LINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "LINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity"

Transcription

1 LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 9 Multicolliearity Dr Shalabh Departmet of Mathematics ad Statistics Idia Istitute of Techology Kapur

2 Multicolliearity diagostics A importat questio that arises is how to diagose the presece of multicolliearity i the data o the basis of give sample iformatio Several diagostic measures are available ad each of them is based o a particular approach It is difficult to say that which of the diagostic is the best or ultimate Some of the popular ad importat diagostics are described further The detectio of multicolliearity ivolves 3 aspects: (i) Determiig its presece (ii) Determiig its severity (iii) Determiig its form or locatio Determiat of X ' X ( X ' X ) This measure is based o the fact that the matrix X ' X becomes ill coditioed i the presece of multicolliearity The value of determiat of X ' X, ie, X ' X declies as degree of multicolliearity icreases If ra ( X ' X) < the X ' X will be sigular ad so X ' X 0 So as X ' X 0, the degree of multicolliearity icreases ad it becomes exact or perfect at X ' X 0 Thus X ' X serves as a measure of multicolliearity ad X ' X 0 idicates that perfect multicolliearity exists

3 3 Limitatios: This measure has followig limitatios i It is ot bouded as ii 0 < X ' X < It is affected by dispersio of explaatory variables For example, if, the X ' X x i xx i i i i xix i xi i i ( ) x i xi r i i where r is the correlatio coefficiet betwee X ad X So X ' X variability of explaatory variable If explaatory variables have very low variability, the zero which will idicate the presece of multicolliearity ad which is ot the case so depeds o correlatio coefficiet ad X ' X may ted to iii It gives o idea about the relative effects o idividual coefficiets If multicolliearity is preset, the it will ot idicate that which variable i X ' X is causig multicolliearity ad is hard to determie

4 4 Ispectio of correlatio matrix The ispectio of off-diagoal elemets r i i gives a idea about the presece of multicolliearity If X i ad X are early liearly depedet the r i will be close to Note that the observatios i X are stadardized i the sese that each observatio is subtracted from mea of that variable ad divided by the square root of corrected sum of squares of that variable X ' X Whe more tha two explaatory variables are cosidered ad if they are ivolved i ear-liear depedecy, the it is ot ecessary that ay of the r i will be large Geerally, pairwise ispectio of correlatio coefficiets is ot sufficiet for detectig multicolliearity i the data 3 Determiat of correlatio matrix Let D be the determiat of correlatio matrix the 0 D If D 0 the it idicates the existece of exact liear depedece amog explaatory variables If D the the colums of X matrix are orthoormal Thus a value close to 0 is a idicatio of high degree of multicolliearity Ay value of D betwee 0 ad gives a idea of the degree of multicolliearity Limitatio: It gives o iformatio about the umber of liear depedecies amog explaatory variables

5 5 Advatages over X ' X (i) It is a bouded measure 0 D (ii) It is ot affected by the dispersio of explaatory variables For example, whe, xi xx i i i i r xx i i xi i i X ' X ( ) 4 Measure based o partial regressio A measure of multicolliearity ca be obtaied o the basis of coefficiets of determiatio based o partial regressio Let R be the coefficiet of determiatio i the full model, ie, based o all explaatory variables ad determiatio i the model whe i th explaatory variable is dropped, i,,,, ad R L R Max( R, R,, R ) be the coefficiet of

6 6 Procedure: i Drop oe of the explaatory variable amog variables, say X ii Ru regressio of y over rest of the ( - ) variables X, X 3,, X iii Calculate R iv Similarly calculate 3 v Fid R Max( R, R,, R ) vi L Determie R, R,, R R R L ( ) R R L R L The quatity provides a measure of multicolliearity If multicolliearity is preset, will be high Higher the R L ( R R L ) degree of multicolliearity, higher the value of So i the presece of multicolliearity, be low Thus if R R L ( ) is close to 0, it idicates the high degree of multicolliearity Limitatios: i It gives o iformatio about the uderlyig relatios about explaatory variables, ie, how may relatioships are preset or how may explaatory variables are resposible for the multicolliearity ii Small value of ( R R L ) may occur because of poor specificatio of the model also ad it may be iferred i such situatio that multicolliearity is preset

7 5 Variace iflatio factors (VIF) The matrix C ( X ' X) X ' X becomes ill-coditioed i the presece of multicolliearity i the data So the diagoal elemets of helps i the detectio of multicolliearity If deotes the coefficiet of determiatio obtaied whe X is regressed o the remaiig ( - ) variables excludig X, the the th diagoal elemet of C is R 7 C R If X is early orthogoal to remaiig explaatory variables, the is small ad cosequetly C is close to R If X is early liearly depedet o a subset of remaiig explaatory variables, the C is large Sice the variace of th OLSE of β is Var b σ ( ) C is close to ad cosequetly So C is the factor by which the variace of b icreases whe the explaatory variables are ear liear depedet Based o this cocept, the variace iflatio factor for the th explaatory variable is defied as VIF R This is the factor which is resposible for iflatig the samplig variace The combied effect of depedecies amog the explaatory variables o the variace of a term is measured by the VIF of that term i the model Oe or more large VIFs idicate the presece of multicolliearity i the data R I practice, usually a VIF > 5 or 0 idicates that the associated regressio coefficiets are poorly estimated because of multicolliearity If regressio coefficiets are estimated by OLSE ad its variace is part of this variace is give by VIF σ ( X ' X) So VIF idicates that a

8 8 Limitatios: (i) (ii) It sheds o light o the umber of depedecies amog the explaatory variables The rule of VIF > 5 or 0 is a rule of thumb which may differ from oe situatio to aother situatio Aother iterpretatio of VIF The VIFs ca also be viewed as follows The cofidece iterval of th OLSE of b± ˆ σ C t α, The legth of the cofidece iterval is β is give by L ˆ σ C t α, Now cosider a situatio where X is a orthogoal matrix, ie, X ' X I so that C, sample size is same as earlier ad same root mea squares ( x ), the the legth of cofidece iterval becomes i x L* ˆ σt α, i L Cosider the ratio C L * Thus VIF idicates the icrease i the legth of cofidece iterval of th regressio coefficiet due to the presece of multicolliearity

9 6 Coditio umber ad coditio idex λ, λ,, λ X ' X Let be the eigevalues (or characteristic roots) of Let 9 λ Max( λ, λ,, λ ) max λ Mi( λ, λ,, λ ) mi The coditio umber (CN) is defied as CN λ λ max < < mi,0 CN The small values of characteristic roots idicates the presece of ear-liear depedecies i the data The CN provides a measure of spread i the spectrum of characteristic roots of X X The coditio umber provides a measure of multicolliearity If CN < 00, the it is cosidered as o-harmful multicolliearity If 00 < CN < 000, the it idicates that the multicolliearity is moderate to severe (or strog) This rage is referred to as dager level If CN > 000, the it idicates a severe (or strog) multicolliearity The coditio umber is based oly or two eigevalues: use iformatio o other eigevalues as well The coditio idices of X X are defied as I fact, largest C CN λ mi ad λ Aother measures are coditio idices which The umber of coditio idices that are large, say more tha 000, idicate the umber of ear-liear depedecies i X X A limitatio of CN ad C is that they are ubouded measures as 0 < CN <, 0 < C < max max C λ,,,, λ

10 0 7 Measure based o characteristic roots ad proportio of variaces λ, λ,, λ X ' X, Λ diag( λ, λ,, λ ) Let be the eigevalues of is matrix ad V is a matrix costructed by the eigevectors of X X Obviously, V is a orthogoal matrix The X X ca be decomposed as V, V,, V λ X ' X VΛV ' Let be the colum of V If there is ear-liear depedecy i the data, the is close to zero ad the ature of liear depedecy is described by the elemets of associated eigevector V The covariace matrix of OLSE is Vb ( ) σ ( X' X) σ ( VΛV ') σ VΛ V ' vi vi v i Var( bi ) σ λ λ λ where v, v,, v are the elemets i V i i i The coditio idices are max C λ,,,, λ

11 Procedure: i Fid coditio idex C, C,, C ii (a) Idetify those λ ' s for which C is greater tha the dager level 000 (b) This gives the umber of liear depedecies (c) Do t cosider those C which are below the dager level ' s iii For such λ 's with coditio idex above the dager level, choose oe such eigevalue, say iv Fid the value of proportio of variace correspodig to i Var( b ), Var( b ),, Var( b ) as v i Note that ca be foud from the expressio λ vi vi v i Var( bi ) σ λ λ λ ie, correspodig to factor The proportio of variace i th p i ( vi / λ ) vi / λ VIF vi p i ( / λ ) λ provides a measure of multicolliearity λ p > 05, If it idicates that is adversely affected by the multicolliearity, ie, estimate of is iflueced by the i presece of multicolliearity b i β i It is a good diagostic tool i the sese that it tells about the presece of harmful multicolliearity as well as also idicates the umber of liear depedecies resposible for multicolliearity This diagostic is better tha other diagostics

12 The coditio idices are also defied by the sigular value decompositio of X matrix as follows: where U is matrix, V is matrix, is matrix UU ' I, VV ' I, D is matrix, D diag ( µ, µ,, µ ) ad µ, µ,, µ are the sigular values of X, V is a matrix whose colums are eigevectors correspodig to eigevalues of X X ad U is a matrix whose colums are the eigevectors associated with the ozero eigevalues of X X X UDV ' The coditio idices of X matrix are defied as µ max η,,,, µ where µ max Max( µ, µ,, µ ) If λ, λ,, λ are the eigevalues of X X the p p X X UDV UDV VD V V V ' ( ') ' ' ' Λ ', so,,,, µ λ Note that with µ λ Var( b ) σ VIF p i i, v i i i ( v i / µ i ) VIF µ v µ i i

13 3 The ill-coditioig i X is reflected i the size of sigular values There will be oe small sigular value for each oliear depedecy The extet of ill coditioig is described by how small is µ µ max relative to It is suggested that the explaatory variables should be scaled to uit legth but should ot be cetered whe computig p i This will help i diagosig the role of itercept term i ear-liear depedece No uique guidace is available i literature o the issue of ceterig the explaatory variables The ceterig maes the itercept orthogoal to explaatory variables So this may remove the ill coditioig due to itercept term i the model

Probability, Expectation Value and Uncertainty

Probability, Expectation Value and Uncertainty Chapter 1 Probability, Expectatio Value ad Ucertaity We have see that the physically observable properties of a quatum system are represeted by Hermitea operators (also referred to as observables ) such

More information

II. Descriptive Statistics D. Linear Correlation and Regression. 1. Linear Correlation

II. 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 information

SIMPLE LINEAR REGRESSION AND CORRELATION ANALYSIS

SIMPLE LINEAR REGRESSION AND CORRELATION ANALYSIS SIMPLE LINEAR REGRESSION AND CORRELATION ANALSIS INTRODUCTION There are lot of statistical ivestigatio to kow whether there is a relatioship amog variables Two aalyses: (1) regressio aalysis; () correlatio

More information

7-1. Chapter 4. Part I. Sampling Distributions and Confidence Intervals

7-1. Chapter 4. Part I. Sampling Distributions and Confidence Intervals 7-1 Chapter 4 Part I. Samplig Distributios ad Cofidece Itervals 1 7- Sectio 1. Samplig Distributio 7-3 Usig Statistics Statistical Iferece: Predict ad forecast values of populatio parameters... Test hypotheses

More information

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + 62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of

More information

Zeros of Polynomials

Zeros 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 information

REGRESSION (Physics 1210 Notes, Partial Modified Appendix A)

REGRESSION (Physics 1210 Notes, Partial Modified Appendix A) REGRESSION (Physics 0 Notes, Partial Modified Appedix A) HOW TO PERFORM A LINEAR REGRESSION Cosider the followig data poits ad their graph (Table I ad Figure ): X Y 0 3 5 3 7 4 9 5 Table : Example Data

More information

Assessment and Modeling of Forests. FR 4218 Spring Assignment 1 Solutions

Assessment 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 information

The variance of a sum of independent variables is the sum of their variances, since covariances are zero. Therefore. V (xi )= n n 2 σ2 = σ2.

The variance of a sum of independent variables is the sum of their variances, since covariances are zero. Therefore. V (xi )= n n 2 σ2 = σ2. SAMPLE STATISTICS A radom sample x 1,x,,x from a distributio f(x) is a set of idepedetly ad idetically variables with x i f(x) for all i Their joit pdf is f(x 1,x,,x )=f(x 1 )f(x ) f(x )= f(x i ) The sample

More information

Correlation and Covariance

Correlation 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 information

Cov(aX, cy ) Var(X) Var(Y ) It is completely invariant to affine transformations: for any a, b, c, d R, ρ(ax + b, cy + d) = a.s. X i. as n.

Cov(aX, cy ) Var(X) Var(Y ) It is completely invariant to affine transformations: for any a, b, c, d R, ρ(ax + b, cy + d) = a.s. X i. as n. CS 189 Itroductio to Machie Learig Sprig 218 Note 11 1 Caoical Correlatio Aalysis The Pearso Correlatio Coefficiet ρ(x, Y ) is a way to measure how liearly related (i other words, how well a liear model

More information

Number of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day

Number of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day LECTURE # 8 Mea Deviatio, Stadard Deviatio ad Variace & Coefficiet of variatio Mea Deviatio Stadard Deviatio ad Variace Coefficiet of variatio First, we will discuss it for the case of raw data, ad the

More information

STA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to:

STA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to: STA 2023 Module 10 Comparig Two Proportios Learig Objectives Upo completig this module, you should be able to: 1. Perform large-sample ifereces (hypothesis test ad cofidece itervals) to compare two populatio

More information

Chapter 11 Output Analysis for a Single Model. Banks, Carson, Nelson & Nicol Discrete-Event System Simulation

Chapter 11 Output Analysis for a Single Model. Banks, Carson, Nelson & Nicol Discrete-Event System Simulation Chapter Output Aalysis for a Sigle Model Baks, Carso, Nelso & Nicol Discrete-Evet System Simulatio Error Estimatio If {,, } are ot statistically idepedet, the S / is a biased estimator of the true variace.

More information

The DOA Estimation of Multiple Signals based on Weighting MUSIC Algorithm

The DOA Estimation of Multiple Signals based on Weighting MUSIC Algorithm , pp.10-106 http://dx.doi.org/10.1457/astl.016.137.19 The DOA Estimatio of ultiple Sigals based o Weightig USIC Algorithm Chagga Shu a, Yumi Liu State Key Laboratory of IPOC, Beijig Uiversity of Posts

More information

Lecture 3 The Lebesgue Integral

Lecture 3 The Lebesgue Integral Lecture 3: The Lebesgue Itegral 1 of 14 Course: Theory of Probability I Term: Fall 2013 Istructor: Gorda Zitkovic Lecture 3 The Lebesgue Itegral The costructio of the itegral Uless expressly specified

More information

Simple Linear Regression

Simple Linear Regression Simple Liear Regressio 1. Model ad Parameter Estimatio (a) Suppose our data cosist of a collectio of pairs (x i, y i ), where x i is a observed value of variable X ad y i is the correspodig observatio

More information

Section 11.8: Power Series

Section 11.8: Power Series Sectio 11.8: Power Series 1. Power Series I this sectio, we cosider geeralizig the cocept of a series. Recall that a series is a ifiite sum of umbers a. We ca talk about whether or ot it coverges ad i

More information

Median and IQR The median is the value which divides the ordered data values in half.

Median and IQR The median is the value which divides the ordered data values in half. STA 666 Fall 2007 Web-based Course Notes 4: Describig Distributios Numerically Numerical summaries for quatitative variables media ad iterquartile rage (IQR) 5-umber summary mea ad stadard deviatio Media

More information

(all terms are scalars).the minimization is clearer in sum notation:

(all terms are scalars).the minimization is clearer in sum notation: 7 Multiple liear regressio: with predictors) Depedet data set: y i i = 1, oe predictad, predictors x i,k i = 1,, k = 1, ' The forecast equatio is ŷ i = b + Use matrix otatio: k =1 b k x ik Y = y 1 y 1

More information

, then cv V. Differential Equations Elements of Lineaer Algebra Name: Consider the differential equation. and y2 cos( kx)

, then cv V. Differential Equations Elements of Lineaer Algebra Name: Consider the differential equation. and y2 cos( kx) Cosider the differetial equatio y '' k y 0 has particular solutios y1 si( kx) ad y cos( kx) I geeral, ay liear combiatio of y1 ad y, cy 1 1 cy where c1, c is also a solutio to the equatio above The reaso

More information

The standard deviation of the mean

The standard deviation of the mean Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider

More information

5.1. The Rayleigh s quotient. Definition 49. Let A = A be a self-adjoint matrix. quotient is the function. R(x) = x,ax, for x = 0.

5.1. The Rayleigh s quotient. Definition 49. Let A = A be a self-adjoint matrix. quotient is the function. R(x) = x,ax, for x = 0. 40 RODICA D. COSTIN 5. The Rayleigh s priciple ad the i priciple for the eigevalues of a self-adjoit matrix Eigevalues of self-adjoit matrices are easy to calculate. This sectio shows how this is doe usig

More information

Section 14. Simple linear regression.

Section 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 information

Chapter 6 Principles of Data Reduction

Chapter 6 Principles of Data Reduction Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a

More information

o <Xln <X2n <... <X n < o (1.1)

o <Xln <X2n <... <X n < o (1.1) Metrika, Volume 28, 1981, page 257-262. 9 Viea. Estimatio Problems for Rectagular Distributios (Or the Taxi Problem Revisited) By J.S. Rao, Sata Barbara I ) Abstract: The problem of estimatig the ukow

More information

3/3/2014. CDS M Phil Econometrics. Types of Relationships. Types of Relationships. Types of Relationships. Vijayamohanan Pillai N.

3/3/2014. 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 information

CTL.SC0x Supply Chain Analytics

CTL.SC0x Supply Chain Analytics CTL.SC0x Supply Chai Aalytics Key Cocepts Documet V1.1 This documet cotais the Key Cocepts documets for week 6, lessos 1 ad 2 withi the SC0x course. These are meat to complemet, ot replace, the lesso videos

More information

TENSOR PRODUCTS AND PARTIAL TRACES

TENSOR PRODUCTS AND PARTIAL TRACES Lecture 2 TENSOR PRODUCTS AND PARTIAL TRACES Stéphae ATTAL Abstract This lecture cocers special aspects of Operator Theory which are of much use i Quatum Mechaics, i particular i the theory of Quatum Ope

More information

SALES AND MARKETING Department MATHEMATICS. 2nd Semester. Bivariate statistics LESSONS

SALES AND MARKETING Department MATHEMATICS. 2nd Semester. Bivariate statistics LESSONS SALES AND MARKETING Departmet MATHEMATICS d Semester Bivariate statistics LESSONS Olie documet: http://jff-dut-tc.weebly.com sectio DUT Maths S. IUT de Sait-Etiee Départemet TC J.F.Ferraris Math S StatVar

More information

arxiv: v1 [math.pr] 13 Oct 2011

arxiv: v1 [math.pr] 13 Oct 2011 A tail iequality for quadratic forms of subgaussia radom vectors Daiel Hsu, Sham M. Kakade,, ad Tog Zhag 3 arxiv:0.84v math.pr] 3 Oct 0 Microsoft Research New Eglad Departmet of Statistics, Wharto School,

More information

Singular Continuous Measures by Michael Pejic 5/14/10

Singular Continuous Measures by Michael Pejic 5/14/10 Sigular Cotiuous Measures by Michael Peic 5/4/0 Prelimiaries Give a set X, a σ-algebra o X is a collectio of subsets of X that cotais X ad ad is closed uder complemetatio ad coutable uios hece, coutable

More information

Eksamen 2006 H Utsatt SENSORVEILEDNING. Problem 1. Settet består av 9 delspørsmål som alle anbefales å telle likt. Svar er gitt i <<.. >>.

Eksamen 2006 H Utsatt SENSORVEILEDNING. Problem 1. Settet består av 9 delspørsmål som alle anbefales å telle likt. Svar er gitt i <<.. >>. Eco 43 Eksame 6 H Utsatt SENSORVEILEDNING Settet består av 9 delspørsmål som alle abefales å telle likt. Svar er gitt i . Problem a. Let the radom variable (rv.) X be expoetially distributed with

More information

17 Phonons and conduction electrons in solids (Hiroshi Matsuoka)

17 Phonons and conduction electrons in solids (Hiroshi Matsuoka) 7 Phoos ad coductio electros i solids Hiroshi Matsuoa I this chapter we will discuss a miimal microscopic model for phoos i a solid ad a miimal microscopic model for coductio electros i a simple metal.

More information

A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence

A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as

More information

Discrete probability distributions

Discrete probability distributions Discrete probability distributios I the chapter o probability we used the classical method to calculate the probability of various values of a radom variable. I some cases, however, we may be able to develop

More information

The Choquet Integral with Respect to Fuzzy-Valued Set Functions

The Choquet Integral with Respect to Fuzzy-Valued Set Functions The Choquet Itegral with Respect to Fuzzy-Valued Set Fuctios Weiwei Zhag Abstract The Choquet itegral with respect to real-valued oadditive set fuctios, such as siged efficiecy measures, has bee used i

More information

Some Properties of the Exact and Score Methods for Binomial Proportion and Sample Size Calculation

Some Properties of the Exact and Score Methods for Binomial Proportion and Sample Size Calculation Some Properties of the Exact ad Score Methods for Biomial Proportio ad Sample Size Calculatio K. KRISHNAMOORTHY AND JIE PENG Departmet of Mathematics, Uiversity of Louisiaa at Lafayette Lafayette, LA 70504-1010,

More information

Output Analysis and Run-Length Control

Output Analysis and Run-Length Control IEOR E4703: Mote Carlo Simulatio Columbia Uiversity c 2017 by Marti Haugh Output Aalysis ad Ru-Legth Cotrol I these otes we describe how the Cetral Limit Theorem ca be used to costruct approximate (1 α%

More information

Question 1: Exercise 8.2

Question 1: Exercise 8.2 Questio 1: Exercise 8. (a) Accordig to the regressio results i colum (1), the house price is expected to icrease by 1% ( 100% 0.0004 500 ) with a additioal 500 square feet ad other factors held costat.

More information

Supplementary Appendix for. Mandatory Portfolio Disclosure, Stock Liquidity, and Mutual Fund Performance

Supplementary Appendix for. Mandatory Portfolio Disclosure, Stock Liquidity, and Mutual Fund Performance Supplemetary Appedix for Madatory Portfolio Disclosure, Stock Liquidity, ad Mutual Fud Performace This Supplemetary Appedix cosists of two sectios Sectio I provides the propositios ad their proofs about

More information

Introduction to Machine Learning DIS10

Introduction to Machine Learning DIS10 CS 189 Fall 017 Itroductio to Machie Learig DIS10 1 Fu with Lagrage Multipliers (a) Miimize the fuctio such that f (x,y) = x + y x + y = 3. Solutio: The Lagragia is: L(x,y,λ) = x + y + λ(x + y 3) Takig

More information

Introducing Sample Proportions

Introducing Sample Proportions Itroducig Sample Proportios Probability ad statistics Aswers & Notes TI-Nspire Ivestigatio Studet 60 mi 7 8 9 0 Itroductio A 00 survey of attitudes to climate chage, coducted i Australia by the CSIRO,

More information

Chapter 3. Strong convergence. 3.1 Definition of almost sure convergence

Chapter 3. Strong convergence. 3.1 Definition of almost sure convergence Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i

More information

Chapter Objectives. Bivariate Data. Terminology. Lurking Variable. Types of Relations. Chapter 3 Linear Regression and Correlation

Chapter Objectives. Bivariate Data. Terminology. Lurking Variable. Types of Relations. Chapter 3 Linear Regression and Correlation Chapter Objectives Chapter 3 Liear Regressio ad Correlatio Descriptive Aalysis & Presetatio of Two Quatitative Data To be able to preset two-variables data i tabular ad graphic form Display the relatioship

More information

SINGLE-CHANNEL QUEUING PROBLEMS APPROACH

SINGLE-CHANNEL QUEUING PROBLEMS APPROACH SINGLE-CHANNEL QUEUING ROBLEMS AROACH Abdurrzzag TAMTAM, Doctoral Degree rogramme () Dept. of Telecommuicatios, FEEC, BUT E-mail: xtamta@stud.feec.vutbr.cz Supervised by: Dr. Karol Molár ABSTRACT The paper

More information

Chapter VII Measures of Correlation

Chapter VII Measures of Correlation Chapter VII Measures of Correlatio A researcher may be iterested i fidig out whether two variables are sigificatly related or ot. For istace, he may be iterested i kowig whether metal ability is sigificatly

More information

R. van Zyl 1, A.J. van der Merwe 2. Quintiles International, University of the Free State

R. van Zyl 1, A.J. van der Merwe 2. Quintiles International, University of the Free State Bayesia Cotrol Charts for the Two-parameter Expoetial Distributio if the Locatio Parameter Ca Take o Ay Value Betwee Mius Iity ad Plus Iity R. va Zyl, A.J. va der Merwe 2 Quitiles Iteratioal, ruaavz@gmail.com

More information

Instructor: Judith Canner Spring 2010 CONFIDENCE INTERVALS How do we make inferences about the population parameters?

Instructor: Judith Canner Spring 2010 CONFIDENCE INTERVALS How do we make inferences about the population parameters? CONFIDENCE INTERVALS How do we make ifereces about the populatio parameters? The samplig distributio allows us to quatify the variability i sample statistics icludig how they differ from the parameter

More information

Decoupling Zeros of Positive Discrete-Time Linear Systems*

Decoupling Zeros of Positive Discrete-Time Linear Systems* Circuits ad Systems,,, 4-48 doi:.436/cs..7 Published Olie October (http://www.scirp.org/oural/cs) Decouplig Zeros of Positive Discrete-Time Liear Systems* bstract Tadeusz Kaczorek Faculty of Electrical

More information

On the Linear Complexity of Feedback Registers

On the Linear Complexity of Feedback Registers O the Liear Complexity of Feedback Registers A. H. Cha M. Goresky A. Klapper Northeaster Uiversity Abstract I this paper, we study sequeces geerated by arbitrary feedback registers (ot ecessarily feedback

More information

Tables and Formulas for Sullivan, Fundamentals of Statistics, 2e Pearson Education, Inc.

Tables and Formulas for Sullivan, Fundamentals of Statistics, 2e Pearson Education, Inc. Table ad Formula for Sulliva, Fudametal of Statitic, e. 008 Pearo Educatio, Ic. CHAPTER Orgaizig ad Summarizig Data Relative frequecy frequecy um of all frequecie Cla midpoit: The um of coecutive lower

More information

Generalization of Samuelson s inequality and location of eigenvalues

Generalization of Samuelson s inequality and location of eigenvalues Proc. Idia Acad. Sci. Math. Sci.) Vol. 5, No., February 05, pp. 03. c Idia Academy of Scieces Geeralizatio of Samuelso s iequality ad locatio of eigevalues R SHARMA ad R SAINI Departmet of Mathematics,

More information

Regression with an Evaporating Logarithmic Trend

Regression with an Evaporating Logarithmic Trend Regressio with a Evaporatig Logarithmic Tred Peter C. B. Phillips Cowles Foudatio, Yale Uiversity, Uiversity of Aucklad & Uiversity of York ad Yixiao Su Departmet of Ecoomics Yale Uiversity October 5,

More information

Binomial Distribution

Binomial Distribution 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Overview Example: coi tossed three times Defiitio Formula Recall that a r.v. is discrete if there are either a fiite umber of possible

More information

2.4 - Sequences and Series

2.4 - Sequences and Series 2.4 - Sequeces ad Series Sequeces A sequece is a ordered list of elemets. Defiitio 1 A sequece is a fuctio from a subset of the set of itegers (usually either the set 80, 1, 2, 3,... < or the set 81, 2,

More information

Definition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4.

Definition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4. 4. BASES I BAACH SPACES 39 4. BASES I BAACH SPACES Sice a Baach space X is a vector space, it must possess a Hamel, or vector space, basis, i.e., a subset {x γ } γ Γ whose fiite liear spa is all of X ad

More information

KLMED8004 Medical statistics. Part I, autumn Estimation. We have previously learned: Population and sample. New questions

KLMED8004 Medical statistics. Part I, autumn Estimation. We have previously learned: Population and sample. New questions We have previously leared: KLMED8004 Medical statistics Part I, autum 00 How kow probability distributios (e.g. biomial distributio, ormal distributio) with kow populatio parameters (mea, variace) ca give

More information

Most text will write ordinary derivatives using either Leibniz notation 2 3. y + 5y= e and y y. xx tt t

Most text will write ordinary derivatives using either Leibniz notation 2 3. y + 5y= e and y y. xx tt t Itroductio to Differetial Equatios Defiitios ad Termiolog Differetial Equatio: A equatio cotaiig the derivatives of oe or more depedet variables, with respect to oe or more idepedet variables, is said

More information

Why learn matrix algebra? Vectors & Matrices with statistical applications. Brief history of linear algebra

Why learn matrix algebra? Vectors & Matrices with statistical applications. Brief history of linear algebra R Vectors & Matrices with statistical applicatios x RXX RXY y RYX RYY Why lear matrix algebra? Simple way to express liear combiatios of variables ad geeral solutios of equatios. Liear statistical models

More information

a for a 1 1 matrix. a b a b 2 2 matrix: We define det ad bc 3 3 matrix: We define a a a a a a a a a a a a a a a a a a

a for a 1 1 matrix. a b a b 2 2 matrix: We define det ad bc 3 3 matrix: We define a a a a a a a a a a a a a a a a a a Math S-b Lecture # Notes This wee is all about determiats We ll discuss how to defie them, how to calculate them, lear the allimportat property ow as multiliearity, ad show that a square matrix A is ivertible

More information

Regression. Correlation vs. regression. The parameters of linear regression. Regression assumes... Random sample. Y = α + β X.

Regression. Correlation vs. regression. The parameters of linear regression. Regression assumes... Random sample. Y = α + β X. Regressio Correlatio vs. regressio Predicts Y from X Liear regressio assumes that the relatioship betwee X ad Y ca be described by a lie Regressio assumes... Radom sample Y is ormally distributed with

More information

CALCULATING FIBONACCI VECTORS

CALCULATING FIBONACCI VECTORS THE GENERALIZED BINET FORMULA FOR CALCULATING FIBONACCI VECTORS Stuart D Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithacaedu ad Dai Novak Departmet

More information

Notes on iteration and Newton s method. Iteration

Notes 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 information

MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND.

MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. XI-1 (1074) MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. R. E. D. WOOLSEY AND H. S. SWANSON XI-2 (1075) STATISTICAL DECISION MAKING Advaced

More information

w (1) ˆx w (1) x (1) /ρ and w (2) ˆx w (2) x (2) /ρ.

w (1) ˆx w (1) x (1) /ρ and w (2) ˆx w (2) x (2) /ρ. 2 5. Weighted umber of late jobs 5.1. Release dates ad due dates: maximimizig the weight of o-time jobs Oce we add release dates, miimizig the umber of late jobs becomes a sigificatly harder problem. For

More information

First Year Quantitative Comp Exam Spring, Part I - 203A. f X (x) = 0 otherwise

First Year Quantitative Comp Exam Spring, Part I - 203A. f X (x) = 0 otherwise First Year Quatitative Comp Exam Sprig, 2012 Istructio: There are three parts. Aswer every questio i every part. Questio I-1 Part I - 203A A radom variable X is distributed with the margial desity: >

More information

Joint Probability Distributions and Random Samples. Jointly Distributed Random Variables. Chapter { }

Joint Probability Distributions and Random Samples. Jointly Distributed Random Variables. Chapter { } UCLA STAT A Applied Probability & Statistics for Egieers Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology Teachig Assistat: Neda Farziia, UCLA Statistics Uiversity of Califoria, Los Ageles, Sprig

More information

Asymptotic Results for the Linear Regression Model

Asymptotic Results for the Linear Regression Model Asymptotic Results for the Liear Regressio Model C. Fli November 29, 2000 1. Asymptotic Results uder Classical Assumptios The followig results apply to the liear regressio model y = Xβ + ε, where X is

More information

Varanasi , India. Corresponding author

Varanasi , India. Corresponding author A Geeral Family of Estimators for Estimatig Populatio Mea i Systematic Samplig Usig Auxiliary Iformatio i the Presece of Missig Observatios Maoj K. Chaudhary, Sachi Malik, Jayat Sigh ad Rajesh Sigh Departmet

More information

WHAT IS THE PROBABILITY FUNCTION FOR LARGE TSUNAMI WAVES? ABSTRACT

WHAT IS THE PROBABILITY FUNCTION FOR LARGE TSUNAMI WAVES? ABSTRACT WHAT IS THE PROBABILITY FUNCTION FOR LARGE TSUNAMI WAVES? Harold G. Loomis Hoolulu, HI ABSTRACT Most coastal locatios have few if ay records of tsuami wave heights obtaied over various time periods. Still

More information

Dimensionality Reduction vs. Clustering

Dimensionality Reduction vs. Clustering Dimesioality Reductio vs. Clusterig Lecture 9: Cotiuous Latet Variable Models Sam Roweis Traiig such factor models (e.g. FA, PCA, ICA) is called dimesioality reductio. You ca thik of this as (o)liear regressio

More information

Spectral Graph Theory and its Applications. Lillian Dai Oct. 20, 2004

Spectral Graph Theory and its Applications. Lillian Dai Oct. 20, 2004 Spectral raph Theory ad its Applicatios Lillia Dai 6.454 Oct. 0, 004 Outlie Basic spectral graph theory raph partitioig usig spectral methods D. Spielma ad S. Teg, Spectral Partitioig Works: Plaar raphs

More information

Math 2784 (or 2794W) University of Connecticut

Math 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 information

a for a 1 1 matrix. a b a b 2 2 matrix: We define det ad bc 3 3 matrix: We define a a a a a a a a a a a a a a a a a a

a for a 1 1 matrix. a b a b 2 2 matrix: We define det ad bc 3 3 matrix: We define a a a a a a a a a a a a a a a a a a Math E-2b Lecture #8 Notes This week is all about determiats. We ll discuss how to defie them, how to calculate them, lear the allimportat property kow as multiliearity, ad show that a square matrix A

More information

REVIEW OF SIMPLE LINEAR REGRESSION SIMPLE LINEAR REGRESSION

REVIEW OF SIMPLE LINEAR REGRESSION SIMPLE LINEAR REGRESSION REVIEW OF SIMPLE LINEAR REGRESSION SIMPLE LINEAR REGRESSION I liear regreio, we coider the frequecy ditributio of oe variable (Y) at each of everal level of a ecod variable (X). Y i kow a the depedet variable.

More information

Element sampling: Part 2

Element sampling: Part 2 Chapter 4 Elemet samplig: Part 2 4.1 Itroductio We ow cosider uequal probability samplig desigs which is very popular i practice. I the uequal probability samplig, we ca improve the efficiecy of the resultig

More information

Proof of Fermat s Last Theorem by Algebra Identities and Linear Algebra

Proof of Fermat s Last Theorem by Algebra Identities and Linear Algebra Proof of Fermat s Last Theorem by Algebra Idetities ad Liear Algebra Javad Babaee Ragai Youg Researchers ad Elite Club, Qaemshahr Brach, Islamic Azad Uiversity, Qaemshahr, Ira Departmet of Civil Egieerig,

More information

MA131 - Analysis 1. Workbook 2 Sequences I

MA131 - Analysis 1. Workbook 2 Sequences I MA3 - Aalysis Workbook 2 Sequeces I Autum 203 Cotets 2 Sequeces I 2. Itroductio.............................. 2.2 Icreasig ad Decreasig Sequeces................ 2 2.3 Bouded Sequeces..........................

More information

Measures of Spread: Variance and Standard Deviation

Measures of Spread: Variance and Standard Deviation Lesso 1-6 Measures of Spread: Variace ad Stadard Deviatio BIG IDEA Variace ad stadard deviatio deped o the mea of a set of umbers. Calculatig these measures of spread depeds o whether the set is a sample

More information

Supplementary Appendix for. Mandatory Portfolio Disclosure, Stock Liquidity, and Mutual Fund Performance

Supplementary Appendix for. Mandatory Portfolio Disclosure, Stock Liquidity, and Mutual Fund Performance Supplemetary Appedix for Madatory Portfolio Disclosure, Stock Liquidity, ad Mutual Fud Performace This Supplemetary Appedix cosists of two sectios Sectio I provides the propositios ad their proofs about

More information

Chapter 4 - Summarizing Numerical Data

Chapter 4 - Summarizing Numerical Data Chapter 4 - Summarizig Numerical Data 15.075 Cythia Rudi Here are some ways we ca summarize data umerically. Sample Mea: i=1 x i x :=. Note: i this class we will work with both the populatio mea µ ad the

More information

Statisticians use the word population to refer the total number of (potential) observations under consideration

Statisticians use the word population to refer the total number of (potential) observations under consideration 6 Samplig Distributios Statisticias use the word populatio to refer the total umber of (potetial) observatios uder cosideratio The populatio is just the set of all possible outcomes i our sample space

More information

Sequences of Definite Integrals, Factorials and Double Factorials

Sequences of Definite Integrals, Factorials and Double Factorials 47 6 Joural of Iteger Sequeces, Vol. 8 (5), Article 5.4.6 Sequeces of Defiite Itegrals, Factorials ad Double Factorials Thierry Daa-Picard Departmet of Applied Mathematics Jerusalem College of Techology

More information

DISTRIBUTION LAW Okunev I.V.

DISTRIBUTION LAW Okunev I.V. 1 DISTRIBUTION LAW Okuev I.V. Distributio law belogs to a umber of the most complicated theoretical laws of mathematics. But it is also a very importat practical law. Nothig ca help uderstad complicated

More information

Elementary Statistics

Elementary Statistics Elemetary Statistics M. Ghamsary, Ph.D. Sprig 004 Chap 0 Descriptive Statistics Raw Data: Whe data are collected i origial form, they are called raw data. The followig are the scores o the first test of

More information

Statistical inference: example 1. Inferential Statistics

Statistical inference: example 1. Inferential Statistics Statistical iferece: example 1 Iferetial Statistics POPULATION SAMPLE A clothig store chai regularly buys from a supplier large quatities of a certai piece of clothig. Each item ca be classified either

More information

PAijpam.eu ON TENSOR PRODUCT DECOMPOSITION

PAijpam.eu ON TENSOR PRODUCT DECOMPOSITION Iteratioal Joural of Pure ad Applied Mathematics Volume 103 No 3 2015, 537-545 ISSN: 1311-8080 (prited versio); ISSN: 1314-3395 (o-lie versio) url: http://wwwijpameu doi: http://dxdoiorg/1012732/ijpamv103i314

More information

FIR Filters. Lecture #7 Chapter 5. BME 310 Biomedical Computing - J.Schesser

FIR Filters. Lecture #7 Chapter 5. BME 310 Biomedical Computing - J.Schesser FIR Filters Lecture #7 Chapter 5 8 What Is this Course All About? To Gai a Appreciatio of the Various Types of Sigals ad Systems To Aalyze The Various Types of Systems To Lear the Skills ad Tools eeded

More information

CH19 Confidence Intervals for Proportions. Confidence intervals Construct confidence intervals for population proportions

CH19 Confidence Intervals for Proportions. Confidence intervals Construct confidence intervals for population proportions CH19 Cofidece Itervals for Proportios Cofidece itervals Costruct cofidece itervals for populatio proportios Motivatio Motivatio We are iterested i the populatio proportio who support Mr. Obama. This sample

More information

Computing the output response of LTI Systems.

Computing the output response of LTI Systems. Computig the output respose of LTI Systems. By breaig or decomposig ad represetig the iput sigal to the LTI system ito terms of a liear combiatio of a set of basic sigals. Usig the superpositio property

More information

Chapter 3 Inner Product Spaces. Hilbert Spaces

Chapter 3 Inner Product Spaces. Hilbert Spaces Chapter 3 Ier Product Spaces. Hilbert Spaces 3. Ier Product Spaces. Hilbert Spaces 3.- Defiitio. A ier product space is a vector space X with a ier product defied o X. A Hilbert space is a complete ier

More information

On an axiomatization of the quasi-arithmetic mean values without the symmetry axiom

On an axiomatization of the quasi-arithmetic mean values without the symmetry axiom O a axiomatizatio of the quasi-arithmetic mea values without the symmetry axiom Jea-Luc Marichal Departmet of Maagemet, FEGSS, Uiversity of Liège Boulevard du Rectorat 7 - B31, B-4000 Liège, Belgium Email:

More information

ESTIMATION AND PREDICTION BASED ON K-RECORD VALUES FROM NORMAL DISTRIBUTION

ESTIMATION AND PREDICTION BASED ON K-RECORD VALUES FROM NORMAL DISTRIBUTION STATISTICA, ao LXXIII,. 4, 013 ESTIMATION AND PREDICTION BASED ON K-RECORD VALUES FROM NORMAL DISTRIBUTION Maoj Chacko Departmet of Statistics, Uiversity of Kerala, Trivadrum- 695581, Kerala, Idia M. Shy

More information

CALCULATION OF FIBONACCI VECTORS

CALCULATION OF FIBONACCI VECTORS CALCULATION OF FIBONACCI VECTORS Stuart D. Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithaca.edu ad Dai Novak Departmet of Mathematics, Ithaca College

More information

ESTIMATION BIAS IN SPATIAL MODELS WITH STRONGLY CONNECTED WEIGHT MATRICES

ESTIMATION BIAS IN SPATIAL MODELS WITH STRONGLY CONNECTED WEIGHT MATRICES ESTIMATION BIAS IN SPATIAL MODELS ITH STRONGLY CONNECTED EIGHT MATRICES Toy E. Smith Departmet of Systems ad Electrical Egieerig Uiversity of Pesylvaia Jauary, 008 (Revised Jue 8, 008) Abstract I this

More information

Applying least absolute deviation regression to regressiontype estimation of the index of a stable distribution using the characteristic function

Applying least absolute deviation regression to regressiontype estimation of the index of a stable distribution using the characteristic function Applyig least absolute deviatio regressio to regressiotype estimatio of the idex of a stable distributio usig the characteristic fuctio J. MARTIN VAN ZYL Departmet of Mathematical Statistics ad Actuarial

More information

ON THE CONNECTION BETWEEN THE DISTRIBUTION OF EIGENVALUES IN MULTIPLE CORRESPONDENCE ANALYSIS AND LOG-LINEAR MODELS

ON THE CONNECTION BETWEEN THE DISTRIBUTION OF EIGENVALUES IN MULTIPLE CORRESPONDENCE ANALYSIS AND LOG-LINEAR MODELS ON THE CONNECTION BETWEEN THE DISTRIBUTION OF EIGENVALUES IN MULTIPLE CORRESPONDENCE ANALYSIS AND LOG-LINEAR MODELS Authors: S Be Ammou Departmet of Quatitative Methods, Faculté de Droit et des Scieces

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 6 9/24/2008 DISCRETE RANDOM VARIABLES AND THEIR EXPECTATIONS

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 6 9/24/2008 DISCRETE RANDOM VARIABLES AND THEIR EXPECTATIONS MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 6 9/24/2008 DISCRETE RANDOM VARIABLES AND THEIR EXPECTATIONS Cotets 1. A few useful discrete radom variables 2. Joit, margial, ad

More information

SEQUENCES AND SERIES

SEQUENCES AND SERIES 9 SEQUENCES AND SERIES INTRODUCTION Sequeces have may importat applicatios i several spheres of huma activities Whe a collectio of objects is arraged i a defiite order such that it has a idetified first

More information