# Least Squares Fitting of Data

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Least Squares Fttng of Data Davd Eberly Geoetrc Tools, LLC Copyrght c All Rghts Reserved. Created: July 15, 1999 Last Modfed: January 5, 2015 Contents 1 Lnear Fttng of 2D Ponts of For x, fx)) 2 2 Lnear Fttng of nd Ponts Usng Orthogonal Regresson 2 3 Planar Fttng of 3D Ponts of For x, y, fx, y)) 3 4 Hyperplanar Fttng of nd Ponts Usng Orthogonal Regresson 4 5 Fttng a Crcle to 2D Ponts 5 6 Fttng a Sphere to 3D Ponts 6 7 Fttng an Ellpse to 2D Ponts 7 8 Fttng an Ellpsod to 3D Ponts 8 9 Fttng a Parabolod to 3D Ponts of the For x, y, fx, y)) 8 1

2 Ths docuent descrbes soe algorths for fttng 2D or 3D pont sets by lnear or quadratc structures usng least squares nzaton. 1 Lnear Fttng of 2D Ponts of For x, fx)) Ths s the usual ntroducton to least squares ft by a lne when the data represents easureents where the y-coponent s assued to be functonally dependent on the x-coponent. Gven a set of saples {x, y )}, deterne A and B so that the lne y = Ax + B best fts the saples n the sense that the su of the squared errors between the y and the lne values Ax + B s nzed. Note that the error s easured only n the y-drecton. Defne EA, B) = [Ax + B) y ] 2. Ths functon s nonnegatve and ts graph s a parabolod whose vertex occurs when the gradent satstfes E = 0, 0). Ths leads to a syste of two lnear equatons n A and B whch can be easly solved. Precsely, 0, 0) = E = 2 [Ax + B) y ]x, 1) and so x2 x x 1 A B = x y y. The soluton provdes the least squares soluton y = Ax + B. If pleented drectly, ths forulaton can lead to an ll-condtoned lnear syste. To avod ths, you should frst copute the averages x = x and ȳ = y and subtract the fro the data, x x x and y y ȳ. The ftted lne s of the for y ȳ = Ax x). 2 Lnear Fttng of nd Ponts Usng Orthogonal Regresson It s also possble to ft a lne usng least squares where the errors are easured orthogonally to the proposed lne rather than easured vertcally. The followng arguent holds for saple ponts and lnes n n densons. Let the lne be Lt) = td + A where D s unt length. Defne X to be the saple ponts; then X = A + d D + p D where d = D X A) and D s soe unt length vector perpendcular to D wth approprate coeffcent p. Defne Y = X A. The vector fro X to ts projecton onto the lne s Y d D = p D. The squared length of ths vector s p 2 = Y d D) 2. The energy functon for the least squares nzaton s EA, D) = p2. Two alternate fors for ths functon are EA, D) = Y T [I DD T] ) Y 2

3 and EA, D) = D T [ ] ) Y Y )I Y Y T D = D T MA)D. Usng the frst for of E n the prevous equaton, take the dervatve wth respect to A to get [ A = 2 I DD T] Y. Ths partal dervatve s zero whenever Y = 0 n whch case A = 1/) X the average of the saple ponts). Gven A, the atrx MA) s deterned n the second for of the energy functon. The quantty D T MA)D s a quadratc for whose nu s the sallest egenvalue of MA). Ths can be found by standard egensyste solvers. A correspondng unt length egenvector D copletes our constructon of the least squares lne. For n = 2, f A = a, b), then atrx MA) s gven by ) n MA) = x a) 2 + y b) x a) 2 x a)y b) x. a)y b) y b) 2 For n = 3, f A = a, b, c), then atrx MA) s gven by x a) 2 x a)y b) x a)z c) MA) = δ x a)y b) y b) 2 y b)z c) x a)z c) y b)z c) z c) 2 where δ = x a) 2 + y b) 2 + z c) 2. 3 Planar Fttng of 3D Ponts of For x, y, fx, y)) The assupton s that the z-coponent of the data s functonally dependent on the x- and y-coponents. Gven a set of saples {x, y, z )}, deterne A, B, and C so that the plane z = Ax + By + C best fts the saples n the sense that the su of the squared errors between the z and the plane values Ax +By +C s nzed. Note that the error s easured only n the z-drecton. Defne EA, B, C) = [Ax + By + C) z ] 2. Ths functon s nonnegatve and ts graph s a hyperparabolod whose vertex occurs when the gradent satstfes E = 0, 0, 0). Ths leads to a syste of three lnear equatons n A, B, and C whch can be easly solved. Precsely, 0, 0, 0) = E = 2 [Ax + By + C) z ]x, y, 1) 3

4 and so x2 x y x x y y2 y x y 1 A B C = x z y z z. The soluton provdes the least squares soluton z = Ax + By + C. If pleented drectly, ths forulaton can lead to an ll-condtoned lnear syste. To avod ths, you should frst copute the averages x = x, ȳ = y, and z = z, and then subtract the fro the data, x x x, y y ȳ, and z z z. The ftted plane s of the for z z = Ax x)+by ȳ). 4 Hyperplanar Fttng of nd Ponts Usng Orthogonal Regresson It s also possble to ft a plane usng least squares where the errors are easured orthogonally to the proposed plane rather than easured vertcally. The followng arguent holds for saple ponts and hyperplanes n n densons. Let the hyperplane be N X A) = 0 where N s a unt length noral to the hyperplane and A s a pont on the hyperplane. Defne X to be the saple ponts; then X = A + λ N + p N where λ = N X A) and N s soe unt length vector perpendcular to N wth approprate coeffcent p. Defne Y = X A. The vector fro X to ts projecton onto the hyperplane s λ N. The squared length of ths vector s λ 2 = N Y ) 2. The energy functon for the least squares nzaton s EA, N) = λ2. Two alternate fors for ths functon are EA, N) = Y T [NN T] ) Y and ) EA, N) = N T Y Y T N = N T MA)N. Usng the frst for of E n the prevous equaton, take the dervatve wth respect to A to get [ A = 2 NN T] Y. Ths partal dervatve s zero whenever Y = 0 n whch case A = 1/) X the average of the saple ponts). Gven A, the atrx MA) s deterned n the second for of the energy functon. The quantty N T MA)N s a quadratc for whose nu s the sallest egenvalue of MA). Ths can be found by standard egensyste solvers. A correspondng unt length egenvector N copletes our constructon of the least squares hyperplane. 4

5 For n = 3, f A = a, b, c), then atrx MA) s gven by x a) 2 x a)y b) x a)z c) MA) = x a)y b) y b) 2 y b)z c) x a)z c) y b)z c) z c) 2. 5 Fttng a Crcle to 2D Ponts Gven a set of ponts {x, y )}, 3, ft the wth a crcle x a)2 + y b) 2 = r 2 where a, b) s the crcle center and r s the crcle radus. An assupton of ths algorth s that not all the ponts are collnear. The energy functon to be nzed s Ea, b, r) = L r) 2 where L = x a) 2 + y b) 2. Take the partal dervatve wth respect to r to obtan Settng equal to zero yelds r = 2 L r). r = 1 L. Take the partal dervatve wth respect to a to obtan a = 2 L r) L a = 2 x a) + r L ) a and take the partal dervatve wth respect to b to obtan b = 2 L r) L b = 2 y b) + r L ). b Settng these two dervatves equal to zero yelds a = 1 x + r 1 and b = 1 y + r 1 Replacng r by ts equvalent fro / r = 0 and usng L / a = a x )/L and L / b = b y )/L, we get two nonlnear equatons n a and b: a = x + L L a =: F a, b) b = ȳ + L L b =: Ga, b) L a L b. 5

6 where x = 1 x ȳ = 1 y L = 1 L L a = 1 L b = 1 Fxed pont teraton can be appled to solvng these equatons: a 0 = x, b 0 = ȳ, and a +1 = F a, b ) and b +1 = Ga, b ) for 0. Warnng. I have not analyzed the convergence propertes of ths algorth. In a few experents t sees to converge just fne. a x L b y L 6 Fttng a Sphere to 3D Ponts Gven a set of ponts {x, y, z )}, 4, ft the wth a sphere x a)2 + y b) 2 + z c) 2 = r 2 where a, b, c) s the sphere center and r s the sphere radus. An assupton of ths algorth s that not all the ponts are coplanar. The energy functon to be nzed s Ea, b, c, r) = L r) 2 where L = x a) 2 + y b) 2 + z c). Take the partal dervatve wth respect to r to obtan Settng equal to zero yelds r = 2 L r). r = 1 L. Take the partal dervatve wth respect to a to obtan a = 2 L r) L a = 2 x a) + r L ), a take the partal dervatve wth respect to b to obtan b = 2 L r) L b = 2 y b) + r L ), b and take the partal dervatve wth respect to c to obtan c = 2 L r) L c = 2 z c) + r L ). c 6

7 Settng these three dervatves equal to zero yelds and and a = 1 b = 1 c = 1 x + r 1 y + r 1 z + r 1 Replacng r by ts equvalent fro / r = 0 and usng L / a = a x )/L, L / b = b y )/L, and L / c = c z )/L, we get three nonlnear equatons n a, b, and c: where L a L b. L c. a = x + L L a =: F a, b, c) b = ȳ + L L b =: Ga, b, c) c = z + L L c =: Ha, b, c) x = 1 x ȳ = 1 y z = 1 z L = 1 L L a = 1 L b = 1 L c = 1 Fxed pont teraton can be appled to solvng these equatons: a 0 = x, b 0 = ȳ, c 0 = z, and a +1 = F a, b, c ), b +1 = Ga, b, c ), and c +1 = Ha, b, c ) for 0. Warnng. I have not analyzed the convergence propertes of ths algorth. In a few experents t sees to converge just fne. a x L b y L c z L 7 Fttng an Ellpse to 2D Ponts Gven a set of ponts {X }, 3, ft the wth an ellpse X U)T R T DRX U) = 1 where U s the ellpse center, R s an orthonoral atrx representng the ellpse orentaton, and D s a dagonal atrx whose dagonal entres represent the recprocal of the squares of the half-lengths lengths of the axes of the ellpse. An axs-algned ellpse wth center at the orgn has equaton x/a) 2 + y/b) 2 = 1. In ths settng, U = 0, 0), R = I the dentty atrx), and D = dag1/a 2, 1/b 2 ). The energy functon to be nzed s EU, R, D) = 7 L r) 2

8 where L s the dstance fro X to the ellpse wth the gven paraeters. Ths proble s ore dffcult than that of fttng crcles. The dstance L s coputed accordng to the algorth descrbed n Dstance fro a Pont to an Ellpse, an Ellpsod, or a Hyperellpsod. The functon E s nzed teratvely usng Powell s drecton-set ethod to search for a nu. An pleentaton s GteApprEllpse2.h. 8 Fttng an Ellpsod to 3D Ponts Gven a set of ponts {X }, 3, ft the wth an ellpsod X U)T R T DRX U) = 1 where U s the ellpsod center and R s an orthonoral atrx representng the ellpsod orentaton. The atrx D s a dagonal atrx whose dagonal entres represent the recprocal of the squares of the half-lengths of the axes of the ellpsod. An axs-algned ellpsod wth center at the orgn has equaton x/a) 2 + y/b) 2 + z/c) 2 = 1. In ths settng, U = 0, 0, 0), R = I the dentty atrx), and D = dag1/a 2, 1/b 2, 1/c 2 ). The energy functon to be nzed s EU, R, D) = L r) 2 where L s the dstance fro X to the ellpse wth the gven paraeters. Ths proble s ore dffcult than that of fttng spheres. The dstance L s coputed accordng to the algorth descrbed n Dstance fro a Pont to an Ellpse, an Ellpsod, or a Hyperellpsod. The functon E s nzed teratvely usng Powell s drecton-set ethod to search for a nu. An pleentaton s GteApprEllpsod3.h. 9 Fttng a Parabolod to 3D Ponts of the For x, y, fx, y)) Gven a set of saples {x, y, z )} and assung that the true values le on a parabolod z = fx, y) = p 1 x 2 + p 2 xy + p 3 y 2 + p 4 x + p 5 y + p 6 = P Qx, y) where P = p 1, p 2, p 3, p 4, p 5, p 6 ) and Qx, y) = x 2, xy, y 2, x, y, 1), select P to nze the su of squared errors EP) = P Q z ) 2 where Q = Qx, y ). The nu occurs when the gradent of E s the zero vector, E = 2 P Q z )Q = 0. Soe algebra converts ths to a syste of 6 equatons n 6 unknowns: ) Q Q T P = z Q. 8

9 The product Q Q T s a product of the 6 1 atrx Q wth the 1 6 atrx Q T, the result beng a 6 6 atrx. Defne the 6 6 syetrc atrx A = Q Q T and the 6 1 vector B = z Q. The choce for P s the soluton to the lnear syste of equatons AP = B. The entres of A and B ndcate suatons over the approprate product of varables. For exaple, sx 3 y) = x3 y : sx 4 ) sx 3 y) sx 2 y 2 ) sx 3 ) sx 2 y) sx 2 ) p 1 szx 2 ) sx 3 y) sx 2 y 2 ) sxy 3 ) sx 2 y) sxy 2 ) sxy) p 2 szxy) sx 2 y 2 ) sxy 3 ) sy 4 ) sxy 2 ) sy 3 ) sy 2 ) p 3 szy sx 3 ) sx 2 y) sxy 2 ) sx 2 = 2 ) ) sxy) sx) p 4 szx) sx 2 y) sxy 2 ) sy 3 ) sxy) sy 2 ) sy) p 5 szy) sx 2 ) sxy) sy 2 ) sx) sy) s1) sz) p 6 9

### where the coordinates are related to those in the old frame as follows.

Chapter 2 - Cartesan Vectors and Tensors: Ther Algebra Defnton of a vector Examples of vectors Scalar multplcaton Addton of vectors coplanar vectors Unt vectors A bass of non-coplanar vectors Scalar product

### 8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by

6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng

### Support Vector Machines

Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.

### v a 1 b 1 i, a 2 b 2 i,..., a n b n i.

SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are

### Journal of Advanced Mechanical Design, Systems, and Manufacturing

Journal of Advanced Mechancal Desgn, Systes, and Manufacturng Vol.4, No., Robust Method for Poston Measureent of Verte of Polyhedron Usng Shape fro Focus * akash HARADA ** **Departent of Mechancal Engneerng,

### State function: eigenfunctions of hermitian operators-> normalization, orthogonality completeness

Schroednger equaton Basc postulates of quantum mechancs. Operators: Hermtan operators, commutators State functon: egenfunctons of hermtan operators-> normalzaton, orthogonalty completeness egenvalues and

### Description of the Force Method Procedure. Indeterminate Analysis Force Method 1. Force Method con t. Force Method con t

Indeternate Analyss Force Method The force (flexblty) ethod expresses the relatonshps between dsplaceents and forces that exst n a structure. Prary objectve of the force ethod s to deterne the chosen set

### ELE427 - Testing Linear Sensors. Linear Regression, Accuracy, and Resolution.

ELE47 - Testng Lnear Sensors Lnear Regresson, Accurac, and Resoluton. Introducton: In the frst three la eperents we wll e concerned wth the characterstcs of lnear sensors. The asc functon of these sensors

### 5. Simultaneous eigenstates: Consider two operators that commute: Â η = a η (13.29)

5. Smultaneous egenstates: Consder two operators that commute: [ Â, ˆB ] = 0 (13.28) Let Â satsfy the followng egenvalue equaton: Multplyng both sdes by ˆB Â η = a η (13.29) ˆB [ Â η ] = ˆB [a η ] = a

### Chapter 6 Reed-Solomon Codes

Chapter 6 Reed-oloon Codes. Introducton The Reed-oloon codes R codes are nonbnary cyclc codes wth code sybols fro a Galos feld. They were dscovered n 96 by I. Reed and G. oloon. The work was done when

### HYPOTHESIS TESTING OF PARAMETERS FOR ORDINARY LINEAR CIRCULAR REGRESSION

HYPOTHESIS TESTING OF PARAMETERS FOR ORDINARY LINEAR CIRCULAR REGRESSION Abdul Ghapor Hussn Centre for Foundaton Studes n Scence Unversty of Malaya 563 KUALA LUMPUR E-mal: ghapor@umedumy Abstract Ths paper

### Introduction to Regression

Introducton to Regresson Regresson a means of predctng a dependent varable based one or more ndependent varables. -Ths s done by fttng a lne or surface to the data ponts that mnmzes the total error. -

### A machine vision approach for detecting and inspecting circular parts

A machne vson approach for detectng and nspectng crcular parts Du-Mng Tsa Machne Vson Lab. Department of Industral Engneerng and Management Yuan-Ze Unversty, Chung-L, Tawan, R.O.C. E-mal: edmtsa@saturn.yzu.edu.tw

### QUANTUM MECHANICS, BRAS AND KETS

PH575 SPRING QUANTUM MECHANICS, BRAS AND KETS The followng summares the man relatons and defntons from quantum mechancs that we wll be usng. State of a phscal sstem: The state of a phscal sstem s represented

### 6. EIGENVALUES AND EIGENVECTORS 3 = 3 2

EIGENVALUES AND EIGENVECTORS The Characterstc Polynomal If A s a square matrx and v s a non-zero vector such that Av v we say that v s an egenvector of A and s the correspondng egenvalue Av v Example :

### BERNSTEIN POLYNOMIALS

On-Lne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful

### Ganesh Subramaniam. American Solutions Inc., 100 Commerce Dr Suite # 103, Newark, DE 19713, USA

238 Int. J. Sulaton and Process Modellng, Vol. 3, No. 4, 2007 Sulaton-based optsaton for ateral dspatchng n Vendor-Managed Inventory systes Ganesh Subraana Aercan Solutons Inc., 100 Coerce Dr Sute # 103,

### L10: Linear discriminants analysis

L0: Lnear dscrmnants analyss Lnear dscrmnant analyss, two classes Lnear dscrmnant analyss, C classes LDA vs. PCA Lmtatons of LDA Varants of LDA Other dmensonalty reducton methods CSCE 666 Pattern Analyss

### n + d + q = 24 and.05n +.1d +.25q = 2 { n + d + q = 24 (3) n + 2d + 5q = 40 (2)

MATH 16T Exam 1 : Part I (In-Class) Solutons 1. (0 pts) A pggy bank contans 4 cons, all of whch are nckels (5 ), dmes (10 ) or quarters (5 ). The pggy bank also contans a con of each denomnaton. The total

### The Magnetic Field. Concepts and Principles. Moving Charges. Permanent Magnets

. The Magnetc Feld Concepts and Prncples Movng Charges All charged partcles create electrc felds, and these felds can be detected by other charged partcles resultng n electrc force. However, a completely

### SIX WAYS TO SOLVE A SIMPLE PROBLEM: FITTING A STRAIGHT LINE TO MEASUREMENT DATA

SIX WAYS TO SOLVE A SIMPLE PROBLEM: FITTING A STRAIGHT LINE TO MEASUREMENT DATA E. LAGENDIJK Department of Appled Physcs, Delft Unversty of Technology Lorentzweg 1, 68 CJ, The Netherlands E-mal: e.lagendjk@tnw.tudelft.nl

### FORCED CONVECTION HEAT TRANSFER IN A DOUBLE PIPE HEAT EXCHANGER

FORCED CONVECION HEA RANSFER IN A DOUBLE PIPE HEA EXCHANGER Dr. J. Mchael Doster Department of Nuclear Engneerng Box 7909 North Carolna State Unversty Ralegh, NC 27695-7909 Introducton he convectve heat

### greatest common divisor

4. GCD 1 The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no

Introducton Faraday's Law o Inducton In ths lab, you wll study Faraday's Law o nducton usng a wand wth col whch swngs through a magnetc eld. You wll also examne converson o mechanc energy nto electrc energy

### 1 Example 1: Axis-aligned rectangles

COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton

### A Fuzzy Optimization Framework for COTS Products Selection of Modular Software Systems

Internatonal Journal of Fuy Systes, Vol. 5, No., June 0 9 A Fuy Optaton Fraework for COTS Products Selecton of Modular Software Systes Pankaj Gupta, Hoang Pha, Mukesh Kuar Mehlawat, and Shlp Vera Abstract

### The eigenvalue derivatives of linear damped systems

Control and Cybernetcs vol. 32 (2003) No. 4 The egenvalue dervatves of lnear damped systems by Yeong-Jeu Sun Department of Electrcal Engneerng I-Shou Unversty Kaohsung, Tawan 840, R.O.C e-mal: yjsun@su.edu.tw

### Homework: 49, 56, 67, 60, 64, 74 (p. 234-237)

Hoework: 49, 56, 67, 60, 64, 74 (p. 34-37) 49. bullet o ass 0g strkes a ballstc pendulu o ass kg. The center o ass o the pendulu rses a ertcal dstance o c. ssung that the bullet reans ebedded n the pendulu,

### We are now ready to answer the question: What are the possible cardinalities for finite fields?

Chapter 3 Fnte felds We have seen, n the prevous chapters, some examples of fnte felds. For example, the resdue class rng Z/pZ (when p s a prme) forms a feld wth p elements whch may be dentfed wth the

### z(t) = z 1 (t) + t(z 2 z 1 ) z(t) = 1 + i + t( 2 3i (1 + i)) z(t) = 1 + i + t( 3 4i); 0 t 1

(4.): ontours. Fnd an admssble parametrzaton. (a). the lne segment from z + to z 3. z(t) z (t) + t(z z ) z(t) + + t( 3 ( + )) z(t) + + t( 3 4); t (b). the crcle jz j 4 traversed once clockwse startng at

### Elastic Systems for Static Balancing of Robot Arms

. th World ongress n Mechans and Machne Scence, Guanajuato, Méco, 9- June, 0 _ lastc Sstes for Statc alancng of Robot rs I.Sonescu L. uptu Lucana Ionta I.Ion M. ne Poltehnca Unverst Poltehnca Unverst Poltehnca

### Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting

Causal, Explanatory Forecastng Assumes cause-and-effect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of

### Calibration Curves. Lecture #5 - Overview. Construction of Calibration Curves. Construction of Calibration Curves. Construction of Calibration Curves

Lecture #5 - Overvew Statstcs - Part 3 Statstcal Tools n Quanttatve Analyss The Method of Least Squares Calbraton Curves Usng a Spreadsheet for Least Squares Analytcal Response Measure of Unknown Calbraton

### Multiple discount and forward curves

Multple dscount and forward curves TopQuants presentaton 21 ovember 2012 Ton Broekhuzen, Head Market Rsk and Basel coordnator, IBC Ths presentaton reflects personal vews and not necessarly the vews of

### Graph Theory and Cayley s Formula

Graph Theory and Cayley s Formula Chad Casarotto August 10, 2006 Contents 1 Introducton 1 2 Bascs and Defntons 1 Cayley s Formula 4 4 Prüfer Encodng A Forest of Trees 7 1 Introducton In ths paper, I wll

### 21 Vectors: The Cross Product & Torque

21 Vectors: The Cross Product & Torque Do not use our left hand when applng ether the rght-hand rule for the cross product of two vectors dscussed n ths chapter or the rght-hand rule for somethng curl

### Point cloud to point cloud rigid transformations. Minimizing Rigid Registration Errors

Pont cloud to pont cloud rgd transformatons Russell Taylor 600.445 1 600.445 Fall 000-014 Copyrght R. H. Taylor Mnmzng Rgd Regstraton Errors Typcally, gven a set of ponts {a } n one coordnate system and

### x f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60

BIVARIATE DISTRIBUTIONS Let be a varable that assumes the values { 1,,..., n }. Then, a functon that epresses the relatve frequenc of these values s called a unvarate frequenc functon. It must be true

### Quantization Effects in Digital Filters

Quantzaton Effects n Dgtal Flters Dstrbuton of Truncaton Errors In two's complement representaton an exact number would have nfntely many bts (n general). When we lmt the number of bts to some fnte value

### GENERALIZED PROCRUSTES ANALYSIS AND ITS APPLICATIONS IN PHOTOGRAMMETRY

SWISS FEDERL INSIUE OF ECHNOLOGY Insttute of Geodes and Photograetr EH-Hoenggerberg, Zuerch GENERLIZED PROCRUSES NLYSIS ND IS PPLICIONS IN PHOOGRMMERY Prepared for: Praktku n Photograetre, Fernerkundung

### Technical Report, SFB 475: Komplexitätsreduktion in Multivariaten Datenstrukturen, Universität Dortmund, No. 1998,04

econstor www.econstor.eu Der Open-Access-Publkatonsserver der ZBW Lebnz-Inforatonszentru Wrtschaft The Open Access Publcaton Server of the ZBW Lebnz Inforaton Centre for Econocs Becka, Mchael Workng Paper

### BANDWIDTH ALLOCATION AND PRICING PROBLEM FOR A DUOPOLY MARKET

Yugoslav Journal of Operatons Research (0), Nuber, 65-78 DOI: 0.98/YJOR0065Y BANDWIDTH ALLOCATION AND PRICING PROBLEM FOR A DUOPOLY MARKET Peng-Sheng YOU Graduate Insttute of Marketng and Logstcs/Transportaton,

### 2. Linear Algebraic Equations

2. Lnear Algebrac Equatons Many physcal systems yeld smultaneous algebrac equatons when mathematcal functons are requred to satsfy several condtons smultaneously. Each condton results n an equaton that

### High Order Reverse Mode of AD Theory and Implementation

Hgh Order Reverse Mode of AD Theory and Implementaton Mu Wang and Alex Pothen Department of Computer Scence Purdue Unversty September 30, 2016 Mu Wang and Alex Pothen Hgh Order Reverse AD September 30,

### Recurrence. 1 Definitions and main statements

Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.

### INTRODUCTION. governed by a differential equation Need systematic approaches to generate FE equations

WEIGHTED RESIDUA METHOD INTRODUCTION Drect stffness method s lmted for smple D problems PMPE s lmted to potental problems FEM can be appled to many engneerng problems that are governed by a dfferental

### Semiconductor sensors of temperature

Semconductor sensors of temperature he measurement objectve 1. Identfy the unknown bead type thermstor. Desgn the crcutry for lnearzaton of ts transfer curve.. Fnd the dependence of forward voltage drop

### Math 131: Homework 4 Solutions

Math 3: Homework 4 Solutons Greg Parker, Wyatt Mackey, Chrstan Carrck October 6, 05 Problem (Munkres 3.) Let {A n } be a sequence of connected subspaces of X such that A n \ A n+ 6= ; for all n. Then S

### Multivariate EWMA Control Chart

Multvarate EWMA Control Chart Summary The Multvarate EWMA Control Chart procedure creates control charts for two or more numerc varables. Examnng the varables n a multvarate sense s extremely mportant

### THE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES

The goal: to measure (determne) an unknown quantty x (the value of a RV X) Realsaton: n results: y 1, y 2,..., y j,..., y n, (the measured values of Y 1, Y 2,..., Y j,..., Y n ) every result s encumbered

### Questions that we may have about the variables

Antono Olmos, 01 Multple Regresson Problem: we want to determne the effect of Desre for control, Famly support, Number of frends, and Score on the BDI test on Perceved Support of Latno women. Dependent

### Lecture 2: Single Layer Perceptrons Kevin Swingler

Lecture 2: Sngle Layer Perceptrons Kevn Sngler kms@cs.str.ac.uk Recap: McCulloch-Ptts Neuron Ths vastly smplfed model of real neurons s also knon as a Threshold Logc Unt: W 2 A Y 3 n W n. A set of synapses

### Section 5.4 Annuities, Present Value, and Amortization

Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today

### CS 2750 Machine Learning. Lecture 17a. Clustering. CS 2750 Machine Learning. Clustering

Lecture 7a Clusterng Mlos Hauskrecht mlos@cs.ptt.edu 539 Sennott Square Clusterng Groups together smlar nstances n the data sample Basc clusterng problem: dstrbute data nto k dfferent groups such that

### GRAVITY DATA VALIDATION AND OUTLIER DETECTION USING L 1 -NORM

GRAVITY DATA VALIDATION AND OUTLIER DETECTION USING L 1 -NORM BARRIOT Jean-Perre, SARRAILH Mchel BGI/CNES 18.av.E.Beln 31401 TOULOUSE Cedex 4 (France) Emal: jean-perre.barrot@cnes.fr 1/Introducton The

### Maximizing profit using recommender systems

Maxzng proft usng recoender systes Aparna Das Brown Unversty rovdence, RI aparna@cs.brown.edu Clare Matheu Brown Unversty rovdence, RI clare@cs.brown.edu Danel Rcketts Brown Unversty rovdence, RI danel.bore.rcketts@gal.co

### Goals Rotational quantities as vectors. Math: Cross Product. Angular momentum

Physcs 106 Week 5 Torque and Angular Momentum as Vectors SJ 7thEd.: Chap 11.2 to 3 Rotatonal quanttes as vectors Cross product Torque expressed as a vector Angular momentum defned Angular momentum as a

### Nonlinear data mapping by neural networks

Nonlnear data mappng by neural networks R.P.W. Dun Delft Unversty of Technology, Netherlands Abstract A revew s gven of the use of neural networks for nonlnear mappng of hgh dmensonal data on lower dmensonal

### I. SCOPE, APPLICABILITY AND PARAMETERS Scope

D Executve Board Annex 9 Page A/R ethodologcal Tool alculaton of the number of sample plots for measurements wthn A/R D project actvtes (Verson 0) I. SOPE, PIABIITY AD PARAETERS Scope. Ths tool s applcable

### How Much to Bet on Video Poker

How Much to Bet on Vdeo Poker Trstan Barnett A queston that arses whenever a gae s favorable to the player s how uch to wager on each event? Whle conservatve play (or nu bet nzes large fluctuatons, t lacks

### Software Alignment for Tracking Detectors

Software Algnment for Trackng Detectors V. Blobel Insttut für Expermentalphysk, Unverstät Hamburg, Germany Abstract Trackng detectors n hgh energy physcs experments requre an accurate determnaton of a

### benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).

REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or

### General Physics (PHY 2130)

General Physcs (PHY 30) Lecture 8 Moentu Collsons Elastc and nelastc collsons http://www.physcs.wayne.edu/~apetro/phy30/ Lghtnng Reew Last lecture:. Moentu: oentu and pulse oentu conseraton Reew Proble:

### Time Series Analysis in Studies of AGN Variability. Bradley M. Peterson The Ohio State University

Tme Seres Analyss n Studes of AGN Varablty Bradley M. Peterson The Oho State Unversty 1 Lnear Correlaton Degree to whch two parameters are lnearly correlated can be expressed n terms of the lnear correlaton

### Regression Models for a Binary Response Using EXCEL and JMP

SEMATECH 997 Statstcal Methods Symposum Austn Regresson Models for a Bnary Response Usng EXCEL and JMP Davd C. Trndade, Ph.D. STAT-TECH Consultng and Tranng n Appled Statstcs San Jose, CA Topcs Practcal

### What is Candidate Sampling

What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble

### SIMPLE LINEAR CORRELATION

SIMPLE LINEAR CORRELATION Smple lnear correlaton s a measure of the degree to whch two varables vary together, or a measure of the ntensty of the assocaton between two varables. Correlaton often s abused.

### Experiment 8 Two Types of Pendulum

Experment 8 Two Types of Pendulum Preparaton For ths week's quz revew past experments and read about pendulums and harmonc moton Prncples Any object that swngs back and forth can be consdered a pendulum

### Physics problem solving (Key)

Name: Date: Name: Physcs problem solvng (Key) Instructons: 1.. Fnd one partner to work together. You can use your textbook, calculator and may also want to have scratch paper. 3. Work through the problems

### MANY machine learning and pattern recognition applications

1 Trace Rato Problem Revsted Yangqng Ja, Fepng Ne, and Changshu Zhang Abstract Dmensonalty reducton s an mportant ssue n many machne learnng and pattern recognton applcatons, and the trace rato problem

### Examples of Multiple Linear Regression Models

ECON *: Examples of Multple Regresson Models Examples of Multple Lnear Regresson Models Data: Stata tutoral data set n text fle autoraw or autotxt Sample data: A cross-sectonal sample of 7 cars sold n

### II. THE QUALITY AND REGULATION OF THE DISTRIBUTION COMPANIES I. INTRODUCTION

Fronter Methodology to fx Qualty goals n Electrcal Energy Dstrbuton Copanes R. Rarez 1, A. Sudrà 2, A. Super 3, J.Bergas 4, R.Vllafáfla 5 1-2 -3-4-5 - CITCEA - UPC UPC., Unversdad Poltécnca de Cataluña,

### Yves Genin, Yurii Nesterov, Paul Van Dooren. CESAME, Universite Catholique de Louvain. B^atiment Euler, Avenue G. Lema^tre 4-6

Submtted to ECC 99 as a regular paper n Lnear Systems Postve transfer functons and convex optmzaton 1 Yves Genn, Yur Nesterov, Paul Van Dooren CESAME, Unverste Catholque de Louvan B^atment Euler, Avenue

### Naglaa Raga Said Assistant Professor of Operations. Egypt.

Volue, Issue, Deceer ISSN: 77 8X Internatonal Journal of Adanced Research n Coputer Scence and Software Engneerng Research Paper Aalale onlne at: www.jarcsse.co Optal Control Theory Approach to Sole Constraned

### Linear Algebra for Quantum Mechanics

prevous ndex next Lnear Algebra for Quantum Mechancs Mchael Fowler 0/4/08 Introducton We ve seen that n quantum mechancs, the state of an electron n some potental s gven by a ψ x t, and physcal varables

### 2.4 Bivariate distributions

page 28 2.4 Bvarate dstrbutons 2.4.1 Defntons Let X and Y be dscrete r.v.s defned on the same probablty space (S, F, P). Instead of treatng them separately, t s often necessary to thnk of them actng together

### A NOTE ON THE PREDICTION AND TESTING OF SYSTEM RELIABILITY UNDER SHOCK MODELS C. Bouza, Departamento de Matemática Aplicada, Universidad de La Habana

REVISTA INVESTIGACION OPERACIONAL Vol., No. 3, 000 A NOTE ON THE PREDICTION AND TESTING OF SYSTEM RELIABILITY UNDER SHOCK MODELS C. Bouza, Departaento de Mateátca Aplcada, Unversdad de La Habana ABSTRACT

### Hedging Interest-Rate Risk with Duration

FIXED-INCOME SECURITIES Chapter 5 Hedgng Interest-Rate Rsk wth Duraton Outlne Prcng and Hedgng Prcng certan cash-flows Interest rate rsk Hedgng prncples Duraton-Based Hedgng Technques Defnton of duraton

### 9.1 The Cumulative Sum Control Chart

Learnng Objectves 9.1 The Cumulatve Sum Control Chart 9.1.1 Basc Prncples: Cusum Control Chart for Montorng the Process Mean If s the target for the process mean, then the cumulatve sum control chart s

### Rotation Kinematics, Moment of Inertia, and Torque

Rotaton Knematcs, Moment of Inerta, and Torque Mathematcally, rotaton of a rgd body about a fxed axs s analogous to a lnear moton n one dmenson. Although the physcal quanttes nvolved n rotaton are qute

### On the Solution of Indefinite Systems Arising in Nonlinear Optimization

On the Soluton of Indefnte Systems Arsng n Nonlnear Optmzaton Slva Bonettn, Valera Ruggero and Federca Tnt Dpartmento d Matematca, Unverstà d Ferrara Abstract We consder the applcaton of the precondtoned

### PHYS 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

### Chapter 7: Answers to Questions and Problems

19. Based on the nformaton contaned n Table 7-3 of the text, the food and apparel ndustres are most compettve and therefore probably represent the best match for the expertse of these managers. Chapter

### Passive Filters. References: Barbow (pp 265-275), Hayes & Horowitz (pp 32-60), Rizzoni (Chap. 6)

Passve Flters eferences: Barbow (pp 6575), Hayes & Horowtz (pp 360), zzon (Chap. 6) Frequencyselectve or flter crcuts pass to the output only those nput sgnals that are n a desred range of frequences (called

### PLANAR GRAPHS. Plane graph (or embedded graph) A graph that is drawn on the plane without edge crossing, is called a Plane graph

PLANAR GRAPHS Basc defntons Isomorphc graphs Two graphs G(V,E) and G2(V2,E2) are somorphc f there s a one-to-one correspondence F of ther vertces such that the followng holds: - u,v V, uv E, => F(u)F(v)

### Loop Parallelization

- - Loop Parallelzaton C-52 Complaton steps: nested loops operatng on arrays, sequentell executon of teraton space DECLARE B[..,..+] FOR I :=.. FOR J :=.. I B[I,J] := B[I-,J]+B[I-,J-] ED FOR ED FOR analyze

### Linear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits

Lnear Crcuts Analyss. Superposton, Theenn /Norton Equalent crcuts So far we hae explored tmendependent (resste) elements that are also lnear. A tmendependent elements s one for whch we can plot an / cure.

### Logistic Regression. Steve Kroon

Logstc Regresson Steve Kroon Course notes sectons: 24.3-24.4 Dsclamer: these notes do not explctly ndcate whether values are vectors or scalars, but expects the reader to dscern ths from the context. Scenaro

### Chapter 7. Random-Variate Generation 7.1. Prof. Dr. Mesut Güneş Ch. 7 Random-Variate Generation

Chapter 7 Random-Varate Generaton 7. Contents Inverse-transform Technque Acceptance-Rejecton Technque Specal Propertes 7. Purpose & Overvew Develop understandng of generatng samples from a specfed dstrbuton

### Logistic Regression. Lecture 4: More classifiers and classes. Logistic regression. Adaboost. Optimization. Multiple class classification

Lecture 4: More classfers and classes C4B Machne Learnng Hlary 20 A. Zsserman Logstc regresson Loss functons revsted Adaboost Loss functons revsted Optmzaton Multple class classfcaton Logstc Regresson

### A Novel Dynamic Role-Based Access Control Scheme in User Hierarchy

Journal of Coputatonal Inforaton Systes 6:7(200) 2423-2430 Avalable at http://www.jofcs.co A Novel Dynac Role-Based Access Control Schee n User Herarchy Xuxa TIAN, Zhongqn BI, Janpng XU, Dang LIU School

### Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic

Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange

### CHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES

CHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES In ths chapter, we wll learn how to descrbe the relatonshp between two quanttatve varables. Remember (from Chapter 2) that the terms quanttatve varable

### 1. Measuring association using correlation and regression

How to measure assocaton I: Correlaton. 1. Measurng assocaton usng correlaton and regresson We often would lke to know how one varable, such as a mother's weght, s related to another varable, such as a

### CHAPTER 7 VECTOR BUNDLES

CHAPTER 7 VECTOR BUNDLES We next begn addressng the queston: how do we assemble the tangent spaces at varous ponts of a manfold nto a coherent whole? In order to gude the decson, consder the case of U

### Conversion between the vector and raster data structures using Fuzzy Geographical Entities

Converson between the vector and raster data structures usng Fuzzy Geographcal Enttes Cdála Fonte Department of Mathematcs Faculty of Scences and Technology Unversty of Combra, Apartado 38, 3 454 Combra,

### Calculating the high frequency transmission line parameters of power cables

< ' Calculatng the hgh frequency transmsson lne parameters of power cables Authors: Dr. John Dcknson, Laboratory Servces Manager, N 0 RW E B Communcatons Mr. Peter J. Ncholson, Project Assgnment Manager,

### 8 Algorithm for Binary Searching in Trees

8 Algorthm for Bnary Searchng n Trees In ths secton we present our algorthm for bnary searchng n trees. A crucal observaton employed by the algorthm s that ths problem can be effcently solved when the