REGRESSION (Physics 1210 Notes, Partial Modified Appendix A)
|
|
- Berenice Dennis
- 6 years ago
- Views:
Transcription
1 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 Table : Example Data Y X Figure : Graph of Example Data I this example, the poits are o a perfect straight lie. The formula of a geeral straight lie is Y=a*X+b where a is the slope of the lie ad b is the itercept of that lie with the y-axis. I this example, it is easy to verify that a= ad b= I. I geeral, with the data poits you obtai i your experimets, fidig a ad b is ot so easy. We wat to use a computer to calculate a ad b for us. For this, we use the regressio fuctio of Excel. Whe you are i Excel, type i your data poits as show i Table I. Now, we wat to do a liear regressio o these data poits. That will hopefully give us the value for a ad b. To do this, look i the meu for [Tools], the select [Data Aalysis] ad fially select [Regressio]. You are the faced with a dialogue-box "Regressio". For "iput Y rage", select the Y colum of your data For "iput X rage", select the X colum of your data, the click o "OK". After a few secods, you will see a ew Excel sheet with a overkill of umbers called the "Summary Output". Of this Summary Output, the oly part you eed is: Coefficiets Stadard Error t Stat Itercept X Variable From this, you ca read the coefficiet values for a ad b as follows: b=itercept= a=x-variable= which is what we expected. The equatio for the lie i this case would be: Y=aX+b=X+ The stadard errors i a ad b are zero here because the poits are o a perfect straight lie. I geeral, this will ot be the case, because experimets are ot perfect, ufortuately. For example, if you were to use the followig data poits (they are the same poits as before, except for the last oe) ad do a liear regressio o them, you will get:
2 X Y Table : Not-so-perfect Example Data Coefficiets Stadard Error t Stat Itercept X Variable Table 3: Regressio Results of Not-so-perfect Data Ad you ow have stadard errors, which are ot zero. You would quote your results at the 95% cofidece level as: b=itercept=o.38±0.999 a= x -variable=.574±0.399 Of course, you must decide for yourself each time how may decimals are realistic ad what the uit is. Liear regressio is a very useful tool, ad you will eed it frequetly durig this course. I your report, DO NOT iclude the "regressio-summary" Excel produces. Istead, whe you do a liear regressio o your data, all you have to give is the equatio of the lie (icludig errors) Excel calculated, ad state that the calculatio was a liear regressio. Velocity (m/s) Measured Table 4. Free Fall Data HOW TO PERFORM NONLINEAR TRENDLINE Velocity (m/s) Besides liear tredlie, Excel has the capability of fittig logarithmic, polyomial of arbitrary order, power or expoetial fuctios to data. For the data preseted i Table 4, it appears that a quadratic Quadradic Tredlie (Show) F = -0.07V V R = Figure. Velocity vs. Force Liear Tredlie F = V R = 0.96
3 relatioship should produce a excellet fit. Figure substatiates this i that this quadratic tredlie has a r of as compared to a value of 0.96 for the liear fit whe the itercept value is set to 4.90 (See Curve Fittig.xls example). Higher order polyomials may be used but ay icrease i r that is obtaied by this icreased complexity is rather superficial. HOW TO PERFORM NONLINEAR OPTIMIZING SOLVER If we start over o this problem ad apply some basic dyamics to the free fall problem, the summatio of forces i this case must be equal to the gravitatioal body force (m-g) i the dowward directio plus a drag force i the upward directio that is some ukow fuctio of velocity. Therefore theory implies that the force versus velocity relatioship must have the followig geeral form: F ( V ) = mg Drag () but it does ot supply ay iformatio about how the drag varies with velocity. Our ow persoal experiece idicates that the drag force icreases with velocity ad extesive experimetal testig over the years has show that power laws ca be used frequetly to correlate velocity-drag data over limited F = mg av b () velocity rages. If this is assumed to be the case here, the Theory ad some empirical isight has therefore bee combied to obtai a possible fuctio form betwee velocity ad force i terms of two arbitrary costats (a, b) that is based upo the physics of pheomea ad ot just blid curve fittig as was doe i the liear ad quadratic (Figure ) curve fit examples. The values of a ad b that give the best fit with the experimetal data ca be determie through the use of the Excel oliear optimizig solver. The fust requiremet of usig the oliear optimizig solver is the developmet of a regressio fuctio that you what to optimize i terms of miimizig or maximizig its value or obtaiig a specified value. The tredlies that are preseted i the previous two curve fits are based upo least square regressio i which the followig regressio fuctio is miimized ( F i F i ) i= (3) where F i is the measured force ad F i is the correspodig predicted value i the data set that cotais values. I this case Equatio would be subsituted for F i (F i =mg-av b ). Istead of doig this, lets miimize r ( F i F ) i = i= (4) ( Fi Fi ) i= r. That is where F is the mea force of the experimetal data set. Excel provides a oliear optimizig solver for miimizig fuctios such as Equatio 4. However, the problem must be prepared properly to obtai a
4 appropriate solutio. Table 5 presets a copy of the spreadsheet (see file Curve Fittig.xls for the actual spreadsheet) that was used to determie a & b. This table cotais six colums: colum is the idepedet variable (velocity); colum is the measured variable (acceleratio a i ); colum 3 is the depedet variable (force F i ) calculated from the measured variable, a i ; colum 4 is the predicted depedet variable (the force calculated from Equatio,F i ); colum 5 is the square of the differece betwee colums 3 ad 4; ad colum 6 is the square of the differece betwee colum 3 ad the average force which is calculated at the ed of colum 3. The colums 5 & 6 are the summed ad these values are used to calculate the r value for a guess set of coefficiets (a, b). For istace, the guess of (,) produces a very poor r value of Velocity (m/s) a = N/(m/s)^b g = 9.8 (m/s^) b =.6635 m = 0.5 (kg) Accel. (m/s^) Measured Fi Predicted* Fi (Fi - Fi)^ N^ Table 5. Excel Table Used to Perform Noliear Regressio (Fav - Fi)^ N^ E E E E E E E E E E E E E E E E E E E Fav =.9 Sum = 5.80E R^ = = - SUM(Fi - Fi)^/SUM(Fav - Fi)^ * Fi = Force(m,g,V,a,b) see Module Force(m,g,V,a,b) Excel uses a iterative approach to solve the oliear regressio problem oce it has a iitial guess set to start this iterative process. I this case, the program will systematically vary aad b to determie the local gradiet of' r ad thereby determie how the (a, b) set should be varied to maximize r. I order to use the solver tool, the tool must be loaded ito Excel. The solver ca be loaded by: () Click o Tools i the mai meu bar () Click o Solver i the pull dow meu If Solver is ot a optio, the (a) Click o Add-Is i the pull dow meu (b) click o Solver Add-I i the Add-Is dialog box (the check box must be checked) (c) Click OK (d) Click Solver The Solver dialog box is ow visible. The first meu item is the target cell which is r i this case. The secod item delieates what actio is to be perform o the target cell. I this example we wish to miimize
5 the target cell. The third item specifies which cells may have their values varied to accomplish the objective which i this case are cells cotaiig the guess values of the regressio parameters a ad b. Note that amed cells ca be utilized i specifyig the cell locatios of the target cell ad the adjustable cells. As a optio, you ca set umerical costraits o the adjustable cells. A little thought about the physics of this problem idicates that both a ad b are positive ad these costraits may be added. I some problems you may wish to chage the default Precisio ad Tolerace values by first clickig the Optios butto. Now click OK, ad Excel will attempt to fid the optimum solutio ad replace the guess values of the regressio parameters with the optimum values. Table 5 idicates that the combied theoretical/empirical correlatio produces a r of which is slightly better tha the quadratic. This correlatio is also simpler tha the quadratic fit ad it is more physically sigificat. Istead of basig the curve fit o r, try usig the least squares regressio method to compute the coefficiets ad compare your results. This example also illustrates the use of a fuctio module. To see it, click Tools, Macro ad Visual Basic Editor. Oe word of cautio: oliear fuctios ofte cotai more tha oe solutio ad that a give guess set may produce a local solutio (i this case, a local miimum) istead of a global solutio. Highly oliear problems may also require a fairly accurate iitial guess to obtai a global solutio or ay solutio. You may have to resort to plots to produce a accurate iitial guess. See Noliear Regressio.xls for aother example. Referece F = V.663 (5) Physics 0: Egieerig Physics. Lab Maual, Appedix A, Uiversity of Wyomig Physics ad Astroomy, Sprig, 999.
Response Variable denoted by y it is the variable that is to be predicted measure of the outcome of an experiment also called the dependent variable
Statistics Chapter 4 Correlatio ad Regressio If we have two (or more) variables we are usually iterested i the relatioship betwee the variables. Associatio betwee Variables Two variables are associated
More informationt distribution [34] : used to test a mean against an hypothesized value (H 0 : µ = µ 0 ) or the difference
EXST30 Backgroud material Page From the textbook The Statistical Sleuth Mea [0]: I your text the word mea deotes a populatio mea (µ) while the work average deotes a sample average ( ). Variace [0]: The
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 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 informationII. Descriptive Statistics D. Linear Correlation and Regression. 1. Linear Correlation
II. Descriptive Statistics D. Liear Correlatio ad Regressio I this sectio Liear Correlatio Cause ad Effect Liear Regressio 1. Liear Correlatio Quatifyig Liear Correlatio The Pearso product-momet correlatio
More informationAssessment and Modeling of Forests. FR 4218 Spring Assignment 1 Solutions
Assessmet ad Modelig of Forests FR 48 Sprig Assigmet Solutios. The first part of the questio asked that you calculate the average, stadard deviatio, coefficiet of variatio, ad 9% cofidece iterval of the
More informationCorrelation Regression
Correlatio Regressio While correlatio methods measure the stregth of a liear relatioship betwee two variables, we might wish to go a little further: How much does oe variable chage for a give chage i aother
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationRegression, Inference, and Model Building
Regressio, Iferece, ad Model Buildig Scatter Plots ad Correlatio Correlatio coefficiet, r -1 r 1 If r is positive, the the scatter plot has a positive slope ad variables are said to have a positive relatioship
More informationThe Pendulum. Purpose
The Pedulum Purpose To carry out a example illustratig how physics approaches ad solves problems. The example used here is to explore the differet factors that determie the period of motio of a pedulum.
More informationCorrelation and Covariance
Correlatio ad Covariace Tom Ilveto FREC 9 What is Next? Correlatio ad Regressio Regressio We specify a depedet variable as a liear fuctio of oe or more idepedet variables, based o co-variace Regressio
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 informationLinear Regression Models
Liear Regressio Models Dr. Joh Mellor-Crummey Departmet of Computer Sciece Rice Uiversity johmc@cs.rice.edu COMP 528 Lecture 9 15 February 2005 Goals for Today Uderstad how to Use scatter diagrams to ispect
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 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 informationST 305: Exam 3 ( ) = P(A)P(B A) ( ) = P(A) + P(B) ( ) = 1 P( A) ( ) = P(A) P(B) σ X 2 = σ a+bx. σ ˆp. σ X +Y. σ X Y. σ X. σ Y. σ n.
ST 305: Exam 3 By hadig i this completed exam, I state that I have either give or received assistace from aother perso durig the exam period. I have used o resources other tha the exam itself ad the basic
More informationLinear Regression Analysis. Analysis of paired data and using a given value of one variable to predict the value of the other
Liear Regressio Aalysis Aalysis of paired data ad usig a give value of oe variable to predict the value of the other 5 5 15 15 1 1 5 5 1 3 4 5 6 7 8 1 3 4 5 6 7 8 Liear Regressio Aalysis E: The chirp rate
More informationChapter If n is odd, the median is the exact middle number If n is even, the median is the average of the two middle numbers
Chapter 4 4-1 orth Seattle Commuity College BUS10 Busiess Statistics Chapter 4 Descriptive Statistics Summary Defiitios Cetral tedecy: The extet to which the data values group aroud a cetral value. Variatio:
More informationRegression, Part I. A) Correlation describes the relationship between two variables, where neither is independent or a predictor.
Regressio, Part I I. Differece from correlatio. II. Basic idea: A) Correlatio describes the relatioship betwee two variables, where either is idepedet or a predictor. - I correlatio, it would be irrelevat
More informationKinetics of Complex Reactions
Kietics of Complex Reactios by Flick Colema Departmet of Chemistry Wellesley College Wellesley MA 28 wcolema@wellesley.edu Copyright Flick Colema 996. All rights reserved. You are welcome to use this documet
More informationa is some real number (called the coefficient) other
Precalculus Notes for Sectio.1 Liear/Quadratic Fuctios ad Modelig http://www.schooltube.com/video/77e0a939a3344194bb4f Defiitios: A moomial is a term of the form tha zero ad is a oegative iteger. a where
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 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 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 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 informationStatistical Inference (Chapter 10) Statistical inference = learn about a population based on the information provided by a sample.
Statistical Iferece (Chapter 10) Statistical iferece = lear about a populatio based o the iformatio provided by a sample. Populatio: The set of all values of a radom variable X of iterest. Characterized
More informationS Y Y = ΣY 2 n. Using the above expressions, the correlation coefficient is. r = SXX S Y Y
1 Sociology 405/805 Revised February 4, 004 Summary of Formulae for Bivariate Regressio ad Correlatio Let X be a idepedet variable ad Y a depedet variable, with observatios for each of the values of these
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 information11 Correlation and Regression
11 Correlatio Regressio 11.1 Multivariate Data Ofte we look at data where several variables are recorded for the same idividuals or samplig uits. For example, at a coastal weather statio, we might record
More informationNCSS Statistical Software. Tolerance Intervals
Chapter 585 Itroductio This procedure calculates oe-, ad two-, sided tolerace itervals based o either a distributio-free (oparametric) method or a method based o a ormality assumptio (parametric). A two-sided
More informationFirst, note that the LS residuals are orthogonal to the regressors. X Xb X y = 0 ( normal equations ; (k 1) ) So,
0 2. OLS Part II The OLS residuals are orthogoal to the regressors. If the model icludes a itercept, the orthogoality of the residuals ad regressors gives rise to three results, which have limited practical
More informationPrinciple Of Superposition
ecture 5: PREIMINRY CONCEP O RUCUR NYI Priciple Of uperpositio Mathematically, the priciple of superpositio is stated as ( a ) G( a ) G( ) G a a or for a liear structural system, the respose at a give
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 c the remainder is Pc ().
Algebra, Polyomial ad Ratioal Fuctios Page 1 K.Paulk Notes Chapter 3, Sectio 3.1 to 3.4 Summary Sectio Theorem Notes 3.1 Zeros of a Fuctio Set the fuctio to zero ad solve for x. The fuctio is zero at these
More informationSimple Linear Regression
Chapter 2 Simple Liear Regressio 2.1 Simple liear model The simple liear regressio model shows how oe kow depedet variable is determied by a sigle explaatory variable (regressor). Is is writte as: Y i
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 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 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 informationOptimization Methods MIT 2.098/6.255/ Final exam
Optimizatio Methods MIT 2.098/6.255/15.093 Fial exam Date Give: December 19th, 2006 P1. [30 pts] Classify the followig statemets as true or false. All aswers must be well-justified, either through a short
More informationGoodness-of-Fit Tests and Categorical Data Analysis (Devore Chapter Fourteen)
Goodess-of-Fit Tests ad Categorical Data Aalysis (Devore Chapter Fourtee) MATH-252-01: Probability ad Statistics II Sprig 2019 Cotets 1 Chi-Squared Tests with Kow Probabilities 1 1.1 Chi-Squared Testig................
More informationStatistical Properties of OLS estimators
1 Statistical Properties of OLS estimators Liear Model: Y i = β 0 + β 1 X i + u i OLS estimators: β 0 = Y β 1X β 1 = Best Liear Ubiased Estimator (BLUE) Liear Estimator: β 0 ad β 1 are liear fuctio of
More informationQuadratic Functions. Before we start looking at polynomials, we should know some common terminology.
Quadratic Fuctios 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 i mathematical
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 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 informationThe Phi Power Series
The Phi Power Series I did this work i about 0 years while poderig the relatioship betwee the golde mea ad the Madelbrot set. I have fially decided to make it available from my blog at http://semresearch.wordpress.com/.
More informationSTP 226 ELEMENTARY STATISTICS
TP 6 TP 6 ELEMENTARY TATITIC CHAPTER 4 DECRIPTIVE MEAURE IN REGREION AND CORRELATION Liear Regressio ad correlatio allows us to examie the relatioship betwee two or more quatitative variables. 4.1 Liear
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 informationCastiel, Supernatural, Season 6, Episode 18
13 Differetial Equatios the aswer to your questio ca best be epressed as a series of partial differetial equatios... Castiel, Superatural, Seaso 6, Episode 18 A differetial equatio is a mathematical equatio
More informationNonlinear regression
oliear regressio How to aalyse data? How to aalyse data? Plot! How to aalyse data? Plot! Huma brai is oe the most powerfull computatioall tools Works differetly tha a computer What if data have o liear
More informationPH 425 Quantum Measurement and Spin Winter SPINS Lab 1
PH 425 Quatum Measuremet ad Spi Witer 23 SPIS Lab Measure the spi projectio S z alog the z-axis This is the experimet that is ready to go whe you start the program, as show below Each atom is measured
More informationLeast-Squares Regression
MATH 482 Least-Squares Regressio Dr. Neal, WKU As well as fidig the correlatio of paired sample data {{ x 1, y 1 }, { x 2, y 2 },..., { x, y }}, we also ca plot the data with a scatterplot ad fid the least
More informationMCT242: Electronic Instrumentation Lecture 2: Instrumentation Definitions
Faculty of Egieerig MCT242: Electroic Istrumetatio Lecture 2: Istrumetatio Defiitios Overview Measuremet Error Accuracy Precisio ad Mea Resolutio Mea Variace ad Stadard deviatio Fiesse Sesitivity Rage
More informationPaired Data and Linear Correlation
Paired Data ad Liear Correlatio Example. A group of calculus studets has take two quizzes. These are their scores: Studet st Quiz Score ( data) d Quiz Score ( data) 7 5 5 0 3 0 3 4 0 5 5 5 5 6 0 8 7 0
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationA NOTE ON THE TOTAL LEAST SQUARES FIT TO COPLANAR POINTS
A NOTE ON THE TOTAL LEAST SQUARES FIT TO COPLANAR POINTS STEVEN L. LEE Abstract. The Total Least Squares (TLS) fit to the poits (x,y ), =1,,, miimizes the sum of the squares of the perpedicular distaces
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 informationOutline. Linear regression. Regularization functions. Polynomial curve fitting. Stochastic gradient descent for regression. MLE for regression
REGRESSION 1 Outlie Liear regressio Regularizatio fuctios Polyomial curve fittig Stochastic gradiet descet for regressio MLE for regressio Step-wise forward regressio Regressio methods Statistical techiques
More informationOn a Smarandache problem concerning the prime gaps
O a Smaradache problem cocerig the prime gaps Felice Russo Via A. Ifate 7 6705 Avezzao (Aq) Italy felice.russo@katamail.com Abstract I this paper, a problem posed i [] by Smaradache cocerig the prime gaps
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 information3/3/2014. CDS M Phil Econometrics. Types of Relationships. Types of Relationships. Types of Relationships. Vijayamohanan Pillai N.
3/3/04 CDS M Phil Old Least Squares (OLS) Vijayamohaa Pillai N CDS M Phil Vijayamoha CDS M Phil Vijayamoha Types of Relatioships Oly oe idepedet variable, Relatioship betwee ad is Liear relatioships Curviliear
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More information6.867 Machine learning, lecture 7 (Jaakkola) 1
6.867 Machie learig, lecture 7 (Jaakkola) 1 Lecture topics: Kerel form of liear regressio Kerels, examples, costructio, properties Liear regressio ad kerels Cosider a slightly simpler model where we omit
More informationAlgebra of Least Squares
October 19, 2018 Algebra of Least Squares Geometry of Least Squares Recall that out data is like a table [Y X] where Y collects observatios o the depedet variable Y ad X collects observatios o the k-dimesioal
More informationA.1 Algebra Review: Polynomials/Rationals. Definitions:
MATH 040 Notes: Uit 0 Page 1 A.1 Algera Review: Polyomials/Ratioals Defiitios: A polyomial is a sum of polyomial terms. Polyomial terms are epressios formed y products of costats ad variales with whole
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 informationRevision Topic 1: Number and algebra
Revisio Topic : Number ad algebra Chapter : Number Differet types of umbers You eed to kow that there are differet types of umbers ad recogise which group a particular umber belogs to: Type of umber Symbol
More informationUnit 4: Polynomial and Rational Functions
48 Uit 4: Polyomial ad Ratioal Fuctios Polyomial Fuctios A polyomial fuctio y px ( ) is a fuctio of the form p( x) ax + a x + a x +... + ax + ax+ a 1 1 1 0 where a, a 1,..., a, a1, a0are real costats ad
More informationChapter 12 Correlation
Chapter Correlatio Correlatio is very similar to regressio with oe very importat differece. Regressio is used to explore the relatioship betwee a idepedet variable ad a depedet variable, whereas correlatio
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 informationIn algebra one spends much time finding common denominators and thus simplifying rational expressions. For example:
74 The Method of Partial Fractios I algebra oe speds much time fidig commo deomiators ad thus simplifyig ratioal epressios For eample: + + + 6 5 + = + = = + + + + + ( )( ) 5 It may the seem odd to be watig
More informationCEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering
CEE 5 Autum 005 Ucertaity Cocepts for Geotechical Egieerig Basic Termiology Set A set is a collectio of (mutually exclusive) objects or evets. The sample space is the (collectively exhaustive) collectio
More informationNumerical Methods in Fourier Series Applications
Numerical Methods i Fourier Series Applicatios Recall that the basic relatios i usig the Trigoometric Fourier Series represetatio were give by f ( x) a o ( a x cos b x si ) () where the Fourier coefficiets
More informationMachine Learning Regression I Hamid R. Rabiee [Slides are based on Bishop Book] Spring
Machie Learig Regressio I Hamid R. Rabiee [Slides are based o Bishop Book] Sprig 015 http://ce.sharif.edu/courses/93-94//ce717-1 Liear Regressio Liear regressio: ivolves a respose variable ad a sigle predictor
More informationLinear regression. Daniel Hsu (COMS 4771) (y i x T i β)2 2πσ. 2 2σ 2. 1 n. (x T i β y i ) 2. 1 ˆβ arg min. β R n d
Liear regressio Daiel Hsu (COMS 477) Maximum likelihood estimatio Oe of the simplest liear regressio models is the followig: (X, Y ),..., (X, Y ), (X, Y ) are iid radom pairs takig values i R d R, ad Y
More informationEXPERIMENT OF SIMPLE VIBRATION
EXPERIMENT OF SIMPLE VIBRATION. PURPOSE The purpose of the experimet is to show free vibratio ad damped vibratio o a system havig oe degree of freedom ad to ivestigate the relatioship betwee the basic
More informationChapter Vectors
Chapter 4. Vectors fter readig this chapter you should be able to:. defie a vector. add ad subtract vectors. fid liear combiatios of vectors ad their relatioship to a set of equatios 4. explai what it
More informationMechatronics. Time Response & Frequency Response 2 nd -Order Dynamic System 2-Pole, Low-Pass, Active Filter
Time Respose & Frequecy Respose d -Order Dyamic System -Pole, Low-Pass, Active Filter R 4 R 7 C 5 e i R 1 C R 3 - + R 6 - + e out Assigmet: Perform a Complete Dyamic System Ivestigatio of the Two-Pole,
More informationProblem Cosider the curve give parametrically as x = si t ad y = + cos t for» t» ß: (a) Describe the path this traverses: Where does it start (whe t =
Mathematics Summer Wilso Fial Exam August 8, ANSWERS Problem 1 (a) Fid the solutio to y +x y = e x x that satisfies y() = 5 : This is already i the form we used for a first order liear differetial equatio,
More informationIP Reference guide for integer programming formulations.
IP Referece guide for iteger programmig formulatios. by James B. Orli for 15.053 ad 15.058 This documet is iteded as a compact (or relatively compact) guide to the formulatio of iteger programs. For more
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 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 informationChapter 9: Numerical Differentiation
178 Chapter 9: Numerical Differetiatio Numerical Differetiatio Formulatio of equatios for physical problems ofte ivolve derivatives (rate-of-chage quatities, such as velocity ad acceleratio). Numerical
More informationECE 8527: Introduction to Machine Learning and Pattern Recognition Midterm # 1. Vaishali Amin Fall, 2015
ECE 8527: Itroductio to Machie Learig ad Patter Recogitio Midterm # 1 Vaishali Ami Fall, 2015 tue39624@temple.edu Problem No. 1: Cosider a two-class discrete distributio problem: ω 1 :{[0,0], [2,0], [2,2],
More informationREVISION SHEET FP1 (MEI) ALGEBRA. Identities In mathematics, an identity is a statement which is true for all values of the variables it contains.
The mai ideas are: Idetities REVISION SHEET FP (MEI) ALGEBRA Before the exam you should kow: If a expressio is a idetity the it is true for all values of the variable it cotais The relatioships betwee
More informationFinal Examination Solutions 17/6/2010
The Islamic Uiversity of Gaza Faculty of Commerce epartmet of Ecoomics ad Political Scieces A Itroductio to Statistics Course (ECOE 30) Sprig Semester 009-00 Fial Eamiatio Solutios 7/6/00 Name: I: Istructor:
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 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 informationMath 2784 (or 2794W) University of Connecticut
ORDERS OF GROWTH PAT SMITH Math 2784 (or 2794W) Uiversity of Coecticut Date: Mar. 2, 22. ORDERS OF GROWTH. Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really
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 informationStat 139 Homework 7 Solutions, Fall 2015
Stat 139 Homework 7 Solutios, Fall 2015 Problem 1. I class we leared that the classical simple liear regressio model assumes the followig distributio of resposes: Y i = β 0 + β 1 X i + ɛ i, i = 1,...,,
More information1 of 7 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 6. Order Statistics Defiitios Suppose agai that we have a basic radom experimet, ad that X is a real-valued radom variable
More informationOpen book and notes. 120 minutes. Cover page and six pages of exam. No calculators.
IE 330 Seat # Ope book ad otes 120 miutes Cover page ad six pages of exam No calculators Score Fial Exam (example) Schmeiser Ope book ad otes No calculator 120 miutes 1 True or false (for each, 2 poits
More informationComplex Numbers Solutions
Complex Numbers Solutios Joseph Zoller February 7, 06 Solutios. (009 AIME I Problem ) There is a complex umber with imagiary part 64 ad a positive iteger such that Fid. [Solutio: 697] 4i + + 4i. 4i 4i
More informationNUMERICAL METHODS FOR SOLVING EQUATIONS
Mathematics Revisio Guides Numerical Methods for Solvig Equatios Page 1 of 11 M.K. HOME TUITION Mathematics Revisio Guides Level: GCSE Higher Tier NUMERICAL METHODS FOR SOLVING EQUATIONS Versio:. Date:
More informationWe will conclude the chapter with the study a few methods and techniques which are useful
Chapter : Coordiate geometry: I this chapter we will lear about the mai priciples of graphig i a dimesioal (D) Cartesia system of coordiates. We will focus o drawig lies ad the characteristics of the graphs
More informationStudy on Coal Consumption Curve Fitting of the Thermal Power Based on Genetic Algorithm
Joural of ad Eergy Egieerig, 05, 3, 43-437 Published Olie April 05 i SciRes. http://www.scirp.org/joural/jpee http://dx.doi.org/0.436/jpee.05.34058 Study o Coal Cosumptio Curve Fittig of the Thermal Based
More informationLINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity
LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 9 Multicolliearity Dr Shalabh Departmet of Mathematics ad Statistics Idia Istitute of Techology Kapur Multicolliearity diagostics A importat questio that
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 informationCurve Sketching Handout #5 Topic Interpretation Rational Functions
Curve Sketchig Hadout #5 Topic Iterpretatio Ratioal Fuctios A ratioal fuctio is a fuctio f that is a quotiet of two polyomials. I other words, p ( ) ( ) f is a ratioal fuctio if p ( ) ad q ( ) are polyomials
More informationENGI 9420 Engineering Analysis Assignment 3 Solutions
ENGI 9 Egieerig Aalysis Assigmet Solutios Fall [Series solutio of ODEs, matri algebra; umerical methods; Chapters, ad ]. Fid a power series solutio about =, as far as the term i 7, to the ordiary differetial
More information