Relations to Other Statistical Methods Statistical Data Analysis with Positive Definite Kernels
|
|
- Grace Peters
- 5 years ago
- Views:
Transcription
1 Relatos to Other Statstcal Methods Statstcal Data Aalyss wth Postve Defte Kerels Kej Fukuzu Isttute of Statstcal Matheatcs, ROIS Departet of Statstcal Scece, Graduate Uversty for Advaced Studes October 6-0, 008, Kyushu Uversty
2 Outle. Sple soothg ad RKHS. Relato to rado process
3 Outle. Sple soothg ad RKHS. Relato to rado process 3
4 Sple soothg X,Y ),..., X, Y ) : X R, Y R P: dfferetal operator o R Sple soothg: = ) + λ Y f X ) Pf x) dx f Roughess pealty 4
5 Laplaca ad Gree fucto Laplaca Self-adjot: f f f Δ f = + + L+ x x x f f x), g x) 0 Δ f x) g xdx ) = f x) Δg xdx ) [partal tegral] Gree fucto for Laplaca Δ Gx, ξ ) = δ x ξ).e. Gree fucto solves a dfferetal equato: ) f x) = G x, y) ϕ y) dy Δ f = ϕ gve ϕ. 5
6 Soothg pealty Regularzato ter Cosder fuctos o R for splcty o boudary)! α J f) = D f α! α! Lα! = α + L+ α =! f α α α!!! α + L+ α = α α Lα x x L x L L or of -th dervatve dx exaple = = ) f f f J f) = + + dx x x x x 6
7 Soothg ) + λ f = = 0 Y f X ) a J f) a 0) Expresso by Laplaca Partal tegral shows ) ) J f) = f, Δ f The soothg proble s expressed by = Y f X ) + λ f Af ) ), f L L where A = ) a Δ = 0 7
8 Case e.g. Two cases The Gree fucto s a postve defte kerel. The pealty ter s equal to the squared RKHS or. Case e.g. a 0 0 a = 0 0 f x) dx + f x) ) dx = f, f ) + f, Δf ) x ) f x) dx = f, Δ f ) x Sple soothg The fuctoal space s RKHS + polyoal of soe order The pealty ter s equal to the squared RKHS or of the projecto of f oto the RKHS. L L L 8
9 a 0 0 : RKHS regularzato Soluto = Varatoal calculus Y f X ) + λ f Af ) ), f = ) Y f x) δ x X ) + λaf = 0 Af = Y f x) δ x X ) λ = = = ) If we have the Gree fucto G for A.e. f ξ ) = Y f x)) δ x X ) G x, ξ) dx λ = Y f X )) G ξ, X ) λ ote: fx ) ukow L 9
10 The soluto s to have the for: f = cg, X ) = Plug t to the orgal proble: c R = ) j j j = j + λ, j= j Y c G X, X ) cc G X, X ) Q) By dfferetato, c= G+ λi) Y where G j = G X, X ) j Y = Y, K, Y ) T The soluto: T Y where g ), x = G x X ) f x) = G+λI) g x) 0
11 Gree fucto Theore If a0 0, a j 0 j ), the Gree fucto of A s a postve defte kerel. Proof. Sce Α s shft varat, so s G Gx, y) = Gx-y) ). Thus, = 0 By Fourer trasfor ) a Δ G z) = δ z) If a0 0, a j 0 j ), the Fourer verso s possble. Use Bocher s theore.
12 Regularzato by RKHS or Assue a0 0, a 0 G: Gree fucto of A. H G : RKHS w.r.t. G. ) + λ f = = 0 Y f X ) a J f) The soluto s gve by The pealty ter s, the, f = = cg, X ) The above regularzato s equvalet to the kerel rdge regresso = ) Y f X + λ f ) f H G
13 a 0 = 0: Sple soothg Th-plate sple = ) Y f X ) + λj f) f! α J f ) = D f α + L+ α = α! α! Lα! L The Gree fucto of s ot ecessarly postve defte, but codtoally postve defte). The fucto space for f s ad α B : D f L R ) α = ) J f) = 0 f P P - : Polyoals of degree at ost - 3
14 B = P H Let be decoposto by drect su. * Theore Meguet 979) If > /,the subspace H * s a RKHS wth er product! α = α! Lα! I partcular, the or s gve by H f = J f) * α α ) ) f, g = D f, D g = ) Δ f, g H * L L = ) Y f X ) + λj f) f g H* p P, = ) Y g X + p X + λ g H* ) )) 4
15 Soluto of sple soothg By the represeter theore, the soluto s to be of the for: By pluggg t, f x) = ck x X ) + bφ x) M l l = l= The soluto: 5
16 Outle. Sple soothg. Relato to rado process 6
17 Gaussa process A Gaussa process s a rado process rado varables wth dex Ω) such that for ay fte subset {t,..., t } of Ω, the rado vector X, K, ) s a Gaussa rado vector. Mea fucto Covarace fucto t X t A Gaussa process s uquely detered by the ea ad covarace fucto. X = X t,..., X ) t Rt, t) Rt, t) L Rt, t) Rt, t) Rt, t) L Rt, t) μx = μ t), K, μ t )), Σ X = M M O M Rt, t) Rt, t) L Rt, t) 7
18 Exaples σ = σ = 0.3 ea zero covarace fucto Rst,) = exp s t) σ Geerated by Matlab gpl toolbox Rasusse ad Wllas) 8
19 Rado process ad postve defte kerel Covarace fucto s a postve defte kerel Theore The covarace fucto Rs, t) of a rado process s a postve defte kerel. Q) For splcty, ea = 0., j= cc jr t, tj ) =, j= cc je[ Xt, X ] tj E cxt j cjxt E = = = cxt =, = ) 0 j A rado process o Ω deteres a RKHS o Ω. 9
20 Postve defte kerel defes Gaussa process ks,t): postve defte kerel o Ω. For ay fte subset t = t,, t ) of Ω, the Gra atrx Σ t = kt, t j ) ) s always postve sedefte. By Kologorov exteso theore, there s a Gaussa process wth dex set Ω such that X = X,..., X ) t t The covarace fucto = ks,t). 0
21 Statoary process ad shftvarat kerel Statoary case : rado process o R statoary process EX [ + X + ] = EXX [ ] tsh,, R ) t h s h t s covarace fucto s gve by Rts, ) Rt s) Postve defte kerel for a statoary process s gve by Kts, ) = Kt s) Bocher s theore Weer-Khche s theore covarace fucto of a statoary process o R s the verse Fourer trasfor of the power spectral.)
22 Refereces Wahba, G. Sple Models for Observatoal Data. CBMS-SF Regoal Coferece Seres Appled Matheatcs 59. SIAM Meguet, J. 979) Multvarate Iterpolato at Arbtrary Pots Made Sple. J. Appled Matheatcs ad Physcs ZAMP) 30, Berlet, A. ad C. Thoas-Aga. Reproducg Kerel Hlbert Spaces Probablty ad Statstcs. Kluwer Acadec Publshers, 003. Gel fad, I.M. ad Vlek,.Ya. Geeralzed Fuctos Vol.4: Applcatos of Haroc Aalyss. Acadec Press. 964.
Other Topics in Kernel Method Statistical Inference with Reproducing Kernel Hilbert Space
Oher Topcs Kerel Mehod Sascal Iferece wh Reproducg Kerel Hlber Space Kej Fukumzu Isue of Sascal Mahemacs, ROIS Deparme of Sascal Scece, Graduae Uversy for Advaced Sudes Sepember 6, 008 / Sascal Learg Theory
More informationSome Different Perspectives on Linear Least Squares
Soe Dfferet Perspectves o Lear Least Squares A stadard proble statstcs s to easure a respose or depedet varable, y, at fed values of oe or ore depedet varables. Soetes there ests a deterstc odel y f (,,
More informationMultivariate Transformation of Variables and Maximum Likelihood Estimation
Marquette Uversty Multvarate Trasformato of Varables ad Maxmum Lkelhood Estmato Dael B. Rowe, Ph.D. Assocate Professor Departmet of Mathematcs, Statstcs, ad Computer Scece Copyrght 03 by Marquette Uversty
More informationCOV. Violation of constant variance of ε i s but they are still independent. The error term (ε) is said to be heteroscedastic.
c Pogsa Porchawseskul, Faculty of Ecoomcs, Chulalogkor Uversty olato of costat varace of s but they are stll depedet. C,, he error term s sad to be heteroscedastc. c Pogsa Porchawseskul, Faculty of Ecoomcs,
More informationThe Mathematics of Portfolio Theory
The Matheatcs of Portfolo Theory The rates of retur of stocks, ad are as follows Market odtos state / scearo) earsh Neutral ullsh Probablty 0. 0.5 0.3 % 5% 9% -3% 3% % 5% % -% Notato: R The retur of stock
More informationQualifying Exam Statistical Theory Problem Solutions August 2005
Qualfyg Exam Statstcal Theory Problem Solutos August 5. Let X, X,..., X be d uform U(,),
More informationA Remark on the Uniform Convergence of Some Sequences of Functions
Advaces Pure Mathematcs 05 5 57-533 Publshed Ole July 05 ScRes. http://www.scrp.org/joural/apm http://dx.do.org/0.436/apm.05.59048 A Remark o the Uform Covergece of Some Sequeces of Fuctos Guy Degla Isttut
More informationX ε ) = 0, or equivalently, lim
Revew for the prevous lecture Cocepts: order statstcs Theorems: Dstrbutos of order statstcs Examples: How to get the dstrbuto of order statstcs Chapter 5 Propertes of a Radom Sample Secto 55 Covergece
More informationSpecial Instructions / Useful Data
JAM 6 Set of all real umbers P A..d. B, p Posso Specal Istructos / Useful Data x,, :,,, x x Probablty of a evet A Idepedetly ad detcally dstrbuted Bomal dstrbuto wth parameters ad p Posso dstrbuto wth
More information. The set of these sums. be a partition of [ ab, ]. Consider the sum f( x) f( x 1)
Chapter 7 Fuctos o Bouded Varato. Subject: Real Aalyss Level: M.Sc. Source: Syed Gul Shah (Charma, Departmet o Mathematcs, US Sargodha Collected & Composed by: Atq ur Rehma (atq@mathcty.org, http://www.mathcty.org
More informationUNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS
UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Postpoed exam: ECON430 Statstcs Date of exam: Jauary 0, 0 Tme for exam: 09:00 a.m. :00 oo The problem set covers 5 pages Resources allowed: All wrtte ad prted
More informationAnalysis of Lagrange Interpolation Formula
P IJISET - Iteratoal Joural of Iovatve Scece, Egeerg & Techology, Vol. Issue, December 4. www.jset.com ISS 348 7968 Aalyss of Lagrage Iterpolato Formula Vjay Dahya PDepartmet of MathematcsMaharaja Surajmal
More information7.0 Equality Contraints: Lagrange Multipliers
Systes Optzato 7.0 Equalty Cotrats: Lagrage Multplers Cosder the zato of a o-lear fucto subject to equalty costrats: g f() R ( ) 0 ( ) (7.) where the g ( ) are possbly also olear fuctos, ad < otherwse
More informationA Conventional Approach for the Solution of the Fifth Order Boundary Value Problems Using Sixth Degree Spline Functions
Appled Matheatcs, 1, 4, 8-88 http://d.do.org/1.4/a.1.448 Publshed Ole Aprl 1 (http://www.scrp.org/joural/a) A Covetoal Approach for the Soluto of the Ffth Order Boudary Value Probles Usg Sth Degree Sple
More informationCS 2750 Machine Learning. Lecture 7. Linear regression. CS 2750 Machine Learning. Linear regression. is a linear combination of input components x
CS 75 Mache Learg Lecture 7 Lear regresso Mlos Hauskrecht los@cs.ptt.edu 59 Seott Square CS 75 Mache Learg Lear regresso Fucto f : X Y s a lear cobato of put copoets f + + + K d d K k - paraeters eghts
More informationSTK4011 and STK9011 Autumn 2016
STK4 ad STK9 Autum 6 Pot estmato Covers (most of the followg materal from chapter 7: Secto 7.: pages 3-3 Secto 7..: pages 3-33 Secto 7..: pages 35-3 Secto 7..3: pages 34-35 Secto 7.3.: pages 33-33 Secto
More informationDIFFERENTIAL GEOMETRIC APPROACH TO HAMILTONIAN MECHANICS
DIFFERENTIAL GEOMETRIC APPROACH TO HAMILTONIAN MECHANICS Course Project: Classcal Mechacs (PHY 40) Suja Dabholkar (Y430) Sul Yeshwath (Y444). Itroducto Hamltoa mechacs s geometry phase space. It deals
More informationSUBCLASS OF HARMONIC UNIVALENT FUNCTIONS ASSOCIATED WITH SALAGEAN DERIVATIVE. Sayali S. Joshi
Faculty of Sceces ad Matheatcs, Uversty of Nš, Serba Avalable at: http://wwwpfacyu/float Float 3:3 (009), 303 309 DOI:098/FIL0903303J SUBCLASS OF ARMONIC UNIVALENT FUNCTIONS ASSOCIATED WIT SALAGEAN DERIVATIVE
More informationChapter 5 Properties of a Random Sample
Lecture 6 o BST 63: Statstcal Theory I Ku Zhag, /0/008 Revew for the prevous lecture Cocepts: t-dstrbuto, F-dstrbuto Theorems: Dstrbutos of sample mea ad sample varace, relatoshp betwee sample mea ad sample
More informationCoherent Potential Approximation
Coheret Potetal Approxato Noveber 29, 2009 Gree-fucto atrces the TB forals I the tght bdg TB pcture the atrx of a Haltoa H s the for H = { H j}, where H j = δ j ε + γ j. 2 Sgle ad double uderles deote
More informationThe Arithmetic-Geometric mean inequality in an external formula. Yuki Seo. October 23, 2012
Sc. Math. Japocae Vol. 00, No. 0 0000, 000 000 1 The Arthmetc-Geometrc mea equalty a exteral formula Yuk Seo October 23, 2012 Abstract. The classcal Jese equalty ad ts reverse are dscussed by meas of terally
More informationAN EULER-MC LAURIN FORMULA FOR INFINITE DIMENSIONAL SPACES
AN EULER-MC LAURIN FORMULA FOR INFINITE DIMENSIONAL SPACES Jose Javer Garca Moreta Graduate Studet of Physcs ( Sold State ) at UPV/EHU Address: P.O 6 890 Portugalete, Vzcaya (Spa) Phoe: (00) 3 685 77 16
More informationA NEW LOG-NORMAL DISTRIBUTION
Joural of Statstcs: Advaces Theory ad Applcatos Volume 6, Number, 06, Pages 93-04 Avalable at http://scetfcadvaces.co. DOI: http://dx.do.org/0.864/jsata_700705 A NEW LOG-NORMAL DISTRIBUTION Departmet of
More informationA Bivariate Distribution with Conditional Gamma and its Multivariate Form
Joural of Moder Appled Statstcal Methods Volue 3 Issue Artcle 9-4 A Bvarate Dstrbuto wth Codtoal Gaa ad ts Multvarate For Sue Se Old Doo Uversty, sxse@odu.edu Raja Lachhae Texas A&M Uversty, raja.lachhae@tauk.edu
More informationTraining Sample Model: Given n observations, [[( Yi, x i the sample model can be expressed as (1) where, zero and variance σ
Stat 74 Estmato for Geeral Lear Model Prof. Goel Broad Outle Geeral Lear Model (GLM): Trag Samle Model: Gve observatos, [[( Y, x ), x = ( x,, xr )], =,,, the samle model ca be exressed as Y = µ ( x, x,,
More informationOn Convergence a Variation of the Converse of Fabry Gap Theorem
Scece Joural of Appled Matheatcs ad Statstcs 05; 3(): 58-6 Pulshed ole Aprl 05 (http://www.scecepulshggroup.co//sas) do: 0.648/.sas.05030.5 ISSN: 376-949 (Prt); ISSN: 376-953 (Ole) O Covergece a Varato
More informationA Characterization of Jacobson Radical in Γ-Banach Algebras
Advaces Pure Matheatcs 43-48 http://dxdoorg/436/ap66 Publshed Ole Noveber (http://wwwscrporg/joural/ap) A Characterzato of Jacobso Radcal Γ-Baach Algebras Nlash Goswa Departet of Matheatcs Gauhat Uversty
More information5 Short Proofs of Simplified Stirling s Approximation
5 Short Proofs of Smplfed Strlg s Approxmato Ofr Gorodetsky, drtymaths.wordpress.com Jue, 20 0 Itroducto Strlg s approxmato s the followg (somewhat surprsg) approxmato of the factoral,, usg elemetary fuctos:
More informationPoint Estimation: definition of estimators
Pot Estmato: defto of estmators Pot estmator: ay fucto W (X,..., X ) of a data sample. The exercse of pot estmato s to use partcular fuctos of the data order to estmate certa ukow populato parameters.
More informationNew Schedule. Dec. 8 same same same Oct. 21. ^2 weeks ^1 week ^1 week. Pattern Recognition for Vision
ew Schedule Dec. 8 same same same Oct. ^ weeks ^ week ^ week Fall 004 Patter Recogto for Vso 9.93 Patter Recogto for Vso Classfcato Berd Hesele Fall 004 Overvew Itroducto Lear Dscrmat Aalyss Support Vector
More informationA New Method for Solving Fuzzy Linear. Programming by Solving Linear Programming
ppled Matheatcal Sceces Vol 008 o 50 7-80 New Method for Solvg Fuzzy Lear Prograg by Solvg Lear Prograg S H Nasser a Departet of Matheatcs Faculty of Basc Sceces Mazadara Uversty Babolsar Ira b The Research
More informationOrder Nonlinear Vector Differential Equations
It. Joural of Math. Aalyss Vol. 3 9 o. 3 39-56 Coverget Power Seres Solutos of Hgher Order Nolear Vector Dfferetal Equatos I. E. Kougas Departet of Telecoucato Systes ad Networs Techologcal Educatoal Isttute
More informationAn Introduction to. Support Vector Machine
A Itroducto to Support Vector Mache Support Vector Mache (SVM) A classfer derved from statstcal learg theory by Vapk, et al. 99 SVM became famous whe, usg mages as put, t gave accuracy comparable to eural-etwork
More informationCubic Nonpolynomial Spline Approach to the Solution of a Second Order Two-Point Boundary Value Problem
Joural of Amerca Scece ;6( Cubc Nopolyomal Sple Approach to the Soluto of a Secod Order Two-Pot Boudary Value Problem W.K. Zahra, F.A. Abd El-Salam, A.A. El-Sabbagh ad Z.A. ZAk * Departmet of Egeerg athematcs
More informationTHE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA
THE ROYAL STATISTICAL SOCIETY 3 EXAMINATIONS SOLUTIONS GRADUATE DIPLOMA PAPER I STATISTICAL THEORY & METHODS The Socety provdes these solutos to assst caddates preparg for the examatos future years ad
More informationTaylor s Series and Interpolation. Interpolation & Curve-fitting. CIS Interpolation. Basic Scenario. Taylor Series interpolates at a specific
CIS 54 - Iterpolato Roger Crawfs Basc Scearo We are able to prod some fucto, but do ot kow what t really s. Ths gves us a lst of data pots: [x,f ] f(x) f f + x x + August 2, 25 OSU/CIS 54 3 Taylor s Seres
More informationMULTIOBJECTIVE NONLINEAR FRACTIONAL PROGRAMMING PROBLEMS INVOLVING GENERALIZED d - TYPE-I n -SET FUNCTIONS
THE PUBLIHING HOUE PROCEEDING OF THE ROMANIAN ACADEMY, eres A OF THE ROMANIAN ACADEMY Volue 8, Nuber /27,.- MULTIOBJECTIVE NONLINEAR FRACTIONAL PROGRAMMING PROBLEM INVOLVING GENERALIZED d - TYPE-I -ET
More informationEVALUATION OF FUNCTIONAL INTEGRALS BY MEANS OF A SERIES AND THE METHOD OF BOREL TRANSFORM
EVALUATION OF FUNCTIONAL INTEGRALS BY MEANS OF A SERIES AND THE METHOD OF BOREL TRANSFORM Jose Javer Garca Moreta Ph. D research studet at the UPV/EHU (Uversty of Basque coutry) Departmet of Theoretcal
More informationMATH 247/Winter Notes on the adjoint and on normal operators.
MATH 47/Wter 00 Notes o the adjot ad o ormal operators I these otes, V s a fte dmesoal er product space over, wth gve er * product uv, T, S, T, are lear operators o V U, W are subspaces of V Whe we say
More informationThe Mathematical Appendix
The Mathematcal Appedx Defto A: If ( Λ, Ω, where ( λ λ λ whch the probablty dstrbutos,,..., Defto A. uppose that ( Λ,,..., s a expermet type, the σ-algebra o λ λ λ are defed s deoted by ( (,,...,, σ Ω.
More informationLecture 3. Sampling, sampling distributions, and parameter estimation
Lecture 3 Samplg, samplg dstrbutos, ad parameter estmato Samplg Defto Populato s defed as the collecto of all the possble observatos of terest. The collecto of observatos we take from the populato s called
More informationArithmetic Mean and Geometric Mean
Acta Mathematca Ntresa Vol, No, p 43 48 ISSN 453-6083 Arthmetc Mea ad Geometrc Mea Mare Varga a * Peter Mchalča b a Departmet of Mathematcs, Faculty of Natural Sceces, Costate the Phlosopher Uversty Ntra,
More informationBERNSTEIN COLLOCATION METHOD FOR SOLVING NONLINEAR DIFFERENTIAL EQUATIONS. Aysegul Akyuz Dascioglu and Nese Isler
Mathematcal ad Computatoal Applcatos, Vol. 8, No. 3, pp. 293-300, 203 BERNSTEIN COLLOCATION METHOD FOR SOLVING NONLINEAR DIFFERENTIAL EQUATIONS Aysegul Ayuz Dascoglu ad Nese Isler Departmet of Mathematcs,
More informationLecture Notes Types of economic variables
Lecture Notes 3 1. Types of ecoomc varables () Cotuous varable takes o a cotuum the sample space, such as all pots o a le or all real umbers Example: GDP, Polluto cocetrato, etc. () Dscrete varables fte
More informationNon-degenerate Perturbation Theory
No-degeerate Perturbato Theory Proble : H E ca't solve exactly. But wth H H H' H" L H E Uperturbed egevalue proble. Ca solve exactly. E Therefore, kow ad. H ' H" called perturbatos Copyrght Mchael D. Fayer,
More informationThe Geometric Least Squares Fitting Of Ellipses
IOSR Joural of Matheatcs (IOSR-JM) e-issn: 78-578, p-issn: 39-765X. Volue 4, Issue 3 Ver.I (May - Jue 8), PP -8 www.osrourals.org Abdellatf Bettayeb Departet of Geeral Studes, Jubal Idustral College, Jubal
More informationIntegral Equation Methods. Jacob White. Thanks to Deepak Ramaswamy, Michal Rewienski, Xin Wang and Karen Veroy
Itroducto to Smulato - Lecture 22 Itegral Equato ethods Jacob Whte Thaks to Deepak Ramaswamy, chal Rewesk, X Wag ad Kare Veroy Outle Itegral Equato ethods Exteror versus teror problems Start wth usg pot
More informationX X X E[ ] E X E X. is the ()m n where the ( i,)th. j element is the mean of the ( i,)th., then
Secto 5 Vectors of Radom Varables Whe workg wth several radom varables,,..., to arrage them vector form x, t s ofte coveet We ca the make use of matrx algebra to help us orgaze ad mapulate large umbers
More informationDepartment of Mathematics UNIVERSITY OF OSLO. FORMULAS FOR STK4040 (version 1, September 12th, 2011) A - Vectors and matrices
Deartet of Matheatcs UNIVERSITY OF OSLO FORMULAS FOR STK4040 (verso Seteber th 0) A - Vectors ad atrces A) For a x atrx A ad a x atrx B we have ( AB) BA A) For osgular square atrces A ad B we have ( )
More informationQuantization in Dynamic Smarandache Multi-Space
Quatzato Dyamc Smaradache Mult-Space Fu Yuhua Cha Offshore Ol Research Ceter, Beg, 7, Cha (E-mal: fuyh@cooc.com.c ) Abstract: Dscussg the applcatos of Dyamc Smaradache Mult-Space (DSMS) Theory. Supposg
More informationMidterm Exam 1, section 1 (Solution) Thursday, February hour, 15 minutes
coometrcs, CON Sa Fracsco State Uversty Mchael Bar Sprg 5 Mdterm am, secto Soluto Thursday, February 6 hour, 5 mutes Name: Istructos. Ths s closed book, closed otes eam.. No calculators of ay kd are allowed..
More informationStrong Laws of Large Numbers for Fuzzy Set-Valued Random Variables in Gα Space
Advaces Pure Matheatcs 26 6 583-592 Publshed Ole August 26 ScRes http://wwwscrporg/oural/ap http://dxdoorg/4236/ap266947 Strog Laws of Large Nubers for uzzy Set-Valued Rado Varables G Space Lae She L Gua
More informationA Family of Non-Self Maps Satisfying i -Contractive Condition and Having Unique Common Fixed Point in Metrically Convex Spaces *
Advaces Pure Matheatcs 0 80-84 htt://dxdoorg/0436/a04036 Publshed Ole July 0 (htt://wwwscrporg/oural/a) A Faly of No-Self Mas Satsfyg -Cotractve Codto ad Havg Uque Coo Fxed Pot Metrcally Covex Saces *
More informationC-1: Aerodynamics of Airfoils 1 C-2: Aerodynamics of Airfoils 2 C-3: Panel Methods C-4: Thin Airfoil Theory
ROAD MAP... AE301 Aerodyamcs I UNIT C: 2-D Arfols C-1: Aerodyamcs of Arfols 1 C-2: Aerodyamcs of Arfols 2 C-3: Pael Methods C-4: Th Arfol Theory AE301 Aerodyamcs I Ut C-3: Lst of Subects Problem Solutos?
More informationLinear Regression Linear Regression with Shrinkage. Some slides are due to Tommi Jaakkola, MIT AI Lab
Lear Regresso Lear Regresso th Shrkage Some sldes are due to Tomm Jaakkola, MIT AI Lab Itroducto The goal of regresso s to make quattatve real valued predctos o the bass of a vector of features or attrbutes.
More informationFunctions of Random Variables
Fuctos of Radom Varables Chapter Fve Fuctos of Radom Varables 5. Itroducto A geeral egeerg aalyss model s show Fg. 5.. The model output (respose) cotas the performaces of a system or product, such as weght,
More informationOn the convergence of derivatives of Bernstein approximation
O the covergece of dervatves of Berste approxmato Mchael S. Floater Abstract: By dfferetatg a remader formula of Stacu, we derve both a error boud ad a asymptotc formula for the dervatves of Berste approxmato.
More informationA New Measure of Probabilistic Entropy. and its Properties
Appled Mathematcal Sceces, Vol. 4, 200, o. 28, 387-394 A New Measure of Probablstc Etropy ad ts Propertes Rajeesh Kumar Departmet of Mathematcs Kurukshetra Uversty Kurukshetra, Ida rajeesh_kuk@redffmal.com
More informationKURODA S METHOD FOR CONSTRUCTING CONSISTENT INPUT-OUTPUT DATA SETS. Peter J. Wilcoxen. Impact Research Centre, University of Melbourne.
KURODA S METHOD FOR CONSTRUCTING CONSISTENT INPUT-OUTPUT DATA SETS by Peter J. Wlcoxe Ipact Research Cetre, Uversty of Melboure Aprl 1989 Ths paper descrbes a ethod that ca be used to resolve cossteces
More informationLecture 3 Probability review (cont d)
STATS 00: Itroducto to Statstcal Iferece Autum 06 Lecture 3 Probablty revew (cot d) 3. Jot dstrbutos If radom varables X,..., X k are depedet, the ther dstrbuto may be specfed by specfyg the dvdual dstrbuto
More informationCHAPTER VI Statistical Analysis of Experimental Data
Chapter VI Statstcal Aalyss of Expermetal Data CHAPTER VI Statstcal Aalyss of Expermetal Data Measuremets do ot lead to a uque value. Ths s a result of the multtude of errors (maly radom errors) that ca
More information3. Basic Concepts: Consequences and Properties
: 3. Basc Cocepts: Cosequeces ad Propertes Markku Jutt Overvew More advaced cosequeces ad propertes of the basc cocepts troduced the prevous lecture are derved. Source The materal s maly based o Sectos.6.8
More information4 Inner Product Spaces
11.MH1 LINEAR ALGEBRA Summary Notes 4 Ier Product Spaces Ier product s the abstracto to geeral vector spaces of the famlar dea of the scalar product of two vectors or 3. I what follows, keep these key
More informationCHAPTER 3 POSTERIOR DISTRIBUTIONS
CHAPTER 3 POSTERIOR DISTRIBUTIONS If scece caot measure the degree of probablt volved, so much the worse for scece. The practcal ma wll stck to hs apprecatve methods utl t does, or wll accept the results
More informationDr. Shalabh. Indian Institute of Technology Kanpur
Aalyss of Varace ad Desg of Expermets-I MODULE -I LECTURE - SOME RESULTS ON LINEAR ALGEBRA, MATRIX THEORY AND DISTRIBUTIONS Dr. Shalabh Departmet t of Mathematcs t ad Statstcs t t Ida Isttute of Techology
More informationPROJECTION PROBLEM FOR REGULAR POLYGONS
Joural of Mathematcal Sceces: Advaces ad Applcatos Volume, Number, 008, Pages 95-50 PROJECTION PROBLEM FOR REGULAR POLYGONS College of Scece Bejg Forestry Uversty Bejg 0008 P. R. Cha e-mal: sl@bjfu.edu.c
More informationENGI 4421 Propagation of Error Page 8-01
ENGI 441 Propagato of Error Page 8-01 Propagato of Error [Navd Chapter 3; ot Devore] Ay realstc measuremet procedure cotas error. Ay calculatos based o that measuremet wll therefore also cota a error.
More informationρ < 1 be five real numbers. The
Lecture o BST 63: Statstcal Theory I Ku Zhag, /0/006 Revew for the prevous lecture Deftos: covarace, correlato Examples: How to calculate covarace ad correlato Theorems: propertes of correlato ad covarace
More informationResearch Article Multidimensional Hilbert-Type Inequalities with a Homogeneous Kernel
Hdaw Publshg Corporato Joural of Iequaltes ad Applcatos Volume 29, Artcle ID 3958, 2 pages do:.55/29/3958 Research Artcle Multdmesoal Hlbert-Type Iequaltes wth a Homogeeous Kerel Predrag Vuovć Faculty
More informationGeneralized Convex Functions on Fractal Sets and Two Related Inequalities
Geeralzed Covex Fuctos o Fractal Sets ad Two Related Iequaltes Huxa Mo, X Su ad Dogya Yu 3,,3School of Scece, Bejg Uversty of Posts ad Telecommucatos, Bejg,00876, Cha, Correspodece should be addressed
More informationStationary states of atoms and molecules
Statoary states of atos ad olecules I followg wees the geeral aspects of the eergy level structure of atos ad olecules that are essetal for the terpretato ad the aalyss of spectral postos the rotatoal
More information2/20/2013. Topics. Power Flow Part 1 Text: Power Transmission. Power Transmission. Power Transmission. Power Transmission
/0/0 Topcs Power Flow Part Text: 0-0. Power Trassso Revsted Power Flow Equatos Power Flow Proble Stateet ECEGR 45 Power Systes Power Trassso Power Trassso Recall that for a short trassso le, the power
More informationMaximum Likelihood Estimation
Marquette Uverst Maxmum Lkelhood Estmato Dael B. Rowe, Ph.D. Professor Departmet of Mathematcs, Statstcs, ad Computer Scece Coprght 08 b Marquette Uverst Maxmum Lkelhood Estmato We have bee sag that ~
More informationto the estimation of total sensitivity indices
Applcato of the cotrol o varate ate techque to the estmato of total sestvty dces S KUCHERENKO B DELPUECH Imperal College Lodo (UK) skuchereko@mperalacuk B IOOSS Electrcté de Frace (Frace) S TARANTOLA Jot
More informationInitial-boundary value problem for second order degenerate pseudohyperbolic equations
PROCEEDINGS OF TE YEREVAN STATE UNIVERSITY Physcal ad Mathematcal Sceces,, p 3 Mathematcs Ital-boudary value problem for secod order degeerate pseudohyperbolc equatos G S akobya, Savash Ghorbaa Char of
More informationRecall MLR 5 Homskedasticity error u has the same variance given any values of the explanatory variables Var(u x1,...,xk) = 2 or E(UU ) = 2 I
Chapter 8 Heterosedastcty Recall MLR 5 Homsedastcty error u has the same varace gve ay values of the eplaatory varables Varu,..., = or EUU = I Suppose other GM assumptos hold but have heterosedastcty.
More informationEstimation of Stress- Strength Reliability model using finite mixture of exponential distributions
Iteratoal Joural of Computatoal Egeerg Research Vol, 0 Issue, Estmato of Stress- Stregth Relablty model usg fte mxture of expoetal dstrbutos K.Sadhya, T.S.Umamaheswar Departmet of Mathematcs, Lal Bhadur
More informationA practical threshold estimation for jump processes
A practcal threshold estmato for jump processes Yasutaka Shmzu (Osaka Uversty, Japa) WORKSHOP o Face ad Related Mathematcal ad Statstcal Issues @ Kyoto, JAPAN, 3 6 Sept., 2008. Itroducto O (Ω, F,P; {F
More informationSome results and conjectures about recurrence relations for certain sequences of binomial sums.
Soe results ad coectures about recurrece relatos for certa sequeces of boal sus Joha Cgler Faultät für Matheat Uverstät We A-9 We Nordbergstraße 5 Joha Cgler@uveacat Abstract I a prevous paper [] I have
More informationON WEIGHTED INTEGRAL AND DISCRETE OPIAL TYPE INEQUALITIES
M atheatcal I equaltes & A pplcatos Volue 19, Nuber 4 16, 195 137 do:1.7153/a-19-95 ON WEIGHTED INTEGRAL AND DISCRETE OPIAL TYPE INEQUALITIES MAJA ANDRIĆ, JOSIP PEČARIĆ AND IVAN PERIĆ Coucated by C. P.
More informationThe theoretical background of
he theoretcal backgroud of -echologes he theoretcal backgroud of FactSage he followg sldes gve a abrdged overvew of the ajor uderlyg prcples of the calculatoal odules of FactSage. -echologes he bbs Eergy
More informationNon-uniform Turán-type problems
Joural of Combatoral Theory, Seres A 111 2005 106 110 wwwelsevercomlocatecta No-uform Turá-type problems DhruvMubay 1, Y Zhao 2 Departmet of Mathematcs, Statstcs, ad Computer Scece, Uversty of Illos at
More informationSummary of the lecture in Biostatistics
Summary of the lecture Bostatstcs Probablty Desty Fucto For a cotuos radom varable, a probablty desty fucto s a fucto such that: 0 dx a b) b a dx A probablty desty fucto provdes a smple descrpto of the
More information( ) ( ) ( ( )) ( ) ( ) ( ) ( ) ( ) = ( ) ( ) + ( ) ( ) = ( ( )) ( ) + ( ( )) ( ) Review. Second Derivatives for f : y R. Let A be an m n matrix.
Revew + v, + y = v, + v, + y, + y, Cato! v, + y, + v, + y geeral Let A be a atr Let f,g : Ω R ( ) ( ) R y R Ω R h( ) f ( ) g ( ) ( ) ( ) ( ( )) ( ) dh = f dg + g df A, y y A Ay = = r= c= =, : Ω R he Proof
More informationDATA DOMAIN DATA DOMAIN
3//6 Coprght otce: Most ages these sldes are Gozalez ad oods Pretce-Hall Note: ages are [spatall] ostatoar sgals. e eed tools to aalze the locall at dfferet resolutos e ca do ths the data doa or sutable
More informationFaculty Research Interest Seminar Department of Biostatistics, GSPH University of Pittsburgh. Gong Tang Feb. 18, 2005
Faculty Research Iterest Semar Departmet of Bostatstcs, GSPH Uversty of Pttsburgh Gog ag Feb. 8, 25 Itroducto Joed the departmet 2. each two courses: Elemets of Stochastc Processes (Bostat 24). Aalyss
More informationGENERALIZED METHOD OF MOMENTS CHARACTERISTICS AND ITS APPLICATION ON PANELDATA
Sc.It.(Lahore),26(3),985-990,2014 ISSN 1013-5316; CODEN: SINTE 8 GENERALIZED METHOD OF MOMENTS CHARACTERISTICS AND ITS APPLICATION ON PANELDATA Beradhta H. S. Utam 1, Warsoo 1, Da Kurasar 1, Mustofa Usma
More informationExtreme Value Theory: An Introduction
(correcto d Extreme Value Theory: A Itroducto by Laures de Haa ad Aa Ferrera Wth ths webpage the authors ted to form the readers of errors or mstakes foud the book after publcato. We also gve extesos for
More informationEstimation and Testing in Type-II Generalized Half Logistic Distribution
Joural of Moder Appled Statstcal Methods Volume 13 Issue 1 Artcle 17 5-1-014 Estmato ad Testg Type-II Geeralzed Half Logstc Dstrbuto R R. L. Katam Acharya Nagarjua Uversty, Ida, katam.rrl@gmal.com V Ramakrsha
More informationGlobal Optimization for Solving Linear Non-Quadratic Optimal Control Problems
Joural of Appled Matheatcs ad Physcs 06 4 859-869 http://wwwscrporg/joural/jap ISSN Ole: 37-4379 ISSN Prt: 37-435 Global Optzato for Solvg Lear No-Quadratc Optal Cotrol Probles Jghao Zhu Departet of Appled
More informationJournal Of Inequalities And Applications, 2008, v. 2008, p
Ttle O verse Hlbert-tye equaltes Authors Chagja, Z; Cheug, WS Ctato Joural Of Iequaltes Ad Alcatos, 2008, v. 2008,. 693248 Issued Date 2008 URL htt://hdl.hadle.et/0722/56208 Rghts Ths work s lcesed uder
More informationOrdinary Least Squares Regression. Simple Regression. Algebra and Assumptions.
Ordary Least Squares egresso. Smple egresso. Algebra ad Assumptos. I ths part of the course we are gog to study a techque for aalysg the lear relatoshp betwee two varables Y ad X. We have pars of observatos
More informationStrong Convergence of Weighted Averaged Approximants of Asymptotically Nonexpansive Mappings in Banach Spaces without Uniform Convexity
BULLETIN of the MALAYSIAN MATHEMATICAL SCIENCES SOCIETY Bull. Malays. Math. Sc. Soc. () 7 (004), 5 35 Strog Covergece of Weghted Averaged Appromats of Asymptotcally Noepasve Mappgs Baach Spaces wthout
More informationConstruction of Composite Indices in Presence of Outliers
Costructo of Coposte dces Presece of Outlers SK Mshra Dept. of Ecoocs North-Easter Hll Uversty Shllog (da). troducto: Oftetes we requre costructg coposte dces by a lear cobato of a uber of dcator varables.
More informationSolving the fuzzy shortest path problem on networks by a new algorithm
Proceedgs of the 0th WSEAS Iteratoal Coferece o FUZZY SYSTEMS Solvg the fuzzy shortest path proble o etworks by a ew algorth SADOAH EBRAHIMNEJAD a, ad REZA TAVAKOI-MOGHADDAM b a Departet of Idustral Egeerg,
More informationTHE PROBABILISTIC STABILITY FOR THE GAMMA FUNCTIONAL EQUATION
Joural of Scece ad Arts Year 12, No. 3(2), pp. 297-32, 212 ORIGINAL AER THE ROBABILISTIC STABILITY FOR THE GAMMA FUNCTIONAL EQUATION DOREL MIHET 1, CLAUDIA ZAHARIA 1 Mauscrpt receved: 3.6.212; Accepted
More informationPRACTICAL CONSIDERATIONS IN HUMAN-INDUCED VIBRATION
PRACTICAL CONSIDERATIONS IN HUMAN-INDUCED VIBRATION Bars Erkus, 4 March 007 Itroducto Ths docuet provdes a revew of fudaetal cocepts structural dyacs ad soe applcatos hua-duced vbrato aalyss ad tgato of
More informationMidterm Exam 1, section 2 (Solution) Thursday, February hour, 15 minutes
coometrcs, CON Sa Fracsco State Uverst Mchael Bar Sprg 5 Mdterm xam, secto Soluto Thursda, Februar 6 hour, 5 mutes Name: Istructos. Ths s closed book, closed otes exam.. No calculators of a kd are allowed..
More informationComparing Different Estimators of three Parameters for Transmuted Weibull Distribution
Global Joural of Pure ad Appled Mathematcs. ISSN 0973-768 Volume 3, Number 9 (207), pp. 55-528 Research Ida Publcatos http://www.rpublcato.com Comparg Dfferet Estmators of three Parameters for Trasmuted
More informationLecture 1: Introduction to Regression
Lecture : Itroducto to Regresso A Eample: Eplag State Homcde Rates What kds of varables mght we use to epla/predct state homcde rates? Let s cosder just oe predctor for ow: povert Igore omtted varables,
More information