPACES
REGULAR MULTILINEAR OPERATORS ON C(K) SPACES FERNANDO BOMBAL AND IGNACIO VILLANUEVA Abstract. The purpose of ths paper s to characterze the class of regular contnuous multlnear operators on a product of
More informationEmbedding lattices in the Kleene degrees
F U N D A M E N T A MATHEMATICAE 62 (999) Embeddng lattces n the Kleene degrees by Hsato M u r a k (Nagoya) Abstract. Under ZFC+CH, we prove that some lattces whose cardnaltes do not exceed ℵ can be embedded
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 informationQUESTIONS, How can quantum computers do the amazing things that they are able to do, such. cryptography quantum computers
2O cryptography quantum computers cryptography quantum computers QUESTIONS, Quantum Computers, and Cryptography A mathematcal metaphor for the power of quantum algorthms Mark Ettnger How can quantum computers
More informationCHAPTER 7 VECTOR BUNDLES
CHAPTER 7 VECTOR BUNDLES We next begn addressng the queston: how do we assemble the tangent spaces at varous ponts of a manfold nto a coherent whole? In order to gude the decson, consder the case of U
More informationVRT012 User s guide V0.1. Address: Žirmūnų g. 27, Vilnius LT09105, Phone: (3705) 2127472, Fax: (3705) 276 1380, Email: info@teltonika.
VRT012 User s gude V0.1 Thank you for purchasng our product. We hope ths userfrendly devce wll be helpful n realsng your deas and brngng comfort to your lfe. Please take few mnutes to read ths manual
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 informationSimple Interest Loans (Section 5.1) :
Chapter 5 Fnance The frst part of ths revew wll explan the dfferent nterest and nvestment equatons you learned n secton 5.1 through 5.4 of your textbook and go through several examples. The second part
More informationEnforcement in Private vs. Public Externalities
Enforcement n Prvate vs. Publc Externaltes Hakan Inal Department of Economcs, School of Busness and L. Douglas Wlder School of Government and Publc Affars Vrgna Commonwealth Unversty hnal@vcu.edu November
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 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 information1. Measuring association using correlation and regression
How to measure assocaton I: Correlaton. 1. Measurng assocaton usng correlaton and regresson We often would lke to know how one varable, such as a mother's weght, s related to another varable, such as a
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 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 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 informationThe descriptive complexity of the family of Banach spaces with the πproperty
Arab. J. Math. (2015) 4:35 39 DOI 10.1007/s4006501401163 Araban Journal of Mathematcs Ghadeer Ghawadrah The descrptve complexty of the famly of Banach spaces wth the πproperty Receved: 25 March 2014
More informationThe University of Texas at Austin. Austin, Texas 78712. December 1987. Abstract. programs in which operations of dierent processes mayoverlap.
Atomc Semantcs of Nonatomc Programs James H. Anderson Mohamed G. Gouda Department of Computer Scences The Unversty of Texas at Austn Austn, Texas 78712 December 1987 Abstract We argue that t s possble,
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 informationCS 2750 Machine Learning. Lecture 17a. Clustering. CS 2750 Machine Learning. Clustering
Lecture 7a Clusterng Mlos Hauskrecht mlos@cs.ptt.edu 539 Sennott Square Clusterng Groups together smlar nstances n the data sample Basc clusterng problem: dstrbute data nto k dfferent groups such that
More informationStochastic Games on a Multiple Access Channel
Stochastc Games on a Multple Access Channel Prashant N and Vnod Sharma Department of Electrcal Communcaton Engneerng Indan Insttute of Scence, Bangalore 560012, Inda Emal: prashant2406@gmal.com, vnod@ece.sc.ernet.n
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 informationSurvey on Virtual Machine Placement Techniques in Cloud Computing Environment
Survey on Vrtual Machne Placement Technques n Cloud Computng Envronment Rajeev Kumar Gupta and R. K. Paterya Department of Computer Scence & Engneerng, MANIT, Bhopal, Inda ABSTRACT In tradtonal data center
More informationReflective Navigation
Reflectve Navgaton Bors Kluge InMach Intellgente Maschnen GmbH Helmholtzstr. 16, 89081 Ulm, Germany kluge@nmach.de Erwn Prassler BonnRhenSeg Unversty of Appled Scence GranthamAllee 20, 53757 Sankt Augustn,
More informationRing structure of splines on triangulations
www.oeaw.ac.at Rng structure of splnes on trangulatons N. Vllamzar RICAMReport 201448 www.rcam.oeaw.ac.at RING STRUCTURE OF SPLINES ON TRIANGULATIONS NELLY VILLAMIZAR Introducton For a trangulated regon
More informationRate Monotonic (RM) Disadvantages of cyclic. TDDB47 Real Time Systems. Lecture 2: RM & EDF. Prioritybased scheduling. States of a process
Dsadvantages of cyclc TDDB47 Real Tme Systems Manual scheduler constructon Cannot deal wth any runtme changes What happens f we add a task to the set? RealTme Systems Laboratory Department of Computer
More informationA Model of Private Equity Fund Compensation
A Model of Prvate Equty Fund Compensaton Wonho Wlson Cho Andrew Metrck Ayako Yasuda KAIST Yale School of Management Unversty of Calforna at Davs June 26, 2011 Abstract: Ths paper analyzes the economcs
More informationLecture 7 March 20, 2002
MIT 8.996: Topc n TCS: Internet Research Problems Sprng 2002 Lecture 7 March 20, 2002 Lecturer: Bran Dean Global Load Balancng Scrbe: John Kogel, Ben Leong In today s lecture, we dscuss global load balancng
More informationProductForm Stationary Distributions for Deficiency Zero Chemical Reaction Networks
Bulletn of Mathematcal Bology (21 DOI 1.17/s11538195174 ORIGINAL ARTICLE ProductForm Statonary Dstrbutons for Defcency Zero Chemcal Reacton Networks Davd F. Anderson, Gheorghe Cracun, Thomas G. Kurtz
More informationTo manage leave, meeting institutional requirements and treating individual staff members fairly and consistently.
Corporate Polces & Procedures Human Resources  Document CPP216 Leave Management Frst Produced: Current Verson: Past Revsons: Revew Cycle: Apples From: 09/09/09 26/10/12 09/09/09 3 years Immedately Authorsaton:
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 information1.1 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 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 information6. EIGENVALUES AND EIGENVECTORS 3 = 3 2
EIGENVALUES AND EIGENVECTORS The Characterstc Polynomal If A s a square matrx and v s a nonzero vector such that Av v we say that v s an egenvector of A and s the correspondng egenvalue Av v Example :
More informationTexas Instruments 30X IIS Calculator
Texas Instruments 30X IIS Calculator Keystrokes for the TI30X IIS are shown for a few topcs n whch keystrokes are unque. Start by readng the Quk Start secton. Then, before begnnng a specfc unt of the
More information8 Algorithm for Binary Searching in Trees
8 Algorthm for Bnary Searchng n Trees In ths secton we present our algorthm for bnary searchng n trees. A crucal observaton employed by the algorthm s that ths problem can be effcently solved when the
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 informationWhch one should I mtate? Karl H. Schlag Projektberech B Dscusson Paper No. B365 March, 996 I wsh to thank Avner Shaked for helpful comments. Fnancal support from the Deutsche Forschungsgemenschaft, Sonderforschungsberech
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 informationStrategy Machines. Representation and Complexity of Strategies in Infinite Games
Strategy Machnes Representaton and Complexty of Strateges n Infnte Games Von der Fakultät für Mathematk, Informatk und Naturwssenschaften der RWTH Aachen Unversty zur Erlangung des akademschen Grades enes
More informationComplete Fairness in Secure TwoParty Computation
Complete Farness n Secure TwoParty Computaton S. Dov Gordon Carmt Hazay Jonathan Katz Yehuda Lndell Abstract In the settng of secure twoparty computaton, two mutually dstrustng partes wsh to compute
More information8.4. Annuities: Future Value. INVESTIGATE the Math. 504 8.4 Annuities: Future Value
8. Annutes: Future Value YOU WILL NEED graphng calculator spreadsheet software GOAL Determne the future value of an annuty earnng compound nterest. INVESTIGATE the Math Chrstne decdes to nvest $000 at
More informationTrafficlight extended with stress test for insurance and expense risks in life insurance
PROMEMORIA Datum 0 July 007 FI Dnr 07117130 Fnansnspetonen Författare Bengt von Bahr, Göran Ronge Traffclght extended wth stress test for nsurance and expense rss n lfe nsurance Summary Ths memorandum
More information21 Vectors: The Cross Product & Torque
21 Vectors: The Cross Product & Torque Do not use our left hand when applng ether the rghthand rule for the cross product of two vectors dscussed n ths chapter or the rghthand rule for somethng curl
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 informationSeries Solutions of ODEs 2 the Frobenius method. The basic idea of the Frobenius method is to look for solutions of the form 3
Royal Holloway Unversty of London Department of Physs Seres Solutons of ODEs the Frobenus method Introduton to the Methodology The smple seres expanson method works for dfferental equatons whose solutons
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 informationOn Lockett pairs and Lockett conjecture for πsoluble Fitting classes
On Lockett pars and Lockett conjecture for πsoluble Fttng classes Lujn Zhu Department of Mathematcs, Yangzhou Unversty, Yangzhou 225002, P.R. Chna Emal: ljzhu@yzu.edu.cn Nanyng Yang School of Mathematcs
More information) of the Cell class is created containing information about events associated with the cell. Events are added to the Cell instance
Calbraton Method Instances of the Cell class (one nstance for each FMS cell) contan ADC raw data and methods assocated wth each partcular FMS cell. The calbraton method ncludes event selecton (Class Cell
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 informationNew bounds in BalogSzemerédiGowers theorem
New bounds n BalogSzemerédGowers theorem By Tomasz Schoen Abstract We prove, n partcular, that every fnte subset A of an abelan group wth the addtve energy κ A 3 contans a set A such that A κ A and A
More informationCredit Limit Optimization (CLO) for Credit Cards
Credt Lmt Optmzaton (CLO) for Credt Cards Vay S. Desa CSCC IX, Ednburgh September 8, 2005 Copyrght 2003, SAS Insttute Inc. All rghts reserved. SAS Propretary Agenda Background Tradtonal approaches to credt
More informationGeneral Auction Mechanism for Search Advertising
General Aucton Mechansm for Search Advertsng Gagan Aggarwal S. Muthukrshnan Dávd Pál Martn Pál Keywords game theory, onlne auctons, stable matchngs ABSTRACT Internet search advertsng s often sold by an
More informationVision Mouse. Saurabh Sarkar a* University of Cincinnati, Cincinnati, USA ABSTRACT 1. INTRODUCTION
Vson Mouse Saurabh Sarkar a* a Unversty of Cncnnat, Cncnnat, USA ABSTRACT The report dscusses a vson based approach towards trackng of eyes and fngers. The report descrbes the process of locatng the possble
More informationProbabilities and Probabilistic Models
Probabltes and Probablstc Models Probablstc models A model means a system that smulates an obect under consderaton. A probablstc model s a model that produces dfferent outcomes wth dfferent probabltes
More informationA Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy Scurve Regression
Novel Methodology of Workng Captal Management for Large Publc Constructons by Usng Fuzzy Scurve Regresson ChengWu Chen, Morrs H. L. Wang and TngYa Hseh Department of Cvl Engneerng, Natonal Central Unversty,
More informationBERNSTEIN POLYNOMIALS
OnLne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful
More informationBrigid Mullany, Ph.D University of North Carolina, Charlotte
Evaluaton And Comparson Of The Dfferent Standards Used To Defne The Postonal Accuracy And Repeatablty Of Numercally Controlled Machnng Center Axes Brgd Mullany, Ph.D Unversty of North Carolna, Charlotte
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 informationCALL ADMISSION CONTROL IN WIRELESS MULTIMEDIA NETWORKS
CALL ADMISSION CONTROL IN WIRELESS MULTIMEDIA NETWORKS Novella Bartoln 1, Imrch Chlamtac 2 1 Dpartmento d Informatca, Unverstà d Roma La Sapenza, Roma, Italy novella@ds.unroma1.t 2 Center for Advanced
More informationDscreteTme Approxmatons of the HolmstromMlgrom BrownanMoton Model of Intertemporal Incentve Provson 1 Martn Hellwg Unversty of Mannhem Klaus M. Schmdt Unversty of Munch and CEPR Ths verson: May 5, 1998
More informationClustering Gene Expression Data. (Slides thanks to Dr. Mark Craven)
Clusterng Gene Epresson Data Sldes thanks to Dr. Mark Craven Gene Epresson Proles we ll assume we have a D matr o gene epresson measurements rows represent genes columns represent derent eperments tme
More informationLinear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits
Lnear Crcuts Analyss. Superposton, Theenn /Norton Equalent crcuts So far we hae explored tmendependent (resste) elements that are also lnear. A tmendependent elements s one for whch we can plot an / cure.
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 informationSolving Factored MDPs with Continuous and Discrete Variables
Solvng Factored MPs wth Contnuous and screte Varables Carlos Guestrn Berkeley Research Center Intel Corporaton Mlos Hauskrecht epartment of Computer Scence Unversty of Pttsburgh Branslav Kveton Intellgent
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 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 informationThe Greedy Method. Introduction. 0/1 Knapsack Problem
The Greedy Method Introducton We have completed data structures. We now are gong to look at algorthm desgn methods. Often we are lookng at optmzaton problems whose performance s exponental. For an optmzaton
More informationRealistic Image Synthesis
Realstc Image Synthess  Combned Samplng and Path Tracng  Phlpp Slusallek Karol Myszkowsk Vncent Pegoraro Overvew: Today Combned Samplng (Multple Importance Samplng) Renderng and Measurng Equaton Random
More informationA DYNAMIC CRASHING METHOD FOR PROJECT MANAGEMENT USING SIMULATIONBASED OPTIMIZATION. Michael E. Kuhl Radhamés A. TolentinoPeña
Proceedngs of the 2008 Wnter Smulaton Conference S. J. Mason, R. R. Hll, L. Mönch, O. Rose, T. Jefferson, J. W. Fowler eds. A DYNAMIC CRASHING METHOD FOR PROJECT MANAGEMENT USING SIMULATIONBASED OPTIMIZATION
More information