Fall 2012 Analysis of Experimental Measurements B. Eisenstein/rev. S. Errede. ) with a symmetric Pcovariance matrix of the y( x ) measurements V
|
|
- Verity Lloyd
- 6 years ago
- Views:
Transcription
1 Fall Analyss o Experental Measureents B Esensten/rev S Errede General Least Squares wth General Constrants: Suppose we have easureents y( x ( y( x, y( x,, y( x wth a syetrc covarance atrx o the y( x easureents y( x Suppose the theory predcton ( ; λ ( ( ; λ, ( ; λ,, ( ; λ nvolves M (< paraeters λ ( λ, λ,, λm y x y x y x y x n soe general (e not necessarly lnear anner Addtonally, suppose there are unctons ( λ ( λ, ( λ,, ( λ that relate (e constran the M λ -paraeters n soe general (but not necessarly lnear anner va use o Lagrange Multplers α ( α α α he χ ( ; λα s dened as:,,, y( x χ λα ; χ λ + α λ y x y x; λ y x y x; λ + α λ where y( x s the easureents, and the syetrc nverse o the covarance atrx o the y ( x colun vector {nb In the lnear constrant case ( λ Bλ b ( λ ay be non-lnear unctons o the M λ -paraeters} We nze the χ ( λα ; by takng dervatves wrt ( ; λ contans the constrant equatons However, n general the constrant equatons λ α We (agan use the teraton technque here too Suppose that ater ν teratons, we have obtaned a set o approxate values o the M λ -paraeters and Lagrange Multplers α : ν λ ν ν λ λ ν λ M and: ν α ν ν α α ν α We then expand (e lnearze χ ( λα ; n a aylor seres around these ponts ( ν ν ; ν ν ν ν ν ν ν ν then solve or Δλ ( Δλ, Δλ,, ΔλM, Δα ( Δα, Δα,, Δα and terate urther λ α, slar to the dscusson n 598AEM Lect Notes (p 5-9 For addtonal detals, see eg ndvdual progra wrte-ups or eg advanced texts on ths subject * * Let us assue that we have deterned the best values ( ; the Lagrange Multpler constraned LSQ t ethod λ α o these paraeters usng We can obtan a better estate, we wsh, o the easured rando varables y x hs procedure goes by the nae Adjustent o Observatons : 598AEM Lecture Notes
2 Fall Analyss o Experental Measureents B Esensten/rev S Errede We dene a colun vector o easured values o the rando varables (nb these ay not necessarly be ndependent, wth correspondng syetrc covarance atrx o the easureents We want to know the true values (e expectaton values o the easureents: E ˆ ˆ ˆ ˆ [ ],,, We wll estate the usng a LSQ ttng ethod, and call the estates the tted values o the easureents We obtan the tted values o the easureents by adjustng the easureents so that: Each easureent s allowed to ove by an aount deterned ro the sze o the uncertanty on the easureent, σ he resultng tted values o the easureents satsy one or ore constrants We dene a colun vector: o tted values o, e the estates o ˆ Let there be constrants whch can be expressed n the or: ( (,,,,,,,,, ( or, denng a nb In general, these wll be non-lnear equatons colun vector: ( (,,, (,,, Reeberng the teratve χ nzaton ethod(s, we choose to work wth lnearzed correctons : (,,, c c c or, denng a colun vector: c In ters o χ nzaton, snce the s are just constants, nzng c s equvalent to nzng χ wth respect to χ wth respect to 598AEM Lecture Notes
3 Fall Analyss o Experental Measureents B Esensten/rev S Errede What should we actually nze? I we use χ ( ( c c χ ( c the soluton s (obvously, e the best estate o ˆ s, tsel In order to do better, we ust add n soe new noraton n ths case, the requreent that the constrants be satsed by the s hus, we nstead nze: ( c χ ; α χ + α where: c c+ α akng dervatves o χ ( c; α α α α α, we obtan: s a colun vector o Lagrange Multplers ( χ α ( c; α ( ( c; χ ( α χ α c ; (e the constrants wll be satsed ( c+ α Note that s a atrx ( B( wth jk th k eleent: B jk, j where j,,, ranges over the tted varables and k,,, ranges over the constrants hus, the equatons that we need to solve n order to accoplsh ths χ ( c; ( and: c + B α α nzaton are: For the general non-lnear case, we ust resort to approxaton ethods We aylor seres expand (e lnearze the constrant equatons around, an ntal estate o the tted values o the easureents hen we requre that: As usual, we assue that ( ( + + s sall enough so that we can saely neglect/gnore the and the hgher- ters n the aylor seres expanson nvolvng hgher powers o ( order dervatves o ( (hs step s known as lnearzng the constrants 598AEM Lecture Notes 3
4 Fall Analyss o Experental Measureents B Esensten/rev S Errede hen: ( + + B A neat trck exsts or solvng ths convenently We wrte: hen: ( ( ( ( c c where: c ( We rewrte ths as: + B ( cc and: c ( + c c + B cc where t s plctly understood that the dervatves and the constrants are evaluated at the an ntal estate he other equaton we ust solve s: yelds: c Bα hus: ( r B c B B α Hα where: H B B hen: B c B c r c+ Bα, whch, ultplyng on the LHS by By constructon, H B B s a square, syetrc (and real atrx, and thereore, t has a square, syetrc (and real nverse H ( B B hus, ultplyng r B c ( B B α Hα on the LHS by H ( B B Lagrange Multplers: α H r and Fnally, the result o ths step s: + c c B BH r gves the α + gves the correcton We explctly need to check/very whether or not ths new satses the constrants: ( I t does, then we re done I not, then we use ths as a new and repeat (e terate the above procedure untl ( s satsed I ( s satsed, then ( also 598AEM Lecture Notes 4
5 Fall Analyss o Experental Measureents B Esensten/rev S Errede Now let us calculate χ ( c; I α ro the quanttes that we have obtaned s satsed, recallng that and H B B are syetrc atrces, then: hus, ( c; χ α r α χ c; α c c+ α c c ( BH r ( BH r r H B ( BH r r H ( B B H r H r H H H r H r α r α hs s the value o r χ ater the step to + c Next, we deterne the covarance atrx o the tted values usng error propagaton: Now t s just algebra But: r B c B ( ( BH r + c c r BH, thus: ( r B B B and thus: ( BH B hen: BH B BH B ( ( BH B BH B Multplyng ths out on the RHS and agan usng H ths sples to: B B B H B 598AEM Lecture Notes 5
6 Fall Analyss o Experental Measureents B Esensten/rev S Errede As beore, snce atrx s syetrc, t has postve dagonal eleents Lkewse, the syetrc H B B H B B also has postve dagonal eleents, and so does hereore, ro ( B H ( B ( B( B B ( B, we see that the dagonal eleents o are saller than the dagonal eleents o hus, the -standard devaton uncertantes assocated the adjusted (e tted easureents are less than the -standard devaton uncertantes on the orgnal easureents ull Quanttes: ull quanttes are dstrbutons o noralzed/ractonal derences between the tted easured quanttes whch can be very helpul n veryng the valdty o the LSQ ttng procedure We dene the th c pull quantty as the noralzed correcton: p c where the brackets are synonyous wth the expectaton value, e: c E[ c ] E[ ] Note that there s no bas, then: c I everythng s nce e the nput easureents are Gaussan/norally-dstrbuted and ther uncertantes, as contaned n the ndvdual eleents o the covarance atrx o the easureents have all been correctly / properly assgned and the varous approxatons and assuptons are all vald, then the N, p should be dstrbuted as By explctly lookng at the dstrbutons (eg hstogras o the p or any ndependent easureents o each o the, we can turn ths around and check the ngredents lsted above, especally whether the uncertantes on the ndvdual have ndeed been correctly assgned or not, by seeng whether the pull dstrbuton p or each N, or not s ndeed dstrbuted as Let us suppose that we have perored the Adjustent o Observatons, startng wth our ntal easureents and arrvng at nal adjusted/tted values It s not trval to evaluate the c he colun vector correcton c We also have the covarance atrx o the easureents and that o the adjusted/tted easureents E[( ( ] ( ( Forally: ˆ ˆ ˆ ˆ j j j j j and: ( E ˆ ˆ ˆ ˆ [] j j j j j 598AEM Lecture Notes 6
7 Fall Analyss o Experental Measureents B Esensten/rev S Errede I the easureents are truly unbased, then: ˆ ˆ, e E [ ] E[ ] hus: c ( ˆ ( ˆ For convenence, we dene the colun vectors: ( ˆ hen: ( ˆ ( ˆ and: δ and: ( ˆ c δ δ or: δ δ c E[ δδ ] δδ δ c δ c δ δ + c c cδ or: + c c cδ or: c c + cδ c δ hs s what we need, snce the dagonal eleents o the covarance atrx are the c But we need to evaluate cδ n order to nsh the job c c c Let us evaluate c δ or the case where δ Dδ Note that ths s a lnear relatonshp, wth D beng a square atrx hen: δ Dδ ( ˆ D( ˆ Or: ( ˆ ( ( D Dˆ D Dˆ ˆ D D ˆ Now: ( ˆ ( ˆ c δ δ cδ δ δ δ δ δ δδ δδ But ro δ Dδ we get: D ( δ, and ro: D ( D ˆ ( δ we get: D cδ δδ δδ D δδ D D Or: c ( D δ, snce s a syetrc atrx hus: δ c D 598AEM Lecture Notes 7
8 Fall Analyss o Experental Measureents B Esensten/rev S Errede r But we earler derved: + BH ( BH B and: ( B H ( B cδ ( B H B BH B ( B H ( B ( BH B + hus or the lnear case where δ Dδ : c c + cδ c c p c ( p c ( c ( or: p σ σ Navely, one ght expect between and Snce ( B H ( B c σ σ + σ, but ths gnores/neglects the correlaton, then σ calculatng the p pulls > σ and thus we won t get nto trouble n Exaples o LSQ t pulls are shown n the gures below or a oy Monte Carlo progra that carres out LSQ ts to branchng ratos o neutral and charged chared D esons, ro a paper by Werner M Sun, Sultaneous least-squares treatent o statstcal and systeatc uncertantes, Nucl Inst Meth hys Res A (6 598AEM Lecture Notes 8
Fall 2012 Analysis of Experimental Measurements B. Eisenstein/rev. S. Errede
Fall 0 Analyss of Expermental easurements B. Esensten/rev. S. Errede We now reformulate the lnear Least Squares ethod n more general terms, sutable for (eventually extendng to the non-lnear case, and also
More informationWhat is LP? LP is an optimization technique that allocates limited resources among competing activities in the best possible manner.
(C) 998 Gerald B Sheblé, all rghts reserved Lnear Prograng Introducton Contents I. What s LP? II. LP Theor III. The Splex Method IV. Refneents to the Splex Method What s LP? LP s an optzaton technque that
More informationXII.3 The EM (Expectation-Maximization) Algorithm
XII.3 The EM (Expectaton-Maxzaton) Algorth Toshnor Munaata 3/7/06 The EM algorth s a technque to deal wth varous types of ncoplete data or hdden varables. It can be appled to a wde range of learnng probles
More informationLeast Squares Fitting of Data
Least Squares Fttng of Data Davd Eberly Geoetrc Tools, LLC http://www.geoetrctools.co/ Copyrght c 1998-2014. All Rghts Reserved. Created: July 15, 1999 Last Modfed: February 9, 2008 Contents 1 Lnear Fttng
More informationLECTURE :FACTOR ANALYSIS
LCUR :FACOR ANALYSIS Rta Osadchy Based on Lecture Notes by A. Ng Motvaton Dstrbuton coes fro MoG Have suffcent aount of data: >>n denson Use M to ft Mture of Gaussans nu. of tranng ponts If
More informationy new = M x old Feature Selection: Linear Transformations Constraint Optimization (insertion)
Feature Selecton: Lnear ransforatons new = M x old Constrant Optzaton (nserton) 3 Proble: Gven an objectve functon f(x) to be optzed and let constrants be gven b h k (x)=c k, ovng constants to the left,
More information1 Definition of Rademacher Complexity
COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture #9 Scrbe: Josh Chen March 5, 2013 We ve spent the past few classes provng bounds on the generalzaton error of PAClearnng algorths for the
More informationOn Pfaff s solution of the Pfaff problem
Zur Pfaff scen Lösung des Pfaff scen Probles Mat. Ann. 7 (880) 53-530. On Pfaff s soluton of te Pfaff proble By A. MAYER n Lepzg Translated by D. H. Delpenc Te way tat Pfaff adopted for te ntegraton of
More informationInternational Journal of Mathematical Archive-9(3), 2018, Available online through ISSN
Internatonal Journal of Matheatcal Archve-9(3), 208, 20-24 Avalable onlne through www.ja.nfo ISSN 2229 5046 CONSTRUCTION OF BALANCED INCOMPLETE BLOCK DESIGNS T. SHEKAR GOUD, JAGAN MOHAN RAO M AND N.CH.
More informationCOMP th April, 2007 Clement Pang
COMP 540 12 th Aprl, 2007 Cleent Pang Boostng Cobnng weak classers Fts an Addtve Model Is essentally Forward Stagewse Addtve Modelng wth Exponental Loss Loss Functons Classcaton: Msclasscaton, Exponental,
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 informationLeast Squares Fitting of Data
Least Squares Fttng of Data Davd Eberly Geoetrc Tools, LLC http://www.geoetrctools.co/ Copyrght c 1998-2015. All Rghts Reserved. Created: July 15, 1999 Last Modfed: January 5, 2015 Contents 1 Lnear Fttng
More informationLecture 3 Stat102, Spring 2007
Lecture 3 Stat0, Sprng 007 Chapter 3. 3.: Introducton to regresson analyss Lnear regresson as a descrptve technque The least-squares equatons Chapter 3.3 Samplng dstrbuton of b 0, b. Contnued n net lecture
More informationExcess Error, Approximation Error, and Estimation Error
E0 370 Statstcal Learnng Theory Lecture 10 Sep 15, 011 Excess Error, Approxaton Error, and Estaton Error Lecturer: Shvan Agarwal Scrbe: Shvan Agarwal 1 Introducton So far, we have consdered the fnte saple
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 informationFall 2012 Analysis of Experimental Measurements B. Eisenstein/rev. S. Errede. . For P such independent random variables (aka degrees of freedom): 1 =
Fall Analss of Epermental Measurements B. Esensten/rev. S. Errede More on : The dstrbuton s the.d.f. for a (normalzed sum of squares of ndependent random varables, each one of whch s dstrbuted as N (,.
More informationElastic Collisions. Definition: two point masses on which no external forces act collide without losing any energy.
Elastc Collsons Defnton: to pont asses on hch no external forces act collde thout losng any energy v Prerequstes: θ θ collsons n one denson conservaton of oentu and energy occurs frequently n everyday
More informationFermi-Dirac statistics
UCC/Physcs/MK/EM/October 8, 205 Fer-Drac statstcs Fer-Drac dstrbuton Matter partcles that are eleentary ostly have a type of angular oentu called spn. hese partcles are known to have a agnetc oent whch
More informationOur focus will be on linear systems. A system is linear if it obeys the principle of superposition and homogenity, i.e.
SSTEM MODELLIN In order to solve a control syste proble, the descrptons of the syste and ts coponents ust be put nto a for sutable for analyss and evaluaton. The followng ethods can be used to odel physcal
More informationSlobodan Lakić. Communicated by R. Van Keer
Serdca Math. J. 21 (1995), 335-344 AN ITERATIVE METHOD FOR THE MATRIX PRINCIPAL n-th ROOT Slobodan Lakć Councated by R. Van Keer In ths paper we gve an teratve ethod to copute the prncpal n-th root and
More informationApplied Mathematics Letters
Appled Matheatcs Letters 2 (2) 46 5 Contents lsts avalable at ScenceDrect Appled Matheatcs Letters journal hoepage: wwwelseverco/locate/al Calculaton of coeffcents of a cardnal B-splne Gradr V Mlovanovć
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 informationarxiv: v2 [math.co] 3 Sep 2017
On the Approxate Asyptotc Statstcal Independence of the Peranents of 0- Matrces arxv:705.0868v2 ath.co 3 Sep 207 Paul Federbush Departent of Matheatcs Unversty of Mchgan Ann Arbor, MI, 4809-043 Septeber
More information4DVAR, according to the name, is a four-dimensional variational method.
4D-Varatonal Data Assmlaton (4D-Var) 4DVAR, accordng to the name, s a four-dmensonal varatonal method. 4D-Var s actually a drect generalzaton of 3D-Var to handle observatons that are dstrbuted n tme. The
More informationSystem in Weibull Distribution
Internatonal Matheatcal Foru 4 9 no. 9 94-95 Relablty Equvalence Factors of a Seres-Parallel Syste n Webull Dstrbuton M. A. El-Dacese Matheatcs Departent Faculty of Scence Tanta Unversty Tanta Egypt eldacese@yahoo.co
More information10-701/ Machine Learning, Fall 2005 Homework 3
10-701/15-781 Machne Learnng, Fall 2005 Homework 3 Out: 10/20/05 Due: begnnng of the class 11/01/05 Instructons Contact questons-10701@autonlaborg for queston Problem 1 Regresson and Cross-valdaton [40
More informationCHAPTER 7 CONSTRAINED OPTIMIZATION 1: THE KARUSH-KUHN-TUCKER CONDITIONS
CHAPER 7 CONSRAINED OPIMIZAION : HE KARUSH-KUHN-UCKER CONDIIONS 7. Introducton We now begn our dscusson of gradent-based constraned optzaton. Recall that n Chapter 3 we looked at gradent-based unconstraned
More informationFinite Vector Space Representations Ross Bannister Data Assimilation Research Centre, Reading, UK Last updated: 2nd August 2003
Fnte Vector Space epresentatons oss Bannster Data Asslaton esearch Centre, eadng, UK ast updated: 2nd August 2003 Contents What s a lnear vector space?......... 1 About ths docuent............ 2 1. Orthogonal
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 informationGeneral Tips on How to Do Well in Physics Exams. 1. Establish a good habit in keeping track of your steps. For example, when you use the equation
General Tps on How to Do Well n Physcs Exams 1. Establsh a good habt n keepng track o your steps. For example when you use the equaton 1 1 1 + = d d to solve or d o you should rst rewrte t as 1 1 1 = d
More informationCHAPTER 6 CONSTRAINED OPTIMIZATION 1: K-T CONDITIONS
Chapter 6: Constraned Optzaton CHAPER 6 CONSRAINED OPIMIZAION : K- CONDIIONS Introducton We now begn our dscusson of gradent-based constraned optzaton. Recall that n Chapter 3 we looked at gradent-based
More informationQuantum Mechanics for Scientists and Engineers
Quantu Mechancs or Scentsts and Engneers Sangn K Advanced Coputatonal Electroagnetcs Lab redkd@yonse.ac.kr Nov. 4 th, 26 Outlne Quantu Mechancs or Scentsts and Engneers Blnear expanson o lnear operators
More informationPROBABILITY AND STATISTICS Vol. III - Analysis of Variance and Analysis of Covariance - V. Nollau ANALYSIS OF VARIANCE AND ANALYSIS OF COVARIANCE
ANALYSIS OF VARIANCE AND ANALYSIS OF COVARIANCE V. Nollau Insttute of Matheatcal Stochastcs, Techncal Unversty of Dresden, Gerany Keywords: Analyss of varance, least squares ethod, odels wth fxed effects,
More information,..., k N. , k 2. ,..., k i. The derivative with respect to temperature T is calculated by using the chain rule: & ( (5) dj j dt = "J j. k i.
Suppleentary Materal Dervaton of Eq. 1a. Assue j s a functon of the rate constants for the N coponent reactons: j j (k 1,,..., k,..., k N ( The dervatve wth respect to teperature T s calculated by usng
More informationChapter 3 Differentiation and Integration
MEE07 Computer Modelng Technques n Engneerng Chapter Derentaton and Integraton Reerence: An Introducton to Numercal Computatons, nd edton, S. yakowtz and F. zdarovsky, Mawell/Macmllan, 990. Derentaton
More informationXiangwen Li. March 8th and March 13th, 2001
CS49I Approxaton Algorths The Vertex-Cover Proble Lecture Notes Xangwen L March 8th and March 3th, 00 Absolute Approxaton Gven an optzaton proble P, an algorth A s an approxaton algorth for P f, for an
More information36.1 Why is it important to be able to find roots to systems of equations? Up to this point, we have discussed how to find the solution to
ChE Lecture Notes - D. Keer, 5/9/98 Lecture 6,7,8 - Rootndng n systems o equatons (A) Theory (B) Problems (C) MATLAB Applcatons Tet: Supplementary notes rom Instructor 6. Why s t mportant to be able to
More informationBAYESIAN CURVE FITTING USING PIECEWISE POLYNOMIALS. Dariusz Biskup
BAYESIAN CURVE FITTING USING PIECEWISE POLYNOMIALS Darusz Bskup 1. Introducton The paper presents a nonparaetrc procedure for estaton of an unknown functon f n the regresson odel y = f x + ε = N. (1) (
More informationPreference and Demand Examples
Dvson of the Huantes and Socal Scences Preference and Deand Exaples KC Border October, 2002 Revsed Noveber 206 These notes show how to use the Lagrange Karush Kuhn Tucker ultpler theores to solve the proble
More information1 GSW Iterative Techniques for y = Ax
1 for y = A I m gong to cheat here. here are a lot of teratve technques that can be used to solve the general case of a set of smultaneous equatons (wrtten n the matr form as y = A), but ths chapter sn
More informationPHYS 450 Spring semester Lecture 02: Dealing with Experimental Uncertainties. Ron Reifenberger Birck Nanotechnology Center Purdue University
PHYS 45 Sprng semester 7 Lecture : Dealng wth Expermental Uncertantes Ron Refenberger Brck anotechnology Center Purdue Unversty Lecture Introductory Comments Expermental errors (really expermental uncertantes)
More informationDenote the function derivatives f(x) in given points. x a b. Using relationships (1.2), polynomials (1.1) are written in the form
SET OF METHODS FO SOUTION THE AUHY POBEM FO STIFF SYSTEMS OF ODINAY DIFFEENTIA EUATIONS AF atypov and YuV Nulchev Insttute of Theoretcal and Appled Mechancs SB AS 639 Novosbrs ussa Introducton A constructon
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 informationHidden Markov Models & The Multivariate Gaussian (10/26/04)
CS281A/Stat241A: Statstcal Learnng Theory Hdden Markov Models & The Multvarate Gaussan (10/26/04) Lecturer: Mchael I. Jordan Scrbes: Jonathan W. Hu 1 Hdden Markov Models As a bref revew, hdden Markov models
More informationSolutions Homework 4 March 5, 2018
1 Solutons Homework 4 March 5, 018 Soluton to Exercse 5.1.8: Let a IR be a translaton and c > 0 be a re-scalng. ˆb1 (cx + a) cx n + a (cx 1 + a) c x n x 1 cˆb 1 (x), whch shows ˆb 1 s locaton nvarant and
More informationBy M. O'Neill,* I. G. Sinclairf and Francis J. Smith
52 Polynoal curve fttng when abscssas and ordnates are both subject to error By M. O'Nell,* I. G. Snclarf and Francs J. Sth Departents of Coputer Scence and Appled Matheatcs, School of Physcs and Appled
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 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 informationModified parallel multisplitting iterative methods for non-hermitian positive definite systems
Adv Coput ath DOI 0.007/s0444-0-9262-8 odfed parallel ultsplttng teratve ethods for non-hertan postve defnte systes Chuan-Long Wang Guo-Yan eng Xue-Rong Yong Receved: Septeber 20 / Accepted: 4 Noveber
More informationP A = (P P + P )A = P (I P T (P P ))A = P (A P T (P P )A) Hence if we let E = P T (P P A), We have that
Backward Error Analyss for House holder Reectors We want to show that multplcaton by householder reectors s backward stable. In partcular we wsh to show fl(p A) = P (A) = P (A + E where P = I 2vv T s the
More informationChapter 11: Simple Linear Regression and Correlation
Chapter 11: Smple Lnear Regresson and Correlaton 11-1 Emprcal Models 11-2 Smple Lnear Regresson 11-3 Propertes of the Least Squares Estmators 11-4 Hypothess Test n Smple Lnear Regresson 11-4.1 Use of t-tests
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 informationQuantum Particle Motion in Physical Space
Adv. Studes Theor. Phys., Vol. 8, 014, no. 1, 7-34 HIKARI Ltd, www.-hkar.co http://dx.do.org/10.1988/astp.014.311136 Quantu Partcle Moton n Physcal Space A. Yu. Saarn Dept. of Physcs, Saara State Techncal
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 informationSalmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2
Salmon: Lectures on partal dfferental equatons 5. Classfcaton of second-order equatons There are general methods for classfyng hgher-order partal dfferental equatons. One s very general (applyng even to
More informationb ), which stands for uniform distribution on the interval a x< b. = 0 elsewhere
Fall Analyss of Epermental Measurements B. Esensten/rev. S. Errede Some mportant probablty dstrbutons: Unform Bnomal Posson Gaussan/ormal The Unform dstrbuton s often called U( a, b ), hch stands for unform
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 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 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 informationHopfield Training Rules 1 N
Hopfeld Tranng Rules To memorse a sngle pattern Suppose e set the eghts thus - = p p here, s the eght beteen nodes & s the number of nodes n the netor p s the value requred for the -th node What ll the
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 informationMinimization of l 2 -Norm of the KSOR Operator
ournal of Matheatcs and Statstcs 8 (): 6-70, 0 ISSN 59-36 0 Scence Publcatons do:0.38/jssp.0.6.70 Publshed Onlne 8 () 0 (http://www.thescpub.co/jss.toc) Mnzaton of l -Nor of the KSOR Operator Youssef,
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 informationprinceton univ. F 17 cos 521: Advanced Algorithm Design Lecture 7: LP Duality Lecturer: Matt Weinberg
prnceton unv. F 17 cos 521: Advanced Algorthm Desgn Lecture 7: LP Dualty Lecturer: Matt Wenberg Scrbe: LP Dualty s an extremely useful tool for analyzng structural propertes of lnear programs. Whle there
More informationCOS 511: Theoretical Machine Learning
COS 5: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture #0 Scrbe: José Sões Ferrera March 06, 203 In the last lecture the concept of Radeacher coplexty was ntroduced, wth the goal of showng that
More informationGradient Descent Learning and Backpropagation
Artfcal Neural Networks (art 2) Chrstan Jacob Gradent Descent Learnng and Backpropagaton CSC 533 Wnter 200 Learnng by Gradent Descent Defnton of the Learnng roble Let us start wth the sple case of lnear
More informationChat eld, C. and A.J.Collins, Introduction to multivariate analysis. Chapman & Hall, 1980
MT07: Multvarate Statstcal Methods Mke Tso: emal mke.tso@manchester.ac.uk Webpage for notes: http://www.maths.manchester.ac.uk/~mkt/new_teachng.htm. Introducton to multvarate data. Books Chat eld, C. and
More informationEconomics 130. Lecture 4 Simple Linear Regression Continued
Economcs 130 Lecture 4 Contnued Readngs for Week 4 Text, Chapter and 3. We contnue wth addressng our second ssue + add n how we evaluate these relatonshps: Where do we get data to do ths analyss? How do
More informationAPPROXIMATE PRICES OF BASKET AND ASIAN OPTIONS DUPONT OLIVIER. Premia 14
APPROXIMAE PRICES OF BASKE AND ASIAN OPIONS DUPON OLIVIER Prema 14 Contents Introducton 1 1. Framewor 1 1.1. Baset optons 1.. Asan optons. Computng the prce 3. Lower bound 3.1. Closed formula for the prce
More informationThe Prncpal Component Transform The Prncpal Component Transform s also called Karhunen-Loeve Transform (KLT, Hotellng Transform, oregenvector Transfor
Prncpal Component Transform Multvarate Random Sgnals A real tme sgnal x(t can be consdered as a random process and ts samples x m (m =0; ;N, 1 a random vector: The mean vector of X s X =[x0; ;x N,1] T
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 informationQuantum Mechanics for Scientists and Engineers. David Miller
Quantum Mechancs for Scentsts and Engneers Davd Mller Types of lnear operators Types of lnear operators Blnear expanson of operators Blnear expanson of lnear operators We know that we can expand functons
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 informationOPTIMISATION. Introduction Single Variable Unconstrained Optimisation Multivariable Unconstrained Optimisation Linear Programming
OPTIMIATION Introducton ngle Varable Unconstraned Optmsaton Multvarable Unconstraned Optmsaton Lnear Programmng Chapter Optmsaton /. Introducton In an engneerng analss, sometmes etremtes, ether mnmum or
More informationImplicit scaling of linear least squares problems
RAL-TR-98-07 1 Iplct scalng of lnear least squares probles by J. K. Red Abstract We consder the soluton of weghted lnear least squares probles by Householder transforatons wth plct scalng, that s, wth
More informationCSci 6974 and ECSE 6966 Math. Tech. for Vision, Graphics and Robotics Lecture 21, April 17, 2006 Estimating A Plane Homography
CSc 6974 and ECSE 6966 Math. Tech. for Vson, Graphcs and Robotcs Lecture 21, Aprl 17, 2006 Estmatng A Plane Homography Overvew We contnue wth a dscusson of the major ssues, usng estmaton of plane projectve
More informationReview: Fit a line to N data points
Revew: Ft a lne to data ponts Correlated parameters: L y = a x + b Orthogonal parameters: J y = a (x ˆ x + b For ntercept b, set a=0 and fnd b by optmal average: ˆ b = y, Var[ b ˆ ] = For slope a, set
More informationLecture 4: Universal Hash Functions/Streaming Cont d
CSE 5: Desgn and Analyss of Algorthms I Sprng 06 Lecture 4: Unversal Hash Functons/Streamng Cont d Lecturer: Shayan Oves Gharan Aprl 6th Scrbe: Jacob Schreber Dsclamer: These notes have not been subjected
More informationChapter 12 Lyes KADEM [Thermodynamics II] 2007
Chapter 2 Lyes KDEM [Therodynacs II] 2007 Gas Mxtures In ths chapter we wll develop ethods for deternng therodynac propertes of a xture n order to apply the frst law to systes nvolvng xtures. Ths wll be
More informationITERATIVE ESTIMATION PROCEDURE FOR GEOSTATISTICAL REGRESSION AND GEOSTATISTICAL KRIGING
ESE 5 ITERATIVE ESTIMATION PROCEDURE FOR GEOSTATISTICAL REGRESSION AND GEOSTATISTICAL KRIGING Gven a geostatstcal regresson odel: k Y () s x () s () s x () s () s, s R wth () unknown () E[ ( s)], s R ()
More informationUniversity of Washington Department of Chemistry Chemistry 452/456 Summer Quarter 2013
Lecture 8/8/3 Unversty o Washngton Departent o Chestry Chestry 45/456 Suer Quarter 3 A. The Gbbs-Duhe Equaton Fro Lecture 7 and ro the dscusson n sectons A and B o ths lecture, t s clear that the actvty
More informationUniversity of Washington Department of Chemistry Chemistry 452/456 Summer Quarter 2014
Lecture 16 8/4/14 Unversty o Washngton Department o Chemstry Chemstry 452/456 Summer Quarter 214. Real Vapors and Fugacty Henry s Law accounts or the propertes o extremely dlute soluton. s shown n Fgure
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 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 informationC/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1
C/CS/Phy9 Problem Set 3 Solutons Out: Oct, 8 Suppose you have two qubts n some arbtrary entangled state ψ You apply the teleportaton protocol to each of the qubts separately What s the resultng state obtaned
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationMAE140 - Linear Circuits - Winter 16 Midterm, February 5
Instructons ME140 - Lnear Crcuts - Wnter 16 Mdterm, February 5 () Ths exam s open book. You may use whatever wrtten materals you choose, ncludng your class notes and textbook. You may use a hand calculator
More informationRelevance Vector Machines Explained
October 19, 2010 Relevance Vector Machnes Explaned Trstan Fletcher www.cs.ucl.ac.uk/staff/t.fletcher/ Introducton Ths document has been wrtten n an attempt to make Tppng s [1] Relevance Vector Machnes
More informationChapter 12 Analysis of Covariance
Chapter Analyss of Covarance Any scentfc experment s performed to know somethng that s unknown about a group of treatments and to test certan hypothess about the correspondng treatment effect When varablty
More informationSTAT 3008 Applied Regression Analysis
STAT 3008 Appled Regresson Analyss Tutoral : Smple Lnear Regresson LAI Chun He Department of Statstcs, The Chnese Unversty of Hong Kong 1 Model Assumpton To quantfy the relatonshp between two factors,
More informationPHYS 2211L - Principles of Physics Laboratory I
PHYS L - Prncples of Physcs Laboratory I Laboratory Adanced Sheet Ballstc Pendulu. Objecte. The objecte of ths laboratory s to use the ballstc pendulu to predct the ntal elocty of a projectle usn the prncples
More informationIntroducing Entropy Distributions
Graubner, Schdt & Proske: Proceedngs of the 6 th Internatonal Probablstc Workshop, Darstadt 8 Introducng Entropy Dstrbutons Noel van Erp & Peter van Gelder Structural Hydraulc Engneerng and Probablstc
More informationSupplement: Proofs and Technical Details for The Solution Path of the Generalized Lasso
Supplement: Proofs and Techncal Detals for The Soluton Path of the Generalzed Lasso Ryan J. Tbshran Jonathan Taylor In ths document we gve supplementary detals to the paper The Soluton Path of the Generalzed
More informationQuantum Mechanics I - Session 4
Quantum Mechancs I - Sesson 4 Aprl 3, 05 Contents Operators Change of Bass 4 3 Egenvectors and Egenvalues 5 3. Denton....................................... 5 3. Rotaton n D....................................
More informationTime-Varying Systems and Computations Lecture 6
Tme-Varyng Systems and Computatons Lecture 6 Klaus Depold 14. Januar 2014 The Kalman Flter The Kalman estmaton flter attempts to estmate the actual state of an unknown dscrete dynamcal system, gven nosy
More informationModeling and Simulation NETW 707
Modelng and Smulaton NETW 707 Lecture 5 Tests for Random Numbers Course Instructor: Dr.-Ing. Magge Mashaly magge.ezzat@guc.edu.eg C3.220 1 Propertes of Random Numbers Random Number Generators (RNGs) must
More informationComplex Variables. Chapter 18 Integration in the Complex Plane. March 12, 2013 Lecturer: Shih-Yuan Chen
omplex Varables hapter 8 Integraton n the omplex Plane March, Lecturer: Shh-Yuan hen Except where otherwse noted, content s lcensed under a BY-N-SA. TW Lcense. ontents ontour ntegrals auchy-goursat theorem
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 information, are assumed to fluctuate around zero, with E( i) 0. Now imagine that this overall random effect, , is composed of many independent factors,
Part II. Contnuous Spatal Data Analyss 3. Spatally-Dependent Rando Effects Observe that all regressons n the llustratons above [startng wth expresson (..3) n the Sudan ranfall exaple] have reled on an
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 information