Code_Aster. Identification of the model of Weibull
|
|
- Cuthbert Perkins
- 6 years ago
- Views:
Transcription
1 Verson Ttre : Identfcaton du modèle de Webull Date : 2/09/2009 Page : /8 Responsable : PARROT Aurore Clé : R70209 Révson : Identfcaton of the model of Webull Summary One tackles here the problem of the dentfcaton of the parameters of the model of WEIBULL on a sample of tests representatve of behavour wth rupture of a fragle materal (typcally, ferrtc steel of low temperature) The method of regresson lnear and the method of the maxmum of probablty are the two adopted methods One detals of t the prncple as well as the assocated methods of resoluton, beng based n both cases on an teratve process Lastly, one shows ther extenson f one of the two parameters of ths model (the constrant of cleavage) depends on the temperature Warnng : The translaton process used on ths webste s a "Machne Translaton" It may be mprecse and naccurate n whole or n part and s provded as a convenence Copyrght 207 EDF R&D - Lcensed under the terms of the GNU FDL (
2 Verson Ttre : Identfcaton du modèle de Webull Date : 2/09/2009 Page : 2/8 Responsable : PARROT Aurore Clé : R70209 Révson : Contents Introducton 3 2 Recalls 3 2 The model of WEIBULL 3 22 Identfcaton of the parameters 3 3 Method of the lnear regresson 4 3 Prncple 4 32 Resoluton 5 4 Method of the maxmum of probablty 6 4 Prncple 6 42 Resoluton 6 5 Dependence of the parameters wth the temperature 7 5 Lnear regresson 7 52 Maxmum of probablty 8 6 Concluson 8 7 Bblography 8 Descrpton of the versons of the document 8 Warnng : The translaton process used on ths webste s a "Machne Translaton" It may be mprecse and naccurate n whole or n part and s provded as a convenence Copyrght 207 EDF R&D - Lcensed under the terms of the GNU FDL (
3 Verson Ttre : Identfcaton du modèle de Webull Date : 2/09/2009 Page : 3/8 Responsable : PARROT Aurore Clé : R70209 Révson : Introducton When they call on the model of WEIBULL (cf POST_ELEM [U4822]), the study of modelng of the brttle fracture of steels n general requre a prelmnary dentfcaton of the parameters of ths model In order to avod a hard dentfcaton wth the hand of these parameters whch would requre to start agan the operaton repeatedly POST_ELEM wth the opton WEIBULL, an automatc procedure of retmng was establshed n Code_Aster In ths document, one brefly ponts out the equatons of the model of WEIBULL then one defnes the problem of dentfcaton posed One then descrbes the prncple of the two methods of resoluton adopted (lnear and maxmum regresson of probablty) by ncludng the case where one of the two parameters of the model depends on the temperature 2 Recalls 2 The model of WEIBULL One consders a structure of behavor elastoplastc subjected to a thermomechancal request It s supposed that the m] probablty of cumulated rupture of ths structure follows the law of WEIBULL [bb] wth two parameters followng: P f w = exp[ w éq 2- u expresson n whch the module of WEIBULL m descrbed the tal of the statstcal dstrbuton of the szes of the defects at the orgn of cleavage, u s the constrant of cleavage and w s the constrant of WEIBULL whch depends on the hstory of the prncpal stress feld n the plastczed zone of the structure For example, n the case of a monotonous way of loadng, t s wrtten: w = m p The summaton relates to volumes of matter V p plastczed, I p Ip m V p V 0 éq 2-2 constrant n each one of these volumes ( V 0 s a volume characterstc of materal) 22 Identfcaton of the parameters ndcatng the maxmum prncpal In a very general way, one consders an expermental base made up of tests of varous natures (type 2,, N), each type of test beng carred out n j tme so that the full number of tests rses wth: j=n N= n j j = Ths expermental base could for example be made up of tests on notched axsymmetrc test-tubes of dfferent rays of notch led to varous temperatures Takng nto account the random nature of the propertes wth rupture of materal consdered, ths base consttutes only one sample The more mportant the number of these samples wll be, the more t wll be representatve of the behavor of materal consdered Warnng : The translaton process used on ths webste s a "Machne Translaton" It may be mprecse and naccurate n whole or n part and s provded as a convenence Copyrght 207 EDF R&D - Lcensed under the terms of the GNU FDL (
4 Verson Ttre : Identfcaton du modèle de Webull Date : 2/09/2009 Page : 4/8 Responsable : PARROT Aurore Clé : R70209 Révson : Among the varous methods of dentfcaton suggested n the lterature (see for example [bb2]), we retan two of them: method of regresson lnear, often used, lke that of the maxmum of probablty recommended by the Structural European Integrty Socety (ESIS) [bb3] Notce : A comparatve systematc study of the results gven by these two methods [bb2] accordng to the number of sample taken by chance on a theoretcal dstrbuton showed that the method of the maxmum of probablty led to a better estmate of the parameters of the model of WEIBULL The method of regresson lnear remanng nevertheless very much used, we ntegrated t nto our developments In the two adopted methods of retmng, one typcally carres out the frst calculaton of the constrants of WEIBULL wth a game of parameter gven (, m=20, s u =3000 MPa ) One classfes these NR tests usng ther constrant of WEIBULL reached at the nstant of the falure One thus has an ncreasng lst of constrants of WEIBULL w,, w,, N w, such as for each, the number of test-tubes broken wth a constrant of lower WEIBULL or equalzes wth w s n w ) Among the varous possble estmators of the probablty of cumulated rupture P f wth w [bb2], we choose that generally recommended: P f = N Notce : (n general n w = correspondent In the typcal case where the constrant of WEIBULL depends on the temperature, the precedng rankng must be made temperature by temperature, each temperature correspondng to a dfferent statstcal law The estmator of the probablty of rupture precedent thus becomes: P f = N T temperature T, for whch there was N T tests, f the test-tube was broken at the The two adopted methods of retmng are vald as long as [éq 2-] remans true If the dentfcaton s carred out on test results ansothermes whereas the constrant of cleavage s supposed to depend on the temperature, ths condton s not checked any more (cf POST_ELEM [U4822]) In ths case, typcal case one wll not be able to thus apply the developments whch follow 3 Method of the lnear regresson 3 Prncple The varaton theory-experment s measured by the expresson: 2 LogLog P f W éq 3- LogLog P f ( Log ndcates the Naperan logarthm) One wants to mnmze ths varaton compared to (m, u ) Warnng : The translaton process used on ths webste s a "Machne Translaton" It may be mprecse and naccurate n whole or n part and s provded as a convenence Copyrght 207 EDF R&D - Lcensed under the terms of the GNU FDL (
5 Verson Ttre : Identfcaton du modèle de Webull Date : 2/09/2009 Page : 5/8 Responsable : PARROT Aurore Clé : R70209 Révson : 32 Resoluton The method of retmng usually used s based on successve lnear regressons: wth the teraton k, values ( m k, u k ) module and constrant of cleavage are known It s thus possble, wth these values, to calculate the constrants of WEIBULL W k at the varous moments of rupture thanks to [éq 2-] One then classfes these new constrants of WEIBULL by ncreasng ampltude and one from of deduced the new estmates from the probablty of rupture P f k wth the teraton k For these values of constrants of fxed WEIBULL, the mnmzaton of [éq 3-] s brought back to a smple lnear regresson on the group of dots ( Log W k, LogLog P f k ) snce f one defers LogLog P f accordng to Log W, one obtans a lne of slope m whch cuts the x-axs n ( Log u ) New values ( m k, u k ) of these parameters are thus gven by (cancellaton of the dervatve partal of [éq 3-] compared to each parameter): X N k Y j k Y k X k, j m k = éq 32- X N k X j k X 2 k, j u k =exp N X k m wth X k =Log W k Y k and Y k =LogLog, éq 32-2 P f k These teratons are repeated as long as the dfference between the games of parameter obtaned wth the teratons (K) and (k+) s sgnfcant (typcally, fve teratons) The measurement of ths varaton s gven by: Max[ m m k k m k, ] uk u k uk Notce : If m s fxed, u k s always gven by [éq 32-2] On the other hand, f u s fxed, X k Y k m k s not gven any more by [eq 32-] but: m k = X k 2 log u X k Warnng : The translaton process used on ths webste s a "Machne Translaton" It may be mprecse and naccurate n whole or n part and s provded as a convenence Copyrght 207 EDF R&D - Lcensed under the terms of the GNU FDL (
6 Verson Ttre : Identfcaton du modèle de Webull Date : 2/09/2009 Page : 6/8 Responsable : PARROT Aurore Clé : R70209 Révson : 4 Method of the maxmum of probablty 4 Prncple Let us note p f w densty of probablty assocated wth the probablty of cumulated rupture P f w : p f w = m s m w u u exp[ w u Quantty p f w d W s equal to the probablty of breakng a test-tube subjected to a request correspondng to a constrant of WEIBULL understood n the nterval [ W, W d W ] The probablty so that all the test-tubes of the base broke thus rases wth: p f W d w, éq 4- pm, u d w = p beng the functon of probablty The method of the maxmum of probablty then conssts n choosng the parameters of the model of WEIBULL so that the functon of probablty defned by [éq 4-] (n practce rather ts Naperan logarthm) that s to say maxmum m] 42 Resoluton An teratve process agan s used There stll, wth the teraton W k m k = N k = m k k, ( m k, u k ) as well as are known For these values of constrants of fxed WEIBULL, the maxmzaton of Log p condut wth a new couple ( m k, u k ) gven by: =N m k = N = = N Log W k N f = N = W k =N = m k Log W k W k m k W k m k éq 42-2 =0 éq 42- Wth each step, the resoluton of [éq 42-] can be realzed usng the method of Newton, the gradent of f m beng gven by: df m =- N dm m 2 =N W = m Log 2 W =N W = = N W = = N m W = m2 m Log W 2 Warnng : The translaton process used on ths webste s a "Machne Translaton" It may be mprecse and naccurate n whole or n part and s provded as a convenence Copyrght 207 EDF R&D - Lcensed under the terms of the GNU FDL (
7 Verson Ttre : Identfcaton du modèle de Webull Date : 2/09/2009 Page : 7/8 Responsable : PARROT Aurore Clé : R70209 Révson : Notce : If m s fxed, u k s gven by [42-2] On the other hand, f u s fxed, m k s not any more soluton of [42-] but of: f m k = N = N Log W k W k m k =0 m k = u u Ths equaton can be agan solved usng the method of Newton, the gradent beng now gven by: = N df dm m =- N m 2 = W m Log 2 W u u 5 Dependence of the parameters wth the temperature If one wshes to fx ndependently the two parameters temperature by temperature, t s enough to break up the base of tests nto as much of under - bases by temperature and to apply to each one of these subbases the precedng methods If, on the other hand, one only wshes to vary the constrant of cleavage u wth the temperature, one proceeds n the followng way 5 Lnear regresson The estmate of the probabltes of rupture beng now carred out temperature by temperature (cf notces [ 22]), t s enough to fx the constrant of cleavage on each group of dots assocated wth the varous temperatures (T) The equaton [éq 32-2] thus becomes: u k =exp N T T X k m T Y k ( N T ndcatng the number of tests for the subbase correspondng to the temperature (T)), the module of WEIBULL beng gven by: m k = T N T T, j T T N T X k Y j k X k X j k T, j T Y k X k X 2 k Warnng : The translaton process used on ths webste s a "Machne Translaton" It may be mprecse and naccurate n whole or n part and s provded as a convenence Copyrght 207 EDF R&D - Lcensed under the terms of the GNU FDL (
8 Verson Ttre : Identfcaton du modèle de Webull Date : 2/09/2009 Page : 8/8 Responsable : PARROT Aurore Clé : R70209 Révson : 52 Maxmum of probablty The constrant of cleavage s gven for each temperature (T) consdered by: m k beng soluton of: f m k = N m k u k T = m k = N = N T T Log W k T W k T m k, N T T W k T m k Log W k W k m k =0 6 Concluson The order RECA_WEIBULL Code_Aster allows to carry out the chock of the parameters of the model of WEIBULL [U48206] The user gves as starter ths order the concepts results assocated wth varous nonlnear calculatons carred out The possble dependence of the constrant of cleavage wth the temperature s mplctly specfed when dfferent temperatures are assocated wth each one of these concepts results (f all these temperatures are dentcal or f they are not specfed, t does not have there dependence wth the temperature of ths parameter) The user can carry out ths retmng by the method of the maxmum of probablty ( METHOD: MAXI_VRAI ) or that of the lnear regresson (METHOD: REGR_LIN ) Szes determned by the order RECA_WEIBULL are deferred n a table n whch one fnds the value of the dentfed parameters, probabltes of rupture estmated startng from the expermental results as well as the probabltes of theoretcal rupture calculated wth the dentfed parameters 7 Bblography [] F BEREMIN, wth room crteron for cleavage fracture of has nuclear presses vessel steel, Metall Trans 4A, pp , 98 [2] A KHALILI, K KROMP, Statstcal propertes of webull estmators, Newspaper of Materal Scence, 26, pp , 99 [3] ESIS, TC one Local Approach, Procedure to local measure and calculate approach crtera usng notched tensle specmens, P6, 998 Descrpton of the versons of the document Verson Aster Author (S) Organzaton (S) 0/05/00 R MASSON, W LEFEVRE (EDF/RNE/MTC) Descrpton of the modfcatons Intal text Warnng : The translaton process used on ths webste s a "Machne Translaton" It may be mprecse and naccurate n whole or n part and s provded as a convenence Copyrght 207 EDF R&D - Lcensed under the terms of the GNU FDL (
Code_Aster. Identification of the Summarized
Verson Ttre : Identfcaton du modèle de Webull Date : 2/09/2009 Page : /8 Responsable : Aurore PARROT Clé : R70209 Révson : 609 Identfcaton of the Summarzed Webull model One tackles here the problem of
More informationThe Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction
ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also
More informationIntroduction to Vapor/Liquid Equilibrium, part 2. Raoult s Law:
CE304, Sprng 2004 Lecture 4 Introducton to Vapor/Lqud Equlbrum, part 2 Raoult s Law: The smplest model that allows us do VLE calculatons s obtaned when we assume that the vapor phase s an deal gas, and
More informationKernel Methods and SVMs Extension
Kernel Methods and SVMs Extenson The purpose of ths document s to revew materal covered n Machne Learnng 1 Supervsed Learnng regardng support vector machnes (SVMs). Ths document also provdes a general
More informationLecture Notes on Linear Regression
Lecture Notes on Lnear Regresson Feng L fl@sdueducn Shandong Unversty, Chna Lnear Regresson Problem In regresson problem, we am at predct a contnuous target value gven an nput feature vector We assume
More informationNUMERICAL DIFFERENTIATION
NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More informationChapter 13: Multiple Regression
Chapter 13: Multple Regresson 13.1 Developng the multple-regresson Model The general model can be descrbed as: It smplfes for two ndependent varables: The sample ft parameter b 0, b 1, and b are used to
More informationLinear Regression Analysis: Terminology and Notation
ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page ) Lnear Regresson Analyss: Termnology and Notaton Consder the generc verson of the smple (two-varable) lnear regresson model. It s represented
More informationModule 2. Random Processes. Version 2 ECE IIT, Kharagpur
Module Random Processes Lesson 6 Functons of Random Varables After readng ths lesson, ou wll learn about cdf of functon of a random varable. Formula for determnng the pdf of a random varable. Let, X be
More informationThe Geometry of Logit and Probit
The Geometry of Logt and Probt Ths short note s meant as a supplement to Chapters and 3 of Spatal Models of Parlamentary Votng and the notaton and reference to fgures n the text below s to those two chapters.
More informationELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationx = , so that calculated
Stat 4, secton Sngle Factor ANOVA notes by Tm Plachowsk n chapter 8 we conducted hypothess tests n whch we compared a sngle sample s mean or proporton to some hypotheszed value Chapter 9 expanded ths to
More informationComparison of Regression Lines
STATGRAPHICS Rev. 9/13/2013 Comparson of Regresson Lnes Summary... 1 Data Input... 3 Analyss Summary... 4 Plot of Ftted Model... 6 Condtonal Sums of Squares... 6 Analyss Optons... 7 Forecasts... 8 Confdence
More informationPhysics 5153 Classical Mechanics. Principle of Virtual Work-1
P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal
More informationLecture 10 Support Vector Machines II
Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the fake-test data; fxed
More informationUncertainty in measurements of power and energy on power networks
Uncertanty n measurements of power and energy on power networks E. Manov, N. Kolev Department of Measurement and Instrumentaton, Techncal Unversty Sofa, bul. Klment Ohrdsk No8, bl., 000 Sofa, Bulgara Tel./fax:
More informationCOMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
Avalable onlne at http://sck.org J. Math. Comput. Sc. 3 (3), No., 6-3 ISSN: 97-537 COMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
More informationLINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity
LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 31 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 6. Rdge regresson The OLSE s the best lnear unbased
More informationSome modelling aspects for the Matlab implementation of MMA
Some modellng aspects for the Matlab mplementaton of MMA Krster Svanberg krlle@math.kth.se Optmzaton and Systems Theory Department of Mathematcs KTH, SE 10044 Stockholm September 2004 1. Consdered optmzaton
More informationNegative Binomial Regression
STATGRAPHICS Rev. 9/16/2013 Negatve Bnomal Regresson Summary... 1 Data Input... 3 Statstcal Model... 3 Analyss Summary... 4 Analyss Optons... 7 Plot of Ftted Model... 8 Observed Versus Predcted... 10 Predctons...
More informationNumerical Solution of Ordinary Differential Equations
Numercal Methods (CENG 00) CHAPTER-VI Numercal Soluton of Ordnar Dfferental Equatons 6 Introducton Dfferental equatons are equatons composed of an unknown functon and ts dervatves The followng are examples
More informationLecture 12: Classification
Lecture : Classfcaton g Dscrmnant functons g The optmal Bayes classfer g Quadratc classfers g Eucldean and Mahalanobs metrcs g K Nearest Neghbor Classfers Intellgent Sensor Systems Rcardo Guterrez-Osuna
More informationLimited Dependent Variables
Lmted Dependent Varables. What f the left-hand sde varable s not a contnuous thng spread from mnus nfnty to plus nfnty? That s, gven a model = f (, β, ε, where a. s bounded below at zero, such as wages
More informationDETERMINATION OF UNCERTAINTY ASSOCIATED WITH QUANTIZATION ERRORS USING THE BAYESIAN APPROACH
Proceedngs, XVII IMEKO World Congress, June 7, 3, Dubrovn, Croata Proceedngs, XVII IMEKO World Congress, June 7, 3, Dubrovn, Croata TC XVII IMEKO World Congress Metrology n the 3rd Mllennum June 7, 3,
More informationMath 426: Probability MWF 1pm, Gasson 310 Homework 4 Selected Solutions
Exercses from Ross, 3, : Math 26: Probablty MWF pm, Gasson 30 Homework Selected Solutons 3, p. 05 Problems 76, 86 3, p. 06 Theoretcal exercses 3, 6, p. 63 Problems 5, 0, 20, p. 69 Theoretcal exercses 2,
More informationWeek3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity
Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle
More information2E Pattern Recognition Solutions to Introduction to Pattern Recognition, Chapter 2: Bayesian pattern classification
E395 - Pattern Recognton Solutons to Introducton to Pattern Recognton, Chapter : Bayesan pattern classfcaton Preface Ths document s a soluton manual for selected exercses from Introducton to Pattern Recognton
More informationLinear Approximation with Regularization and Moving Least Squares
Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...
More informationTransfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system
Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng
More informationOn an Extension of Stochastic Approximation EM Algorithm for Incomplete Data Problems. Vahid Tadayon 1
On an Extenson of Stochastc Approxmaton EM Algorthm for Incomplete Data Problems Vahd Tadayon Abstract: The Stochastc Approxmaton EM (SAEM algorthm, a varant stochastc approxmaton of EM, s a versatle tool
More informationStatistics Chapter 4
Statstcs Chapter 4 "There are three knds of les: les, damned les, and statstcs." Benjamn Dsrael, 1895 (Brtsh statesman) Gaussan Dstrbuton, 4-1 If a measurement s repeated many tmes a statstcal treatment
More informationANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U)
Econ 413 Exam 13 H ANSWERS Settet er nndelt 9 deloppgaver, A,B,C, som alle anbefales å telle lkt for å gøre det ltt lettere å stå. Svar er gtt . Unfortunately, there s a prntng error n the hnt of
More informationPhysics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1
P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the
More informationCHAPTER 14 GENERAL PERTURBATION THEORY
CHAPTER 4 GENERAL PERTURBATION THEORY 4 Introducton A partcle n orbt around a pont mass or a sphercally symmetrc mass dstrbuton s movng n a gravtatonal potental of the form GM / r In ths potental t moves
More informationTemperature. Chapter Heat Engine
Chapter 3 Temperature In prevous chapters of these notes we ntroduced the Prncple of Maxmum ntropy as a technque for estmatng probablty dstrbutons consstent wth constrants. In Chapter 9 we dscussed the
More informationECONOMICS 351*-A Mid-Term Exam -- Fall Term 2000 Page 1 of 13 pages. QUEEN'S UNIVERSITY AT KINGSTON Department of Economics
ECOOMICS 35*-A Md-Term Exam -- Fall Term 000 Page of 3 pages QUEE'S UIVERSITY AT KIGSTO Department of Economcs ECOOMICS 35* - Secton A Introductory Econometrcs Fall Term 000 MID-TERM EAM ASWERS MG Abbott
More informationAn identification algorithm of model kinetic parameters of the interfacial layer growth in fiber composites
IOP Conference Seres: Materals Scence and Engneerng PAPER OPE ACCESS An dentfcaton algorthm of model knetc parameters of the nterfacal layer growth n fber compostes o cte ths artcle: V Zubov et al 216
More informationLaboratory 1c: Method of Least Squares
Lab 1c, Least Squares Laboratory 1c: Method of Least Squares Introducton Consder the graph of expermental data n Fgure 1. In ths experment x s the ndependent varable and y the dependent varable. Clearly
More informationCHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE
CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng
More informationSolution Thermodynamics
Soluton hermodynamcs usng Wagner Notaton by Stanley. Howard Department of aterals and etallurgcal Engneerng South Dakota School of nes and echnology Rapd Cty, SD 57701 January 7, 001 Soluton hermodynamcs
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationLINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity
LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 30 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 2 Remedes for multcollnearty Varous technques have
More informationTurbulence classification of load data by the frequency and severity of wind gusts. Oscar Moñux, DEWI GmbH Kevin Bleibler, DEWI GmbH
Turbulence classfcaton of load data by the frequency and severty of wnd gusts Introducton Oscar Moñux, DEWI GmbH Kevn Blebler, DEWI GmbH Durng the wnd turbne developng process, one of the most mportant
More information4 Analysis of Variance (ANOVA) 5 ANOVA. 5.1 Introduction. 5.2 Fixed Effects ANOVA
4 Analyss of Varance (ANOVA) 5 ANOVA 51 Introducton ANOVA ANOVA s a way to estmate and test the means of multple populatons We wll start wth one-way ANOVA If the populatons ncluded n the study are selected
More informationPHYS 705: Classical Mechanics. Calculus of Variations II
1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary
More informationLaboratory 3: Method of Least Squares
Laboratory 3: Method of Least Squares Introducton Consder the graph of expermental data n Fgure 1. In ths experment x s the ndependent varable and y the dependent varable. Clearly they are correlated wth
More informationCIS526: Machine Learning Lecture 3 (Sept 16, 2003) Linear Regression. Preparation help: Xiaoying Huang. x 1 θ 1 output... θ M x M
CIS56: achne Learnng Lecture 3 (Sept 6, 003) Preparaton help: Xaoyng Huang Lnear Regresson Lnear regresson can be represented by a functonal form: f(; θ) = θ 0 0 +θ + + θ = θ = 0 ote: 0 s a dummy attrbute
More informationMMA and GCMMA two methods for nonlinear optimization
MMA and GCMMA two methods for nonlnear optmzaton Krster Svanberg Optmzaton and Systems Theory, KTH, Stockholm, Sweden. krlle@math.kth.se Ths note descrbes the algorthms used n the author s 2007 mplementatons
More informationFREQUENCY DISTRIBUTIONS Page 1 of The idea of a frequency distribution for sets of observations will be introduced,
FREQUENCY DISTRIBUTIONS Page 1 of 6 I. Introducton 1. The dea of a frequency dstrbuton for sets of observatons wll be ntroduced, together wth some of the mechancs for constructng dstrbutons of data. Then
More informationLossy Compression. Compromise accuracy of reconstruction for increased compression.
Lossy Compresson Compromse accuracy of reconstructon for ncreased compresson. The reconstructon s usually vsbly ndstngushable from the orgnal mage. Typcally, one can get up to 0:1 compresson wth almost
More informationMaximum Likelihood Estimation of Binary Dependent Variables Models: Probit and Logit. 1. General Formulation of Binary Dependent Variables Models
ECO 452 -- OE 4: Probt and Logt Models ECO 452 -- OE 4 Maxmum Lkelhood Estmaton of Bnary Dependent Varables Models: Probt and Logt hs note demonstrates how to formulate bnary dependent varables models
More informationEVALUATION OF THE VISCO-ELASTIC PROPERTIES IN ASPHALT RUBBER AND CONVENTIONAL MIXES
EVALUATION OF THE VISCO-ELASTIC PROPERTIES IN ASPHALT RUBBER AND CONVENTIONAL MIXES Manuel J. C. Mnhoto Polytechnc Insttute of Bragança, Bragança, Portugal E-mal: mnhoto@pb.pt Paulo A. A. Perera and Jorge
More informationChapter 2 - The Simple Linear Regression Model S =0. e i is a random error. S β2 β. This is a minimization problem. Solution is a calculus exercise.
Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where y + = β + β e for =,..., y and are observable varables e s a random error How can an estmaton rule be constructed for the
More informationSome Comments on Accelerating Convergence of Iterative Sequences Using Direct Inversion of the Iterative Subspace (DIIS)
Some Comments on Acceleratng Convergence of Iteratve Sequences Usng Drect Inverson of the Iteratve Subspace (DIIS) C. Davd Sherrll School of Chemstry and Bochemstry Georga Insttute of Technology May 1998
More informationOn the Interval Zoro Symmetric Single-step Procedure for Simultaneous Finding of Polynomial Zeros
Appled Mathematcal Scences, Vol. 5, 2011, no. 75, 3693-3706 On the Interval Zoro Symmetrc Sngle-step Procedure for Smultaneous Fndng of Polynomal Zeros S. F. M. Rusl, M. Mons, M. A. Hassan and W. J. Leong
More informationCode_Aster. Graphic charter for the realization of the formulas mathematics in documentation Code_Aster
Ttre : Charte graphque pour la réalsaton des formules [...] Date : /1/008 Page : 1/15 Organzaton (S):EDF/IMA/MMN Data-processng Manuel de Descrptf D8.01 booklet: Presentaton of documentaton D8.01.0 document
More informationDepartment of Quantitative Methods & Information Systems. Time Series and Their Components QMIS 320. Chapter 6
Department of Quanttatve Methods & Informaton Systems Tme Seres and Ther Components QMIS 30 Chapter 6 Fall 00 Dr. Mohammad Zanal These sldes were modfed from ther orgnal source for educatonal purpose only.
More informationCanonical transformations
Canoncal transformatons November 23, 2014 Recall that we have defned a symplectc transformaton to be any lnear transformaton M A B leavng the symplectc form nvarant, Ω AB M A CM B DΩ CD Coordnate transformatons,
More informationChapter 12. Ordinary Differential Equation Boundary Value (BV) Problems
Chapter. Ordnar Dfferental Equaton Boundar Value (BV) Problems In ths chapter we wll learn how to solve ODE boundar value problem. BV ODE s usuall gven wth x beng the ndependent space varable. p( x) q(
More informationChapter 14 Simple Linear Regression
Chapter 4 Smple Lnear Regresson Chapter 4 - Smple Lnear Regresson Manageral decsons often are based on the relatonshp between two or more varables. Regresson analss can be used to develop an equaton showng
More informationNumerical Heat and Mass Transfer
Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06-Fnte-Dfference Method (One-dmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analyss of Varance and Desgn of Experment-I MODULE VII LECTURE - 3 ANALYSIS OF COVARIANCE Dr Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur Any scentfc experment s performed
More informationModule 9. Lecture 6. Duality in Assignment Problems
Module 9 1 Lecture 6 Dualty n Assgnment Problems In ths lecture we attempt to answer few other mportant questons posed n earler lecture for (AP) and see how some of them can be explaned through the concept
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analyss of Varance and Desgn of Exerments-I MODULE III LECTURE - 2 EXPERIMENTAL DESIGN MODELS Dr. Shalabh Deartment of Mathematcs and Statstcs Indan Insttute of Technology Kanur 2 We consder the models
More informationOpen Systems: Chemical Potential and Partial Molar Quantities Chemical Potential
Open Systems: Chemcal Potental and Partal Molar Quanttes Chemcal Potental For closed systems, we have derved the followng relatonshps: du = TdS pdv dh = TdS + Vdp da = SdT pdv dg = VdP SdT For open systems,
More informationA Modified Neuber Method Avoiding Artefacts Under Random Loads
A Modfed Neuber Method Avodng Artefacts Under Random Loads T. Herbland a,b, G. Calletaud a, J. L. Chaboche c, S. Qulc a, F. Gallerneau c a Mnes Pars Pars Tech, CNRS UMR 7633, P 87, 91003 vry cedex, France
More informationCredit Card Pricing and Impact of Adverse Selection
Credt Card Prcng and Impact of Adverse Selecton Bo Huang and Lyn C. Thomas Unversty of Southampton Contents Background Aucton model of credt card solctaton - Errors n probablty of beng Good - Errors n
More informationComparative Studies of Law of Conservation of Energy. and Law Clusters of Conservation of Generalized Energy
Comparatve Studes of Law of Conservaton of Energy and Law Clusters of Conservaton of Generalzed Energy No.3 of Comparatve Physcs Seres Papers Fu Yuhua (CNOOC Research Insttute, E-mal:fuyh1945@sna.com)
More informationUsing T.O.M to Estimate Parameter of distributions that have not Single Exponential Family
IOSR Journal of Mathematcs IOSR-JM) ISSN: 2278-5728. Volume 3, Issue 3 Sep-Oct. 202), PP 44-48 www.osrjournals.org Usng T.O.M to Estmate Parameter of dstrbutons that have not Sngle Exponental Famly Jubran
More informationLecture 21: Numerical methods for pricing American type derivatives
Lecture 21: Numercal methods for prcng Amercan type dervatves Xaoguang Wang STAT 598W Aprl 10th, 2014 (STAT 598W) Lecture 21 1 / 26 Outlne 1 Fnte Dfference Method Explct Method Penalty Method (STAT 598W)
More informationEcon107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4)
I. Classcal Assumptons Econ7 Appled Econometrcs Topc 3: Classcal Model (Studenmund, Chapter 4) We have defned OLS and studed some algebrac propertes of OLS. In ths topc we wll study statstcal propertes
More informationRockefeller College University at Albany
Rockefeller College Unverst at Alban PAD 705 Handout: Maxmum Lkelhood Estmaton Orgnal b Davd A. Wse John F. Kenned School of Government, Harvard Unverst Modfcatons b R. Karl Rethemeer Up to ths pont n
More informationEntropy of Markov Information Sources and Capacity of Discrete Input Constrained Channels (from Immink, Coding Techniques for Digital Recorders)
Entropy of Marov Informaton Sources and Capacty of Dscrete Input Constraned Channels (from Immn, Codng Technques for Dgtal Recorders). Entropy of Marov Chans We have already ntroduced the noton of entropy
More informationThermodynamics and statistical mechanics in materials modelling II
Course MP3 Lecture 8/11/006 (JAE) Course MP3 Lecture 8/11/006 Thermodynamcs and statstcal mechancs n materals modellng II A bref résumé of the physcal concepts used n materals modellng Dr James Ellott.1
More informationChapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems
Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons
More informationReport on Image warping
Report on Image warpng Xuan Ne, Dec. 20, 2004 Ths document summarzed the algorthms of our mage warpng soluton for further study, and there s a detaled descrpton about the mplementaton of these algorthms.
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More informationInductance Calculation for Conductors of Arbitrary Shape
CRYO/02/028 Aprl 5, 2002 Inductance Calculaton for Conductors of Arbtrary Shape L. Bottura Dstrbuton: Internal Summary In ths note we descrbe a method for the numercal calculaton of nductances among conductors
More informationUniversity of Washington Department of Chemistry Chemistry 453 Winter Quarter 2015
Lecture 2. 1/07/15-1/09/15 Unversty of Washngton Department of Chemstry Chemstry 453 Wnter Quarter 2015 We are not talkng about truth. We are talkng about somethng that seems lke truth. The truth we want
More informationLecture 16 Statistical Analysis in Biomaterials Research (Part II)
3.051J/0.340J 1 Lecture 16 Statstcal Analyss n Bomaterals Research (Part II) C. F Dstrbuton Allows comparson of varablty of behavor between populatons usng test of hypothess: σ x = σ x amed for Brtsh statstcan
More informationCHAPTER IV RESEARCH FINDING AND DISCUSSIONS
CHAPTER IV RESEARCH FINDING AND DISCUSSIONS A. Descrpton of Research Fndng. The Implementaton of Learnng Havng ganed the whole needed data, the researcher then dd analyss whch refers to the statstcal data
More informationis the calculated value of the dependent variable at point i. The best parameters have values that minimize the squares of the errors
Multple Lnear and Polynomal Regresson wth Statstcal Analyss Gven a set of data of measured (or observed) values of a dependent varable: y versus n ndependent varables x 1, x, x n, multple lnear regresson
More informationNON-CENTRAL 7-POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS
IJRRAS 8 (3 September 011 www.arpapress.com/volumes/vol8issue3/ijrras_8_3_08.pdf NON-CENTRAL 7-POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS H.O. Bakodah Dept. of Mathematc
More informationDO NOT OPEN THE QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY THE CHIEF INVIGILATOR. Introductory Econometrics 1 hour 30 minutes
25/6 Canddates Only January Examnatons 26 Student Number: Desk Number:...... DO NOT OPEN THE QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY THE CHIEF INVIGILATOR Department Module Code Module Ttle Exam Duraton
More informationPsychology 282 Lecture #24 Outline Regression Diagnostics: Outliers
Psychology 282 Lecture #24 Outlne Regresson Dagnostcs: Outlers In an earler lecture we studed the statstcal assumptons underlyng the regresson model, ncludng the followng ponts: Formal statement of assumptons.
More information8/25/17. Data Modeling. Data Modeling. Data Modeling. Patrice Koehl Department of Biological Sciences National University of Singapore
8/5/17 Data Modelng Patrce Koehl Department of Bologcal Scences atonal Unversty of Sngapore http://www.cs.ucdavs.edu/~koehl/teachng/bl59 koehl@cs.ucdavs.edu Data Modelng Ø Data Modelng: least squares Ø
More informationSTAT 405 BIOSTATISTICS (Fall 2016) Handout 15 Introduction to Logistic Regression
STAT 45 BIOSTATISTICS (Fall 26) Handout 5 Introducton to Logstc Regresson Ths handout covers materal found n Secton 3.7 of your text. You may also want to revew regresson technques n Chapter. In ths handout,
More informationProblem Points Score Total 100
Physcs 450 Solutons of Sample Exam I Problem Ponts Score 1 8 15 3 17 4 0 5 0 Total 100 All wor must be shown n order to receve full credt. Wor must be legble and comprehensble wth answers clearly ndcated.
More informationImplicit Integration Henyey Method
Implct Integraton Henyey Method In realstc stellar evoluton codes nstead of a drect ntegraton usng for example the Runge-Kutta method one employs an teratve mplct technque. Ths s because the structure
More informationP R. Lecture 4. Theory and Applications of Pattern Recognition. Dept. of Electrical and Computer Engineering /
Theory and Applcatons of Pattern Recognton 003, Rob Polkar, Rowan Unversty, Glassboro, NJ Lecture 4 Bayes Classfcaton Rule Dept. of Electrcal and Computer Engneerng 0909.40.0 / 0909.504.04 Theory & Applcatons
More informationCS-433: Simulation and Modeling Modeling and Probability Review
CS-433: Smulaton and Modelng Modelng and Probablty Revew Exercse 1. (Probablty of Smple Events) Exercse 1.1 The owner of a camera shop receves a shpment of fve cameras from a camera manufacturer. Unknown
More informationTensor Smooth Length for SPH Modelling of High Speed Impact
Tensor Smooth Length for SPH Modellng of Hgh Speed Impact Roman Cherepanov and Alexander Gerasmov Insttute of Appled mathematcs and mechancs, Tomsk State Unversty 634050, Lenna av. 36, Tomsk, Russa RCherepanov82@gmal.com,Ger@npmm.tsu.ru
More informationStatistics for Business and Economics
Statstcs for Busness and Economcs Chapter 11 Smple Regresson Copyrght 010 Pearson Educaton, Inc. Publshng as Prentce Hall Ch. 11-1 11.1 Overvew of Lnear Models n An equaton can be ft to show the best lnear
More informationComposite Hypotheses testing
Composte ypotheses testng In many hypothess testng problems there are many possble dstrbutons that can occur under each of the hypotheses. The output of the source s a set of parameters (ponts n a parameter
More informationCHARACTERISTICS OF COMPLEX SEPARATION SCHEMES AND AN ERROR OF SEPARATION PRODUCTS OUTPUT DETERMINATION
Górnctwo Geonżynera Rok 0 Zeszyt / 006 Igor Konstantnovch Mladetskj * Petr Ivanovch Plov * Ekaterna Nkolaevna Kobets * Tasya Igorevna Markova * CHARACTERISTICS OF COMPLEX SEPARATION SCHEMES AND AN ERROR
More informationChapter - 2. Distribution System Power Flow Analysis
Chapter - 2 Dstrbuton System Power Flow Analyss CHAPTER - 2 Radal Dstrbuton System Load Flow 2.1 Introducton Load flow s an mportant tool [66] for analyzng electrcal power system network performance. Load
More informationChapter 8 Indicator Variables
Chapter 8 Indcator Varables In general, e explanatory varables n any regresson analyss are assumed to be quanttatve n nature. For example, e varables lke temperature, dstance, age etc. are quanttatve n
More informatione i is a random error
Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where + β + β e for,..., and are observable varables e s a random error How can an estmaton rule be constructed for the unknown
More information