The Mathematical Derivation of Least Squares
|
|
- Benedict George
- 7 years ago
- Views:
Transcription
1 Pscholog 885 Prof. Federco The Mathematcal Dervaton of Least Squares Back when the powers that e forced ou to learn matr algera and calculus, I et ou all asked ourself the age-old queston: When the hell wll I use ths stuff? Well, at long last, that when s now! Gven the centralt of the lnear regresson model to research n the socal and ehavoral scences, our decson to ecome a pschologst more or less ensures that ou wll regularl use a tool that s crtcall dependent on matr algera and dfferental calculus n order to do some quanttatve heav lftng. As ou know, oth varate and multple OLS regresson requres us to estmate values for a crtcal set of parameters: a regresson constant and one regresson coeffcent for each ndependent varale n our model. The regresson constant tells us the predcted value of the dependent varale (DV, hereafter) when all of the ndependent varales (IVs, hereafter) equal. The unstandardzed regresson coeffcent for each IV tells us how much the predcted value of the DV would change wth a one-unt ncrease n the IV, when all other IVs are at. OLS estmates these parameters fndng the values for the constant and coeffcents that mnmze the sum of the squared errors of predcton,.e., the dfferences etween a case s actual score on the DV and the score we predct for them usng actual scores on the IVs. For oth the varate and multple regresson cases, ths handout wll show how ths s done hopefull sheddng lght on the conceptual underpnnngs of regresson tself. The Bvarate Case For the case n whch there s onl one IV, the classcal OLS regresson model can e epressed as follows: + + e () where s case s score on the DV, s case s score on the IV, s the regresson constant, s the regresson coeffcent for the effect of, and e s the error we make n predctng from. Page
2 Pscholog 885 Prof. Federco ow, n runnng the regresson model, what are trng to do s to mnmze the sum of the squared errors of predcton.e., of the e values across all cases. Mathematcall, ths quantt can e epressed as: SSE e () Specfcall, what we want to do s fnd the values of and that mnmze the quantt n Equaton aove. So, how do we do ths? The ke s to thnk ack to dfferental calculus and rememer how one goes aout fndng the mnmum value of a mathematcal functon. Ths nvolves takng the dervatve of that functon. As ou ma recall, f s some mathematcal functon of varale, the dervatve of wth respect to s the amount of change n that occurs wth a tn change n. Roughl, t s the nstantaneous rate of change n wth respect to changes n. So, what does ths have to do wth the mnmum of a mathematcal functon? Well, the dervatve of functon wth respect to the etent to whch changes wth a tn change n equals zero when s at ts mnmum value. If we fnd the value of for whch the dervatve of equals zero, then we have found the value of for whch s nether ncreasng nor decreasng wth respect to. Thus, f we want to fnd the values of and that mnmze SSE, we need to epress SSE n terms of and, take the dervatves of SSE wth respect to and, set these dervatves to zero, and solve for and. Formall, for the mathematcall nclned, the dervatve of wth respect to d/d s defned as: d d Δ lm Δ Δ In plan Englsh, t s the value that the change n Δ relatve to the change n Δ converges on as the sze of Δ approaches zero. It s an nstantaneous rate of change n. ote that the value of for whch the dervatve of equals zero can also ndcate a mamum. However, we can e sure that we have found a mnmum f the second dervatve of wth respect to.e., the dervatve of the dervatve of wth respect to has a postve value at the value of for whch the dervatve of equals zero. As we wll see elow, ths s the case wth regard to the dervatves of SSE wth respect to the regresson constant and coeffcent. Page
3 Pscholog 885 Prof. Federco However, snce SSE s a functon of two crtcal varales and we wll need to take the partal dervatves of SSE wth respect to and. In practce, ths means we wll need to take the dervatve of SSE wth regard to each of these crtcal varales one at a tme, whle treatng the other crtcal varale as a constant (keepng n mnd that the dervatve of a constant alwas equals zero). In effect, what ths does s take the dervatve of SSE wth respect to one varale whle holdng the other constant. We egn rearrangng the asc OLS equaton for the varate case so that we can epress e n terms of,,, and. Ths gves us: e (3) Susttutng ths epresson ack nto Equaton (), we get SSE ( ) (4) where the sample sze for the data. It s ths epresson that we actuall need to dfferentate wth respect to and. Let s start takng the partal dervatve of SSE wth respect to the regresson constant,,.e., ( ) In dong ths, we can move the summaton operator (Σ) out front, snce the dervatve of a sum s equal to the sum of the dervatves: ( ) We then focus on dfferentatng the squared quantt n parentheses. Snce ths quantt s a composte we do the math n parentheses and then square the result we need to use the chan rule n order to otan the partal dervatve of SSE wth respect to the regresson constant. 3 In order to do ths, we treat,, and as constants. Ths gves us: 3 Use of the chan rule n ths contet s a two-step procedure. In the frst step, we take the partal dervatve of the quantt n parentheses wth respect to. Here, we treat,, and as Page 3
4 Pscholog 885 Prof. Federco [ ( )] Further rearrangement gves us a fnal result of: ( ) (5) For the tme eng, let s put ths result asde and take the partal dervatve of SSE wth respect to the regresson coeffcent,,.e., ( ) Agan, we can move the summaton operator (Σ) out front: ( ) We then dfferentate the squared quantt n parentheses, agan usng the chan rule. Ths tme, however, we treat,, and as constants. Wth some susequent rearrangement, ths gves us: ( ) (6) constants, meanng that the dervatves of frst and last terms n ths quantt equal zero. In order to take the dervatve of the mddle term (- ), we sutract one from the value of the eponent on - (.e., ) and multpl ths result the eponent on - (.e., ) from the orgnal epresson. Snce rasng to the power of zero gves us, the dervatve for the quantt n parentheses s -. In the second step, we take the dervatve of ( ) wth respect to ( ). We do ths sutractng one from the value of the eponent on the quantt n parentheses (.e., ) and multpl ths result the eponent on the quantt n parentheses (.e., ) from the orgnal epresson. Ths gves us ( ). Multplng ths the result from the frst step, we get a fnal result of -( ). Page 4
5 Pscholog 885 Prof. Federco Wth that, we have our two partal dervatves of SSE n Equatons (5) and (6). 4 The net step s to set each one of them to zero: ( ) (7) ( ) (8) Equatons (7) and (8) form a sstem of equatons wth two unknowns our OLS estmates, and. The net step s to solve for these two unknowns. We start solvng Equaton (7) for. Frst, we get rd of the - multplng each sde of the equaton -/: ( ) et, we dstrute the summaton operator though all of the terms n the epresson n parentheses: Then, we add the mddle summaton term on the rght to oth sdes of the equaton, gvng us: Snce and the same for all cases n the orgnal OLS equaton, ths further smplfes to: 4 The second partal dervatves of SSE wth respect to and are and, respectvel. Snce oth of these values are necessarl postve (.e., ecause,, and the square of wll alwas e postve), we can e sure that the values of and that satsf the equatons generated settng each partal dervatve to zero refer to mnmum rather than mamum values of SSE. Page 5
6 Pscholog 885 Prof. Federco Page 6 To solate on the left sde of the equaton, we then dvde oth sdes : (9) Equaton (9) wll come n hand later on, so keep t n mnd. Rght now, though, t s mportant to note that the frst term on the rght of Equaton (9) s smpl the mean of, whle everthng followng n the second term on the rght s the mean of. Ths smplfes the equaton for to the form from lecture: () ow, we need to solve Equaton (8) for. Agan, we get rd of the - multplng each sde of the equaton -/: ( ) et, we dstrute through all of the terms n parentheses: ( ) We then dstrute the summaton operator through all of the terms n the epresson n parentheses: et, we rng all of the constants n these terms (.e., and ) out n front of the summaton operators, as follows:
7 Pscholog 885 Prof. Federco Page 7 We then add the last term on the rght sde of the equaton to oth sdes: et, we go ack to the value for from Equaton (9) and susttute t nto the result we just otaned. Ths gves us: Multplng out the last term on the rght, we get: If we then add the last term on the rght to oth sdes of the equaton, we get: + On the left sde of the equaton, we can then factor out : +
8 Pscholog 885 Prof. Federco Fnall, f we dvde oth sdes of the equaton the quantt n the large rackets on the left sde, we can solate and otan the least-square estmator for the regresson coeffcent n the varate case. Ths s the form from lecture: () The epresson on the rght, as ou wll recall, s the rato of the sum of the crossproducts of and over the sum of squares for. The Multple Regresson Case: Dervng OLS wth Matrces The foregong math s all well and good f ou have onl one ndependent varale n our analss. However, n the socal scences, ths wll rarel e the case: rather, we wll usuall e trng to predct a dependent varale usng scores from several ndependent varales. Dervng a more general form of the least-squares estmator for stuatons lke ths requres the use of matr operatons. As ou wll recall from lecture, the asc OLS regresson equaton can e represented n the followng matr form: Y XB + e () where Y s an column matr of cases scores on the DV, X s an (k+) matr of cases scores on the IVs (where the frst column s a placeholder column of ones for the constant and the remanng columns correspond to each IV), B s a (k+) column matr contanng the regresson constant and coeffcents, and e s an column matr of cases errors of predcton. As efore, what we want to do s fnd the values for the elements of B that mnmze the sum of the squared errors. The quantt that we are trng to mnmze can e epressed as follows: SSE e e (3) Page 8
9 Pscholog 885 Prof. Federco If ou work out the matr operatons for the epresson on the rght, ou ll notce that the result s a scalar a sngle numer consstng of the sum of the squared errors of predcton (.e., multplng a matr a matr produces a matr,.e., a scalar). In order to take the dervatve of the quantt wth regard to the B matr, we frst of all need to epress e n terms of Y, X, and B: e Y XB Susttutng the epresson on the rght sde nto Equaton (3), we get: SSE ( Y XB) ( Y XB) et, the transposton operator on the frst quantt n ths product (Y - XB) can dstruted: 5 ( )( ) SSE Y B X Y XB When ths product s computed, we get the followng: SSE Y Y Y XB B X Y + B X XB ow, f multpled out, the two mddle terms Y XB and B X Y -- are dentcal: the produce the same scalar value. As such, the equaton can e further smplfed to: SSE Y Y Y XB + B X XB (4) We now have an equaton whch epresses SSE n terms of Y, X, and B. The net step as n the varate case s to take the dervatve of SSE wth respect to the matr B. Snce we re reall dealng wth a set of varales n ths dfferentaton prolem the constant and one regresson coeffcent for each IV we agan use the partal dervatve operator: B B ( Y Y Y XB + B X XB) 5 Rememer that for an two matrces A and B that can e multpled together, (AB) B A. Page 9
10 Pscholog 885 Prof. Federco Ths looks lke a comple prolem, ut t s actuall qute smlar to takng the dervatve of a polnomal n the scalar contet. Frst, snce we are treatng all matrces esdes B as the equvalent of constants, the frst term n parentheses ased completel on the Y matr has a dervatve of zero. Second, the mddle term known as a lnear form n B s the equvalent of a scalar term n whch the varale we are dfferentatng wth respect to s rased to the frst power (.e. a lnear term), whch means we otan the dervatve droppng the B and takng the transpose of all the matrces n the epresson whch reman, gvng us -X Y. Fnall, the thrd term known as a quadratc form n B s the equvalent of a scalar term n whch the varale we are dfferentatng wth respect to s rased to the second power (.e., a quadratc term). Ths means we otan the dervatve droppng the B from the term and multplng two, gvng us X XB. Thus, the full partal dervatve s B X Y + X XB (5) The net step s to set ths partal dervatve to zero and solve for the matr B. Ths wll gve us an epresson for the matr of estmates that mnmze the sum of the squared errors of predcton. We start wth the followng: X Y + X XB We then sutract X XB from each sde of the equaton: X XB X Y et, we elmnate the - on each term multplng each sde of the equaton -/: X XB X Y Fnall, we need to solve for B pre-multplng each sde of the equaton the nverse of (X X),.e., (X X) -. Rememer that ths s the matr equvalent of dvdng each sde of the equaton (X X): Page
11 Pscholog 885 Prof. Federco ( X X) X Y B (6) Equaton (6) s, of course, the famlar OLS estmator we dscussed n lecture. To te ths ack to the varate case, note closel what the epresson on the rght does. Whle X Y gves the sum of the cross-products of X and Y, X X gves us the sum of squares for X. Snce pre-multplng X Y (X X) - s the matr equvalent of dvdng X Y X X, ths epresson s ascall dong the same thng as the scalar epresson for n Equaton (): dvdng the sum of the cross products of the IV (or IVs) and the DV the sum of squares for the IV (or IVs). Page
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
More information8.5 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 informationInstitute 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
More informationBERNSTEIN 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
More informationSupport 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.
More informationSimple Interest Loans (Section 5.1) :
Chapter 5 Fnance The frst part of ths revew wll explan the dfferent nterest and nvestment equatons you learned n secton 5.1 through 5.4 of your textbook and go through several examples. The second part
More information1. 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
More informationCalculation of Sampling Weights
Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a two-stage stratfed cluster desgn. 1 The frst stage conssted of a sample
More informationAnswer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy
4.02 Quz Solutons Fall 2004 Multple-Choce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multple-choce questons. For each queston, only one of the answers s correct.
More informationEconomic Interpretation of Regression. Theory and Applications
Economc Interpretaton of Regresson Theor and Applcatons Classcal and Baesan Econometrc Methods Applcaton of mathematcal statstcs to economc data for emprcal support Economc theor postulates a qualtatve
More informationFinite Math Chapter 10: Study Guide and Solution to Problems
Fnte Math Chapter 10: Study Gude and Soluton to Problems Basc Formulas and Concepts 10.1 Interest Basc Concepts Interest A fee a bank pays you for money you depost nto a savngs account. Prncpal P The amount
More informationCausal, 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
More informationWhat 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
More informationv 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
More informationRecurrence. 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.
More informationPERRON FROBENIUS THEOREM
PERRON FROBENIUS THEOREM R. CLARK ROBINSON Defnton. A n n matrx M wth real entres m, s called a stochastc matrx provded () all the entres m satsfy 0 m, () each of the columns sum to one, m = for all, ()
More informationAn Alternative Way to Measure Private Equity Performance
An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate
More informationLinear 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.
More informationTHE 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
More informationForecasting the Direction and Strength of Stock Market Movement
Forecastng the Drecton and Strength of Stock Market Movement Jngwe Chen Mng Chen Nan Ye cjngwe@stanford.edu mchen5@stanford.edu nanye@stanford.edu Abstract - Stock market s one of the most complcated systems
More informationSection 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
More informationLogistic 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+ + + - - This circuit than can be reduced to a planar circuit
MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to
More informationModule 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..
More informationPSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12
14 The Ch-squared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed
More informationHow To Calculate The Accountng Perod Of Nequalty
Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.
More information7.5. Present Value of an Annuity. Investigate
7.5 Present Value of an Annuty Owen and Anna are approachng retrement and are puttng ther fnances n order. They have worked hard and nvested ther earnngs so that they now have a large amount of money on
More informationLogistic 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
More informationNMT 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 informationPRACTICE 1: MUTUAL FUNDS EVALUATION USING MATLAB.
PRACTICE 1: MUTUAL FUNDS EVALUATION USING MATLAB. INDEX 1. Load data usng the Edtor wndow and m-fle 2. Learnng to save results from the Edtor wndow. 3. Computng the Sharpe Rato 4. Obtanng the Treynor Rato
More informationSection 2 Introduction to Statistical Mechanics
Secton 2 Introducton to Statstcal Mechancs 2.1 Introducng entropy 2.1.1 Boltzmann s formula A very mportant thermodynamc concept s that of entropy S. Entropy s a functon of state, lke the nternal energy.
More informationThe OC Curve of Attribute Acceptance Plans
The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4
More informationHÜCKEL MOLECULAR ORBITAL THEORY
1 HÜCKEL MOLECULAR ORBITAL THEORY In general, the vast maorty polyatomc molecules can be thought of as consstng of a collecton of two electron bonds between pars of atoms. So the qualtatve pcture of σ
More informationRotation 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
More informationLecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression.
Lecture 3: Annuty Goals: Learn contnuous annuty and perpetuty. Study annutes whose payments form a geometrc progresson or a arthmetc progresson. Dscuss yeld rates. Introduce Amortzaton Suggested Textbook
More informationJoe Pimbley, unpublished, 2005. Yield Curve Calculations
Joe Pmbley, unpublshed, 005. Yeld Curve Calculatons Background: Everythng s dscount factors Yeld curve calculatons nclude valuaton of forward rate agreements (FRAs), swaps, nterest rate optons, and forward
More informationCHAPTER 14 MORE ABOUT REGRESSION
CHAPTER 14 MORE ABOUT REGRESSION We learned n Chapter 5 that often a straght lne descrbes the pattern of a relatonshp between two quanttatve varables. For nstance, n Example 5.1 we explored the relatonshp
More informationQ3.8: A person trying to throw a ball as far as possible will run forward during the throw. Explain why this increases the distance of the throw.
Problem Set 3 Due: 09/3/, Tuesda Chapter 3: Vectors and Moton n Two Dmensons Questons: 7, 8,, 4, 0 Eercses & Problems:, 7, 8, 33, 37, 44, 46, 65, 73 Q3.7: An athlete performn the lon jump tres to acheve
More information8.4. Annuities: Future Value. INVESTIGATE the Math. 504 8.4 Annuities: Future Value
8. Annutes: Future Value YOU WILL NEED graphng calculator spreadsheet software GOAL Determne the future value of an annuty earnng compound nterest. INVESTIGATE the Math Chrstne decdes to nvest $000 at
More informationLecture 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
More informationProblem Set 3. a) We are asked how people will react, if the interest rate i on bonds is negative.
Queston roblem Set 3 a) We are asked how people wll react, f the nterest rate on bonds s negatve. When
More informationANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING
ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING Matthew J. Lberatore, Department of Management and Operatons, Vllanova Unversty, Vllanova, PA 19085, 610-519-4390,
More informationFINANCIAL MATHEMATICS. A Practical Guide for Actuaries. and other Business Professionals
FINANCIAL MATHEMATICS A Practcal Gude for Actuares and other Busness Professonals Second Edton CHRIS RUCKMAN, FSA, MAAA JOE FRANCIS, FSA, MAAA, CFA Study Notes Prepared by Kevn Shand, FSA, FCIA Assstant
More informationMethods for Calculating Life Insurance Rates
World Appled Scences Journal 5 (4): 653-663, 03 ISSN 88-495 IDOSI Pulcatons, 03 DOI: 0.589/dos.wasj.03.5.04.338 Methods for Calculatng Lfe Insurance Rates Madna Movsarovna Magomadova Chechen State Unversty,
More informationA Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy S-curve Regression
Novel Methodology of Workng Captal Management for Large Publc Constructons by Usng Fuzzy S-curve Regresson Cheng-Wu Chen, Morrs H. L. Wang and Tng-Ya Hseh Department of Cvl Engneerng, Natonal Central Unversty,
More informationSPEE Recommended Evaluation Practice #6 Definition of Decline Curve Parameters Background:
SPEE Recommended Evaluaton Practce #6 efnton of eclne Curve Parameters Background: The producton hstores of ol and gas wells can be analyzed to estmate reserves and future ol and gas producton rates and
More informationTime Value of Money. Types of Interest. Compounding and Discounting Single Sums. Page 1. Ch. 6 - The Time Value of Money. The Time Value of Money
Ch. 6 - The Tme Value of Money Tme Value of Money The Interest Rate Smple Interest Compound Interest Amortzng a Loan FIN21- Ahmed Y, Dasht TIME VALUE OF MONEY OR DISCOUNTED CASH FLOW ANALYSIS Very Important
More informationLecture 3: Force of Interest, Real Interest Rate, Annuity
Lecture 3: Force of Interest, Real Interest Rate, Annuty Goals: Study contnuous compoundng and force of nterest Dscuss real nterest rate Learn annuty-mmedate, and ts present value Study annuty-due, and
More information1 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
More informationAn Overview of Financial Mathematics
An Overvew of Fnancal Mathematcs Wllam Benedct McCartney July 2012 Abstract Ths document s meant to be a quck ntroducton to nterest theory. It s wrtten specfcally for actuaral students preparng to take
More informationLaws of Electromagnetism
There are four laws of electromagnetsm: Laws of Electromagnetsm The law of Bot-Savart Ampere's law Force law Faraday's law magnetc feld generated by currents n wres the effect of a current on a loop of
More informationSIMPLE 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.
More informationn + 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
More informationFigure 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 informationCHAPTER 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
More informationWe 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
More informationFuzzy Regression and the Term Structure of Interest Rates Revisited
Fuzzy Regresson and the Term Structure of Interest Rates Revsted Arnold F. Shapro Penn State Unversty Smeal College of Busness, Unversty Park, PA 68, USA Phone: -84-865-396, Fax: -84-865-684, E-mal: afs@psu.edu
More informationMean Molecular Weight
Mean Molecular Weght The thermodynamc relatons between P, ρ, and T, as well as the calculaton of stellar opacty requres knowledge of the system s mean molecular weght defned as the mass per unt mole of
More informationRegression 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
More informationLuby s Alg. for Maximal Independent Sets using Pairwise Independence
Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent
More informationA Probabilistic Theory of Coherence
A Probablstc Theory of Coherence BRANDEN FITELSON. The Coherence Measure C Let E be a set of n propostons E,..., E n. We seek a probablstc measure C(E) of the degree of coherence of E. Intutvely, we want
More informationThe Greedy Method. Introduction. 0/1 Knapsack Problem
The Greedy Method Introducton We have completed data structures. We now are gong to look at algorthm desgn methods. Often we are lookng at optmzaton problems whose performance s exponental. For an optmzaton
More informationsubstances (among other variables as well). ( ) Thus the change in volume of a mixture can be written as
Mxtures and Solutons Partal Molar Quanttes Partal molar volume he total volume of a mxture of substances s a functon of the amounts of both V V n,n substances (among other varables as well). hus the change
More informationRisk-based Fatigue Estimate of Deep Water Risers -- Course Project for EM388F: Fracture Mechanics, Spring 2008
Rsk-based Fatgue Estmate of Deep Water Rsers -- Course Project for EM388F: Fracture Mechancs, Sprng 2008 Chen Sh Department of Cvl, Archtectural, and Envronmental Engneerng The Unversty of Texas at Austn
More informationSection C2: BJT Structure and Operational Modes
Secton 2: JT Structure and Operatonal Modes Recall that the semconductor dode s smply a pn juncton. Dependng on how the juncton s based, current may easly flow between the dode termnals (forward bas, v
More informationwhere 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
More informationSeries Solutions of ODEs 2 the Frobenius method. The basic idea of the Frobenius method is to look for solutions of the form 3
Royal Holloway Unversty of London Department of Physs Seres Solutons of ODEs the Frobenus method Introduton to the Methodology The smple seres expanson method works for dfferental equatons whose solutons
More informationTexas Instruments 30X IIS Calculator
Texas Instruments 30X IIS Calculator Keystrokes for the TI-30X IIS are shown for a few topcs n whch keystrokes are unque. Start by readng the Quk Start secton. Then, before begnnng a specfc unt of the
More informationChapter 12 Inductors and AC Circuits
hapter Inductors and A rcuts awrence B. ees 6. You may make a sngle copy of ths document for personal use wthout wrtten permsson. Hstory oncepts from prevous physcs and math courses that you wll need for
More informationSolution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt.
Chapter 9 Revew problems 9.1 Interest rate measurement Example 9.1. Fund A accumulates at a smple nterest rate of 10%. Fund B accumulates at a smple dscount rate of 5%. Fnd the pont n tme at whch the forces
More informationtotal A A reag total A A r eag
hapter 5 Standardzng nalytcal Methods hapter Overvew 5 nalytcal Standards 5B albratng the Sgnal (S total ) 5 Determnng the Senstvty (k ) 5D Lnear Regresson and albraton urves 5E ompensatng for the Reagent
More information14.74 Lecture 5: Health (2)
14.74 Lecture 5: Health (2) Esther Duflo February 17, 2004 1 Possble Interventons Last tme we dscussed possble nterventons. Let s take one: provdng ron supplements to people, for example. From the data,
More informationAn Evaluation of the Extended Logistic, Simple Logistic, and Gompertz Models for Forecasting Short Lifecycle Products and Services
An Evaluaton of the Extended Logstc, Smple Logstc, and Gompertz Models for Forecastng Short Lfecycle Products and Servces Charles V. Trappey a,1, Hsn-yng Wu b a Professor (Management Scence), Natonal Chao
More informationChapter 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 informationIS-LM Model 1 C' dy = di
- odel Solow Assumptons - demand rrelevant n long run; assumes economy s operatng at potental GDP; concerned wth growth - Assumptons - supply s rrelevant n short run; assumes economy s operatng below potental
More informationSTATISTICAL DATA ANALYSIS IN EXCEL
Mcroarray Center STATISTICAL DATA ANALYSIS IN EXCEL Lecture 6 Some Advanced Topcs Dr. Petr Nazarov 14-01-013 petr.nazarov@crp-sante.lu Statstcal data analyss n Ecel. 6. Some advanced topcs Correcton for
More informationImplementation of Deutsch's Algorithm Using Mathcad
Implementaton of Deutsch's Algorthm Usng Mathcad Frank Roux The followng s a Mathcad mplementaton of Davd Deutsch's quantum computer prototype as presented on pages - n "Machnes, Logc and Quantum Physcs"
More informationRing 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 informationThe Full-Wave Rectifier
9/3/2005 The Full Wae ectfer.doc /0 The Full-Wae ectfer Consder the followng juncton dode crcut: s (t) Power Lne s (t) 2 Note that we are usng a transformer n ths crcut. The job of ths transformer s to
More informationFeature selection for intrusion detection. Slobodan Petrović NISlab, Gjøvik University College
Feature selecton for ntruson detecton Slobodan Petrovć NISlab, Gjøvk Unversty College Contents The feature selecton problem Intruson detecton Traffc features relevant for IDS The CFS measure The mrmr measure
More informationThe Application of Fractional Brownian Motion in Option Pricing
Vol. 0, No. (05), pp. 73-8 http://dx.do.org/0.457/jmue.05.0..6 The Applcaton of Fractonal Brownan Moton n Opton Prcng Qng-xn Zhou School of Basc Scence,arbn Unversty of Commerce,arbn zhouqngxn98@6.com
More informationLeast Squares Fitting of Data
Least Squares Fttng of Data Davd Eberly Geoetrc Tools, LLC http://www.geoetrctools.co/ Copyrght c 1998-2016. All Rghts Reserved. Created: July 15, 1999 Last Modfed: January 5, 2015 Contents 1 Lnear Fttng
More informationAlthough ordinary least-squares (OLS) regression
egresson through the Orgn Blackwell Oxford, TEST 0141-98X 003 5 31000 Orgnal Joseph Teachng G. UK Artcle Publshng Esenhauer through Statstcs the Ltd Trust Orgn 001 KEYWODS: Teachng; egresson; Analyss of
More informationEvaluating the Effects of FUNDEF on Wages and Test Scores in Brazil *
Evaluatng the Effects of FUNDEF on Wages and Test Scores n Brazl * Naérco Menezes-Flho Elane Pazello Unversty of São Paulo Abstract In ths paper we nvestgate the effects of the 1998 reform n the fundng
More informationVision Mouse. Saurabh Sarkar a* University of Cincinnati, Cincinnati, USA ABSTRACT 1. INTRODUCTION
Vson Mouse Saurabh Sarkar a* a Unversty of Cncnnat, Cncnnat, USA ABSTRACT The report dscusses a vson based approach towards trackng of eyes and fngers. The report descrbes the process of locatng the possble
More informationCharacterization of Assembly. Variation Analysis Methods. A Thesis. Presented to the. Department of Mechanical Engineering. Brigham Young University
Characterzaton of Assembly Varaton Analyss Methods A Thess Presented to the Department of Mechancal Engneerng Brgham Young Unversty In Partal Fulfllment of the Requrements for the Degree Master of Scence
More informationUsing Series to Analyze Financial Situations: Present Value
2.8 Usng Seres to Analyze Fnancal Stuatons: Present Value In the prevous secton, you learned how to calculate the amount, or future value, of an ordnary smple annuty. The amount s the sum of the accumulated
More informationSection 5.3 Annuities, Future Value, and Sinking Funds
Secton 5.3 Annutes, Future Value, and Snkng Funds Ordnary Annutes A sequence of equal payments made at equal perods of tme s called an annuty. The tme between payments s the payment perod, and the tme
More informationFinancial Mathemetics
Fnancal Mathemetcs 15 Mathematcs Grade 12 Teacher Gude Fnancal Maths Seres Overvew In ths seres we am to show how Mathematcs can be used to support personal fnancal decsons. In ths seres we jon Tebogo,
More informationCHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol
CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK Sample Stablty Protocol Background The Cholesterol Reference Method Laboratory Network (CRMLN) developed certfcaton protocols for total cholesterol, HDL
More informationSingle and multiple stage classifiers implementing logistic discrimination
Sngle and multple stage classfers mplementng logstc dscrmnaton Hélo Radke Bttencourt 1 Dens Alter de Olvera Moraes 2 Vctor Haertel 2 1 Pontfíca Unversdade Católca do Ro Grande do Sul - PUCRS Av. Ipranga,
More informationGoals 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 information1. Math 210 Finite Mathematics
1. ath 210 Fnte athematcs Chapter 5.2 and 5.3 Annutes ortgages Amortzaton Professor Rchard Blecksmth Dept. of athematcal Scences Northern Illnos Unversty ath 210 Webste: http://math.nu.edu/courses/math210
More informationIn our example i = r/12 =.0825/12 At the end of the first month after your payment is received your amount in the account, the balance, is
Payout annutes: Start wth P dollars, e.g., P = 100, 000. Over a 30 year perod you receve equal payments of A dollars at the end of each month. The amount of money left n the account, the balance, earns
More informationMultiple-Period Attribution: Residuals and Compounding
Multple-Perod Attrbuton: Resduals and Compoundng Our revewer gave these authors full marks for dealng wth an ssue that performance measurers and vendors often regard as propretary nformaton. In 1994, Dens
More informationNumber of Levels Cumulative Annual operating Income per year construction costs costs ($) ($) ($) 1 600,000 35,000 100,000 2 2,200,000 60,000 350,000
Problem Set 5 Solutons 1 MIT s consderng buldng a new car park near Kendall Square. o unversty funds are avalable (overhead rates are under pressure and the new faclty would have to pay for tself from
More informationMultiple stage amplifiers
Multple stage amplfers Ams: Examne a few common 2-transstor amplfers: -- Dfferental amplfers -- Cascode amplfers -- Darlngton pars -- current mrrors Introduce formal methods for exactly analysng multple
More informationDEFINING %COMPLETE IN MICROSOFT PROJECT
CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMI-SP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,
More informationL10: 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
More informationOptimal Pricing for Integrated-Services Networks. with Guaranteed Quality of Service &
Optmal Prcng for Integrated-Servces Networks wth Guaranteed Qualty of Servce & y Qong Wang * Jon M. Peha^ Marvn A. Sru # Carnege Mellon Unversty Chapter n Internet Economcs, edted y Joseph Baley and Lee
More information