HYPOTHESIS TESTING OF PARAMETERS FOR ORDINARY LINEAR CIRCULAR REGRESSION


 Flora Griffith
 2 years ago
 Views:
Transcription
1 HYPOTHESIS TESTING OF PARAMETERS FOR ORDINARY LINEAR CIRCULAR REGRESSION Abdul Ghapor Hussn Centre for Foundaton Studes n Scence Unversty of Malaya 563 KUALA LUMPUR Emal: Abstract Ths paper presents the hypothess testng of parameters for ordnary lnear crcular regresson model assumng the crcular random error dstrbuted as von Msses dstrbuton The man nterests are n testng of the ntercept and slope parameter of the regresson lne As an llustraton ths hypothess testng wll be used n analyzng the wnd and wave drecton data recorded by two dfferent technques whch are HF radar system and anchored wave buoy Key words: crcular random varable contnuous lnear varables regresson model hypothess testng von Mses dstrbuton 1 Introducton A crcular random varable s a varable whch takes values on the crcumference of a crcle e the angle s n the range ( π ) radans or ( 36 ) Ths random varable must be analysed by technques dfferng from those approprate for the usual Eucldean type varables because the crcumference s a bounded closed space for whch the concept of orgn s arbtrary or undefned A contnuous lnear varable s a random varable wth realsatons on the straght lne whch may be analysed by usual technques Ordnary lnear crcular regresson model s appled when we wsh to determne the relatonshp between a sngle crcular explanatory varable X and a crcular response varable Y As an example s n the relatonshp of the measurements of wnd and wave drectons measured by an anchored buoy (Y) and radar (X) as shown n fgures below Sova (1995) Fgure below gves a smple (or even smplstc) scatter plot for each data set In both cases the observatons (drecton n radans) has been measured smultaneously by an anchored buoy (Y) and by radar (X) Although measurements are n radans we wll often use the more famlar degrees n nformal dscusson of the data In ths case we arbtrarly chose varate Y as the anchored buoy and varate X as radar We would antcpate an deal model y = x as approprate for the data The scatter plots of one drecton aganst the other (as measured n radans antclockwse from North) gve a cluster of ponts along the X=Y dagonal and then a few n the top left and bottom rght corners (359 o aganst 1 o etc) If we thnk of these data as arsng from an ordnary lnear regresson model we would Pak j stat oper res VolII No 6 pp7986
2 Abdul Ghapor Hussn regard those ponts at the top left and bottom rght as outlers but n relaton to an ordnary lnear crcular regresson model ths s not so because the measurements are on the crcle or crcumference not a straght lne Snce 1 o s only o from 359 o the pont (1 o 359 o ) on the smple scatter plot should not really be far from the deal model y = x In ths respect the smple scatter plot s msleadng but t llustrate that an ordnary lnear regresson model whch gnore the crcularty of the data s equally msleadng Perhaps such scatter plots should be drawn on a torus whch mantans the wrappng of the measurements scales Ths shows the chef problem wth ordnary lnear regresson when appled to crcular varables and below we wll propose an ordnary lnear crcular regresson model whch s more suted to ths form of data Hussn (4) 6 5 Anchored buoy Radar (a) Wnd drecton data 6 5 Anchored buoy Radar (b) Wave drecton data Fg 1: Scatter plot for wnd and wave drecton data (n radans) 8 Pak j stat oper res VolII No 6 pp7986
3 Hypothess testng of parameters for ordnary lnear crcular regresson The Model and Parameters Estmaton Suppose for any crcular observatons ( x1 y1 )( xn yn) of crcular varables X and Y we propose a model of y = α + βx + ε (mod π ) (1) where ε s a crcular random error havng a von Mses dstrbuton wth mean crcular and concentraton parameter κ An applcaton when both X and Y are crcular for example n modellng a relatonshp for calbraton between two nstruments for wnd drecton requres β 1 Parameters α and β may be estmate by maxmum lkelhood estmaton Based on the von Mses densty functon the log lkelhood functon for model (1) s gven by log L( α β κ; x1 xn y1 yn) = nlog( π) nlog I ( κ) + κ cos( y α βx ) We dfferentate log L wth respect to α β and κ to get α β and κ whch are gven by 1 tan S > C > C 1 α = tan + π C < () C 1 tan + π S < C > C β β + 1 x sn( y α β x ) x cos( y α β x ) (3) Ths expresson for βˆ may be solved teratvely gven some sutable ntal guesses at the estmate We can then update α and β and proceed teratvely Ths teraton procedure wll contnue untl the convergence crteron satsfed Further the estmate of parameter concentraton s gven by κ = cos( α A β ) y x (4) n 1 1 Pak j stat oper res VolII No 6 pp
4 Abdul Ghapor Hussn We can now proceed to obtan useful approxmatons usng the results of Dobson (1978) who gves varous smple approxmatons for the functon A (rato of the modfed Bessel functons for the frst knd of order one and the frst knd of order zero) for the von Mses concentraton parameter κ For example for 65 < w < 1 the approxmaton s A 1 9 w w w ( ) 81 ( w) Thus by usng maxmum lkelhood estmaton we have shown that αβ and κ can be estmated teratvely Snce both the x and y are measurements of the same quantty unty would be logcal ntal estmate of β and so a possble ntal estmate for teraton s β = 1 n () and (3) 3 The Hypothess Testng of Parameters Ths secton s concerned wth testng hypotheses on α and β e testng the hypothess H :β = (modπ ) vs H 1 :β (modπ ) for the ordnary lnear crcular regresson of the form y = α + βx + ε (mod π ) = 1 n usng observatons ( x1 y1 ) ( xn yn) and assumng that ε are VM( κ ) It follows drectly that the observatons y are VM( α + βx κ ) In partcular our nterest s n testng the hypotheses that H : β = 1 aganst H : β 1 and H 1 : α = (mod π ) aganst H : α (mod π ) In a practcal applcaton 1 α = and β = 1 mples an exact relatonshp between the two varables For large κ Marda (197) has shown that κs has a χ degrees of freedom where { cos( ) } S = n y α β x dstrbuton wth n Further f S and S are the values of S under H and H 1 S1 S( = αβ ) and S = S( α 1) say then for large κ 1 respectvely e κs 1 ~ χ n  8 Pak j stat oper res VolII No 6 pp7986
5 Hypothess testng of parameters for ordnary lnear crcular regresson whch gves κ( S1 S ) ~ χ1 Further κs 1 and κ ( S1 S ) may be assumed to be ndependently dstrbuted Consequently for testng H aganst H 1 the followng Fstatstc may be used ( n ) S S F 1 n = S ( ) 1 In ths secton we wll use the above result to test the hypotheses H :β = 1 aganst H 1 :β 1 and H : α = (mod π ) aganst H 1 :α (mod π ) n order to make a comparson between the ftted model and the deal model of y = x Suppose that ( α 1 ) and ( αβ ) are the maxmum lkelhood estmates of the parameters under H :β = 1 and H 1 :β 1 respectvely Under 1 H :β = 1 the log lkelhood functon s gven by log L = nlog( π) nlog I ( κ) + κ cos( y α x ) Dfferentatng log L wth respect to α gves log L α = κ sn( y α x) Settng ths equal to zero and smplfyng we get Hence tan α = S = C sn( y x ) cos( y x ) say α 1 tan S > C > C 1 = tan π + C < C 1 tan π + S < C > C Therefore ( ) α and S1 = ( n cos( y α β x ) ) S = n cos( y x ) Pak j stat oper res VolII No 6 pp
6 Abdul Ghapor Hussn and the test statstc s gven by F 1 n = ( ) ( n ) S S 1 S 1 The same technque can be used to test the hypothess that H :α = (modπ) aganst H 1 : α (mod π ) Suppose that ( β ) and ( α β) are the maxmum lkelhood estmates of the parameters under H :α = (modπ) and H 1 : α (mod π ) respectvely Under H :α = (modπ) the log lkelhood functon s gven by log L = nlog( π) nlog I ( κ) + κ cos( y β x ) Dfferentatng log L wth respect to β gves log L β = κ x cos( y βx) Ths may be solved teratvely for β and t can be shown that β = β + x sn( y β x) x cos( y β x ) where β s an ntal estmate of β Hence { } β and S1 = { n cos( y x } ) S = n cos( y x ) α β We wll use the same Fstatstc as above to test the hypothess for ntercept α 4 Applcaton and Concluson In ths secton we wll apply the results of the hypothess testng and wll use the wnd and wave drecton data as our examples Suppose varable X s the observaton (drectons) measured by radar and varable Y s the observaton (drectons) measured by anchored buoy The scatter plots of the data set n Fgure 1 ndcate that there s generally a lnear relatonshp between the drectons taken by radar and anchored buoy wth β 1 We assume that each y observaton on Y and x observaton on X can be descrbed by the model y = α + βx+ ε (mod π) 84 Pak j stat oper res VolII No 6 pp7986
7 Hypothess testng of parameters for ordnary lnear crcular regresson where ε represents departures from the straght lne mplct n the model also called crcular random errors and are dstrbuted as a von Mses dstrbuton wth crcular mean and concentraton parameter κ e ε ~ VM( κ ) Estmates for the wnd drecton data are gven n Table 1 together wth ther approxmate standard errors Parameter Estmate St error α x1  β x1  κ Table 1: Parameter estmates for wnd drecton data Testng the null hypothess that α s equal to mod( π ) gave an Fstatstc equal to 575 whch we compare wth F(117) The null hypothess s rejected at the 5% sgnfcance level ndcatng a nonzero α s requred However testng the null hypothess that β equals 1 gves F=17 and we draw the concluson that the null hypothess can not be rejected at the 5% sgnfcance level Table gve the estmates of wave drecton data together wth ther approxmate standard errors Parameter Estmate St error α β x1  κ Table : Parameter estmates for wave drecton data Testng the null hypothess that α s equal to mod( π ) gves F = 1815 and the null hypothess that β s equal to 1 gves F = 77 We compare these F values wth F(1 76) and we draw the concluson that both null hypotheses can not be rejected at 5% sgnfcance level Pak j stat oper res VolII No 6 pp
8 Abdul Ghapor Hussn References 1 Dobson A J Appled Statstcs 7 (1978) pp Marda K V Statstcs of Drectonal Data Academc Press London (197) 3 Hussn AG Feller N R J and Eleanor E C Journal of Appled Scence & Technology 8 Nos 1 & (4) pp Sova M S The samplng varablty and the valdaton of hgh frequency radar measurements of the sea surface PhD Thess School of Mathematcs and Statstcs Unversty of Sheffeld (1995) 86 Pak j stat oper res VolII No 6 pp7986
PSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 12
14 The Chsquared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304C) 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 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 informationThe Analysis of Outliers in Statistical Data
THALES Project No. xxxx The Analyss of Outlers n Statstcal Data Research Team Chrysses Caron, Assocate Professor (P.I.) Vaslk Karot, Doctoral canddate Polychrons Economou, Chrstna Perrakou, Postgraduate
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 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 informationThe Magnetic Field. Concepts and Principles. Moving Charges. Permanent Magnets
. The Magnetc Feld Concepts and Prncples Movng Charges All charged partcles create electrc felds, and these felds can be detected by other charged partcles resultng n electrc force. However, a completely
More informationLinear Regression Analysis for STARDEX
Lnear Regresson Analss for STARDEX Malcolm Halock, Clmatc Research Unt The followng document s an overvew of lnear regresson methods for reference b members of STARDEX. Whle t ams to cover the most common
More informationCalculating the high frequency transmission line parameters of power cables
< ' Calculatng the hgh frequency transmsson lne parameters of power cables Authors: Dr. John Dcknson, Laboratory Servces Manager, N 0 RW E B Communcatons Mr. Peter J. Ncholson, Project Assgnment Manager,
More informationAnalysis of Covariance
Chapter 551 Analyss of Covarance Introducton A common tas n research s to compare the averages of two or more populatons (groups). We mght want to compare the ncome level of two regons, the ntrogen content
More informationThe covariance is the two variable analog to the variance. The formula for the covariance between two variables is
Regresson Lectures So far we have talked only about statstcs that descrbe one varable. What we are gong to be dscussng for much of the remander of the course s relatonshps between two or more varables.
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 informationThe Development of Web Log Mining Based on ImproveKMeans Clustering Analysis
The Development of Web Log Mnng Based on ImproveKMeans Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.
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 informationCausal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting
Causal, Explanatory Forecastng Assumes causeandeffect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of
More information1 Approximation Algorithms
CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons
More informationThe Analysis of Covariance. ERSH 8310 Keppel and Wickens Chapter 15
The Analyss of Covarance ERSH 830 Keppel and Wckens Chapter 5 Today s Class Intal Consderatons Covarance and Lnear Regresson The Lnear Regresson Equaton TheAnalyss of Covarance Assumptons Underlyng the
More informationExhaustive Regression. An Exploration of RegressionBased Data Mining Techniques Using Super Computation
Exhaustve Regresson An Exploraton of RegressonBased Data Mnng Technques Usng Super Computaton Antony Daves, Ph.D. Assocate Professor of Economcs Duquesne Unversty Pttsburgh, PA 58 Research Fellow The
More informationbenefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).
REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or
More information1 Example 1: Axisaligned 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 information) of the Cell class is created containing information about events associated with the cell. Events are added to the Cell instance
Calbraton Method Instances of the Cell class (one nstance for each FMS cell) contan ADC raw data and methods assocated wth each partcular FMS cell. The calbraton method ncludes event selecton (Class Cell
More informationFORCED CONVECTION HEAT TRANSFER IN A DOUBLE PIPE HEAT EXCHANGER
FORCED CONVECION HEA RANSFER IN A DOUBLE PIPE HEA EXCHANGER Dr. J. Mchael Doster Department of Nuclear Engneerng Box 7909 North Carolna State Unversty Ralegh, NC 276957909 Introducton he convectve heat
More informationRiskbased Fatigue Estimate of Deep Water Risers  Course Project for EM388F: Fracture Mechanics, Spring 2008
Rskbased 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 informationBinomial Link Functions. Lori Murray, Phil Munz
Bnomal Lnk Functons Lor Murray, Phl Munz Bnomal Lnk Functons Logt Lnk functon: ( p) p ln 1 p Probt Lnk functon: ( p) 1 ( p) Complentary Log Log functon: ( p) ln( ln(1 p)) Motvatng Example A researcher
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 informationNonlinear data mapping by neural networks
Nonlnear data mappng by neural networks R.P.W. Dun Delft Unversty of Technology, Netherlands Abstract A revew s gven of the use of neural networks for nonlnear mappng of hgh dmensonal data on lower dmensonal
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 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, Hsnyng Wu b a Professor (Management Scence), Natonal Chao
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 informationTesting GOF & Estimating Overdispersion
Testng GOF & Estmatng Overdsperson Your Most General Model Needs to Ft the Dataset It s mportant that the most general (complcated) model n your canddate model lst fts the data well. Ths model s a benchmark
More informationLogical Development Of Vogel s Approximation Method (LDVAM): An Approach To Find Basic Feasible Solution Of Transportation Problem
INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME, ISSUE, FEBRUARY ISSN 77866 Logcal Development Of Vogel s Approxmaton Method (LD An Approach To Fnd Basc Feasble Soluton Of Transportaton
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 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. STATTECH Consultng and Tranng n Appled Statstcs San Jose, CA Topcs Practcal
More informationInequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.
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 information21 Vectors: The Cross Product & Torque
21 Vectors: The Cross Product & Torque Do not use our left hand when applng ether the rghthand rule for the cross product of two vectors dscussed n ths chapter or the rghthand rule for somethng curl
More informationChapter XX More advanced approaches to the analysis of survey data. Gad Nathan Hebrew University Jerusalem, Israel. Abstract
Household Sample Surveys n Developng and Transton Countres Chapter More advanced approaches to the analyss of survey data Gad Nathan Hebrew Unversty Jerusalem, Israel Abstract In the present chapter, we
More informationControl Charts for Means (Simulation)
Chapter 290 Control Charts for Means (Smulaton) Introducton Ths procedure allows you to study the run length dstrbuton of Shewhart (Xbar), Cusum, FIR Cusum, and EWMA process control charts for means usng
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 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 informationCS 2750 Machine Learning. Lecture 3. Density estimation. CS 2750 Machine Learning. Announcements
Lecture 3 Densty estmaton Mlos Hauskrecht mlos@cs.ptt.edu 5329 Sennott Square Next lecture: Matlab tutoral Announcements Rules for attendng the class: Regstered for credt Regstered for audt (only f there
More informationA Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy Scurve Regression
Novel Methodology of Workng Captal Management for Large Publc Constructons by Usng Fuzzy Scurve Regresson ChengWu Chen, Morrs H. L. Wang and TngYa Hseh Department of Cvl Engneerng, Natonal Central Unversty,
More informationNPAR TESTS. OneSample ChiSquare Test. Cell Specification. Observed Frequencies 1O i 6. Expected Frequencies 1EXP i 6
PAR TESTS If a WEIGHT varable s specfed, t s used to replcate a case as many tmes as ndcated by the weght value rounded to the nearest nteger. If the workspace requrements are exceeded and samplng has
More informationPRACTICE 1: MUTUAL FUNDS EVALUATION USING MATLAB.
PRACTICE 1: MUTUAL FUNDS EVALUATION USING MATLAB. INDEX 1. Load data usng the Edtor wndow and mfle 2. Learnng to save results from the Edtor wndow. 3. Computng the Sharpe Rato 4. Obtanng the Treynor Rato
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 informationCalibration and Linear Regression Analysis: A SelfGuided Tutorial
Calbraton and Lnear Regresson Analyss: A SelfGuded Tutoral Part The Calbraton Curve, Correlaton Coeffcent and Confdence Lmts CHM314 Instrumental Analyss Department of Chemstry, Unversty of Toronto Dr.
More informationx f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60
BIVARIATE DISTRIBUTIONS Let be a varable that assumes the values { 1,,..., n }. Then, a functon that epresses the relatve frequenc of these values s called a unvarate frequenc functon. It must be true
More informationCommunication Networks II Contents
8 / 1  Communcaton Networs II (Görg)  www.comnets.unbremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP
More information1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)
6.3 /  Communcaton Networks II (Görg) SS20  www.comnets.unbremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes
More information1 De nitions and Censoring
De ntons and Censorng. Survval Analyss We begn by consderng smple analyses but we wll lead up to and take a look at regresson on explanatory factors., as n lnear regresson part A. The mportant d erence
More informationErrorPropagation.nb 1. Error Propagation
ErrorPropagaton.nb Error Propagaton Suppose that we make observatons of a quantty x that s subject to random fluctuatons or measurement errors. Our best estmate of the true value for ths quantty s then
More informationA machine vision approach for detecting and inspecting circular parts
A machne vson approach for detectng and nspectng crcular parts DuMng Tsa Machne Vson Lab. Department of Industral Engneerng and Management YuanZe Unversty, ChungL, Tawan, R.O.C. Emal: edmtsa@saturn.yzu.edu.tw
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: 84865396, Fax: 84865684, Emal: afs@psu.edu
More informationHow Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence
1 st Internatonal Symposum on Imprecse Probabltes and Ther Applcatons, Ghent, Belgum, 29 June 2 July 1999 How Sets of Coherent Probabltes May Serve as Models for Degrees of Incoherence Mar J. Schervsh
More information2.4 Bivariate distributions
page 28 2.4 Bvarate dstrbutons 2.4.1 Defntons Let X and Y be dscrete r.v.s defned on the same probablty space (S, F, P). Instead of treatng them separately, t s often necessary to thnk of them actng together
More informationLeast Squares Fitting of Data
Least Squares Fttng of Data Davd Eberly Geoetrc Tools, LLC http://www.geoetrctools.co/ Copyrght c 19982016. All Rghts Reserved. Created: July 15, 1999 Last Modfed: January 5, 2015 Contents 1 Lnear Fttng
More informationAn empirical study for credit card approvals in the Greek banking sector
An emprcal study for credt card approvals n the Greek bankng sector Mara Mavr George Ioannou Bergamo, Italy 1721 May 2004 Management Scences Laboratory Department of Management Scence & Technology Athens
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 informationTime Series Analysis in Studies of AGN Variability. Bradley M. Peterson The Ohio State University
Tme Seres Analyss n Studes of AGN Varablty Bradley M. Peterson The Oho State Unversty 1 Lnear Correlaton Degree to whch two parameters are lnearly correlated can be expressed n terms of the lnear correlaton
More informationForecasting the Demand of Emergency Supplies: Based on the CBR Theory and BP Neural Network
700 Proceedngs of the 8th Internatonal Conference on Innovaton & Management Forecastng the Demand of Emergency Supples: Based on the CBR Theory and BP Neural Network Fu Deqang, Lu Yun, L Changbng School
More informationStudy on CET4 Marks in China s Graded English Teaching
Study on CET4 Marks n Chna s Graded Englsh Teachng CHE We College of Foregn Studes, Shandong Insttute of Busness and Technology, P.R.Chna, 264005 Abstract: Ths paper deploys Logt model, and decomposes
More informationNONPARAMETRIC REGRESSION ESTIMATION FOR DATA WITH EQUAL VALUES
European Scentfc Journal February 24 edton vol., No.4 ISSN: 857 788 (Prnt) e  ISSN 857743 NONPARAMETRIC REGRESSION ESTIMATION FOR DATA WITH EQUAL VALUES N. Alp Erll, PhD Department of Econometrcs, Unversty
More informationH 1 : at least one is not zero
Chapter 6 More Multple Regresson Model The Ftest Jont Hypothess Tests Consder the lnear regresson equaton: () y = β + βx + βx + β4x4 + e for =,,..., N The tstatstc gve a test of sgnfcance of an ndvdual
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 informationLatent Class Regression. Statistics for Psychosocial Research II: Structural Models December 4 and 6, 2006
Latent Class Regresson Statstcs for Psychosocal Research II: Structural Models December 4 and 6, 2006 Latent Class Regresson (LCR) What s t and when do we use t? Recall the standard latent class model
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 twostage stratfed cluster desgn. 1 The frst stage conssted of a sample
More informationExamples of Multiple Linear Regression Models
ECON *: Examples of Multple Regresson Models Examples of Multple Lnear Regresson Models Data: Stata tutoral data set n text fle autoraw or autotxt Sample data: A crosssectonal sample of 7 cars sold n
More informationMULTIPLE LINEAR REGRESSION IN MINITAB
MULTIPLE LINEAR REGRESSION IN MINITAB Ths document shows a complcated Mntab multple regresson. It ncludes descrptons of the Mntab commands, and the Mntab output s heavly annotated. Comments n { } are used
More informationProbabilities and Probabilistic Models
Probabltes and Probablstc Models Probablstc models A model means a system that smulates an obect under consderaton. A probablstc model s a model that produces dfferent outcomes wth dfferent probabltes
More informationgreatest common divisor
4. GCD 1 The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no
More informationOn the Optimal Control of a Cascade of HydroElectric Power Stations
On the Optmal Control of a Cascade of HydroElectrc Power Statons M.C.M. Guedes a, A.F. Rbero a, G.V. Smrnov b and S. Vlela c a Department of Mathematcs, School of Scences, Unversty of Porto, Portugal;
More informationRisk Model of LongTerm Production Scheduling in Open Pit Gold Mining
Rsk Model of LongTerm Producton Schedulng n Open Pt Gold Mnng R Halatchev 1 and P Lever 2 ABSTRACT Open pt gold mnng s an mportant sector of the Australan mnng ndustry. It uses large amounts of nvestments,
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 informationCS 2750 Machine Learning. Lecture 17a. Clustering. CS 2750 Machine Learning. Clustering
Lecture 7a Clusterng Mlos Hauskrecht mlos@cs.ptt.edu 539 Sennott Square Clusterng Groups together smlar nstances n the data sample Basc clusterng problem: dstrbute data nto k dfferent groups such that
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 informationQUANTUM MECHANICS, BRAS AND KETS
PH575 SPRING QUANTUM MECHANICS, BRAS AND KETS The followng summares the man relatons and defntons from quantum mechancs that we wll be usng. State of a phscal sstem: The state of a phscal sstem s represented
More informationBERNSTEIN POLYNOMIALS
OnLne 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 informationEstimation of Dispersion Parameters in GLMs with and without Random Effects
Mathematcal Statstcs Stockholm Unversty Estmaton of Dsperson Parameters n GLMs wth and wthout Random Effects Meng Ruoyan Examensarbete 2004:5 Postal address: Mathematcal Statstcs Dept. of Mathematcs Stockholm
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 informationGRAVITY DATA VALIDATION AND OUTLIER DETECTION USING L 1 NORM
GRAVITY DATA VALIDATION AND OUTLIER DETECTION USING L 1 NORM BARRIOT JeanPerre, SARRAILH Mchel BGI/CNES 18.av.E.Beln 31401 TOULOUSE Cedex 4 (France) Emal: jeanperre.barrot@cnes.fr 1/Introducton The
More informationSurvival analysis methods in Insurance Applications in car insurance contracts
Survval analyss methods n Insurance Applcatons n car nsurance contracts Abder OULIDI 1 JeanMare MARION 2 Hervé GANACHAUD 3 Abstract In ths wor, we are nterested n survval models and ther applcatons on
More informationAnalysis of Premium Liabilities for Australian Lines of Business
Summary of Analyss of Premum Labltes for Australan Lnes of Busness Emly Tao Honours Research Paper, The Unversty of Melbourne Emly Tao Acknowledgements I am grateful to the Australan Prudental Regulaton
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 informationCopulas. Modeling dependencies in Financial Risk Management. BMI Master Thesis
Copulas Modelng dependences n Fnancal Rsk Management BMI Master Thess Modelng dependences n fnancal rsk management Modelng dependences n fnancal rsk management 3 Preface Ths paper has been wrtten as part
More informationMarginal Returns to Education For Teachers
The Onlne Journal of New Horzons n Educaton Volume 4, Issue 3 MargnalReturnstoEducatonForTeachers RamleeIsmal,MarnahAwang ABSTRACT FacultyofManagementand Economcs UnverstPenddkanSultan Idrs ramlee@fpe.ups.edu.my
More informationSupport vector domain description
Pattern Recognton Letters 20 (1999) 1191±1199 www.elsever.nl/locate/patrec Support vector doman descrpton Davd M.J. Tax *,1, Robert P.W. Dun Pattern Recognton Group, Faculty of Appled Scence, Delft Unversty
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 annutymmedate, and ts present value Study annutydue, and
More informationSimon Acomb NAG Financial Mathematics Day
1 Why People Who Prce Dervatves Are Interested In Correlaton mon Acomb NAG Fnancal Mathematcs Day Correlaton Rsk What Is Correlaton No lnear relatonshp between ponts Comovement between the ponts Postve
More informationAlthough ordinary leastsquares (OLS) regression
egresson through the Orgn Blackwell Oxford, TEST 014198X 003 5 31000 Orgnal Joseph Teachng G. UK Artcle Publshng Esenhauer through Statstcs the Ltd Trust Orgn 001 KEYWODS: Teachng; egresson; Analyss of
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 informationMarginal Benefit Incidence Analysis Using a Single Crosssection of Data. Mohamed Ihsan Ajwad and Quentin Wodon 1. World Bank.
Margnal Beneft Incdence Analyss Usng a Sngle Crosssecton of Data Mohamed Ihsan Ajwad and uentn Wodon World Bank August 200 Abstract In a recent paper, Lanjouw and Ravallon proposed an attractve and smple
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 informationRESEARCH ON DUALSHAKER SINE VIBRATION CONTROL. Yaoqi FENG 1, Hanping QIU 1. China Academy of Space Technology (CAST) yaoqi.feng@yahoo.
ICSV4 Carns Australa 9 July, 007 RESEARCH ON DUALSHAKER SINE VIBRATION CONTROL Yaoq FENG, Hanpng QIU Dynamc Test Laboratory, BISEE Chna Academy of Space Technology (CAST) yaoq.feng@yahoo.com Abstract
More informationRing structure of splines on triangulations
www.oeaw.ac.at Rng structure of splnes on trangulatons N. Vllamzar RICAMReport 201448 www.rcam.oeaw.ac.at RING STRUCTURE OF SPLINES ON TRIANGULATIONS NELLY VILLAMIZAR Introducton For a trangulated regon
More informationNew bounds in BalogSzemerédiGowers theorem
New bounds n BalogSzemerédGowers theorem By Tomasz Schoen Abstract We prove, n partcular, that every fnte subset A of an abelan group wth the addtve energy κ A 3 contans a set A such that A κ A and A
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, 6105194390,
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 informationMultiple discount and forward curves
Multple dscount and forward curves TopQuants presentaton 21 ovember 2012 Ton Broekhuzen, Head Market Rsk and Basel coordnator, IBC Ths presentaton reflects personal vews and not necessarly the vews of
More informationEvaluating credit risk models: A critique and a new proposal
Evaluatng credt rsk models: A crtque and a new proposal Hergen Frerchs* Gunter Löffler Unversty of Frankfurt (Man) February 14, 2001 Abstract Evaluatng the qualty of credt portfolo rsk models s an mportant
More informationz(t) = z 1 (t) + t(z 2 z 1 ) z(t) = 1 + i + t( 2 3i (1 + i)) z(t) = 1 + i + t( 3 4i); 0 t 1
(4.): ontours. Fnd an admssble parametrzaton. (a). the lne segment from z + to z 3. z(t) z (t) + t(z z ) z(t) + + t( 3 ( + )) z(t) + + t( 3 4); t (b). the crcle jz j 4 traversed once clockwse startng at
More informationOn fourth order simultaneously zerofinding method for multiple roots of complex polynomial equations 1
General Mathematcs Vol. 6, No. 3 (2008), 9 3 On fourth order smultaneously zerofndng method for multple roots of complex polynomal euatons Nazr Ahmad Mr and Khald Ayub Abstract In ths paper, we present
More informationPerformance Analysis of Energy Consumption of Smartphone Running Mobile Hotspot Application
Internatonal Journal of mart Grd and lean Energy Performance Analyss of Energy onsumpton of martphone Runnng Moble Hotspot Applcaton Yun on hung a chool of Electronc Engneerng, oongsl Unversty, 511 angdodong,
More information