Lossless Data Compression


 Sabina Kelly
 1 years ago
 Views:
Transcription
1 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 for now. In practce these two problems are handeled separately,.e. we frst desgn an effcent code for the source (removng source redundancy) and then (f necessary) we desgn a channel code to help us transmt the source code over the channel (addng redundancy). Assumptons (for now):. the channel s perfectly noseless.e. the recever sees exactly the encoder s output. we always requre the output of the decoder to exactly match the orgnal sequence X. 3. X s generated accordng to a fxed probablstc model, p(x) We wll measure the qualty of our compresson scheme (called a code) by examnng the average length of the encoded strng Z, averaged over p(x). Recall: Mathematcal Setup Start wth a sequence of symbols X = X, X,..., X N from a fnte source alphabet A X = {a, a,...}. Examples: A X = {A, B,..., Z, }; A X = {0,,,..., 55}; A X = {C, G, T, A}; A X = {0, }. Encoder: outputs a new sequence Z = Z, Z,..., Z M (usng a possbly dfferent code alphabet A Z ). Decoder tres to convert Z back nto X. In compresson, the encoder tres to remove source redundancy. In nosy channel codng, the encoder tres to protect the message aganst transmsson errors. We almost always use A Z = 0, (e.g. computer fles, dgtal communcaton) but the theory can be generalzed to any fnte set. Encodng One Symbol at a Tme To begn wth, let s thnk about encodng one symbol X at a tme, usng a fxed code that defnes a mappng of each source symbol nto a fnte sequence of code symbols called a codeword. (Later on we wll consder encodng blocks of symbols together.) We wll encode a sequence of source symbols X by concatenatng the codewords of each. Ths s called a symbol code. E.g. source alphabet s A X = {C, G, T, A}. One possble code: C 0; G 0; T 0; A 0 So we would have CCAT We requre that the mappng be such that we can decode ths sequence, no matter what the orgnal symbols were.
2 Notaton for Sequences & Codes A X and A Z are the source and code alphabets. A + X and A+ Z denote sequences of one or more symbols from the source or code alphabets. A symbol code, C, s a mappng A X A + Z. We use c(x) to denote the codeword to whch C maps x. We use concatenaton to extend ths to a mappng for the extended code, C + : A + X A+ Y : c + (x x x N ) = c(x )c(x ) c(x N ).e., we code a strng of symbols by just strngng together the codes for each symbol. I ll sometmes also use C to denote the set of all legal codewords: {w w = C(a) for some a A X }. Unquely Decodable & Instantaneous Codes A code s unquely decodable f the mappng C + : A + X A+ Z s onetoone,.e. x and x n A + X, x x c + (x) c + (x ) A code s obvously not unquely decodable f two symbols have the same codeword e, f c(a ) = c(a j ) for some j so we ll usually assume that ths sn t the case. A code s nstantaneously decodable f any source sequences x and x n A + for whch x s not a prefx of x have encodngs z = C(x) and z = C(x ) for whch z s not a prefx of z. Otherwse, after recevng z, we wouldn t yet know whether the message starts wth z or wth z. Instantaneous codes are also called prefxfree codes or just prefx codes. What Codes are Decodable? We only want to consder codes that can be successfully decoded. To defne what that means, we need to set some rules of the game:. How does the channel termnate the transmsson? (e.g. t could explctly mark the end, t could send only 0s after the end, t could send random garbadge after the end,...). How soon do we requre a decoded symbol to be known? (e.g. nstantaneously as soon as the codeword for the symbol s receved, wthn a fxed delay of when ts codeword s receved, not untl the entre message has been receved,...) Easest case: assume the end of the transmsson s explctly marked, and don t requre any symbols to be decoded untl the entre transmsson has been receved. Hardest case: requre nstantaneous decodng, and thus t doesn t matter what happens at the end of the transmsson. Examples Code A Code B Code C Code D a b c 0 0 Code A: Not unquely decodable Both bbb and cc encode as Code B: Instantaneously decodable End of each codeword marked by 0 Code C: Decodable wth onesymbol delay End of codeword marked by followng 0 Code D: Unquely decodable, but wth unbounded delay: 0 decodes as accccccc 0 decodes as bcccccc
3 More Examples Code E Code F Code G a b c d Code E: Instantaneously decodable All codewords same length Code F: Not unquely decodable e.g. baa,aca,aad all encode as 0000 Code G: Decodable wth sxsymbol delay. (Try to work out why.) A Check for Unque Decodablty The SardnasPatterson Theorem tells us how to check whether a code s unquely decodable. Let C be the set of codewords. Defne C 0 = C. For n > 0, defne C n = Fnally, defne {w A + X uw = v where u C, v C n or u C n, v C} C = C C C 3 Theorem: the code C s unquely decodable f and only f C and C are dsjont. We won t both much wth ths theorem, snce as we ll see t sn t of much practcal use. A Check for Instantaneous Codes A code s nstantaneous f and only f no codeword s a prefx of some other codeword. (e f C s a codeword, C Z cannot be a codeword for any Z). Ths s a prefx code. Proof: ( ) If codeword C(a ) s a prefx of codeword C(a j ), then the encodng of the sequence x = a s obvously a prefx of the encodng of the sequence x = a j. ( ) If the code s not nstantaneous, let z = C(x) be an encodng that s a prefx of another encodng z = C(x ), but wth x not a prefx of x, and let x be as short as possble. The frst symbols of x and x can t be the same, snce f they were, we could drop these symbols and get a shorter nstance. So these two symbols must be dfferent, but one of ther codewords must be a prefx of the other. Exstence of Codes Snce we hope to compress data, we would lke codes that are unquely decodable and whose codewords are short. Also, we d lke to use nstantaneous codes where possble snce they are easest and most effcent to decode. If we could make all the codewords really short, lfe would be really easy. Too easy. Why? Because there are only a few possble short codewords and we can t reuse them or else our code wouldn t be decodable. Instead, makng some codewords short wll requre that other codewords be long, f the code s to be unquely decodable. Queston : What sets of codeword lengths are possble? Queston : Can we always manage to use nstantaneous codes?
4 McMllan s Inequalty There s a unquely decodable bnary code wth codewords havng lengths l,..., l I f and only f I = l E.g. there s a unquely decodable bnary code wth lengths,, 3, 3, snce / + /4 + /8 + /8 = An example of such a code s {0, 0, 0, }. There s no unquely decodable bnary code wth lengths,,,,, snce /4 + /4 + /4 + /4 + /4 > We Can Always Use Instantaneous Codes Snce nstantaneous codes are a proper subset of unquely decodable codes, we mght have expected that the condton for exstence of a u.d. code to be less strngent than that for nstantaneous codes. But combnng Kraft s and McMllan s nequaltes, we conclude that there s an nstantaneous bnary code wth lengths l,..., l I f and only f there s a unquely decodable code wth these lengths. Implcaton: There s probably no practcal beneft to usng unquely decodable codes that aren t nstantaneous. Happy consequence: We don t have to worry about how the encodng s termnated (f at all) or about decodng delays (at least for symbol codes; for block codes ths wll change). Kraft s Inequalty There s an nstantaneous bnary code wth codewords havng lengths l,..., l I f and only f I = l Ths s exactly the same condton as McMllan s nequalty! E.g. there s an nstantaneous bnary code wth lengths,, 3, 3, snce / + /4 + /8 + /8 = An example of such a code s {0, 0, 0, }. There s an nstantaneous bnary code wth lengths,,, snce /4 + /4 + /4 < An example of such a code s {00, 0, 0}. Provng the Two Inequaltes We can prove both Kraft s and McMllan s nequalty by provng that for any set of lengths, l,..., l I, for bnary codewords: A) If I = / l, we can construct an nstantaneous code wth codewords havng these lengths. B) If I = / l >, there s no unquely decodable code wth codewords havng these lengths. (A) s half of Kraft s nequalty. (B) s half of McMllan s nequalty. Usng the fact that nstantaneous codes are unquely decodable, (A) gves the other half of McMllan s nequalty, and (B) gves the other half of Kraft s nequalty. To do ths, we ll ntroduce a helpful way of thnkng about codes as...trees!
5 Vsualzng Prefx Codes as Trees We can vew codewords of an nstantaneous (prefx) code as leaves of a tree. The root represents the null strng; each level corresponds to addng another code symbol. Here s the tree for a code wth codewords 0,, 00, 0: NULL Constructng Instantaneous Codes Suppose that Kraft s Inequalty holds: I = l Order the lengths so l l I. Q: In the bnary tree wth depth l I, how can we allocate subtrees to codewords wth these lengths? A: We go from shortest to longest, =,..., I: ) Pck a node at depth l that sn t n a subtree prevously used, and let the code for codeword be the one at that node. ) Mark all nodes n the subtree headed by the node just pcked as beng used, and not avalable to be pcked later. Let s look at an example... Extendng the Tree to Maxmum Depth We can extend the tree by fllng n the subtree underneath every actual codeword, down to the depth of the longest codeword. Each codeword then corresponds to ether a leaf or a subtree. Prevous tree extended, wth each codeword s leaf or subtree crcled: NULL 0 Short codewords occupy more of the tree. For a bnary code, the fracton of leaves taken by a codeword of length l s / l Buldng an Instantaneous Code Let the lengths of the codewords be {,,3,3}. Frst check: Our fnal code can be read from the leaf nodes: {,00,00,0} NULL
6 Constructon Wll Always Be Possble Q: Wll there always be a node avalable n step () above? If Kraft s nequalty holds, we wll always be able to do ths. To begn, there are l b nodes at depth l b. When we pck a node at depth l a, the number of nodes that become unavalable at depth l b (assumed not less than l a ) s l b l a. When we need to pck a node at depth l j, after havng pcked earler nodes at depths l (wth < j and l l j ), the number of nodes left to pck from wll be j l j l j j l = l j = = > 0 j Snce / l < I / l, by assumpton. = = Ths proves (A). UD Codes Must Obey the Inequalty Let l l I be the codeword lengths. Defne K = I = l. For any postve nteger n, we sum over all possble combnatons of values for,..., n n {,..., I}. K n =,..., n l l n We rewrte ths n terms of possble values for j = l + + l n : K n nl I N j,n = j= j N j,n s the # of sequences of n codewords wth total length j. If the code s unquely decodable, N j,n j, so K n nl I, whch for bg enough n s possble only f K. Ths proves (B). (For nstantaneous codes, the ntuton s that short codes use up ther subtree.)
Module 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 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 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 informationMinimal Coding Network With Combinatorial Structure For Instantaneous Recovery From Edge Failures
Mnmal Codng Network Wth Combnatoral Structure For Instantaneous Recovery From Edge Falures Ashly Joseph 1, Mr.M.Sadsh Sendl 2, Dr.S.Karthk 3 1 Fnal Year ME CSE Student Department of Computer Scence Engneerng
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 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 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 informationIn our example i = r/12 =.0825/12 At the end of the first month after your payment is received your amount owed is. P (1 + i) A
Amortzed loans: Suppose you borrow P dollars, e.g., P = 100, 000 for a house wth a 30 year mortgage wth an nterest rate of 8.25% (compounded monthly). In ths type of loan you make equal payments of A dollars
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 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 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 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 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 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 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 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 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 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 informationTrivial lump sum R5.0
Optons form Once you have flled n ths form, please return t wth your orgnal brth certfcate to: Premer PO Box 2067 Croydon CR90 9ND. Fll n ths form usng BLOCK CAPITALS and black nk. Mark all answers wth
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 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 informationTime Value of Money. Types of Interest. Compounding and Discounting Single Sums. Page 1. Ch. 6  The Time Value of Money. The Time Value of Money
Ch. 6  The Tme Value of Money Tme Value of Money The Interest Rate Smple Interest Compound Interest Amortzng a Loan FIN21 Ahmed Y, Dasht TIME VALUE OF MONEY OR DISCOUNTED CASH FLOW ANALYSIS Very Important
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 informationJoe Pimbley, unpublished, 2005. Yield Curve Calculations
Joe Pmbley, unpublshed, 005. Yeld Curve Calculatons Background: Everythng s dscount factors Yeld curve calculatons nclude valuaton of forward rate agreements (FRAs), swaps, nterest rate optons, and forward
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 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 informationLecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression.
Lecture 3: Annuty Goals: Learn contnuous annuty and perpetuty. Study annutes whose payments form a geometrc progresson or a arthmetc progresson. Dscuss yeld rates. Introduce Amortzaton Suggested Textbook
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 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 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 information7.5. Present Value of an Annuity. Investigate
7.5 Present Value of an Annuty Owen and Anna are approachng retrement and are puttng ther fnances n order. They have worked hard and nvested ther earnngs so that they now have a large amount of money on
More informationProactive Secret Sharing Or: How to Cope With Perpetual Leakage
Proactve Secret Sharng Or: How to Cope Wth Perpetual Leakage Paper by Amr Herzberg Stanslaw Jareck Hugo Krawczyk Mot Yung Presentaton by Davd Zage What s Secret Sharng Basc Idea ((2, 2)threshold scheme):
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 informationActivity Scheduling for CostTime Investment Optimization in Project Management
PROJECT MANAGEMENT 4 th Internatonal Conference on Industral Engneerng and Industral Management XIV Congreso de Ingenería de Organzacón Donosta San Sebastán, September 8 th 10 th 010 Actvty Schedulng
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 informationUsing Series to Analyze Financial Situations: Present Value
2.8 Usng Seres to Analyze Fnancal Stuatons: Present Value In the prevous secton, you learned how to calculate the amount, or future value, of an ordnary smple annuty. The amount s the sum of the accumulated
More informationSection 5.4 Annuities, Present Value, and Amortization
Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today
More informationFinite Math Chapter 10: Study Guide and Solution to Problems
Fnte Math Chapter 10: Study Gude and Soluton to Problems Basc Formulas and Concepts 10.1 Interest Basc Concepts Interest A fee a bank pays you for money you depost nto a savngs account. Prncpal P The amount
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 informationNondegenerate Hilbert Cubes in Random Sets
Journal de Théore des Nombres de Bordeaux 00 (XXXX), 000 000 Nondegenerate Hlbert Cubes n Random Sets par Csaba Sándor Résumé. Une légère modfcaton de la démonstraton du lemme des cubes de Szemeréd donne
More informationMultiplication Algorithms for Radix2 RNCodings and Two s Complement Numbers
Multplcaton Algorthms for Radx RNCodngs and Two s Complement Numbers JeanLuc Beuchat Projet Arénare, LIP, ENS Lyon 46, Allée d Itale F 69364 Lyon Cedex 07 jeanluc.beuchat@enslyon.fr JeanMchel Muller
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 information1. Math 210 Finite Mathematics
1. ath 210 Fnte athematcs Chapter 5.2 and 5.3 Annutes ortgages Amortzaton Professor Rchard Blecksmth Dept. of athematcal Scences Northern Illnos Unversty ath 210 Webste: http://math.nu.edu/courses/math210
More informationNonbinary Quantum ErrorCorrecting Codes from Algebraic Curves
Nonbnary Quantum ErrorCorrectng Codes from Algebrac Curves JonLark Km and Judy Walker Department of Mathematcs Unversty of NebraskaLncoln, Lncoln, NE 685880130 USA emal: {jlkm, jwalker}@math.unl.edu
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 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 informationMean Molecular Weight
Mean Molecular Weght The thermodynamc relatons between P, ρ, and T, as well as the calculaton of stellar opacty requres knowledge of the system s mean molecular weght defned as the mass per unt mole of
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 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 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 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 informationMultiple discount and forward curves
Multple dscount and forward curves TopQuants presentaton 21 ovember 2012 Ton Broekhuzen, Head Market Rsk and Basel coordnator, IBC Ths presentaton reflects personal vews and not necessarly the vews of
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 informationDISTRIBUTED storage systems have been becoming increasingly
268 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 28, NO. 2, FEBRUARY 2010 Cooperatve Recovery of Dstrbuted Storage Systems from Multple Losses wth Network Codng Yuchong Hu, Ynlong Xu, Xaozhao
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 informationIDENTIFICATION AND CONTROL OF A FLEXIBLE TRANSMISSION SYSTEM
Abstract IDENTIFICATION AND CONTROL OF A FLEXIBLE TRANSMISSION SYSTEM Alca Esparza Pedro Dept. Sstemas y Automátca, Unversdad Poltécnca de Valenca, Span alespe@sa.upv.es The dentfcaton and control of a
More informationSection 5.3 Annuities, Future Value, and Sinking Funds
Secton 5.3 Annutes, Future Value, and Snkng Funds Ordnary Annutes A sequence of equal payments made at equal perods of tme s called an annuty. The tme between payments s the payment perod, and the tme
More informationSPEE Recommended Evaluation Practice #6 Definition of Decline Curve Parameters Background:
SPEE Recommended Evaluaton Practce #6 efnton of eclne Curve Parameters Background: The producton hstores of ol and gas wells can be analyzed to estmate reserves and future ol and gas producton rates and
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 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 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 informationThursday, December 10, 2009 Noon  1:50 pm Faraday 143
1. ath 210 Fnte athematcs Chapter 5.2 and 4.3 Annutes ortgages Amortzaton Professor Rchard Blecksmth Dept. of athematcal Scences Northern Illnos Unversty ath 210 Webste: http://math.nu.edu/courses/math210
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 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 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 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 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 informationConferencing protocols and Petri net analysis
Conferencng protocols and Petr net analyss E. ANTONIDAKIS Department of Electroncs, Technologcal Educatonal Insttute of Crete, GREECE ena@chana.tecrete.gr Abstract: Durng a computer conference, users desre
More informationMotivation. Eingebettete Systeme. Terms. Terms. Echtzeitverhalten und Betriebssysteme. 7. Ressourcen
Motvaton Engebettete Systeme Echtzetverhalten und Betrebssysteme 7. Ressourcen 1 2 Terms A resource s any software structure that can be used by a process to advance ts executon, e.g. data structure, a
More informationFormulating & Solving Integer Problems Chapter 11 289
Formulatng & Solvng Integer Problems Chapter 11 289 The Optonal Stop TSP If we drop the requrement that every stop must be vsted, we then get the optonal stop TSP. Ths mght correspond to a ob sequencng
More informationA Performance Analysis of View Maintenance Techniques for Data Warehouses
A Performance Analyss of Vew Mantenance Technques for Data Warehouses Xng Wang Dell Computer Corporaton Round Roc, Texas Le Gruenwald The nversty of Olahoma School of Computer Scence orman, OK 739 Guangtao
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 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 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 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 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 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 informationSmall pots lump sum payment instruction
For customers Small pots lump sum payment nstructon Please read these notes before completng ths nstructon About ths nstructon Use ths nstructon f you re an ndvdual wth Aegon Retrement Choces Self Invested
More informationCompilers. 3 rd year Spring term. Mick O Donnell: Alfonso Ortega: Topic 5: Semantic analysis
Complers 3 rd year Sprng term Mck O Donnell: mchael.odonnell@uam.es Alfonso Ortega: alfonso.ortega@uam.es Topc 5: Semantc analyss 5.0 Introducton 1 Semantc analyss What s the Semantc Analyser? Set of routnes
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 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 informationRequIn, a tool for fast web traffic inference
RequIn, a tool for fast web traffc nference Olver aul, Jean Etenne Kba GET/INT, LOR Department 9 rue Charles Fourer 90 Evry, France Olver.aul@ntevry.fr, JeanEtenne.Kba@ntevry.fr Abstract As networked
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 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 informationFigure 1. Inventory Level vs. Time  EOQ Problem
IEOR 54 Sprng, 009 rof Leahman otes on Eonom Lot Shedulng and Eonom Rotaton Cyles he Eonom Order Quantty (EOQ) Consder an nventory tem n solaton wth demand rate, holdng ost h per unt per unt tme, and replenshment
More informationwww.olr.ccli.com Introducing Online Reporting Your stepbystep guide to the new online copy report Online Reporting
Onlne Reportng Introducng Onlne Reportng www.olr.ccl.com Your stepbystep gude to the new onlne copy report Important nformaton for all lcence holders No more software to download Reportng as you go...
More informationThe Magnetic Field. Concepts and Principles. Moving Charges. Permanent Magnets
. The Magnetc Feld Concepts and Prncples Movng Charges All charged partcles create electrc felds, and these felds can be detected by other charged partcles resultng n electrc force. However, a completely
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 informationLecture 2: Single Layer Perceptrons Kevin Swingler
Lecture 2: Sngle Layer Perceptrons Kevn Sngler kms@cs.str.ac.uk Recap: McCullochPtts Neuron Ths vastly smplfed model of real neurons s also knon as a Threshold Logc Unt: W 2 A Y 3 n W n. A set of synapses
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 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 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 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 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 informationELM for Exchange version 5.5 Exchange Server Migration
ELM for Exchange verson 5.5 Exchange Server Mgraton Copyrght 06 Lexmark. All rghts reserved. Lexmark s a trademark of Lexmark Internatonal, Inc., regstered n the U.S. and/or other countres. All other trademarks
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 informationHollinger Canadian Publishing Holdings Co. ( HCPH ) proceeding under the Companies Creditors Arrangement Act ( CCAA )
February 17, 2011 Andrew J. Hatnay ahatnay@kmlaw.ca Dear Sr/Madam: Re: Re: Hollnger Canadan Publshng Holdngs Co. ( HCPH ) proceedng under the Companes Credtors Arrangement Act ( CCAA ) Update on CCAA Proceedngs
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 informationLevel Annuities with Payments Less Frequent than Each Interest Period
Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annutymmedate 2 Annutydue Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annutymmedate 2 Annutydue Symoblc approach
More information