CHAPTER 5: MULTIVARIATE METHODS

Size: px
Start display at page:

Download "CHAPTER 5: MULTIVARIATE METHODS"

Transcription

1 CHAPER 5: MULIVARIAE MEHODS

2 Mulvarae Daa 3 Mulple measuremens (sensors) npus/feaures/arbues: -varae N nsances/observaons/eamples Each row s an eample Each column represens a feaure X a b correspons o he a-h arbue of he b-h eample N N N X X X X X X X X X X

3 Mulvarae Parameers 4 Mean Mean: E X E[ X,..., X ] [ E[ X],..., E[ X ]] μ,..., hs s a vecor of componens Is -h componen correspons o mean of he -h feaure For nfne sample (populaon) we use epecaon o represen mean, for fne sample we use average Covarance beween -h an -h arbue s Covarance: Cov X, X E[( X )( X )] E[ X X ] When =, becomes he varance of he -h arbue Correlaon I s normalze beween + an - Covarance mar Is (I,)-h enry s Varance: Cov X, X E[( X ) ] E[ X ] X X E X X Cov, μ μ Correlaon: Corr X, X Cov X E X μx μ

4 Parameer Esmaon N N s s s r N m m s N m : :,...,, : R S m Correlaon mar Covarancemar Samplemean 5

5 Esmaon of Mssng Values 6 Wha o o f ceran nsances have mssng arbues? Ignore hose nsances: no a goo ea f he sample s small Impuaon: Fll n he mssng value by esmang her values Mean mpuaon: Use he mos lkely value (e.g., mean) Average value of he same arbue Average value of same arbue comng from same class Impuaon by regresson: Prec base on oher arbues as a soluon o a regresson problem

6 Mulvarae Normal Dsrbuon 7 X s srbue as a - mensonal normal srbuon ~ N μ, Σ I has wo parameers Mean μ, hs s a vecor wh componens Covarance mar Σ, hs s a mar Is ensy funcon s gven as p ep / / μ μ Σ Σ

7 Mulvarae Normal Dsrbuon 8 Mahalanobs sance: ( μ) ( μ) measures he sance from o μ n erms of (normalzes for fference n varances an correlaons) Wha happens f s eny mar? Bvarae: = p, ep z / z z z z

8 Bvarae Normal 9 Suppose here are wo feaures X=(X, X ) In he pcure, X s represene along as an X s represene along y as Relave varance of X an X eermnes he shape of he normal srbuon Crcular vs ellpcal Whenever covarance, Cov(X,X ) s nonzero, s ncae roaon of he shape Posve covarance means roaon n one recon (say clockwse) an negave covarance means roaon n he oher recon When here are more han wo feaures (>) Same hng as above hols for every par of feaures

9 0

10 Inepenen Inpus: Nave Bayes For a -mensonal ranom varable, s covarance mar s Wha happens o hs mar f all parwse covarances are zero? I becomes a agonal mar Why o we care? Each feaure becomes nepenen of anoher Jon probably srbuon can be wren as prouc of margnal srbuons nvolvng separae nvual feaures If are nepenen, offagonals of are 0, Mahalanobs sance reuces o weghe (by /σ ) Euclean sance: p ep / ( ) ep p ( ) hs s eacly Naïve Bayes assumpon f each class cononal probably (lkelhoo) s moele hs way If varances are also equal, reuces o Euclean sance

11 Properes of agonal mar Deermnan Deermnan of a agonal mar s equal o prouc of s agonal enres Inverse Inverse of a agonal mar s a agonal mar whose -h agonal enry s he one over he -h agonal enry of he orgnal mar Why?

12 An mporan ha we wll use laer 3 Suppose hen where N(, ) an w s a mensonal column vecor w w w... w N( w, w w) E[ w ] w E[ ] w Var( w ) E ( w w ) E ( w w )( w w ) E w ( )( ) w w E ( )( ) w w w

13 Paramerc Classfcaon If p ( C ) ~ N ( μ, ) Dscrmnan funcons C p μ μ Σ Σ ep / / C P C P C p g log log log log log μ Σ μ Σ 4

14 Esmaon of Parameers r r r r N r C P m m m S ˆ C P g ˆ log log m m S S 5 Pluggng n hese esmaes, he scrmnan funcon becomes hs s a quarac form, ha s here s a vecor-mar-vecor mulplcaon Such a scrmnan funcon s calle quarac scrmnan funcon

15 Dfferen S Epanng he quarac form, he scrmnan funcon becomes C P w w C P g log ˆ log where log ˆ log S S S S W W S S S S 0 0 m m m w w m m m 6 quarac erm lnear erm scalar erm You can relae o he quarac formula n -, for eample, a +b+c

16 lkelhoos scrmnan: P (C ) = 0.5 poseror for C 7 Quarac scrmnan funcon gves rse o a quarac ecson bounary

17 Why such a moel can be unrealsc? 8 oal number of parameers for a K class classfcaon problem s O(K ) Mean for each class has parameers Covarance mar (symmerc) for each class has (+)/ snc parameers Fnng nverse of a mar akes me O( 3 ) hs s problemac for large scale applcaons, e.g., say when =50,000 So wha can we o? ry o reuce moel parameers by makng assumpons

18 Approach #: common covarance mar S 9 Share common sample covarance S across all K classes Dscrmnan reuces o whch s a lnear scrmnan because S - s same for all classes g w w so ecson bounary s now lnear Number of parameers s sll O( ) k+(+)/ Inverng covarance mar s sll O( 3 ) C S Pˆ S m S m log C g Pˆ 0 where, w S m w m S m log Pˆ C 0

19 0 Common Covarance Mar S

20 Approach #: Share agonal S When =,.., are nepenen, s agonal, we can wre, p ( C ) = p ( C ) (Nave Bayes assumpon) Dscrmnan funcon has he form m g logpˆ C s Classfy base on weghe Euclean sance (n s uns) o he neares mean, when are pror class probables are equal Snce s s same for all classes Dscrmnan funcon s lnear, hence so s he ecson bounary So wha happene o number of parameers an mar nverson oal number of parameer s O(K) K mean vecors of lengh Share agonal mar has parameers (agonal enres) Mar nverson Can be one n O() me Inverse s a agonal mar an each enry s over orgnal agonal enry

21 Dagonal S varances may be fferen

22 Approach # 3: Share agonal S, equal varances 3 S=σ I, ha s S s a agonal mar, where each agonal enry s same Neares mean classfer: Classfy base on Euclean sance o he neares mean when Each mean can be consere a prooype or emplae an hs s emplae machng Dscrmnan funcon s lnear an so s ecson bounary So wha happene o number of parameers an mar nverson g logpˆ C oal number of parameer s O(K) K mean vecors of lengh s m s m logpˆ C Share agonal mar has parameers (common agonal enry) Mar nverson Can be one n O() me (consan me) Inverse s a agonal mar an each enry s over orgnal agonal enry

23 Dagonal S, equal varances 4 *?

24 Moel Selecon 5 Assumpon Covarance mar No of parameers Share, Hyperspherc S =S=s I Share, As-algne S =S, wh s =0 Share, Hyperellpsoal S =S (+)/ Dfferen, Hyperellpsoal S K (+)/ As we ncrease compley (less resrce S), bas ecreases an varance ncreases Assume smple moels (allow some bas) o conrol varance (regularzaon)

25 6

26 Dscree Feaures 7 Bnary feaures: g f are nepenen (Nave Bayes ) he scrmnan s lnear logp C logpc p logp log p logpc Esmae parameers p p C C p p pˆ r r

27 Dscree Feaures 8 Mulnomal (-of-n ) feaures: n {v, v,..., v n } k z C p v C p p k f are nepenen p g pˆ C z logp logpc k z k k k k r r n p z k k k k

28 Mulvarae regresson 9 In mulvarae lnear regresson numerc oupu r s assume o be wren as a lnear funcon ha s, a weghe sum of several npu varables (arbues),,, r g w, w,..., w 0 g w, w,..., w w w w w 0 0 We can mamze he log-lkelhoo of p(r,,, ) an show ha hs s equvalen o mnmzng E w0, w,..., w X r w 0 w w

29 Mulvarae regresson 30 We can mnmze epresson from las sle by akng paral ervave wh respec o each parameer an seng o zero hese are normal equaons All normal equaons can be compacly wren as n mar noaon X Xw X r where, w w0, w,..., w We can solve by pre-mulplyng boh se wh mar nverse o ge Here w X X X r X s N (+) aa mar w s (+) column vecor r s N column vecor of N oupu values

30 Mulvarae polynomal regresson 3 By he way, we can also efne Mulvarae polynomal moel: Defne new hgher-orer varables z =, z =, z 3 =, z 4 =, z 5 = an use he lnear moel n hs new z space (bass funcons, kernel rck: Chaper 3)

Machine Learning 2nd Edition

Machine Learning 2nd Edition INTRODUCTION TO Lecure Sldes for Machne Learnng nd Edon ETHEM ALPAYDIN, modfed by Leonardo Bobadlla and some pars from hp://www.cs.au.ac.l/~aparzn/machnelearnng/ The MIT Press, 00 alpaydn@boun.edu.r hp://www.cmpe.boun.edu.r/~ehem/mle

More information

CS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 4

CS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 4 CS434a/54a: Paern Recognon Prof. Olga Veksler Lecure 4 Oulne Normal Random Varable Properes Dscrmnan funcons Why Normal Random Varables? Analycally racable Works well when observaon comes form a corruped

More information

CHAPTER 10: LINEAR DISCRIMINATION

CHAPTER 10: LINEAR DISCRIMINATION CHAPER : LINEAR DISCRIMINAION Dscrmnan-based Classfcaon 3 In classfcaon h K classes (C,C,, C k ) We defned dscrmnan funcon g j (), j=,,,k hen gven an es eample, e chose (predced) s class label as C f g

More information

Solution in semi infinite diffusion couples (error function analysis)

Solution in semi infinite diffusion couples (error function analysis) Soluon n sem nfne dffuson couples (error funcon analyss) Le us consder now he sem nfne dffuson couple of wo blocks wh concenraon of and I means ha, n a A- bnary sysem, s bondng beween wo blocks made of

More information

Lecture VI Regression

Lecture VI Regression Lecure VI Regresson (Lnear Mehods for Regresson) Conens: Lnear Mehods for Regresson Leas Squares, Gauss Markov heorem Recursve Leas Squares Lecure VI: MLSC - Dr. Sehu Vjayakumar Lnear Regresson Model M

More information

Advanced Machine Learning & Perception

Advanced Machine Learning & Perception Advanced Machne Learnng & Percepon Insrucor: Tony Jebara SVM Feaure & Kernel Selecon SVM Eensons Feaure Selecon (Flerng and Wrappng) SVM Feaure Selecon SVM Kernel Selecon SVM Eensons Classfcaon Feaure/Kernel

More information

Lecture 6: Learning for Control (Generalised Linear Regression)

Lecture 6: Learning for Control (Generalised Linear Regression) Lecure 6: Learnng for Conrol (Generalsed Lnear Regresson) Conens: Lnear Mehods for Regresson Leas Squares, Gauss Markov heorem Recursve Leas Squares Lecure 6: RLSC - Prof. Sehu Vjayakumar Lnear Regresson

More information

Normal Random Variable and its discriminant functions

Normal Random Variable and its discriminant functions Noral Rando Varable and s dscrnan funcons Oulne Noral Rando Varable Properes Dscrnan funcons Why Noral Rando Varables? Analycally racable Works well when observaon coes for a corruped snle prooype 3 The

More information

( ) [ ] MAP Decision Rule

( ) [ ] MAP Decision Rule Announcemens Bayes Decson Theory wh Normal Dsrbuons HW0 due oday HW o be assgned soon Proec descrpon posed Bomercs CSE 90 Lecure 4 CSE90, Sprng 04 CSE90, Sprng 04 Key Probables 4 ω class label X feaure

More information

Bayes rule for a classification problem INF Discriminant functions for the normal density. Euclidean distance. Mahalanobis distance

Bayes rule for a classification problem INF Discriminant functions for the normal density. Euclidean distance. Mahalanobis distance INF 43 3.. Repeon Anne Solberg (anne@f.uo.no Bayes rule for a classfcaon problem Suppose we have J, =,...J classes. s he class label for a pxel, and x s he observed feaure vecor. We can use Bayes rule

More information

In the complete model, these slopes are ANALYSIS OF VARIANCE FOR THE COMPLETE TWO-WAY MODEL. (! i+1 -! i ) + [(!") i+1,q - [(!

In the complete model, these slopes are ANALYSIS OF VARIANCE FOR THE COMPLETE TWO-WAY MODEL. (! i+1 -! i ) + [(!) i+1,q - [(! ANALYSIS OF VARIANCE FOR THE COMPLETE TWO-WAY MODEL The frs hng o es n wo-way ANOVA: Is here neracon? "No neracon" means: The man effecs model would f. Ths n urn means: In he neracon plo (wh A on he horzonal

More information

Ordinary Differential Equations in Neuroscience with Matlab examples. Aim 1- Gain understanding of how to set up and solve ODE s

Ordinary Differential Equations in Neuroscience with Matlab examples. Aim 1- Gain understanding of how to set up and solve ODE s Ordnary Dfferenal Equaons n Neuroscence wh Malab eamples. Am - Gan undersandng of how o se up and solve ODE s Am Undersand how o se up an solve a smple eample of he Hebb rule n D Our goal a end of class

More information

John Geweke a and Gianni Amisano b a Departments of Economics and Statistics, University of Iowa, USA b European Central Bank, Frankfurt, Germany

John Geweke a and Gianni Amisano b a Departments of Economics and Statistics, University of Iowa, USA b European Central Bank, Frankfurt, Germany Herarchcal Markov Normal Mxure models wh Applcaons o Fnancal Asse Reurns Appendx: Proofs of Theorems and Condonal Poseror Dsrbuons John Geweke a and Gann Amsano b a Deparmens of Economcs and Sascs, Unversy

More information

( t) Outline of program: BGC1: Survival and event history analysis Oslo, March-May Recapitulation. The additive regression model

( t) Outline of program: BGC1: Survival and event history analysis Oslo, March-May Recapitulation. The additive regression model BGC1: Survval and even hsory analyss Oslo, March-May 212 Monday May 7h and Tuesday May 8h The addve regresson model Ørnulf Borgan Deparmen of Mahemacs Unversy of Oslo Oulne of program: Recapulaon Counng

More information

Machine Learning Linear Regression

Machine Learning Linear Regression Machne Learnng Lnear Regresson Lesson 3 Lnear Regresson Bascs of Regresson Leas Squares esmaon Polynomal Regresson Bass funcons Regresson model Regularzed Regresson Sascal Regresson Mamum Lkelhood (ML)

More information

Lecture 11 SVM cont

Lecture 11 SVM cont Lecure SVM con. 0 008 Wha we have done so far We have esalshed ha we wan o fnd a lnear decson oundary whose margn s he larges We know how o measure he margn of a lnear decson oundary Tha s: he mnmum geomerc

More information

Clustering with Gaussian Mixtures

Clustering with Gaussian Mixtures Noe o oher eachers and users of hese sldes. Andrew would be delghed f you found hs source maeral useful n gvng your own lecures. Feel free o use hese sldes verbam, or o modfy hem o f your own needs. PowerPon

More information

How about the more general "linear" scalar functions of scalars (i.e., a 1st degree polynomial of the following form with a constant term )?

How about the more general linear scalar functions of scalars (i.e., a 1st degree polynomial of the following form with a constant term )? lmcd Lnear ransformaon of a vecor he deas presened here are que general hey go beyond he radonal mar-vecor ype seen n lnear algebra Furhermore, hey do no deal wh bass and are equally vald for any se of

More information

Department of Economics University of Toronto

Department of Economics University of Toronto Deparmen of Economcs Unversy of Torono ECO408F M.A. Economercs Lecure Noes on Heeroskedascy Heeroskedascy o Ths lecure nvolves lookng a modfcaons we need o make o deal wh he regresson model when some of

More information

Notes on the stability of dynamic systems and the use of Eigen Values.

Notes on the stability of dynamic systems and the use of Eigen Values. Noes on he sabl of dnamc ssems and he use of Egen Values. Source: Macro II course noes, Dr. Davd Bessler s Tme Seres course noes, zarads (999) Ineremporal Macroeconomcs chaper 4 & Techncal ppend, and Hamlon

More information

Pattern Classification (III) & Pattern Verification

Pattern Classification (III) & Pattern Verification Preare by Prof. Hu Jang CSE638 --4 CSE638 3. Seech & Language Processng o.5 Paern Classfcaon III & Paern Verfcaon Prof. Hu Jang Dearmen of Comuer Scence an Engneerng York Unversy Moel Parameer Esmaon Maxmum

More information

F-Tests and Analysis of Variance (ANOVA) in the Simple Linear Regression Model. 1. Introduction

F-Tests and Analysis of Variance (ANOVA) in the Simple Linear Regression Model. 1. Introduction ECOOMICS 35* -- OTE 9 ECO 35* -- OTE 9 F-Tess and Analyss of Varance (AOVA n he Smple Lnear Regresson Model Inroducon The smple lnear regresson model s gven by he followng populaon regresson equaon, or

More information

Clustering (Bishop ch 9)

Clustering (Bishop ch 9) Cluserng (Bshop ch 9) Reference: Daa Mnng by Margare Dunham (a slde source) 1 Cluserng Cluserng s unsupervsed learnng, here are no class labels Wan o fnd groups of smlar nsances Ofen use a dsance measure

More information

Panel Data Regression Models

Panel Data Regression Models Panel Daa Regresson Models Wha s Panel Daa? () Mulple dmensoned Dmensons, e.g., cross-secon and me node-o-node (c) Pongsa Pornchawseskul, Faculy of Economcs, Chulalongkorn Unversy (c) Pongsa Pornchawseskul,

More information

CHAPTER 2: Supervised Learning

CHAPTER 2: Supervised Learning HATER 2: Supervsed Learnng Learnng a lass from Eamples lass of a famly car redcon: Is car a famly car? Knowledge eracon: Wha do people epec from a famly car? Oupu: osve (+) and negave ( ) eamples Inpu

More information

TSS = SST + SSE An orthogonal partition of the total SS

TSS = SST + SSE An orthogonal partition of the total SS ANOVA: Topc 4. Orhogonal conrass [ST&D p. 183] H 0 : µ 1 = µ =... = µ H 1 : The mean of a leas one reamen group s dfferen To es hs hypohess, a basc ANOVA allocaes he varaon among reamen means (SST) equally

More information

Dynamic Team Decision Theory. EECS 558 Project Shrutivandana Sharma and David Shuman December 10, 2005

Dynamic Team Decision Theory. EECS 558 Project Shrutivandana Sharma and David Shuman December 10, 2005 Dynamc Team Decson Theory EECS 558 Proec Shruvandana Sharma and Davd Shuman December 0, 005 Oulne Inroducon o Team Decson Theory Decomposon of he Dynamc Team Decson Problem Equvalence of Sac and Dynamc

More information

Chapter 6: AC Circuits

Chapter 6: AC Circuits Chaper 6: AC Crcus Chaper 6: Oulne Phasors and he AC Seady Sae AC Crcus A sable, lnear crcu operang n he seady sae wh snusodal excaon (.e., snusodal seady sae. Complee response forced response naural response.

More information

CH.3. COMPATIBILITY EQUATIONS. Continuum Mechanics Course (MMC) - ETSECCPB - UPC

CH.3. COMPATIBILITY EQUATIONS. Continuum Mechanics Course (MMC) - ETSECCPB - UPC CH.3. COMPATIBILITY EQUATIONS Connuum Mechancs Course (MMC) - ETSECCPB - UPC Overvew Compably Condons Compably Equaons of a Poenal Vecor Feld Compably Condons for Infnesmal Srans Inegraon of he Infnesmal

More information

CHAPTER 7: CLUSTERING

CHAPTER 7: CLUSTERING CHAPTER 7: CLUSTERING Semparamerc Densy Esmaon 3 Paramerc: Assume a snge mode for p ( C ) (Chapers 4 and 5) Semparamerc: p ( C ) s a mure of denses Mupe possbe epanaons/prooypes: Dfferen handwrng syes,

More information

( ) () we define the interaction representation by the unitary transformation () = ()

( ) () we define the interaction representation by the unitary transformation () = () Hgher Order Perurbaon Theory Mchael Fowler 3/7/6 The neracon Represenaon Recall ha n he frs par of hs course sequence, we dscussed he chrödnger and Hesenberg represenaons of quanum mechancs here n he chrödnger

More information

An introduction to Support Vector Machine

An introduction to Support Vector Machine An nroducon o Suppor Vecor Machne 報告者 : 黃立德 References: Smon Haykn, "Neural Neworks: a comprehensve foundaon, second edon, 999, Chaper 2,6 Nello Chrsann, John Shawe-Tayer, An Inroducon o Suppor Vecor Machnes,

More information

Econ107 Applied Econometrics Topic 5: Specification: Choosing Independent Variables (Studenmund, Chapter 6)

Econ107 Applied Econometrics Topic 5: Specification: Choosing Independent Variables (Studenmund, Chapter 6) Econ7 Appled Economercs Topc 5: Specfcaon: Choosng Independen Varables (Sudenmund, Chaper 6 Specfcaon errors ha we wll deal wh: wrong ndependen varable; wrong funconal form. Ths lecure deals wh wrong ndependen

More information

Appendix H: Rarefaction and extrapolation of Hill numbers for incidence data

Appendix H: Rarefaction and extrapolation of Hill numbers for incidence data Anne Chao Ncholas J Goell C seh lzabeh L ander K Ma Rober K Colwell and Aaron M llson 03 Rarefacon and erapolaon wh ll numbers: a framewor for samplng and esmaon n speces dversy sudes cology Monographs

More information

Anomaly Detection. Lecture Notes for Chapter 9. Introduction to Data Mining, 2 nd Edition by Tan, Steinbach, Karpatne, Kumar

Anomaly Detection. Lecture Notes for Chapter 9. Introduction to Data Mining, 2 nd Edition by Tan, Steinbach, Karpatne, Kumar Anomaly eecon Lecure Noes for Chaper 9 Inroducon o aa Mnng, 2 nd Edon by Tan, Senbach, Karpane, Kumar 2/14/18 Inroducon o aa Mnng, 2nd Edon 1 Anomaly/Ouler eecon Wha are anomales/oulers? The se of daa

More information

Scattering at an Interface: Oblique Incidence

Scattering at an Interface: Oblique Incidence Course Insrucor Dr. Raymond C. Rumpf Offce: A 337 Phone: (915) 747 6958 E Mal: rcrumpf@uep.edu EE 4347 Appled Elecromagnecs Topc 3g Scaerng a an Inerface: Oblque Incdence Scaerng These Oblque noes may

More information

Robustness Experiments with Two Variance Components

Robustness Experiments with Two Variance Components Naonal Insue of Sandards and Technology (NIST) Informaon Technology Laboraory (ITL) Sascal Engneerng Dvson (SED) Robusness Expermens wh Two Varance Componens by Ana Ivelsse Avlés avles@ns.gov Conference

More information

Let s treat the problem of the response of a system to an applied external force. Again,

Let s treat the problem of the response of a system to an applied external force. Again, Page 33 QUANTUM LNEAR RESPONSE FUNCTON Le s rea he problem of he response of a sysem o an appled exernal force. Agan, H() H f () A H + V () Exernal agen acng on nernal varable Hamlonan for equlbrum sysem

More information

January Examinations 2012

January Examinations 2012 Page of 5 EC79 January Examnaons No. of Pages: 5 No. of Quesons: 8 Subjec ECONOMICS (POSTGRADUATE) Tle of Paper EC79 QUANTITATIVE METHODS FOR BUSINESS AND FINANCE Tme Allowed Two Hours ( hours) Insrucons

More information

V.Abramov - FURTHER ANALYSIS OF CONFIDENCE INTERVALS FOR LARGE CLIENT/SERVER COMPUTER NETWORKS

V.Abramov - FURTHER ANALYSIS OF CONFIDENCE INTERVALS FOR LARGE CLIENT/SERVER COMPUTER NETWORKS R&RATA # Vol.) 8, March FURTHER AALYSIS OF COFIDECE ITERVALS FOR LARGE CLIET/SERVER COMPUTER ETWORKS Vyacheslav Abramov School of Mahemacal Scences, Monash Unversy, Buldng 8, Level 4, Clayon Campus, Wellngon

More information

Should Exact Index Numbers have Standard Errors? Theory and Application to Asian Growth

Should Exact Index Numbers have Standard Errors? Theory and Application to Asian Growth Should Exac Index umbers have Sandard Errors? Theory and Applcaon o Asan Growh Rober C. Feensra Marshall B. Rensdorf ovember 003 Proof of Proposon APPEDIX () Frs, we wll derve he convenonal Sao-Vara prce

More information

CHAPTER FOUR REPEATED MEASURES IN TOXICITY TESTING

CHAPTER FOUR REPEATED MEASURES IN TOXICITY TESTING CHAPTER FOUR REPEATED MEASURES IN TOXICITY TESTING 4. Inroducon The repeaed measures sudy s a very commonly used expermenal desgn n oxcy esng because no only allows one o nvesgae he effecs of he oxcans,

More information

Introduction to Compact Dynamical Modeling. III.1 Reducing Linear Time Invariant Systems. Luca Daniel Massachusetts Institute of Technology

Introduction to Compact Dynamical Modeling. III.1 Reducing Linear Time Invariant Systems. Luca Daniel Massachusetts Institute of Technology SF & IH Inroducon o Compac Dynamcal Modelng III. Reducng Lnear me Invaran Sysems Luca Danel Massachuses Insue of echnology Course Oulne Quck Sneak Prevew I. Assemblng Models from Physcal Problems II. Smulang

More information

THE PREDICTION OF COMPETITIVE ENVIRONMENT IN BUSINESS

THE PREDICTION OF COMPETITIVE ENVIRONMENT IN BUSINESS THE PREICTION OF COMPETITIVE ENVIRONMENT IN BUSINESS INTROUCTION The wo dmensonal paral dfferenal equaons of second order can be used for he smulaon of compeve envronmen n busness The arcle presens he

More information

Graduate Macroeconomics 2 Problem set 5. - Solutions

Graduate Macroeconomics 2 Problem set 5. - Solutions Graduae Macroeconomcs 2 Problem se. - Soluons Queson 1 To answer hs queson we need he frms frs order condons and he equaon ha deermnes he number of frms n equlbrum. The frms frs order condons are: F K

More information

. The geometric multiplicity is dim[ker( λi. number of linearly independent eigenvectors associated with this eigenvalue.

. The geometric multiplicity is dim[ker( λi. number of linearly independent eigenvectors associated with this eigenvalue. Lnear Algebra Lecure # Noes We connue wh he dscusson of egenvalues, egenvecors, and dagonalzably of marces We wan o know, n parcular wha condons wll assure ha a marx can be dagonalzed and wha he obsrucons

More information

Motion in Two Dimensions

Motion in Two Dimensions Phys 1 Chaper 4 Moon n Two Dmensons adzyubenko@csub.edu hp://www.csub.edu/~adzyubenko 005, 014 A. Dzyubenko 004 Brooks/Cole 1 Dsplacemen as a Vecor The poson of an objec s descrbed by s poson ecor, r The

More information

The Gaussian classifier. Nuno Vasconcelos ECE Department, UCSD

The Gaussian classifier. Nuno Vasconcelos ECE Department, UCSD he Gaussan classfer Nuno Vasconcelos ECE Department, UCSD Bayesan decson theory recall that we have state of the world X observatons g decson functon L[g,y] loss of predctng y wth g Bayes decson rule s

More information

[Link to MIT-Lab 6P.1 goes here.] After completing the lab, fill in the following blanks: Numerical. Simulation s Calculations

[Link to MIT-Lab 6P.1 goes here.] After completing the lab, fill in the following blanks: Numerical. Simulation s Calculations Chaper 6: Ordnary Leas Squares Esmaon Procedure he Properes Chaper 6 Oulne Cln s Assgnmen: Assess he Effec of Sudyng on Quz Scores Revew o Regresson Model o Ordnary Leas Squares () Esmaon Procedure o he

More information

Math 128b Project. Jude Yuen

Math 128b Project. Jude Yuen Mah 8b Proec Jude Yuen . Inroducon Le { Z } be a sequence of observed ndependen vecor varables. If he elemens of Z have a on normal dsrbuon hen { Z } has a mean vecor Z and a varancecovarance marx z. Geomercally

More information

Outline. Probabilistic Model Learning. Probabilistic Model Learning. Probabilistic Model for Time-series Data: Hidden Markov Model

Outline. Probabilistic Model Learning. Probabilistic Model Learning. Probabilistic Model for Time-series Data: Hidden Markov Model Probablsc Model for Tme-seres Daa: Hdden Markov Model Hrosh Mamsuka Bonformacs Cener Kyoo Unversy Oulne Three Problems for probablsc models n machne learnng. Compung lkelhood 2. Learnng 3. Parsng (predcon

More information

Variants of Pegasos. December 11, 2009

Variants of Pegasos. December 11, 2009 Inroducon Varans of Pegasos SooWoong Ryu bshboy@sanford.edu December, 009 Youngsoo Cho yc344@sanford.edu Developng a new SVM algorhm s ongong research opc. Among many exng SVM algorhms, we wll focus on

More information

ROC Curves for Multivariate Biometric Matching Models

ROC Curves for Multivariate Biometric Matching Models ROC Curves for Mulvarae Bomerc Machng Moels Sung-Hyuk Cha an Charles C. Tapper Absrac The bomerc machng problem s a wo class whn or beween ) classfcaon problem where wo ypes of errors an ) occur. Whle

More information

Robust and Accurate Cancer Classification with Gene Expression Profiling

Robust and Accurate Cancer Classification with Gene Expression Profiling Robus and Accurae Cancer Classfcaon wh Gene Expresson Proflng (Compuaonal ysems Bology, 2005) Auhor: Hafeng L, Keshu Zhang, ao Jang Oulne Background LDA (lnear dscrmnan analyss) and small sample sze problem

More information

[ ] 2. [ ]3 + (Δx i + Δx i 1 ) / 2. Δx i-1 Δx i Δx i+1. TPG4160 Reservoir Simulation 2018 Lecture note 3. page 1 of 5

[ ] 2. [ ]3 + (Δx i + Δx i 1 ) / 2. Δx i-1 Δx i Δx i+1. TPG4160 Reservoir Simulation 2018 Lecture note 3. page 1 of 5 TPG460 Reservor Smulaon 08 page of 5 DISCRETIZATIO OF THE FOW EQUATIOS As we already have seen, fne dfference appromaons of he paral dervaves appearng n he flow equaons may be obaned from Taylor seres

More information

PHYS 1443 Section 001 Lecture #4

PHYS 1443 Section 001 Lecture #4 PHYS 1443 Secon 001 Lecure #4 Monda, June 5, 006 Moon n Two Dmensons Moon under consan acceleraon Projecle Moon Mamum ranges and heghs Reerence Frames and relae moon Newon s Laws o Moon Force Newon s Law

More information

CS 536: Machine Learning. Nonparametric Density Estimation Unsupervised Learning - Clustering

CS 536: Machine Learning. Nonparametric Density Estimation Unsupervised Learning - Clustering CS 536: Machne Learnng Nonparamerc Densy Esmaon Unsupervsed Learnng - Cluserng Fall 2005 Ahmed Elgammal Dep of Compuer Scence Rugers Unversy CS 536 Densy Esmaon - Cluserng - 1 Oulnes Densy esmaon Nonparamerc

More information

. The geometric multiplicity is dim[ker( λi. A )], i.e. the number of linearly independent eigenvectors associated with this eigenvalue.

. The geometric multiplicity is dim[ker( λi. A )], i.e. the number of linearly independent eigenvectors associated with this eigenvalue. Mah E-b Lecure #0 Noes We connue wh he dscusson of egenvalues, egenvecors, and dagonalzably of marces We wan o know, n parcular wha condons wll assure ha a marx can be dagonalzed and wha he obsrucons are

More information

CHAPTER 3: INVERSE METHODS BASED ON LENGTH. 3.1 Introduction. 3.2 Data Error and Model Parameter Vectors

CHAPTER 3: INVERSE METHODS BASED ON LENGTH. 3.1 Introduction. 3.2 Data Error and Model Parameter Vectors eoscences 567: CHAPER 3 (RR/Z) CHAPER 3: IVERSE EHODS BASED O EH 3. Inroucon s caper s concerne w nverse eos base on e leng of varous vecors a arse n a ypcal proble. e wo os coon vecors concerne are e

More information

Fall 2010 Graduate Course on Dynamic Learning

Fall 2010 Graduate Course on Dynamic Learning Fall 200 Graduae Course on Dynamc Learnng Chaper 4: Parcle Flers Sepember 27, 200 Byoung-Tak Zhang School of Compuer Scence and Engneerng & Cognve Scence and Bran Scence Programs Seoul aonal Unversy hp://b.snu.ac.kr/~bzhang/

More information

Delay-Range-Dependent Stability Analysis for Continuous Linear System with Interval Delay

Delay-Range-Dependent Stability Analysis for Continuous Linear System with Interval Delay Inernaonal Journal of Emergng Engneerng esearch an echnology Volume 3, Issue 8, Augus 05, PP 70-76 ISSN 349-4395 (Prn) & ISSN 349-4409 (Onlne) Delay-ange-Depenen Sably Analyss for Connuous Lnear Sysem

More information

Lesson 2 Transmission Lines Fundamentals

Lesson 2 Transmission Lines Fundamentals Lesson Transmsson Lnes Funamenals 楊尚達 Shang-Da Yang Insue of Phooncs Technologes Deparmen of Elecrcal Engneerng Naonal Tsng Hua Unersy Tawan Sec. -1 Inroucon 1. Why o scuss TX lnes srbue crcus?. Crera

More information

New M-Estimator Objective Function. in Simultaneous Equations Model. (A Comparative Study)

New M-Estimator Objective Function. in Simultaneous Equations Model. (A Comparative Study) Inernaonal Mahemacal Forum, Vol. 8, 3, no., 7 - HIKARI Ld, www.m-hkar.com hp://dx.do.org/.988/mf.3.3488 New M-Esmaor Objecve Funcon n Smulaneous Equaons Model (A Comparave Sudy) Ahmed H. Youssef Professor

More information

Advanced time-series analysis (University of Lund, Economic History Department)

Advanced time-series analysis (University of Lund, Economic History Department) Advanced me-seres analss (Unvers of Lund, Economc Hsor Dearmen) 3 Jan-3 Februar and 6-3 March Lecure 4 Economerc echnues for saonar seres : Unvarae sochasc models wh Box- Jenns mehodolog, smle forecasng

More information

Volatility Interpolation

Volatility Interpolation Volaly Inerpolaon Prelmnary Verson March 00 Jesper Andreasen and Bran Huge Danse Mares, Copenhagen wan.daddy@danseban.com brno@danseban.com Elecronc copy avalable a: hp://ssrn.com/absrac=69497 Inro Local

More information

OP = OO' + Ut + Vn + Wb. Material We Will Cover Today. Computer Vision Lecture 3. Multi-view Geometry I. Amnon Shashua

OP = OO' + Ut + Vn + Wb. Material We Will Cover Today. Computer Vision Lecture 3. Multi-view Geometry I. Amnon Shashua Comuer Vson 27 Lecure 3 Mul-vew Geomer I Amnon Shashua Maeral We Wll Cover oa he srucure of 3D->2D rojecon mar omograh Marces A rmer on rojecve geomer of he lane Eolar Geomer an Funamenal Mar ebrew Unvers

More information

A First Guide to Hypothesis Testing in Linear Regression Models. A Generic Linear Regression Model: Scalar Formulation

A First Guide to Hypothesis Testing in Linear Regression Models. A Generic Linear Regression Model: Scalar Formulation ECON 5* -- A rs Gude o Hypoess Tesng MG Abbo A rs Gude o Hypoess Tesng n Lnear Regresson Models A Generc Lnear Regresson Model: Scalar mulaon e - populaon ( sample observaon, e scalar fmulaon of e PRE

More information

UNIVERSITAT AUTÒNOMA DE BARCELONA MARCH 2017 EXAMINATION

UNIVERSITAT AUTÒNOMA DE BARCELONA MARCH 2017 EXAMINATION INTERNATIONAL TRADE T. J. KEHOE UNIVERSITAT AUTÒNOMA DE BARCELONA MARCH 27 EXAMINATION Please answer wo of he hree quesons. You can consul class noes, workng papers, and arcles whle you are workng on he

More information

Discrete Markov Process. Introduction. Example: Balls and Urns. Stochastic Automaton. INTRODUCTION TO Machine Learning 3rd Edition

Discrete Markov Process. Introduction. Example: Balls and Urns. Stochastic Automaton. INTRODUCTION TO Machine Learning 3rd Edition EHEM ALPAYDI he MI Press, 04 Lecure Sldes for IRODUCIO O Machne Learnng 3rd Edon alpaydn@boun.edu.r hp://www.cmpe.boun.edu.r/~ehem/ml3e Sldes from exboo resource page. Slghly eded and wh addonal examples

More information

Lecture 18: The Laplace Transform (See Sections and 14.7 in Boas)

Lecture 18: The Laplace Transform (See Sections and 14.7 in Boas) Lecure 8: The Lalace Transform (See Secons 88- and 47 n Boas) Recall ha our bg-cure goal s he analyss of he dfferenal equaon, ax bx cx F, where we emloy varous exansons for he drvng funcon F deendng on

More information

Improvement in Estimating Population Mean using Two Auxiliary Variables in Two-Phase Sampling

Improvement in Estimating Population Mean using Two Auxiliary Variables in Two-Phase Sampling Rajesh ngh Deparmen of ascs, Banaras Hndu Unvers(U.P.), Inda Pankaj Chauhan, Nrmala awan chool of ascs, DAVV, Indore (M.P.), Inda Florenn marandache Deparmen of Mahemacs, Unvers of New Meco, Gallup, UA

More information

Revision: June 12, E Main Suite D Pullman, WA (509) Voice and Fax

Revision: June 12, E Main Suite D Pullman, WA (509) Voice and Fax .: apacors Reson: June, 5 E Man Sue D Pullman, WA 9963 59 334 636 Voce an Fax Oerew We begn our suy of energy sorage elemens wh a scusson of capacors. apacors, lke ressors, are passe wo-ermnal crcu elemens.

More information

Lecture 2 L n i e n a e r a M od o e d l e s

Lecture 2 L n i e n a e r a M od o e d l e s Lecure Lnear Models Las lecure You have learned abou ha s machne learnng Supervsed learnng Unsupervsed learnng Renforcemen learnng You have seen an eample learnng problem and he general process ha one

More information

CHAPTER 10: LINEAR DISCRIMINATION

CHAPTER 10: LINEAR DISCRIMINATION HAPER : LINEAR DISRIMINAION Dscmnan-based lassfcaon 3 In classfcaon h K classes ( k ) We defned dsmnan funcon g () = K hen gven an es eample e chose (pedced) s class label as f g () as he mamum among g

More information

GMM parameter estimation. Xiaoye Lu CMPS290c Final Project

GMM parameter estimation. Xiaoye Lu CMPS290c Final Project GMM paraeer esaon Xaoye Lu M290c Fnal rojec GMM nroducon Gaussan ure Model obnaon of several gaussan coponens Noaon: For each Gaussan dsrbuon:, s he ean and covarance ar. A GMM h ures(coponens): p ( 2π

More information

J i-1 i. J i i+1. Numerical integration of the diffusion equation (I) Finite difference method. Spatial Discretization. Internal nodes.

J i-1 i. J i i+1. Numerical integration of the diffusion equation (I) Finite difference method. Spatial Discretization. Internal nodes. umercal negraon of he dffuson equaon (I) Fne dfference mehod. Spaal screaon. Inernal nodes. R L V For hermal conducon le s dscree he spaal doman no small fne spans, =,,: Balance of parcles for an nernal

More information

COMPUTER SCIENCE 349A SAMPLE EXAM QUESTIONS WITH SOLUTIONS PARTS 1, 2

COMPUTER SCIENCE 349A SAMPLE EXAM QUESTIONS WITH SOLUTIONS PARTS 1, 2 COMPUTE SCIENCE 49A SAMPLE EXAM QUESTIONS WITH SOLUTIONS PATS, PAT.. a Dene he erm ll-ondoned problem. b Gve an eample o a polynomal ha has ll-ondoned zeros.. Consder evaluaon o anh, where e e anh. e e

More information

by Lauren DeDieu Advisor: George Chen

by Lauren DeDieu Advisor: George Chen b Laren DeDe Advsor: George Chen Are one of he mos powerfl mehods o nmercall solve me dependen paral dfferenal eqaons PDE wh some knd of snglar shock waves & blow-p problems. Fed nmber of mesh pons Moves

More information

Part II CONTINUOUS TIME STOCHASTIC PROCESSES

Part II CONTINUOUS TIME STOCHASTIC PROCESSES Par II CONTINUOUS TIME STOCHASTIC PROCESSES 4 Chaper 4 For an advanced analyss of he properes of he Wener process, see: Revus D and Yor M: Connuous marngales and Brownan Moon Karazas I and Shreve S E:

More information

Chapter 4. Neural Networks Based on Competition

Chapter 4. Neural Networks Based on Competition Chaper 4. Neural Neworks Based on Compeon Compeon s mporan for NN Compeon beween neurons has been observed n bologcal nerve sysems Compeon s mporan n solvng many problems To classfy an npu paern _1 no

More information

HEAT CONDUCTION PROBLEM IN A TWO-LAYERED HOLLOW CYLINDER BY USING THE GREEN S FUNCTION METHOD

HEAT CONDUCTION PROBLEM IN A TWO-LAYERED HOLLOW CYLINDER BY USING THE GREEN S FUNCTION METHOD Journal of Appled Mahemacs and Compuaonal Mechancs 3, (), 45-5 HEAT CONDUCTION PROBLEM IN A TWO-LAYERED HOLLOW CYLINDER BY USING THE GREEN S FUNCTION METHOD Sansław Kukla, Urszula Sedlecka Insue of Mahemacs,

More information

Learning Objectives. Self Organization Map. Hamming Distance(1/5) Introduction. Hamming Distance(3/5) Hamming Distance(2/5) 15/04/2015

Learning Objectives. Self Organization Map. Hamming Distance(1/5) Introduction. Hamming Distance(3/5) Hamming Distance(2/5) 15/04/2015 /4/ Learnng Objecves Self Organzaon Map Learnng whou Exaples. Inroducon. MAXNET 3. Cluserng 4. Feaure Map. Self-organzng Feaure Map 6. Concluson 38 Inroducon. Learnng whou exaples. Daa are npu o he syse

More information

Chapter Lagrangian Interpolation

Chapter Lagrangian Interpolation Chaper 5.4 agrangan Inerpolaon Afer readng hs chaper you should be able o:. dere agrangan mehod of nerpolaon. sole problems usng agrangan mehod of nerpolaon and. use agrangan nerpolans o fnd deraes and

More information

Fitting a Conditional Linear Gaussian Distribution

Fitting a Conditional Linear Gaussian Distribution Fng a Condonal Lnear Gaussan Dsrbuon Kevn P. Murphy 28 Ocober 1998 Revsed 29 January 2003 1 Inroducon We consder he problem of fndng he maxmum lkelhood ML esmaes of he parameers of a condonal Gaussan varable

More information

Chapter 8 Dynamic Models

Chapter 8 Dynamic Models Chaper 8 Dnamc odels 8. Inroducon 8. Seral correlaon models 8.3 Cross-seconal correlaons and me-seres crosssecon models 8.4 me-varng coeffcens 8.5 Kalman fler approach 8. Inroducon When s mporan o consder

More information

Today s focus. Bayes rule explained INF Multivariate classification Anne Solberg

Today s focus. Bayes rule explained INF Multivariate classification Anne Solberg Toay focu INF 4300 006 Mulvarae clafcaon Anne Solberg anne@fuono From a -menonal feaure vecor =[, ] T Gven K fferen clae k=, K Compue he probably ha belong o cla k Pk = p kpk/con How houl he mulvarae eny

More information

SOME NOISELESS CODING THEOREMS OF INACCURACY MEASURE OF ORDER α AND TYPE β

SOME NOISELESS CODING THEOREMS OF INACCURACY MEASURE OF ORDER α AND TYPE β SARAJEVO JOURNAL OF MATHEMATICS Vol.3 (15) (2007), 137 143 SOME NOISELESS CODING THEOREMS OF INACCURACY MEASURE OF ORDER α AND TYPE β M. A. K. BAIG AND RAYEES AHMAD DAR Absrac. In hs paper, we propose

More information

A HIERARCHICAL KALMAN FILTER

A HIERARCHICAL KALMAN FILTER A HIERARCHICAL KALMAN FILER Greg aylor aylor Fry Consulng Acuares Level 8, 3 Clarence Sree Sydney NSW Ausrala Professoral Assocae, Cenre for Acuaral Sudes Faculy of Economcs and Commerce Unversy of Melbourne

More information

Improvement in Estimating Population Mean using Two Auxiliary Variables in Two-Phase Sampling

Improvement in Estimating Population Mean using Two Auxiliary Variables in Two-Phase Sampling Improvemen n Esmang Populaon Mean usng Two Auxlar Varables n Two-Phase amplng Rajesh ngh Deparmen of ascs, Banaras Hndu Unvers(U.P.), Inda (rsnghsa@ahoo.com) Pankaj Chauhan and Nrmala awan chool of ascs,

More information

Supervised Learning in Multilayer Networks

Supervised Learning in Multilayer Networks Copyrgh Cambrdge Unversy Press 23. On-screen vewng permed. Prnng no permed. hp://www.cambrdge.org/521642981 You can buy hs book for 3 pounds or $5. See hp://www.nference.phy.cam.ac.uk/mackay/la/ for lnks.

More information

Displacement, Velocity, and Acceleration. (WHERE and WHEN?)

Displacement, Velocity, and Acceleration. (WHERE and WHEN?) Dsplacemen, Velocy, and Acceleraon (WHERE and WHEN?) Mah resources Append A n your book! Symbols and meanng Algebra Geomery (olumes, ec.) Trgonomery Append A Logarhms Remnder You wll do well n hs class

More information

Filtrage particulaire et suivi multi-pistes Carine Hue Jean-Pierre Le Cadre and Patrick Pérez

Filtrage particulaire et suivi multi-pistes Carine Hue Jean-Pierre Le Cadre and Patrick Pérez Chaînes de Markov cachées e flrage parculare 2-22 anver 2002 Flrage parculare e suv mul-pses Carne Hue Jean-Perre Le Cadre and Parck Pérez Conex Applcaons: Sgnal processng: arge rackng bearngs-onl rackng

More information

Including the ordinary differential of distance with time as velocity makes a system of ordinary differential equations.

Including the ordinary differential of distance with time as velocity makes a system of ordinary differential equations. Soluons o Ordnary Derenal Equaons An ordnary derenal equaon has only one ndependen varable. A sysem o ordnary derenal equaons consss o several derenal equaons each wh he same ndependen varable. An eample

More information

Computing Relevance, Similarity: The Vector Space Model

Computing Relevance, Similarity: The Vector Space Model Compung Relevance, Smlary: The Vecor Space Model Based on Larson and Hears s sldes a UC-Bereley hp://.sms.bereley.edu/courses/s0/f00/ aabase Managemen Sysems, R. Ramarshnan ocumen Vecors v ocumens are

More information

Introduction to Boosting

Introduction to Boosting Inroducon o Boosng Cynha Rudn PACM, Prnceon Unversy Advsors Ingrd Daubeches and Rober Schapre Say you have a daabase of news arcles, +, +, -, -, +, +, -, -, +, +, -, -, +, +, -, + where arcles are labeled

More information

Bayesian Inference of the GARCH model with Rational Errors

Bayesian Inference of the GARCH model with Rational Errors 0 Inernaonal Conference on Economcs, Busness and Markeng Managemen IPEDR vol.9 (0) (0) IACSIT Press, Sngapore Bayesan Inference of he GARCH model wh Raonal Errors Tesuya Takash + and Tng Tng Chen Hroshma

More information

Data Collection Definitions of Variables - Conceptualize vs Operationalize Sample Selection Criteria Source of Data Consistency of Data

Data Collection Definitions of Variables - Conceptualize vs Operationalize Sample Selection Criteria Source of Data Consistency of Data Apply Sascs and Economercs n Fnancal Research Obj. of Sudy & Hypoheses Tesng From framework objecves of sudy are needed o clarfy, hen, n research mehodology he hypoheses esng are saed, ncludng esng mehods.

More information

Chapter 5. The linear fixed-effects estimators: matrix creation

Chapter 5. The linear fixed-effects estimators: matrix creation haper 5 he lnear fed-effecs esmaors: mar creaon In hs chaper hree basc models and he daa marces needed o creae esmaors for hem are defned. he frs s ermed he cross-secon model: alhough ncorporaes some panel

More information

Endogeneity. Is the term given to the situation when one or more of the regressors in the model are correlated with the error term such that

Endogeneity. Is the term given to the situation when one or more of the regressors in the model are correlated with the error term such that s row Endogeney Is he erm gven o he suaon when one or more of he regressors n he model are correlaed wh he error erm such ha E( u 0 The 3 man causes of endogeney are: Measuremen error n he rgh hand sde

More information

Testing a new idea to solve the P = NP problem with mathematical induction

Testing a new idea to solve the P = NP problem with mathematical induction Tesng a new dea o solve he P = NP problem wh mahemacal nducon Bacground P and NP are wo classes (ses) of languages n Compuer Scence An open problem s wheher P = NP Ths paper ess a new dea o compare he

More information