Lecture 19. Curve fitting I. 1 Introduction. 2 Fitting a constant to measured data
|
|
- Ashley Oliver
- 5 years ago
- Views:
Transcription
1 Lecture 9 Curve fittig I Itroductio Suppose we are preseted with eight poits of easured data (x i, y j ). As show i Fig. o the left, we could represet the uderlyig fuctio of which these data are saples by iterpolatig betwee the data poits usig oe of the ethods we have studied previously. Fig. : Measured data with: (left) splie iterpolatio, (right) lie fit. However, aybe the data are saples of the respose of a process that we kow, i theory, is supposed to have the for y= f ( x )=a x+b where a,b are costats. Maybe we also kow that y is a very weak sigal ad the sesor used to easure it is oisy, that is, it adds its ow (rado) sigal i with the true y data. Give this it akes o sese to iterpolate the data because i part we'll be iterpolatig oise, ad we kow that the real sigal should have the for y=ax+b. I a situatio like this we prefer to fit a lie to the data rather tha perfor a iterpolatio (Fig. at right). If doe correctly this ca provide a degree of iuity agaist the effects of easureet errors ad oise. More geerally we wat to develop curve fittig techiques that allow theoretical curves, or odels, with ukow paraeters (such as a ad b i the lie case) to be fit to data poits. Fittig a costat to easured data The siplest curve fittig proble is estiatig a paraeter fro ultiple easureets. Suppose is the ass of a object. We wat to easure this usig a scale. Ufortuately the scales i our laboratory are ot well calibrated. However we have ie scales. We expect that if EE Nuerical Coputig Scott Hudso
2 Lecture 9: Curve fittig I /0 Fig. : Horizotal lie is average of several easureets (dots) we take easureets with all of the ad average the results we should get a better estiate of the true ass that by relyig o the easureet fro a sigle scale. Our results ight look soethig like show i Fig.. Let the easureet of the ith scale be i the the average easureet is give by = i= i () where is the uber of easureets. This is what we should use for our best estiate of the true ass. Averagig is a very basic for of curve fittig. 3 Least-squares lie fit Goig back to the situatio illustrated i Fig., how do we figure out the best fit lie? There does't see to be a straightforward way to average the data like we did i Fig.. Istead, let's suppose we have data poits ( x i, y i ). We are iterested i a liear odel of the for y=a x+b, ad our task is calculate the best values for a ad b. If all our data actually fell o a lie the the best a ad b values would result i y i (a xi +b)=0 for i=,,,. More geerally let's defie the residual ( error of the fit ) for the ith data poit as r i= y i (ax i+b) () A perfect fit would give r i=0 for all i. The residual ca be positive or egative, but what we are ost cocered with is its agitude. Let's defie the ea squared error (MSE) as MSE = r i= ( y i ( a xi +b)) i = i= EE Nuerical Coputig Scott Hudso (3)
3 Lecture 9: Curve fittig I 3/0 We ow seek the values of a ad b that iiize the MSE. These will satisfy MSE MSE =0 ad =0 a b (4) The b derivative is MSE = ( y i (a x i+b))=0 b i = (5) Multiplyig through by / ad rearragig we fid a yi x i b=0 i = i= i= (6) Now defie the average x ad y values as y = y i, x = x i i= i= (7) y a x b=0 (8) a x +b= y (9) Equatio (6) the reads or This tells us that the poit ( x, y ) (the cetroid of the data) falls o the lie. The a derivative of the MSE is MSE = ( y i (a x i+b)) x i =0 a i = (0) Multiplyig through by / ad rearragig we fid a b x i y i x i x i=0 i = i = i = () xy a x b x =0 () or with the additioal defiitios xy = x i y i, x = x i i= i= (3) a x +b x = xy (4) A fial rearrageet gives us We ow have two equatios i the two ukows a,b a x +b= y a x +b x = xy EE Nuerical Coputig Scott Hudso (5)
4 Lecture 9: Curve fittig I 4/0 Fig. 3: Least-squares lie fit to oisy data. Solvig the first equatio for b b= y a x (6) ad substitutig this ito the secod equatio we obtai a x +( y a x ) x = xy (7) Solvig this for a we have a= xy x y x x (8) Equatios (8) ad (6) provide the best-fit values of a ad b. Because we obtaied these paraeters by iiizig the su of squared residuals, this is called a least-squares lie fit. Exaple. The code below geerates six poits o the lie y= x ad adds orally-distributed oise of stadard deviatio 0. to the y values. The (8) ad (6) are used to calculate the best-fit values of a ad b. The data ad fit lie are plotted i Fig. 3. The true values are a=, b=. The fit values are a= 0.9,b=.0. -->x = [0:0.:]'; -->y = -x+rad(x,'oral')*0.; -->a = (ea(x.*y)-ea(x)*ea(y))/(ea(x.^)-ea(x)^) a = >b = ea(y)-a*ea(x) b =.0945 EE Nuerical Coputig Scott Hudso
5 Lecture 9: Curve fittig I 5/0 4 Liear least-squares The least-squares idea ca be applied to a liear cobiatio of ay fuctios f (x), f (x),, f (x). Our odel has the for y= c j f j ( x) (9) j= For exaple, if = ad f (x)=, f ( x)=x the our odel is y=c +c x (0) which is just the liear case we've already dealt with. If we add f 3 (x)=x the the odel is y=c +c x+c 3 x () which is a arbitrary quadratic. Or we could have a odel such as y=c cos (5 x)+c si (5 x)+c 3 cos (0 x)+c 4 si(0 x ) () I ay case we'll cotiue to defie the residuals as the differece betwee the observed ad the odeled y values r i = y i c j f j ( x i) (3) j = ad the ea-squared error as ( ) MSE = r i= y i c j f j (x i ) i = i= j = (4) Let's expad this as ( ) ( [ y i c j f j (x i ) = y i y i c j f j ( x i )+ i = i = j= j = j = ]) c j f j ( x i) (5) Call yi= y i = ad y c f (x )= b j c j i= i j= j j i j = (6) with b j = yi f j ( xi ) i = (7) The last ter i (5) ca be writte EE Nuerical Coputig Scott Hudso
6 Lecture 9: Curve fittig I 6/0 [ j= ] j= k= c j f j ( xi ) = c j f j ( x i ) c k f k (x i) (8) Therefore i = [ ] c j f j ( xi ) = j= i= ( j= k= ) c j f j ( xi ) ck f k (x i ) = a c c j= k= jk j k (9) with a jk =a kj = f (x ) f (x ) i= j i k i (30) Fially we ca write MSE = y bi ci + i = a c c i = j= ij i j (3) This shows that the MSE is a quadratic fuctio of the ukow coefficiets. I the lecture Optiizatio i diesios we calculated the solutio to a syste of this for, except that the secod ter (with the b coefficiets) had a plus rather tha ius sig. Defiig the colu vectors b ad c ad the atrix A as c=[c j ], b=[b j ], A=[a ij ] (3) the coditio for a iiu is (with the ius sig for the b coefficiets) b+a c=0 (33) c=a b (34) ad Aother way arrive at this result is to defie the colu vector y=[ yi ] (35) F=[ f ij] with f ij = f j ( xi ) (36) y=f c (37) ad the atrix The our odel is This is equatios i < ukows ad i geeral will ot have a solutio. Multiplyig both sides o the left by FT results i the syste FT Fc=FT y (38) Sice FT F is ad FT y is this is a syste of equatios i ukows that, i geeral, will have a uique solutio c=( FT F ) FT y (39) The eleets of FT F are EE Nuerical Coputig Scott Hudso
7 Lecture 9: Curve fittig I 7/0 [ F F ] jk = T i= f ij f ik = a jk (40) f ij y i= b j (4) while the eleets of FT y are [ F y ] j = T i = Therefore FT Fc=FT y, whe ultiplied through by /, is equivalet to (4) A c=b The liear syste (38) is called the oral equatio, ad we have the followig algorith Liear least squares fit Give saples (x i, y i ) ad a odel y= c j f j ( x) j= For the atrix F with eleets f ij = f j ( xi ) For the colu vector y with eleets y i Solve the oral equatio FT Fc=FT y for c ^ The odeled y values are y=fc The atrix F is ot square if >, so we caot solve the liear syste y=fc (43) c=f y (44) by writig because F does ot have a iverse. However, as we've see, we ca copute c=( FT F ) FT y (45) ad this c will coe as close as possible (i a least-squares sese) to solvig (43). This leads us to defie the pseudoiverse of F as the atrix F =( F F ) F T + T (46) Our least-squares solutio ca ow be writte c=f + y (47) I Scilab/Matlab the pseudo iverse is coputed by the coad piv(f). However, if we siply apply the backslash operator as we would for a square syste c = F\y Scilab/Matlab returs the least-squares solutio. We do ot have to explicitly for the oral EE Nuerical Coputig Scott Hudso
8 Lecture 9: Curve fittig I 8/0 equatio or the pseudoiverse. Exaple. Noise was added to Eleve saples of y=x x, x=0,0.,0.,,. A least-squares fit of the odel c +c x+c 3 x gave c =0.044, c=.0, c=.039 Code is show below ad results are plotted i Fig >x = [0:0.:]'; -->y0 = x.^-x; -->y = y0+rad(y0,'oral')*0.03; //add oise -->F = [oes(x),x,x.^]; -->c = F\y c = >yf = F*c 5 Goodess of fit Oce we've fit a odel to data we ay woder if the fit is good or ot. It would be helpful to have a easure of goodess of fit. Doig this rigorously requires details fro probability theory. We will preset the followig results without derivatio. Assue our y values are of the for y i=si +ηi where si is the sigal that we are tryig to odel ad ηi is oise. If our odel were to perfectly Fig. 4: f ( x)= x x (dashed curve), saples of f ( x) with oise added (dots) ad least-squares fit of odel c +c x+c3 x (solid lie). EE Nuerical Coputig Scott Hudso
9 Lecture 9: Curve fittig I 9/0 fit the sigal, the the residuals r i= y i c j f j (x i) (48) j = would siply be oise r i=ηi. We ca quatify the goodess of fit by coparig the statistics of our residuals to the (assued kow) statistics of the oise. Specially, for large, ad orally distributed oise, a good fit will result i the uber σ= r i= i (49) beig equal, o average, to the stadard deviatio of the oise, where is the uber of data ad is the uber of odel coefficiets. If it is sigificatly larger tha this it idicates that the odel is ot accoutig for all of the sigal, where a fractioal chage of about /( ) is statistically sigificat. For exaple, /50=0. eas that a chage of aroud 0% is statistically sigificat. If the oise stadard deviatio is 0., a σ larger tha about 0.(.)=0. iplies the sigal is ot beig fully odeled. The followig exaple illustrates the use of this goodess-of-fit easure. Exaple. The followig code was used to geerate 50 saples of the fuctio f (x)=x+x over the iterval 0 x with orally distributed oise of stadard deviatio 0.05 added to each saple. = 50; rad('seed',); x = [lispace(0,,)]'; y = x+x.^+rad(x,'oral')*0.05; These data were the fit by the four odels y=c, y=c +c x, y=c +c x+c 3 x ad y=c +c x+c 3 x +c 4 x 3. The resultig σ values were σ0 =0.608, σ=0.0864, σ = ad σ3 = Sice /50=0. a chage of about 0% is statistically sigificat. The fits iproved sigificatly util the last odel. The data therefore support the odel y=c +c x+c 3 x but ot the cubic odel. The fits are show i Fig. 5. EE Nuerical Coputig Scott Hudso
10 Lecture 9: Curve fittig I 0/0 Fig. 5 Data set fit by polyoials. Top-left: y=c, σ 0= Top-right: y=c +c x, σ = Botto-left: y=c +c x+c3 x, σ = Botto-right: y=c +c x+c3 x +c4 x, σ 3= EE Nuerical Coputig Scott Hudso
Statistics and Data Analysis in MATLAB Kendrick Kay, February 28, Lecture 4: Model fitting
Statistics ad Data Aalysis i MATLAB Kedrick Kay, kedrick.kay@wustl.edu February 28, 2014 Lecture 4: Model fittig 1. The basics - Suppose that we have a set of data ad suppose that we have selected the
More informationLecture 20. Curve fitting II. m y= f (x)= c j. (x) (1) f j. n [ y i )] 2 (2) f (x i. MSE= 1 n i =1
Lecture 20 Curve fittig II Itroductio I the previous lecture we developed a method to solve the geeral liear least-squares problem. Give samples (x i, y i, the coefficiets c j of a model are foud which
More informationECE 901 Lecture 4: Estimation of Lipschitz smooth functions
ECE 9 Lecture 4: Estiatio of Lipschitz sooth fuctios R. Nowak 5/7/29 Cosider the followig settig. Let Y f (X) + W, where X is a rado variable (r.v.) o X [, ], W is a r.v. o Y R, idepedet of X ad satisfyig
More information) is a square matrix with the property that for any m n matrix A, the product AI equals A. The identity matrix has a ii
square atrix is oe that has the sae uber of rows as colus; that is, a atrix. he idetity atrix (deoted by I, I, or [] I ) is a square atrix with the property that for ay atrix, the product I equals. he
More informationA string of not-so-obvious statements about correlation in the data. (This refers to the mechanical calculation of correlation in the data.
STAT-UB.003 NOTES for Wedesday 0.MAY.0 We will use the file JulieApartet.tw. We ll give the regressio of Price o SqFt, show residual versus fitted plot, save residuals ad fitted. Give plot of (Resid, Price,
More informationThe Method of Least Squares. To understand least squares fitting of data.
The Method of Least Squares KEY WORDS Curve fittig, least square GOAL To uderstad least squares fittig of data To uderstad the least squares solutio of icosistet systems of liear equatios 1 Motivatio Curve
More informationLinear Regression Demystified
Liear Regressio Demystified Liear regressio is a importat subject i statistics. I elemetary statistics courses, formulae related to liear regressio are ofte stated without derivatio. This ote iteds to
More information19.1 The dictionary problem
CS125 Lecture 19 Fall 2016 19.1 The dictioary proble Cosider the followig data structural proble, usually called the dictioary proble. We have a set of ites. Each ite is a (key, value pair. Keys are i
More informationWe have also learned that, thanks to the Central Limit Theorem and the Law of Large Numbers,
Cofidece Itervals III What we kow so far: We have see how to set cofidece itervals for the ea, or expected value, of a oral probability distributio, both whe the variace is kow (usig the stadard oral,
More informationInverse Matrix. A meaning that matrix B is an inverse of matrix A.
Iverse Matrix Two square matrices A ad B of dimesios are called iverses to oe aother if the followig holds, AB BA I (11) The otio is dual but we ofte write 1 B A meaig that matrix B is a iverse of matrix
More information5.6 Binomial Multi-section Matching Transformer
4/14/21 5_6 Bioial Multisectio Matchig Trasforers 1/1 5.6 Bioial Multi-sectio Matchig Trasforer Readig Assiget: pp. 246-25 Oe way to axiize badwidth is to costruct a ultisectio Γ f that is axially flat.
More informationECE-S352 Introduction to Digital Signal Processing Lecture 3A Direct Solution of Difference Equations
ECE-S352 Itroductio to Digital Sigal Processig Lecture 3A Direct Solutio of Differece Equatios Discrete Time Systems Described by Differece Equatios Uit impulse (sample) respose h() of a DT system allows
More informationLecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
More informationData Analysis and Statistical Methods Statistics 651
Data Aalysis ad Statistical Methods Statistics 651 http://www.stat.tau.edu/~suhasii/teachig.htl Suhasii Subba Rao Exaple The itroge cotet of three differet clover plats is give below. 3DOK1 3DOK5 3DOK7
More informationLecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
More informationOrthogonal Functions
Royal Holloway Uiversity of odo Departet of Physics Orthogoal Fuctios Motivatio Aalogy with vectors You are probably failiar with the cocept of orthogoality fro vectors; two vectors are orthogoal whe they
More informationTheorem: Let A n n. In this case that A does reduce to I, we search for A 1 as the solution matrix X to the matrix equation A X = I i.e.
Theorem: Let A be a square matrix The A has a iverse matrix if ad oly if its reduced row echelo form is the idetity I this case the algorithm illustrated o the previous page will always yield the iverse
More informationSummer MA Lesson 13 Section 1.6, Section 1.7 (part 1)
Suer MA 1500 Lesso 1 Sectio 1.6, Sectio 1.7 (part 1) I Solvig Polyoial Equatios Liear equatio ad quadratic equatios of 1 variable are specific types of polyoial equatios. Soe polyoial equatios of a higher
More informationX. Perturbation Theory
X. Perturbatio Theory I perturbatio theory, oe deals with a ailtoia that is coposed Ĥ that is typically exactly solvable of two pieces: a referece part ad a perturbatio ( Ĥ ) that is assued to be sall.
More informationChapter 2. Asymptotic Notation
Asyptotic Notatio 3 Chapter Asyptotic Notatio Goal : To siplify the aalysis of ruig tie by gettig rid of details which ay be affected by specific ipleetatio ad hardware. [1] The Big Oh (O-Notatio) : It
More informationSNAP Centre Workshop. Basic Algebraic Manipulation
SNAP Cetre Workshop Basic Algebraic Maipulatio 8 Simplifyig Algebraic Expressios Whe a expressio is writte i the most compact maer possible, it is cosidered to be simplified. Not Simplified: x(x + 4x)
More informationReview Problems 1. ICME and MS&E Refresher Course September 19, 2011 B = C = AB = A = A 2 = A 3... C 2 = C 3 = =
Review Problems ICME ad MS&E Refresher Course September 9, 0 Warm-up problems. For the followig matrices A = 0 B = C = AB = 0 fid all powers A,A 3,(which is A times A),... ad B,B 3,... ad C,C 3,... Solutio:
More information5.6 Binomial Multi-section Matching Transformer
4/14/2010 5_6 Bioial Multisectio Matchig Trasforers 1/1 5.6 Bioial Multi-sectio Matchig Trasforer Readig Assiget: pp. 246-250 Oe way to axiize badwidth is to costruct a ultisectio Γ f that is axially flat.
More informationOptimal Estimator for a Sample Set with Response Error. Ed Stanek
Optial Estiator for a Saple Set wit Respose Error Ed Staek Itroductio We develop a optial estiator siilar to te FP estiator wit respose error tat was cosidered i c08ed63doc Te first 6 pages of tis docuet
More informationAPPENDIX F Complex Numbers
APPENDIX F Complex Numbers Operatios with Complex Numbers Complex Solutios of Quadratic Equatios Polar Form of a Complex Number Powers ad Roots of Complex Numbers Operatios with Complex Numbers Some equatios
More information(3) If you replace row i of A by its sum with a multiple of another row, then the determinant is unchanged! Expand across the i th row:
Math 50-004 Tue Feb 4 Cotiue with sectio 36 Determiats The effective way to compute determiats for larger-sized matrices without lots of zeroes is to ot use the defiitio, but rather to use the followig
More informationAPPLIED MULTIVARIATE ANALYSIS
ALIED MULTIVARIATE ANALYSIS FREQUENTLY ASKED QUESTIONS AMIT MITRA & SHARMISHTHA MITRA DEARTMENT OF MATHEMATICS & STATISTICS INDIAN INSTITUTE OF TECHNOLOGY KANUR X = X X X [] The variace covariace atrix
More informationMa/CS 6a Class 22: Power Series
Ma/CS 6a Class 22: Power Series By Ada Sheffer Power Series Mooial: ax i. Polyoial: a 0 + a 1 x + a 2 x 2 + + a x. Power series: A x = a 0 + a 1 x + a 2 x 2 + Also called foral power series, because we
More informationContents Two Sample t Tests Two Sample t Tests
Cotets 3.5.3 Two Saple t Tests................................... 3.5.3 Two Saple t Tests Setup: Two Saples We ow focus o a sceario where we have two idepedet saples fro possibly differet populatios. Our
More informationZeros of Polynomials
Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree
More informationFind quadratic function which pass through the following points (0,1),(1,1),(2, 3)... 11
Adrew Powuk - http://www.powuk.com- Math 49 (Numerical Aalysis) Iterpolatio... 4. Polyomial iterpolatio (system of equatio)... 4.. Lier iterpolatio... 5... Fid a lie which pass through (,) (,)... 8...
More informationIntegrals of Functions of Several Variables
Itegrals of Fuctios of Several Variables We ofte resort to itegratios i order to deterie the exact value I of soe quatity which we are uable to evaluate by perforig a fiite uber of additio or ultiplicatio
More information3.2 Properties of Division 3.3 Zeros of Polynomials 3.4 Complex and Rational Zeros of Polynomials
Math 60 www.timetodare.com 3. Properties of Divisio 3.3 Zeros of Polyomials 3.4 Complex ad Ratioal Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered
More informationA PROBABILITY PROBLEM
A PROBABILITY PROBLEM A big superarket chai has the followig policy: For every Euros you sped per buy, you ear oe poit (suppose, e.g., that = 3; i this case, if you sped 8.45 Euros, you get two poits,
More informationWe are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n
Review of Power Series, Power Series Solutios A power series i x - a is a ifiite series of the form c (x a) =c +c (x a)+(x a) +... We also call this a power series cetered at a. Ex. (x+) is cetered at
More informationProperties and Hypothesis Testing
Chapter 3 Properties ad Hypothesis Testig 3.1 Types of data The regressio techiques developed i previous chapters ca be applied to three differet kids of data. 1. Cross-sectioal data. 2. Time series data.
More informationPolynomial Functions and Their Graphs
Polyomial Fuctios ad Their Graphs I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively
More information(s)h(s) = K( s + 8 ) = 5 and one finite zero is located at z 1
ROOT LOCUS TECHNIQUE 93 should be desiged differetly to eet differet specificatios depedig o its area of applicatio. We have observed i Sectio 6.4 of Chapter 6, how the variatio of a sigle paraeter like
More informationCHAPTER 5. Theory and Solution Using Matrix Techniques
A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 3 A COLLECTION OF HANDOUTS ON SYSTEMS OF ORDINARY DIFFERENTIAL
More informationLecture 11. Solution of Nonlinear Equations - III
Eiciecy o a ethod Lecture Solutio o Noliear Equatios - III The eiciecy ide o a iterative ethod is deied by / E r r: rate o covergece o the ethod : total uber o uctios ad derivative evaluatios at each step
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More informationAVERAGE MARKS SCALING
TERTIARY INSTITUTIONS SERVICE CENTRE Level 1, 100 Royal Street East Perth, Wester Australia 6004 Telephoe (08) 9318 8000 Facsiile (08) 95 7050 http://wwwtisceduau/ 1 Itroductio AVERAGE MARKS SCALING I
More informationLesson 10: Limits and Continuity
www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals
More informationMost text will write ordinary derivatives using either Leibniz notation 2 3. y + 5y= e and y y. xx tt t
Itroductio to Differetial Equatios Defiitios ad Termiolog Differetial Equatio: A equatio cotaiig the derivatives of oe or more depedet variables, with respect to oe or more idepedet variables, is said
More informationNICK DUFRESNE. 1 1 p(x). To determine some formulas for the generating function of the Schröder numbers, r(x) = a(x) =
AN INTRODUCTION TO SCHRÖDER AND UNKNOWN NUMBERS NICK DUFRESNE Abstract. I this article we will itroduce two types of lattice paths, Schröder paths ad Ukow paths. We will examie differet properties of each,
More informationCHAPTER 10 INFINITE SEQUENCES AND SERIES
CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece
More informationECON 3150/4150, Spring term Lecture 3
Itroductio Fidig the best fit by regressio Residuals ad R-sq Regressio ad causality Summary ad ext step ECON 3150/4150, Sprig term 2014. Lecture 3 Ragar Nymoe Uiversity of Oslo 21 Jauary 2014 1 / 30 Itroductio
More information6 Integers Modulo n. integer k can be written as k = qn + r, with q,r, 0 r b. So any integer.
6 Itegers Modulo I Example 2.3(e), we have defied the cogruece of two itegers a,b with respect to a modulus. Let us recall that a b (mod ) meas a b. We have proved that cogruece is a equivalece relatio
More information1 Inferential Methods for Correlation and Regression Analysis
1 Iferetial Methods for Correlatio ad Regressio Aalysis I the chapter o Correlatio ad Regressio Aalysis tools for describig bivariate cotiuous data were itroduced. The sample Pearso Correlatio Coefficiet
More informationx !1! + 1!2!
4 Euler-Maclauri Suatio Forula 4. Beroulli Nuber & Beroulli Polyoial 4.. Defiitio of Beroulli Nuber Beroulli ubers B (,,3,) are defied as coefficiets of the followig equatio. x e x - B x! 4.. Expreesio
More informationSeptember 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1
September 0 s (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright
More informationIntroduction to Optimization, DIKU Monday 19 November David Pisinger. Duality, motivation
Itroductio to Optiizatio, DIKU 007-08 Moday 9 Noveber David Pisiger Lecture, Duality ad sesitivity aalysis Duality, shadow prices, sesitivity aalysis, post-optial aalysis, copleetary slackess, KKT optiality
More informationOrthogonal Function Solution of Differential Equations
Royal Holloway Uiversity of Loo Departet of Physics Orthogoal Fuctio Solutio of Differetial Equatios trouctio A give oriary ifferetial equatio will have solutios i ters of its ow fuctios Thus, for eaple,
More informationThe Binomial Multi-Section Transformer
4/15/2010 The Bioial Multisectio Matchig Trasforer preset.doc 1/24 The Bioial Multi-Sectio Trasforer Recall that a ulti-sectio atchig etwork ca be described usig the theory of sall reflectios as: where:
More informationNotes on iteration and Newton s method. Iteration
Notes o iteratio ad Newto s method Iteratio Iteratio meas doig somethig over ad over. I our cotet, a iteratio is a sequece of umbers, vectors, fuctios, etc. geerated by a iteratio rule of the type 1 f
More informationRecurrence Relations
Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial(-)); } Let t be the umber of multiplicatios eeded to calculate factorial(). The
More informationTHE SOLUTION OF NONLINEAR EQUATIONS f( x ) = 0.
THE SOLUTION OF NONLINEAR EQUATIONS f( ) = 0. Noliear Equatio Solvers Bracketig. Graphical. Aalytical Ope Methods Bisectio False Positio (Regula-Falsi) Fied poit iteratio Newto Raphso Secat The root of
More informationProperties and Tests of Zeros of Polynomial Functions
Properties ad Tests of Zeros of Polyomial Fuctios The Remaider ad Factor Theorems: Sythetic divisio ca be used to fid the values of polyomials i a sometimes easier way tha substitutio. This is show by
More informationCALCULATION OF FIBONACCI VECTORS
CALCULATION OF FIBONACCI VECTORS Stuart D. Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithaca.edu ad Dai Novak Departmet of Mathematics, Ithaca College
More informationBernoulli Polynomials Talks given at LSBU, October and November 2015 Tony Forbes
Beroulli Polyoials Tals give at LSBU, October ad Noveber 5 Toy Forbes Beroulli Polyoials The Beroulli polyoials B (x) are defied by B (x), Thus B (x) B (x) ad B (x) x, B (x) x x + 6, B (x) dx,. () B 3
More informationMixture models (cont d)
6.867 Machie learig, lecture 5 (Jaakkola) Lecture topics: Differet types of ixture odels (cot d) Estiatig ixtures: the EM algorith Mixture odels (cot d) Basic ixture odel Mixture odels try to capture ad
More informationThe z-transform. 7.1 Introduction. 7.2 The z-transform Derivation of the z-transform: x[n] = z n LTI system, h[n] z = re j
The -Trasform 7. Itroductio Geeralie the complex siusoidal represetatio offered by DTFT to a represetatio of complex expoetial sigals. Obtai more geeral characteristics for discrete-time LTI systems. 7.
More informationExercise 8 CRITICAL SPEEDS OF THE ROTATING SHAFT
Exercise 8 CRITICA SEEDS OF TE ROTATING SAFT. Ai of the exercise Observatio ad easureet of three cosecutive critical speeds ad correspodig odes of the actual rotatig shaft. Copariso of aalytically coputed
More informationBinomial transform of products
Jauary 02 207 Bioial trasfor of products Khristo N Boyadzhiev Departet of Matheatics ad Statistics Ohio Norther Uiversity Ada OH 4580 USA -boyadzhiev@ouedu Abstract Give the bioial trasfors { b } ad {
More informationPARTIAL DIFFERENTIAL EQUATIONS SEPARATION OF VARIABLES
Diola Bagayoko (0 PARTAL DFFERENTAL EQUATONS SEPARATON OF ARABLES. troductio As discussed i previous lectures, partial differetial equatios arise whe the depedet variale, i.e., the fuctio, varies with
More informationDiscrete-Time Systems, LTI Systems, and Discrete-Time Convolution
EEL5: Discrete-Time Sigals ad Systems. Itroductio I this set of otes, we begi our mathematical treatmet of discrete-time s. As show i Figure, a discrete-time operates or trasforms some iput sequece x [
More informationAppendix F: Complex Numbers
Appedix F Complex Numbers F1 Appedix F: Complex Numbers Use the imagiary uit i to write complex umbers, ad to add, subtract, ad multiply complex umbers. Fid complex solutios of quadratic equatios. Write
More informationMatrix Algebra 2.2 THE INVERSE OF A MATRIX Pearson Education, Inc.
2 Matrix Algebra 2.2 THE INVERSE OF A MATRIX MATRIX OPERATIONS A matrix A is said to be ivertible if there is a matrix C such that CA = I ad AC = I where, the idetity matrix. I = I I this case, C is a
More information(3) If you replace row i of A by its sum with a multiple of another row, then the determinant is unchanged! Expand across the i th row:
Math 5-4 Tue Feb 4 Cotiue with sectio 36 Determiats The effective way to compute determiats for larger-sized matrices without lots of zeroes is to ot use the defiitio, but rather to use the followig facts,
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Statistics
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER 1 018/019 DR. ANTHONY BROWN 8. Statistics 8.1. Measures of Cetre: Mea, Media ad Mode. If we have a series of umbers the
More informationA 2nTH ORDER LINEAR DIFFERENCE EQUATION
A 2TH ORDER LINEAR DIFFERENCE EQUATION Doug Aderso Departmet of Mathematics ad Computer Sciece, Cocordia College Moorhead, MN 56562, USA ABSTRACT: We give a formulatio of geeralized zeros ad (, )-discojugacy
More informationCALCULUS BASIC SUMMER REVIEW
CALCULUS BASIC SUMMER REVIEW NAME rise y y y Slope of a o vertical lie: m ru Poit Slope Equatio: y y m( ) The slope is m ad a poit o your lie is, ). ( y Slope-Itercept Equatio: y m b slope= m y-itercept=
More informationECE 901 Lecture 12: Complexity Regularization and the Squared Loss
ECE 90 Lecture : Complexity Regularizatio ad the Squared Loss R. Nowak 5/7/009 I the previous lectures we made use of the Cheroff/Hoeffdig bouds for our aalysis of classifier errors. Hoeffdig s iequality
More informationThe picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled
1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how
More informationSummary: CORRELATION & LINEAR REGRESSION. GC. Students are advised to refer to lecture notes for the GC operations to obtain scatter diagram.
Key Cocepts: 1) Sketchig of scatter diagram The scatter diagram of bivariate (i.e. cotaiig two variables) data ca be easily obtaied usig GC. Studets are advised to refer to lecture otes for the GC operatios
More informationLinear Differential Equations of Higher Order Basic Theory: Initial-Value Problems d y d y dy
Liear Differetial Equatios of Higher Order Basic Theory: Iitial-Value Problems d y d y dy Solve: a( ) + a ( )... a ( ) a0( ) y g( ) + + + = d d d ( ) Subject to: y( 0) = y0, y ( 0) = y,..., y ( 0) = y
More informationDefine a Markov chain on {1,..., 6} with transition probability matrix P =
Pla Group Work 0. The title says it all Next Tie: MCMC ad Geeral-state Markov Chais Midter Exa: Tuesday 8 March i class Hoework 4 due Thursday Uless otherwise oted, let X be a irreducible, aperiodic Markov
More informationAutomated Proofs for Some Stirling Number Identities
Autoated Proofs for Soe Stirlig Nuber Idetities Mauel Kauers ad Carste Scheider Research Istitute for Sybolic Coputatio Johaes Kepler Uiversity Altebergerstraße 69 A4040 Liz, Austria Subitted: Sep 1, 2007;
More informationSection 1.1. Calculus: Areas And Tangents. Difference Equations to Differential Equations
Differece Equatios to Differetial Equatios Sectio. Calculus: Areas Ad Tagets The study of calculus begis with questios about chage. What happes to the velocity of a swigig pedulum as its positio chages?
More informationAlgorithms and Data Structures 2014 Exercises and Solutions Week 13
Algorithms ad Data Structures 204 Exercises ad Solutios Week 3 Toom-Cook (cotiued) Durig the last lecture, two polyomials A(x) a 0 + a x ad B(x) b 0 + b x both of degree were multiplied, first by evaluatig
More informationTEACHER CERTIFICATION STUDY GUIDE
COMPETENCY 1. ALGEBRA SKILL 1.1 1.1a. ALGEBRAIC STRUCTURES Kow why the real ad complex umbers are each a field, ad that particular rigs are ot fields (e.g., itegers, polyomial rigs, matrix rigs) Algebra
More information18.01 Calculus Jason Starr Fall 2005
Lecture 18. October 5, 005 Homework. Problem Set 5 Part I: (c). Practice Problems. Course Reader: 3G 1, 3G, 3G 4, 3G 5. 1. Approximatig Riema itegrals. Ofte, there is o simpler expressio for the atiderivative
More informationLESSON 2: SIMPLIFYING RADICALS
High School: Workig with Epressios LESSON : SIMPLIFYING RADICALS N.RN.. C N.RN.. B 5 5 C t t t t t E a b a a b N.RN.. 4 6 N.RN. 4. N.RN. 5. N.RN. 6. 7 8 N.RN. 7. A 7 N.RN. 8. 6 80 448 4 5 6 48 00 6 6 6
More information1 The Primal and Dual of an Optimization Problem
CS 189 Itroductio to Machie Learig Fall 2017 Note 18 Previously, i our ivestigatio of SVMs, we forulated a costraied optiizatio proble that we ca solve to fid the optial paraeters for our hyperplae decisio
More informationREGRESSION WITH QUADRATIC LOSS
REGRESSION WITH QUADRATIC LOSS MAXIM RAGINSKY Regressio with quadratic loss is aother basic problem studied i statistical learig theory. We have a radom couple Z = X, Y ), where, as before, X is a R d
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationSOLUTION SET VI FOR FALL [(n + 2)(n + 1)a n+2 a n 1 ]x n = 0,
4. Series Solutios of Differetial Equatios:Special Fuctios 4.. Illustrative examples.. 5. Obtai the geeral solutio of each of the followig differetial equatios i terms of Maclauri series: d y (a dx = xy,
More informationDefinitions and Theorems. where x are the decision variables. c, b, and a are constant coefficients.
Defiitios ad Theorems Remember the scalar form of the liear programmig problem, Miimize, Subject to, f(x) = c i x i a 1i x i = b 1 a mi x i = b m x i 0 i = 1,2,, where x are the decisio variables. c, b,
More information6.4 Binomial Coefficients
64 Bioial Coefficiets Pascal s Forula Pascal s forula, aed after the seveteeth-cetury Frech atheaticia ad philosopher Blaise Pascal, is oe of the ost faous ad useful i cobiatorics (which is the foral ter
More informationMAXIMALLY FLAT FIR FILTERS
MAXIMALLY FLAT FIR FILTERS This sectio describes a family of maximally flat symmetric FIR filters first itroduced by Herrma [2]. The desig of these filters is particularly simple due to the availability
More informationREGRESSION (Physics 1210 Notes, Partial Modified Appendix A)
REGRESSION (Physics 0 Notes, Partial Modified Appedix A) HOW TO PERFORM A LINEAR REGRESSION Cosider the followig data poits ad their graph (Table I ad Figure ): X Y 0 3 5 3 7 4 9 5 Table : Example Data
More informationRegression with quadratic loss
Regressio with quadratic loss Maxim Ragisky October 13, 2015 Regressio with quadratic loss is aother basic problem studied i statistical learig theory. We have a radom couple Z = X,Y, where, as before,
More informationSection 14. Simple linear regression.
Sectio 14 Simple liear regressio. Let us look at the cigarette dataset from [1] (available to dowload from joural s website) ad []. The cigarette dataset cotais measuremets of tar, icotie, weight ad carbo
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationRay-triangle intersection
Ray-triagle itersectio ria urless October 2006 I this hadout, we explore the steps eeded to compute the itersectio of a ray with a triagle, ad the to compute the barycetric coordiates of that itersectio.
More informationINEQUALITIES BJORN POONEN
INEQUALITIES BJORN POONEN 1 The AM-GM iequality The most basic arithmetic mea-geometric mea (AM-GM) iequality states simply that if x ad y are oegative real umbers, the (x + y)/2 xy, with equality if ad
More informationName Period ALGEBRA II Chapter 1B and 2A Notes Solving Inequalities and Absolute Value / Numbers and Functions
Nae Period ALGEBRA II Chapter B ad A Notes Solvig Iequalities ad Absolute Value / Nubers ad Fuctios SECTION.6 Itroductio to Solvig Equatios Objectives: Write ad solve a liear equatio i oe variable. Solve
More informationEstimation for Complete Data
Estimatio for Complete Data complete data: there is o loss of iformatio durig study. complete idividual complete data= grouped data A complete idividual data is the oe i which the complete iformatio of
More informationRecursive Algorithms. Recurrences. Recursive Algorithms Analysis
Recursive Algorithms Recurreces Computer Sciece & Egieerig 35: Discrete Mathematics Christopher M Bourke cbourke@cseuledu A recursive algorithm is oe i which objects are defied i terms of other objects
More informationU8L1: Sec Equations of Lines in R 2
MCVU U8L: Sec. 8.9. Equatios of Lies i R Review of Equatios of a Straight Lie (-D) Cosider the lie passig through A (-,) with slope, as show i the diagram below. I poit slope form, the equatio of the lie
More information