Chapter 6 Best Linear Unbiased Estimate (BLUE)

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Chapter 6 Best Linear Unbiased Estimate (BLUE)"

Transcription

1 hpter 6 Bet Lner Unbed Etmte BLUE

2 Motvton for BLUE Except for Lner Model ce, the optml MVU etmtor mght:. not even ext. be dffcult or mpoble to fnd Reort to ub-optml etmte BLUE one uch ub-optml etmte Ide for BLUE:. Retrct etmte to be lner n dt x. Retrct etmte to be unbed 3. Fnd the bet one.e. wth mnmum vrnce dvntge of BLUE:eed only t nd nd moment of PDF Ddvntge of BLUE:. Sub-optml n generl. Sometme totlly npproprte ee bottom of p. 34 Men & ovrnce

3 6.3 Defnton of BLUE clr ce Oberved Dt: x [x[0] x[]... x[ ] ] PDF: px;θ depend on unknown θ BLUE contrned to be lner n dt: hooe to gve: θˆ BLU nx[ n] x n0. unbed etmtor. then mnmze vrnce Vrnce Lner Unbed Etmtor BLUE onlner Unbed Etmtor MVUE ote: h not Fg. 6. 3

4 6.4 Fndng he BLUE Sclr e. ontrn to be Lner: ˆ θ n 0 n x[ n]. ontrn to be Unbed: E { θˆ} θ Ung lner contrnt n 0 n E { x[ n ]} θ Q: When cn we meet both of thee contrnt? : Only for certn obervton model e.g., lner obervton 4

5 Fndng BLUE for Sclr Lner Obervton onder clr-prmeter lner obervton: x[n] θ[n] + w[n] E{x[n]} θ[n] hen for the unbed condton we need: ell how to chooe weght to ue n the BLUE etmtor form { ˆ} E θ θ n ow gven tht thee contrnt re met We need to mnmze the vrnce!! 0 eed ˆ θ n n[ n] θ #"! 0 n x[ n] Gven tht the covrnce mtrx of x we hve: vr θˆ { } { } vr x BLU Lke vr{x} vr{x} 5

6 6 Gol: mnmze ubject to ontrned optmzton ppendx 6: Ue Lgrngn Multpler: Mnmze J + λ 0 : Set J λ λ λ $ #$"$! ˆ vr θ x x ˆ θ BLUE ppendx 6 how tht th cheve globl mnmum

7 pplcblty of BLUE We jut derved the BLUE under the followng:. Lner obervton but wth no contrnt on the noe PDF. o knowledge of the noe PDF other thn t men nd cov!! Wht doe th tell u??? BLUE pplcble to lner obervton But noe need not be Gun!!! w umed n h. 4 Lner Model nd ll we need re the t nd nd moment of the PDF!!! But we ll ee n the Exmple tht we cn often lnerze nonlner model!!! 7

8 6.5 Vector Prmeter e: Gu-Mrkov hm Gu-Mrkov heorem: If dt cn be modeled hvng lner obervton n noe: x Hθ + w Known Mtrx Known Men & ov PDF otherwe rbtrry & unknown ˆ hen the BLUE : θ H H H x BLUE nd t covrnce : θ ˆ H H ote: If noe Gun then BLUE MVUE 8

9 Ex. 4.3: DO-Bed Emtter Locton t x,y t t Rx x,y t t Rx t t 3 x,y Rx 3 x 3,y 3 Hyperbol: τ t t contnt Hyperbol: τ 3 t 3 t contnt DO me-dfference-of-rrvl ume tht the th Rx cn meure t O: t hen from the et of O compute DO hen from the et of DO etmte locton x,y We won t worry bout how they do tht. lo there re DO ytem tht never ctully etmte O! 9

10 O Meurement Model ume meurement of O t recever only 3 hown bove: t 0, t,,t - here re meurement error O meurement model: o me the gnl emtted R Rnge from x to Rx c Speed of Propgton for EM: c 3x0 8 m/ t o + R /c + 0,,..., - Meurement oe zero-men, vrnce σ, ndependent but PDF unknown vrnce determned from etmtor ued to etmte t ow ue: R [ x x + y - y ] / t f x, y o + c x x + y y + onlner Model 0

11 Lnerzton of O Model So we lnerze the model o we cn pply BLUE: ume ome rough etmte vlble x n, y n x x n + δx y y n + δy know etmte know etmte θ [δx δy] ow ue truncted ylor ere to lnerze R x n, y n : R Known R n + xn x δx Rn # $"$! + yn y δy Rn # $"$! B ~ Rn B pply to O: t t o + δ x + δy + c c c known known known hree unknown prmeter to etmte: o, δy, δy

12 O Model v. DO Model wo opton now:. Ue O to etmte 3 prmeter: o, δy, δy. Ue DO to etmte prmeter: δy, δy Generlly the fewer prmeter the better Everythng ele beng the me. But here everythng ele not the me: Opton & hve dfferent noe model Opton h ndependent noe Opton h correlted noe In prctce we d explore both opton nd ee whch bet.

13 3 onveron to DO Model DO rther thn O DO:,,,, ~ ~ t t τ $"$! # #$"$! $"$! # noe correlted known known + + y c B B x c δ δ In mtrx form: x Hθ + w w H B B B B B B c & & & & & & & [ ] τ τ τ ' x [ ] y δx δ θ See book for tructure of mtrx w w } cov{ σ

14 pply BLUE to DO Lnerzed Model θˆ BLUE H H w H w x θˆ H H w H Dependence on σ cncel out!!! H H x σ H Decrbe how lrge the locton error H hng we cn now do:. Explore etmton error cov for dfferent x/rx geometre Plot error ellpe. nlytclly explore mple geometre to fnd trend See next chrt more detl n book 4

15 pply DO Reult to Smple Geometry x R R x α R x α R x3 d d hen cn how: θˆ σ c co 0 α 0 3/ nα Dgonl Error ov lgned Error Ellpe e y nd y-error lwy bgger thn x-error e x 5

16 σ x /cσ or σ y /cσ σ x σ y α degre e Ued Std. Dev. to how unt of X & Y ormlzed by cσ get ctul vlue by multplyng by your pecfc cσ vlue x R R x α R x α R x3 d d For Fxed Rnge R: Increng Rx Spcng d Improve ccurcy For Fxed Spcng d: Decreng Rnge R Improve ccurcy 6

Analysis of Symmetric Structures SYMMETRIC STRUCTURES

Analysis of Symmetric Structures SYMMETRIC STRUCTURES nlyi of Symmetric Structure Mny tructure, becue of ethetic nd/or functionl conidertion, re rrnged in ymmetric pttern. Recognition of uch ymmetry ill be identified nd the ue of thi ymmetry ill be ued to

More information

Linear Open Loop Systems

Linear Open Loop Systems Colordo School of Mne CHEN43 Lner Open Loop Sytem Lner Open Loop Sytem... Trnfer Functon for Smple Proce... Exmple Trnfer Functon Mercury Thermometer...2 Derblty of Devton Vrble...3 Trnfer Functon for

More information

Positive Integral Operators With Analytic Kernels

Positive Integral Operators With Analytic Kernels Çnky Ünverte Fen-Edeyt Fkülte, Journl of Art nd Scence Sy : 6 / Arl k 006 Potve ntegrl Opertor Wth Anlytc Kernel Cn Murt D KMEN Atrct n th pper we contruct exmple of potve defnte ntegrl kernel whch re

More information

Solution to Problem Set 1

Solution to Problem Set 1 CSE 5: Introduction to the Theory o Computtion, Winter A. Hevi nd J. Mo Solution to Prolem Set Jnury, Solution to Prolem Set.4 ). L = {w w egin with nd end with }. q q q q, d). L = {w w h length t let

More information

Chapter Newton-Raphson Method of Solving a Nonlinear Equation

Chapter Newton-Raphson Method of Solving a Nonlinear Equation Chpter.4 Newton-Rphson Method of Solvng Nonlner Equton After redng ths chpter, you should be ble to:. derve the Newton-Rphson method formul,. develop the lgorthm of the Newton-Rphson method,. use the Newton-Rphson

More information

Multiple/Post Hoc Group Comparisons in ANOVA

Multiple/Post Hoc Group Comparisons in ANOVA Multple/Pot Hoc Group Comparon n ANOVA Note: We may ut go over th quckly n cla. The key thng to undertand that, when tryng to dentfy where dfference are between group, there are dfferent way of adutng

More information

Basically, logarithmic transformations ask, a number, to what power equals another number?

Basically, logarithmic transformations ask, a number, to what power equals another number? Wht i logrithm? To nwer thi, firt try to nwer the following: wht i x in thi eqution? 9 = 3 x wht i x in thi eqution? 8 = 2 x Biclly, logrithmic trnformtion k, number, to wht power equl nother number? In

More information

On the Meaning of Regression Coefficients for Categorical and Continuous Variables: Model I and Model II; Effect Coding and Dummy Coding

On the Meaning of Regression Coefficients for Categorical and Continuous Variables: Model I and Model II; Effect Coding and Dummy Coding Dt_nlysisclm On the Mening of Regression for tegoricl nd ontinuous Vribles: I nd II; Effect oding nd Dummy oding R Grdner Deprtment of Psychology This describes the simple cse where there is one ctegoricl

More information

Lecture 3 Basic Probability and Statistics

Lecture 3 Basic Probability and Statistics Lecture 3 Bsic Probbility nd Sttistics The im of this lecture is to provide n extremely speedy introduction to the probbility nd sttistics which will be needed for the rest of this lecture course. The

More information

Optimal Pricing Scheme for Information Services

Optimal Pricing Scheme for Information Services Optml rcng Scheme for Informton Servces Shn-y Wu Opertons nd Informton Mngement The Whrton School Unversty of ennsylvn E-ml: shnwu@whrton.upenn.edu e-yu (Shron) Chen Grdute School of Industrl Admnstrton

More information

Area Between Curves: We know that a definite integral

Area Between Curves: We know that a definite integral Are Between Curves: We know tht definite integrl fx) dx cn be used to find the signed re of the region bounded by the function f nd the x xis between nd b. Often we wnt to find the bsolute re of region

More information

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions. Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd

More information

Lesson 10. Parametric Curves

Lesson 10. Parametric Curves Return to List of Lessons Lesson 10. Prmetric Curves (A) Prmetric Curves If curve fils the Verticl Line Test, it cn t be expressed by function. In this cse you will encounter problem if you try to find

More information

Vectors 2. 1. Recap of vectors

Vectors 2. 1. Recap of vectors Vectors 2. Recp of vectors Vectors re directed line segments - they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms

More information

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

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

More information

Chapter 7 Kinetic energy and work

Chapter 7 Kinetic energy and work Chpter 7 Kc energy nd wor I. Kc energy. II. or. III. or - Kc energy theorem. IV. or done by contnt orce - Grttonl orce V. or done by rble orce. VI. Power - Sprng orce. - Generl. D-Anly 3D-Anly or-kc Energy

More information

Newton-Raphson Method of Solving a Nonlinear Equation Autar Kaw

Newton-Raphson Method of Solving a Nonlinear Equation Autar Kaw Newton-Rphson Method o Solvng Nonlner Equton Autr Kw Ater redng ths chpter, you should be ble to:. derve the Newton-Rphson method ormul,. develop the lgorthm o the Newton-Rphson method,. use the Newton-Rphson

More information

4.0 5-Minute Review: Rational Functions

4.0 5-Minute Review: Rational Functions mth 130 dy 4: working with limits 1 40 5-Minute Review: Rtionl Functions DEFINITION A rtionl function 1 is function of the form y = r(x) = p(x) q(x), 1 Here the term rtionl mens rtio s in the rtio of two

More information

9 CONTINUOUS DISTRIBUTIONS

9 CONTINUOUS DISTRIBUTIONS 9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete

More information

Tests for One Poisson Mean

Tests for One Poisson Mean Chpter 412 Tests for One Poisson Men Introduction The Poisson probbility lw gives the probbility distribution of the number of events occurring in specified intervl of time or spce. The Poisson distribution

More information

x o R n a π(a, x o ) A R n π(a, x o ) π(a, x o ) A R n a a x o x o x n X R n δ(x n, x o ) d(a, x n ) d(, ) δ(, ) R n x n X d(a, x n ) δ(x n, x o ) a = a A π(a, xo ) a a A = X = R π(a, x o ) = (x o + ρ)

More information

2 2 The kinetic energy of a rotating shaft is MUCH higher!!!

2 2 The kinetic energy of a rotating shaft is MUCH higher!!! Hmewrk #6 Slutin Aignment: 4., 4.4, 4.5, 4.8, 4.9, n 4. Bergen & Vittl 4. Prt A: H W S Slutin: kinetic Wkinetic H * S 3 b 5*M 5 3 B Prt B: T fin the kinetic energy f -tn truck (ee imge belw which i ctully

More information

Applications of the Laplace Transform

Applications of the Laplace Transform EE 4G Noe: hper 6 nrucor: heung Applcon of he plce Trnform Applcon n rcu Anly. evew of eve Nework Elemen Superpoon Pge 6 PDF reed wh dekpdf PDF Wrer Trl :: hp://www.docudek.com EE 4G Noe: hper 6 nrucor:

More information

Exponentiation: Theorems, Proofs, Problems Pre/Calculus 11, Veritas Prep.

Exponentiation: Theorems, Proofs, Problems Pre/Calculus 11, Veritas Prep. Exponentition: Theorems, Proofs, Problems Pre/Clculus, Verits Prep. Our Exponentition Theorems Theorem A: n+m = n m Theorem B: ( n ) m = nm Theorem C: (b) n = n b n ( ) n n Theorem D: = b b n Theorem E:

More information

Arc Length. P i 1 P i (1) L = lim. i=1

Arc Length. P i 1 P i (1) L = lim. i=1 Arc Length Suppose tht curve C is defined by the eqution y = f(x), where f is continuous nd x b. We obtin polygonl pproximtion to C by dividing the intervl [, b] into n subintervls with endpoints x, x,...,x

More information

Treatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.

Treatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3. The nlysis of vrince (ANOVA) Although the t-test is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the t-test cn be used to compre the mens of only

More information

and thus, they are similar. If k = 3 then the Jordan form of both matrices is

and thus, they are similar. If k = 3 then the Jordan form of both matrices is Homework ssignment 11 Section 7. pp. 249-25 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If

More information

Control Systems II Lecture 2

Control Systems II Lecture 2 Control Sytem II Lecture 2 ١ Coure Content Modeling of liner ytem Repreenttion uing phe vrible Stte pce uing cnonicl form Propertie of trnition mtri Pole/zero Eigen vlue Pole Plcement

More information

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of

More information

Binary Representation of Numbers Autar Kaw

Binary Representation of Numbers Autar Kaw Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy

More information

Graphs on Logarithmic and Semilogarithmic Paper

Graphs on Logarithmic and Semilogarithmic Paper 0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl

More information

Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the

More information

Resistive Network Analysis. The Node Voltage Method - 1

Resistive Network Analysis. The Node Voltage Method - 1 esste Network Anlyss he nlyss of n electrcl network conssts of determnng ech of the unknown rnch currents nd node oltges. A numer of methods for network nlyss he een deeloped, sed on Ohm s Lw nd Krchoff

More information

Figure 1. Inventory Level vs. Time - EOQ Problem

Figure 1. Inventory Level vs. Time - EOQ Problem IEOR 54 Sprng, 009 rof Leahman otes on Eonom Lot Shedulng and Eonom Rotaton Cyles he Eonom Order Quantty (EOQ) Consder an nventory tem n solaton wth demand rate, holdng ost h per unt per unt tme, and replenshment

More information

Basics of Counting. A note on combinations. Recap. 22C:19, Chapter 6.5, 6.7 Hantao Zhang

Basics of Counting. A note on combinations. Recap. 22C:19, Chapter 6.5, 6.7 Hantao Zhang Bscs of Countng 22C:9, Chpter 6.5, 6.7 Hnto Zhng A note on comntons An lterntve (nd more common) wy to denote n r-comnton: n n C ( n, r) r I ll use C(n,r) whenever possle, s t s eser to wrte n PowerPont

More information

ErrorPropagation.nb 1. Error Propagation

ErrorPropagation.nb 1. Error Propagation ErrorPropagaton.nb Error Propagaton Suppose that we make observatons of a quantty x that s subject to random fluctuatons or measurement errors. Our best estmate of the true value for ths quantty s then

More information

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive

More information

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers. 2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this

More information

Lectures 8 and 9 1 Rectangular waveguides

Lectures 8 and 9 1 Rectangular waveguides 1 Lectures 8 nd 9 1 Rectngulr wveguides y b x z Consider rectngulr wveguide with 0 < x b. There re two types of wves in hollow wveguide with only one conductor; Trnsverse electric wves

More information

Review guide for the final exam in Math 233

Review guide for the final exam in Math 233 Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered

More information

11. Fourier series. sin mx cos nx dx = 0 for any m, n, sin 2 mx dx = π.

11. Fourier series. sin mx cos nx dx = 0 for any m, n, sin 2 mx dx = π. . Fourier series Summry of the bsic ides The following is quick summry of the introductory tretment of Fourier series in MATH. We consider function f with period π, tht is, stisfying f(x + π) = f(x) for

More information

In this section make precise the idea of a matrix inverse and develop a method to find the inverse of a given square matrix when it exists.

In this section make precise the idea of a matrix inverse and develop a method to find the inverse of a given square matrix when it exists. Mth 52 Sec S060/S0602 Notes Mtrices IV 5 Inverse Mtrices 5 Introduction In our erlier work on mtrix multipliction, we sw the ide of the inverse of mtrix Tht is, for squre mtrix A, there my exist mtrix

More information

Your Thoughts. Does the moment of inertia play a part in determining the amount of time it takes an object to get to the bottom of an incline.

Your Thoughts. Does the moment of inertia play a part in determining the amount of time it takes an object to get to the bottom of an incline. Your Thoughts Suppose bll rolls down rmp with coefficient of friction just big enough to keep the bll from slipping. An identicl bll rolls down n identicl rmp with coefficient of friction of. Do both blls

More information

Chapter Solution of Cubic Equations

Chapter Solution of Cubic Equations Chpter. Soluton of Cuc Equtons After redng ths chpter, ou should e le to:. fnd the ect soluton of generl cuc equton. Ho to Fnd the Ect Soluton of Generl Cuc Equton In ths chpter, e re gong to fnd the ect

More information

1. Description of Linear Prediction

1. Description of Linear Prediction Liner Prediction nd Levinson-Durbin lgorithm Cedric Collomb http://ccollomb.free.fr/ Copyright 9. ll ights eserved. Creted: Februry 3, 9 Lst Modified: ovember, 9 Contents. Description of Liner Prediction....

More information

Lecture 3 Gaussian Probability Distribution

Lecture 3 Gaussian Probability Distribution Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike

More information

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is so-clled becuse when the sclr product of two vectors

More information

Continuous Random Variables: Derived Distributions

Continuous Random Variables: Derived Distributions Continuous Rndom Vriles: Derived Distriutions Berlin Chen Deprtment o Computer Science & Inormtion Engineering Ntionl Tiwn Norml Universit Reerence: - D. P. Bertseks, J. N. Tsitsiklis, Introduction to

More information

A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324

A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324 A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................

More information

Net Change and Displacement

Net Change and Displacement mth 11, pplictions motion: velocity nd net chnge 1 Net Chnge nd Displcement We hve seen tht the definite integrl f (x) dx mesures the net re under the curve y f (x) on the intervl [, b] Any prt of the

More information

Uniform convergence and its consequences

Uniform convergence and its consequences Uniform convergence nd its consequences The following issue is centrl in mthemtics: On some domin D, we hve sequence of functions {f n }. This mens tht we relly hve n uncountble set of ordinry sequences,

More information

Lecture 15 - Curve Fitting Techniques

Lecture 15 - Curve Fitting Techniques Lecture 15 - Curve Fitting Techniques Topics curve fitting motivtion liner regression Curve fitting - motivtion For root finding, we used given function to identify where it crossed zero where does fx

More information

not to be republished NCERT POLYNOMIALS CHAPTER 2 (A) Main Concepts and Results (B) Multiple Choice Questions

not to be republished NCERT POLYNOMIALS CHAPTER 2 (A) Main Concepts and Results (B) Multiple Choice Questions POLYNOMIALS (A) Min Concepts nd Results Geometricl mening of zeroes of polynomil: The zeroes of polynomil p(x) re precisely the x-coordintes of the points where the grph of y = p(x) intersects the x-xis.

More information

Section 2.3. Motion Along a Curve. The Calculus of Functions of Several Variables

Section 2.3. Motion Along a Curve. The Calculus of Functions of Several Variables The Clculus of Functions of Severl Vribles Section 2.3 Motion Along Curve Velocity ccelertion Consider prticle moving in spce so tht its position t time t is given by x(t. We think of x(t s moving long

More information

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

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

More information

Communication Networks II Contents

Communication Networks II Contents 8 / 1 -- Communcaton Networs II (Görg) -- www.comnets.un-bremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP

More information

4.11 Inner Product Spaces

4.11 Inner Product Spaces 314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces

More information

Algebra Review. How well do you remember your algebra?

Algebra Review. How well do you remember your algebra? Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then

More information

Ring structure of splines on triangulations

Ring structure of splines on triangulations www.oeaw.ac.at Rng structure of splnes on trangulatons N. Vllamzar RICAM-Report 2014-48 www.rcam.oeaw.ac.at RING STRUCTURE OF SPLINES ON TRIANGULATIONS NELLY VILLAMIZAR Introducton For a trangulated regon

More information

16. Mean Square Estimation

16. Mean Square Estimation 6 Me Sque stmto Gve some fomto tht s elted to uow qutty of teest the poblem s to obt good estmte fo the uow tems of the obseved dt Suppose epeset sequece of dom vbles bout whom oe set of obsevtos e vlble

More information

Plotting and Graphing

Plotting and Graphing Plotting nd Grphing Much of the dt nd informtion used by engineers is presented in the form of grphs. The vlues to be plotted cn come from theoreticl or empiricl (observed) reltionships, or from mesured

More information

MODULE 3. 0, y = 0 for all y

MODULE 3. 0, y = 0 for all y Topics: Inner products MOULE 3 The inner product of two vectors: The inner product of two vectors x, y V, denoted by x, y is (in generl) complex vlued function which hs the following four properties: i)

More information

The Chain Rule. rf dx. t t lim " (x) dt " (0) dx. df dt = df. dt dt. f (r) = rf v (1) df dx

The Chain Rule. rf dx. t t lim  (x) dt  (0) dx. df dt = df. dt dt. f (r) = rf v (1) df dx The Chin Rule The Chin Rule In this section, we generlize the chin rule to functions of more thn one vrible. In prticulr, we will show tht the product in the single-vrible chin rule extends to n inner

More information

Arithmetic Sequences

Arithmetic Sequences Arithmetic equeces A simple wy to geerte sequece is to strt with umber, d dd to it fixed costt d, over d over gi. This type of sequece is clled rithmetic sequece. Defiitio: A rithmetic sequece is sequece

More information

Global Optimization Method for Robust Pricing of Transportation Networks under Uncertain Demand

Global Optimization Method for Robust Pricing of Transportation Networks under Uncertain Demand Interntonl Journl of Trnportton Vol. 2, No. 2 (214), pp.33-48 http://dx.do.org/1.14257/jt.214.2.2.3 Globl Optmzton Method for Robut Prcng of Trnportton Network under Uncertn Demnd hun Wng 1, Luren M. Grdner

More information

CLUSTER SAMPLING DR. SHALABH DEPARTMENT OF MATHEMATICS AND STATISTICS INDIAN INSTITUTE OF TECHNOLOGY KANPUR 1

CLUSTER SAMPLING DR. SHALABH DEPARTMENT OF MATHEMATICS AND STATISTICS INDIAN INSTITUTE OF TECHNOLOGY KANPUR 1 amplng Theory MODULE IX LECTURE - 30 CLUTER AMPLIG DR HALABH DEPARTMET OF MATHEMATIC AD TATITIC IDIA ITITUTE OF TECHOLOGY KAPUR It s one of the asc assumptons n any samplng procedure that the populaton

More information

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

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

More information

Chapter 9: Quadratic Equations

Chapter 9: Quadratic Equations Chpter 9: Qudrtic Equtions QUADRATIC EQUATIONS DEFINITION + + c = 0,, c re constnts (generlly integers) ROOTS Synonyms: Solutions or Zeros Cn hve 0, 1, or rel roots Consider the grph of qudrtic equtions.

More information

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

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

More information

Two special Right-triangles 1. The

Two special Right-triangles 1. The Mth Right Tringle Trigonometry Hndout B (length of ) - c - (length of side ) (Length of side to ) Pythgoren s Theorem: for tringles with right ngle ( side + side = ) + = c Two specil Right-tringles. The

More information

Generalized Inverses: How to Invert a Non-Invertible Matrix

Generalized Inverses: How to Invert a Non-Invertible Matrix Generlized Inverses: How to Invert Non-Invertible Mtrix S. Swyer September 7, 2006 rev August 6, 2008. Introduction nd Definition. Let A be generl m n mtrix. Then nturl question is when we cn solve Ax

More information

Lecture 5. Inner Product

Lecture 5. Inner Product Lecture 5 Inner Product Let us strt with the following problem. Given point P R nd line L R, how cn we find the point on the line closest to P? Answer: Drw line segment from P meeting the line in right

More information

Sequences and Series

Sequences and Series Centre for Eduction in Mthemtics nd Computing Euclid eworkshop # 5 Sequences nd Series c 014 UNIVERSITY OF WATERLOO While the vst mjority of Euclid questions in this topic re use formule for rithmetic

More information

Physics 110 Spring 2006 Forces in 1- and 2-Dimensions Their Solutions

Physics 110 Spring 2006 Forces in 1- and 2-Dimensions Their Solutions Phic 110 Spring 006 orce in 1- nd -Dienion heir Solution 1. wo orce 1 nd ct on 5kg. I the gnitude o 1 nd re 0 nd 15 repectivel wht re the ccelertion o ech o the e elow?.. 0; ( 0 ) + ( 15 ) 1 5kg 15 @ θ

More information

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

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

More information

Brillouin Zones. Physics 3P41 Chris Wiebe

Brillouin Zones. Physics 3P41 Chris Wiebe Brillouin Zones Physics 3P41 Chris Wiebe Direct spce to reciprocl spce * = 2 i j πδ ij Rel (direct) spce Reciprocl spce Note: The rel spce nd reciprocl spce vectors re not necessrily in the sme direction

More information

Math 1105: Calculus II (Math/Sci majors) MWF 11am / 12pm, Campion 235 Written homework 5

Math 1105: Calculus II (Math/Sci majors) MWF 11am / 12pm, Campion 235 Written homework 5 Mth 5: Clculus II Mth/Sci mjos) MWF m / pm, Cmpion 35 Witten homewok 5 6.6, p. 458 3,33), 6.7, p. 467 8,3), 6.875), 7.58,6,6), 7.44,48) Fo pctice not to tun in): 6.6, p. 458,8,,3,4), 6.7, p. 467 4,6,8),

More information

Inventory Aggregation and Discounting

Inventory Aggregation and Discounting Inventory Aggregaton and Dscountng Matchng Supply and Demand utdallas.edu/~metn 1 Outlne Jont fxed costs for multple products Long term quantty dscounts utdallas.edu/~metn Example: Lot Szng wth Multple

More information

10.5 Graphing Quadratic Functions

10.5 Graphing Quadratic Functions 0.5 Grphing Qudrtic Functions Now tht we cn solve qudrtic equtions, we wnt to lern how to grph the function ssocited with the qudrtic eqution. We cll this the qudrtic function. Grphs of Qudrtic Functions

More information

The University of Kansas

The University of Kansas All Greek Summary Rank Chapter Name Total Membership Chapter GPA 1 Beta Theta Pi 3.57 2 Chi Omega 3.42 3 Kappa Alpha Theta 3.36 4 Kappa Kappa Gamma 3.28 *5 Pi Beta Phi 3.27 *5 Gamma Phi Beta 3.27 *7 Alpha

More information

JFET AMPLIFIER CONFIGURATIONS

JFET AMPLIFIER CONFIGURATIONS JFET MPFE CONFGUTON 5 5 5 n G OUT n G OUT n OUT [a] Cn urce plfer [b] Cn ran [urce Fllwer] plfer [c] Cn Gate plfer Cte a: n ce, cure ateral fr 6.0 ntructry nal Electrnc abratry, prn 2007.MT OpenCureWare

More information

Chapter 4 ECONOMIC DISPATCH AND UNIT COMMITMENT

Chapter 4 ECONOMIC DISPATCH AND UNIT COMMITMENT Chapter 4 ECOOMIC DISATCH AD UIT COMMITMET ITRODUCTIO A power system has several power plants. Each power plant has several generatng unts. At any pont of tme, the total load n the system s met by the

More information

Alternatives to an Inefficient International Telephony. Settlement System

Alternatives to an Inefficient International Telephony. Settlement System Alterntve to n Ineffent Interntonl Telephony Settlement Sytem Alterntve to n Ineffent Interntonl Telephony Settlement Sytem Koj Domon Shool of Sol Sene Wed Unverty -6- Nh-Wed Shnjuku-ku Tokyo 69-8050 JAPAN

More information

LECTURE #05. Learning Objective. To describe the geometry in and around a unit cell in terms of directions and planes.

LECTURE #05. Learning Objective. To describe the geometry in and around a unit cell in terms of directions and planes. LECTURE #05 Chpter 3: Lttice Positions, Directions nd Plnes Lerning Objective To describe the geometr in nd round unit cell in terms of directions nd plnes. 1 Relevnt Reding for this Lecture... Pges 64-83.

More information

The Velocity Factor of an Insulated Two-Wire Transmission Line

The Velocity Factor of an Insulated Two-Wire Transmission Line The Velocity Fctor of n Insulted Two-Wire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the

More information

Sect 8.3 Triangles and Hexagons

Sect 8.3 Triangles and Hexagons 13 Objective 1: Sect 8.3 Tringles nd Hexgons Understnding nd Clssifying Different Types of Polygons. A Polygon is closed two-dimensionl geometric figure consisting of t lest three line segments for its

More information

NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING. Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582

NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING. Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582 NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582 7. Root Dynamcs 7.2 Intro to Root Dynamcs We now look at the forces requred to cause moton of the root.e. dynamcs!!

More information

Physics 43 Homework Set 9 Chapter 40 Key

Physics 43 Homework Set 9 Chapter 40 Key Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nm-wide region t x

More information

Lesson 12.1 Trigonometric Ratios

Lesson 12.1 Trigonometric Ratios Lesson 12.1 rigonometric Rtios Nme eriod Dte In Eercises 1 6, give ech nswer s frction in terms of p, q, nd r. 1. sin 2. cos 3. tn 4. sin Q 5. cos Q 6. tn Q p In Eercises 7 12, give ech nswer s deciml

More information

MATLAB Workshop 13 - Linear Systems of Equations

MATLAB Workshop 13 - Linear Systems of Equations MATLAB: Workshop - Liner Systems of Equtions pge MATLAB Workshop - Liner Systems of Equtions Objectives: Crete script to solve commonly occurring problem in engineering: liner systems of equtions. MATLAB

More information

Chapter 04.05 System of Equations

Chapter 04.05 System of Equations hpter 04.05 System of Equtios After redig th chpter, you should be ble to:. setup simulteous lier equtios i mtrix form d vice-vers,. uderstd the cocept of the iverse of mtrix, 3. kow the differece betwee

More information

N V V L. R a L I. Transformer Equation Notes

N V V L. R a L I. Transformer Equation Notes Tnsfome Eqution otes This file conts moe etile eivtion of the tnsfome equtions thn the notes o the expeiment 3 wite-up. t will help you to unestn wht ssumptions wee neee while eivg the iel tnsfome equtions

More information

Anti-derivatives/Indefinite Integrals of Basic Functions

Anti-derivatives/Indefinite Integrals of Basic Functions Anti-derivtives/Indefinite Integrls of Bsic Functions Power Rule: x n+ x n n + + C, dx = ln x + C, if n if n = In prticulr, this mens tht dx = ln x + C x nd x 0 dx = dx = dx = x + C Integrl of Constnt:

More information

Vector Geometry for Computer Graphics

Vector Geometry for Computer Graphics Vector Geometry for Computer Grphcs Bo Getz Jnury, 7 Contents Prt I: Bsc Defntons Coordnte Systems... Ponts nd Vectors Mtrces nd Determnnts.. 4 Prt II: Opertons Vector ddton nd sclr multplcton... 5 The

More information

The University of Kansas

The University of Kansas Fall 2011 Scholarship Report All Greek Summary Rank Chapter Name Chapter GPA 1 Beta Theta Pi 3.57 2 Chi Omega 3.42 3 Kappa Alpha Theta 3.36 *4 Gamma Phi Beta 3.28 4 Kappa Kappa Gamma 3.28 6 Pi Beta Phi

More information

Math 135 Circles and Completing the Square Examples

Math 135 Circles and Completing the Square Examples Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for

More information

Irregular Repeat Accumulate Codes 1

Irregular Repeat Accumulate Codes 1 Irregulr epet Accumulte Codes 1 Hu Jn, Amod Khndekr, nd obert McElece Deprtment of Electrcl Engneerng, Clforn Insttute of Technology Psden, CA 9115 USA E-ml: {hu, mod, rjm}@systems.cltech.edu Abstrct:

More information

Fraternity & Sorority Academic Report Fall 2015

Fraternity & Sorority Academic Report Fall 2015 Fraternity & Sorority Academic Report Organization Lambda Upsilon Lambda 1-1 1 Delta Chi 77 19 96 2 Alpha Delta Chi 30 1 31 3 Alpha Delta Pi 134 62 196 4 Alpha Sigma Phi 37 13 50 5 Sigma Alpha Epsilon

More information

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding 1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde

More information

Physics 2102 Lecture 2. Physics 2102

Physics 2102 Lecture 2. Physics 2102 Physics 10 Jonthn Dowling Physics 10 Lecture Electric Fields Chrles-Augustin de Coulomb (1736-1806) Jnury 17, 07 Version: 1/17/07 Wht re we going to lern? A rod mp Electric chrge Electric force on other

More information