EFFECT OF AUTOCORRELATION ON SPC CHART PERFORMANCE
|
|
- Camron Wright
- 5 years ago
- Views:
Transcription
1 3 rd Research/Exper Conference wih Inernaional Paricipaions QUALIT 3, Zenica, B&H, 13 and 14 ovember, 3. EFFECT OF AUTOCORRELATIO O SPC CHART PERFORMACE Vesna Bucevsa, Ph.D. Universiy S. Cyril and Mehodius, Faculy of Economics Sopje, Republic of Macedonia Keywords: Auocorrelaion, SPC chars, saionariy ABSTRACT Saisical Process Conrol (SPC) aims a qualiy improvemens hrough reducion of variances. The bes nown ool of SPC is he conrol char. Over he years he conrol char has proved o be a successful pracical echnique for monioring process measuremens. However, is usefulness in pracice is limied o hose siuaions where i can be assumed ha successive measuremens are independenly disribued, whereas mos daa ses encounered in pracice exhibi some form of serial correlaion. The quesion ha is considered in his paper is wha conrol char mehods should be used o monior serially correlaed daa and how he signals on such chars should be inerpreed. 1. STOCHASTIC PROCESS Classical conrol chars assume no correlaion beween successive observaions of he qualiy characerisic. In his secion we define in a more precise manner wha is mean by correlaion beween repeaedly observed measuremens of a single qualiy characerisic. We need o now how o esimae he serial correlaion, in case i exiss, and how o sudy is effec on classical SPC chars. To achieve hese goals, he concep of a sochasic process is firs necessary. A sochasic process { (), I} is a family of indexed random variables, where I is called he index se. Someimes we will refer o he mechanism generaing he sochasic process simply as he process, which can be undersood in double sense of he underlying sochasic process ha he qualiy characerisic being modeled follows, or as he producion process iself, which in urn generaes he sochasic process (i.e. he qualiy characerisic). In applicaions in his paper, will relae o he discree poins of ime a which an observaion is obained by sampling (i.e. he index se is I {...,, 1,,1,,... }) and he sochasic process is said o be a discree-ime sochasic process. If I { : < < + }, he process is a coninuos-ime sochasic process. For discree-ime sochasic processes, i is cusomary o denoe hem as { }, ha is, a subscrip is used for discree-ime indices. Discree-ime processes can be I idenified by describing he behavior of is h elemen. Thus, we can wrie, for example, μ + ε Implying ha for he given process (Shewhar's model in his case) he equaion holds for all discree poins in ime. In his case, he ime beween observaion h equals Δ. 71
2 A sochasic process can be hough of as a funcion of wo argumens: namely, { (, w), I, w ς}, where ς is he sample space of he random variables (). I is imporan o poin ou ha for fixed I, (,. ) is a random variable; for fixed w ς, (., w) is a funcion of ime called a realizaion, or rajecory of he process. A ime series is a se of observaions, or realizaion, of a discree-ime sochasic process. In basic probabiliy heory, his is analogous o he noion of a single observed value of a random variable (y) compared o he random variable iself (). The se of all possible realizaions of a sochasic process is called he ensemble of he process. As in many oher areas of saisics, o perform valid saisical inferences from a sochasic process, we need some noion of how repeaable he underlying random experimen is under idenical condiions. For example, one such saisical inference could be o predic o where a qualiy characerisic will move in he near fuure. The only informaion available o us is a single realizaion of a sochasic process, he values recorded of he qualiy characerisic in he pas. If his process is such ha is random variables differ radically a differen poins of ime, no inference could be drawn from a single realizaion since he probabilisic properies of he process during he period of ime when he observaions were aen canno be generalized o oher periods of ime. For he class of models sudied in his paper, he noion ha we need o mae valid inferences is called saionary. A sochasic process is sricly saionary if for any ineger 1, he join disribuion of {,,..., } is idenical o he disribuion of { } 1 + τ, + τ,..., 1 + τ, where i I, i + τ I.Thus he sochasic properies of he process are unaffeced by changes in he ime origin. If in his definiion we loo a he case 1, we noe ha sric saionariy implies ha he disribuion of is he same as he disribuion for for + τ. Therefore, sric saionary implies ha he disribuion of he random variables occurring a differen poins in ime is idenical. 3. MEA AD VARIACE OF A STATIOAR PROCESS { Since for a sricly saionary discree-ime process } he probabiliy densiy funcion f ( ) is he same for all, ha is, we have ha ( ) f ( ) f [ ] yf ( y) [ ] y f ( y) dy μ E dy σ μ Var. Thus μ gives he long-run or asympoic average level of he process a every ime and gives he long run dispersion of he process a every ime. These long-run quaniies are obained over he ensemble of he sochasic process, which is he populaion from which we sample. For example, E [ ] can be hough as he level of he process a ime averaged over an infiniely large number of possible realizaions. The corresponding poin esimaes are given by 1 ˆμ y 1 1 ( ) y ˆ σ 1 where denoes he lengh of he ime series observed. σ 7
3 4. AUTOCOVARIACE AD AUTOCORRELATIO The prefix auo means a reflexive ac upon oneself, hus auocovariance is he covariance ha γ gives a measure of linear a process has "wih iself". The auocovariance funcion ( ), associaion (covariance) beween wo variables of he same process, and +, ha are separaed period of lags, as a funcion of. For any discree-ime process, we define, [ ] E[ ( E[ ])( E[ ])] γ, for, ± 1, ±,... Cov For a sricly saionary process, he mean auocovariance reduces o [ ] E ( μ)( μ) [ ] γ, γ Cov j, j+ + for, ± 1, ±,... μ is consan for all imes, and he so he auocovariance depends only on he lag. Also, noe ha γ E μ Var. [( )] ( ) σ I is usually beer o scale he auocovariance ino a uniless quaniy by dividing by he process variance. For a saionary process, γ ρ for, ± 1, ±,... γ where 1 ρ 1. Given ha γ γ and ρ ρ (i.e., he auocorrelaion and auocovariance are even funcions), i is common o plo hese wo funcions only for posiive lags. Boh he auocovariance and he auocorrelaion a lag give a measure of he degree of linear associaion beween wo random variables of he same process ha are separaed periods. When considering relaions beween random variables, i is useful o recall he following implicaions: 1. If and + are independen, hey are uncorrelaed (i.e., ρ for all ). If and + are correlaed (i.e, ρ for some ), and + are dependen. The direcion of he implicaions is imporan; uncorrelaed random variables may or may no be independen. Correlaion is a measure of linear associaion, so here migh be some nonlinear associaion beween uncorrelaed variables. I should be poined ou ha he sandard error of he average ( ) is grealy affeced by auocorrelaion. Barle (1946) showed ha σ γ γ 1 Thus if he sochasic process is compleely uncorrelaed ( γ for all ), he sandard error of he average is he usual σ /. As he (posiive) auocorrelaion increases, so does he sandard error and he poin esimae becomes less reliable. 73
4 Poin esimaes of he auocovariance funcion are given by he sample auocovariance funcion, compued as c 1 ˆ γ ( y )( y ),1,,... (1.6) + 1 Similarly, he sample auocorrelaion funcion is given by r c ˆ ρ,1,,... c 5. EED FOR STATIOARIT oe ha he poin esimaes, ˆ σ, c, and r are compued based on a single realizaion of he process (i.e., hey are compued based on ime-series daa). Inferences based on a single realizaion are possible because he process is saionary. A differen form of saionariy is called wealy or covariance saionariy. A process is wealy saionary if only is firs wo momens (mean and covariance) are finie and independen of ime (in such case he covariance depends only on he lag (. If each is normally disribued and he process is wealy saionary, he process is sricly saionary. As i urns ou, he ype of sochasic models we sudy are fully characerized by heir firs wo momens, so when we refer o saionariy we are referring o wealy saionariy. Wea saionariy can be beer undersood from an engineering poin of view. If he mean of he process is consan, some process engineers would say ha he process is sable. As will be seen, sabiliy is a propery of a deerminisic dynamic sysem, whereas saionary is a properly relaed o a sochasic process. 6. EFFECT OF AUTOCORRELATIO O SPC CHART PERFORMACE As menioned before, SPC conrol chars assume ha if in conrol, he process has a consan mean and is compleely uncorrelaed. An imporan pracical quesion is o invesigae wha happens wih he performance of SPC chars as he process exhibis more and more serial correlaion. As one process engineer said "almos every producion process exhibis auocorrelaion." The effec and implicaions of auocorrelaion have been opics of frequen discussion and debae in SPC lieraure. For a deailed review please refer o Knoh and Schmid (1). These issues are usually acled numerically: ae for insance he invesigaions by Johnson and Bagshaw (1974), Alwan (199), Maragah and Woodall (199), Wardell, Mosowiz and Plane (1994), Runger, Willemain and Prabhu (1995), Van Bracle III and Reynolds Jr. (1997) and Lu and Reynolds Jr. (1999). These and several oher papers have ables and graphs usually referring o he ARL, o provide evidence ha he performance of he appealing conrol schemes is severely compromised by he presence of serial correlaion. Auocorrelaion ofen reflecs increased variabiliy. Thus, he firs wo opions o consider should be o remove he source of he auocorrelaion or o use some ype of process adjusmen scheme such as hose discussed by Box and Luceno (1997) and Huner (1998). Conrol charing can be used in conducion wih process adjusmen schemes, and Box and Luceno (1997) emphasized ha he wo ypes of ools should be used ogeher. Only if hese 74
5 wo opions prove infeasible should one consider using sand-alone conrol chars for process monioring such as hose discussed by Lu and Reynolds (1999), Lin and Adams (1996) and Adams and Lin (1999). Posiive auocorrelaion a low lags is common because given he advances in sensor echnologies, measuremens are aen closer ogeher in ime. In discree-par manufacuring, his someimes implies ha every par is measured. Observaions ha were generaed close in ime will end o be similar, hence posiive correlaion a low lags will resul. The following wo examples illusrae he effec of auocorrelaion on he performance of SPC chars. Example: Suppose ha a process is described by Z ε + λz + λ λ Z + λ 1 λ ( ) ( ) Z μ + Z where λ 1 and is he qualiy characerisic. There are wo exreme cases: if λ, he process is jus Shewhar's process, where he observaions are compleely uncorrelaed. If λ 1, he process is Z Z μ + μ + ε + 1 which evidenly is highly correlaed wih μ Z Vander Weil considers his process as λ increases from o 1. In he realizaions of Figure 1.13b and d, a susained shif in he mean of magniude 5σ was induced a ime 1; ha is, he mean of he process changes according o μ μ μ + δσ if if > ` where in his example δ 5, 1, and μ 1. This ype of shif suddenly changes he mean of he process a ime. As can be seen from he figures, for λ, deecion should be almos immediae for any SPC char and false alarms should be infrequen. For λ. 8, he shif becomes indisinguishable from he auocorrelaion srucure. Such posiive auocorrelaion will no necessarily worsen he deecion capabiliies of he chars as measured by he ARL ou performance crierion (Goldsmih and Whifield, 1961; Lu and Reynolds, 1999, 1), bu he number of false alarms will cerainly increase. A char ha gives frequen false alarms will soon be abandoned. Example : The impac of auocorrelaion on SPC char performance was sudied by Mragah and Woodall (199), who consider he process μ + φ 1 + ε (1.8) where 1 < < 1 process. If φ in equaion (1.8) we obain a Shewhar's uncorrelaed process. The auocorrelaion funcion of his process is ρ φ, so he process is saionary as long as φ is a parameer ha allow us o define he auocorrelaion in he { } 75
6 φ <1. oe ha for φ 1, he process urns ou o be a random wal. In he case of posiive auocorrelaion ( < φ <1), he movemen of he process is smooher han ha of an uncorrelaed process. In he case of negaive auocorrelaion ( 1 < φ < ), he process shows a sawooh paern, which compared o Shewhar's process is much more crumpled. Maragah and Woodall (199) compued by simulaion he average number of ou-of-conrol poins ha a Shewhar char for individuals will generae i 5 observaions of process (1.8) wih.9 φ. 9 are used o se he char limis. A char for individuals or a char is one in which no subgroups are formed. The conrol limis are compued using he moving-range esimaor 1 ˆ σ MR / d, where MR i i i 1. Table 1 shows some of heir resuls. Table 1: umber of Poins Ouside Limis Generaed by a Shewhar Char for Individuals Used o Monior 5 Observaions of an AR(1) Process, Obained by Simulaion φ Average Sd. Dev Source: Maragah and Woodall (199) oe ha no shif in he mean was inroduced while simulaing he AR(1) process. From he able i can be seen ha he number of ou of conrol signals increases wih increasing posiive auocorrelaion. For φ. 9, for example, an average of abou four signals will occur in 5 samples. This is a much higher signal rae han he adverised average of one false alarm every 37 observaions for Shewhar chars. egaive auocorrelaion reduces he number if signals, bu posiive auocorrelaion is much more common in pracice as menioned before. egaive auocorrelaion a low lags inflaes he variance and hence he conrol limis widh becomes oo large, so deecing acual shifs becomes more difficul. The eviden problem in Example is ha he limis were compued based on a variance esimae, which assumes ha he process is uncorrelaed. If φ > (posiive auocorrelaion), adjacen observaions will end o be similar and he moving-range esimaor will underesimae he variance of he process. This will resul in limis ha are oo narrow, producing many alarms compare o he uncorrelaed vase. This resul is quie general: The wihin-sample variance esimaor will underesimae he process variance in a posiively auocorrelaed process. If φ < (negaive auocorrelaion), he moving-range esimaor will overesimae he rue variance, resuling in limis ha are oo wide, so very few signals will be obained as opposed o he uncorrelaed case. oe ha his will have an impac on he deecion capabiliies of real shifs (i.e., wide limis will mae he char ae much longer o deec shifs in he process mean). 1 This esimaor was used because i is he mos common esimaor in conrol chars for individuals, bu as menioned below, i is no recommended for auocorrelaed daa. 76
7 7. COCLUSIOS In recen years, saisical process conrol (SPC) for auocorrelaed processes has received a grea deal of aenion, due in par o he increasing prevalence of auocorrelaion in process inspecion daa. Wih improvemens in measuremen and daa collecion echnology, processes can be sampled a higher raes, which ofen leads o daa auocorrelaion. I is well nown ha he run lengh properies of common SPC mehods lie CUSUM and X chars are srongly affeced by daa auocorrelaion, and he in-conrol average run lengh (ARL) can be much shorer han inended if he auocorrelaion is posiive. The mos widely researched mehods of SPC for auocorrelaed processes are residual-based conrol chars, which involve fiing some form of auoregressive moving average (ARMA) model o he daa and monioring he model residuals (i.e., he one-sep-ahead predicion errors). If he model is exac, hen he model residuals are independen. Consequenly, sandard SPC conrol chars can be applied o he residuals wih well undersood in-conrol run lengh properies. 8. REFERECES 1. Box, G. E. P., and Luceno, A., Saisical Conrol by Monioring and Feedbac Adjusmen, ew or, Mongomery, D. C., Inroducion o Saisical Qualiy Conrol (3rd ed.), ew or, 3. Rao, C. R., Saisics and Truh: Puing Chance o Wor, Fairland, MD. Inernaional Co- Operaive Publishing House, Ryan, T. P., Saisical Mehods for Qualiy Improvemen (nd ed.), ew or, 5. Shewhar, W. A., Economic Conrol of Qualiy of Manufacured Produc, ew or, Weherill, G.B. and Brown, D.W., Saisical Process Conrol: Theory and Pracice, Chapman and Hall, London, FIGURE 1. SIGLE REALIZATIO (DARKER LIE) COMPARED TO OTHER POSSIBLE REALIZATIOS THAT MAKE UP THE ESEMBLE OF A SIGLE STOCHASTIC PROCESS 77
8 FIGURE. TIME-SERIES PLOT AD SAMPLE AUTOCORRELATIO FUCTIO OF THE DATA I EXAMPLE 1, CHEMICAL PROCESS DATA FIGURE 3. TIME-SERIES PLOT AD SAMPLE AUTOCORRELATIO FUCTIO OF THE DATA I EXAMPLE, MACHIED PARTS DATA FIGURE 4. PROCESS WITH O SUDDE SHIFTS (a,c) AD WITH A SUDDE SHIFT I THE MEA OCCURRIG AT TIME 1 (b, d), EXAMPLE 3. GRAPHS (a) AD (b) DISPLA A UCORRECTED (λ); GRAPHS (c) AD (d) DISPLA A HIGHL AUTOCORRELATED (λ.8) SERIES 78
Diebold, Chapter 7. Francis X. Diebold, Elements of Forecasting, 4th Edition (Mason, Ohio: Cengage Learning, 2006). Chapter 7. Characterizing Cycles
Diebold, Chaper 7 Francis X. Diebold, Elemens of Forecasing, 4h Ediion (Mason, Ohio: Cengage Learning, 006). Chaper 7. Characerizing Cycles Afer compleing his reading you should be able o: Define covariance
More informationR t. C t P t. + u t. C t = αp t + βr t + v t. + β + w t
Exercise 7 C P = α + β R P + u C = αp + βr + v (a) (b) C R = α P R + β + w (c) Assumpions abou he disurbances u, v, w : Classical assumions on he disurbance of one of he equaions, eg. on (b): E(v v s P,
More informationProcess Monitoring versus Process Adjustment
CHAPTER 1 Process Monioring versus Process Adjusmen I can be said ha he birh of modern saisical process conrol Ž SPC. ook place when Waler A. Shewhar, a physicis and saisician working for Bell Laboraories,
More informationExponential Weighted Moving Average (EWMA) Chart Under The Assumption of Moderateness And Its 3 Control Limits
DOI: 0.545/mjis.07.5009 Exponenial Weighed Moving Average (EWMA) Char Under The Assumpion of Moderaeness And Is 3 Conrol Limis KALPESH S TAILOR Assisan Professor, Deparmen of Saisics, M. K. Bhavnagar Universiy,
More informationOBJECTIVES OF TIME SERIES ANALYSIS
OBJECTIVES OF TIME SERIES ANALYSIS Undersanding he dynamic or imedependen srucure of he observaions of a single series (univariae analysis) Forecasing of fuure observaions Asceraining he leading, lagging
More informationEcon Autocorrelation. Sanjaya DeSilva
Econ 39 - Auocorrelaion Sanjaya DeSilva Ocober 3, 008 1 Definiion Auocorrelaion (or serial correlaion) occurs when he error erm of one observaion is correlaed wih he error erm of any oher observaion. This
More informationGINI MEAN DIFFERENCE AND EWMA CHARTS. Muhammad Riaz, Department of Statistics, Quaid-e-Azam University Islamabad,
GINI MEAN DIFFEENCE AND EWMA CHATS Muhammad iaz, Deparmen of Saisics, Quaid-e-Azam Universiy Islamabad, Pakisan. E-Mail: riaz76qau@yahoo.com Saddam Akbar Abbasi, Deparmen of Saisics, Quaid-e-Azam Universiy
More informationLicenciatura de ADE y Licenciatura conjunta Derecho y ADE. Hoja de ejercicios 2 PARTE A
Licenciaura de ADE y Licenciaura conjuna Derecho y ADE Hoja de ejercicios PARTE A 1. Consider he following models Δy = 0.8 + ε (1 + 0.8L) Δ 1 y = ε where ε and ε are independen whie noise processes. In
More information5. Stochastic processes (1)
Lec05.pp S-38.45 - Inroducion o Teleraffic Theory Spring 2005 Conens Basic conceps Poisson process 2 Sochasic processes () Consider some quaniy in a eleraffic (or any) sysem I ypically evolves in ime randomly
More information14 Autoregressive Moving Average Models
14 Auoregressive Moving Average Models In his chaper an imporan parameric family of saionary ime series is inroduced, he family of he auoregressive moving average, or ARMA, processes. For a large class
More informationCOMPUTATION OF THE PERFORMANCE OF SHEWHART CONTROL CHARTS. Pieter Mulder, Julian Morris and Elaine B. Martin
COMUTATION OF THE ERFORMANCE OF SHEWHART CONTROL CHARTS ieer Mulder, Julian Morris and Elaine B. Marin Cenre for rocess Analyics and Conrol Technology, School of Chemical Engineering and Advanced Maerials,
More informationVehicle Arrival Models : Headway
Chaper 12 Vehicle Arrival Models : Headway 12.1 Inroducion Modelling arrival of vehicle a secion of road is an imporan sep in raffic flow modelling. I has imporan applicaion in raffic flow simulaion where
More informationBias in Conditional and Unconditional Fixed Effects Logit Estimation: a Correction * Tom Coupé
Bias in Condiional and Uncondiional Fixed Effecs Logi Esimaion: a Correcion * Tom Coupé Economics Educaion and Research Consorium, Naional Universiy of Kyiv Mohyla Academy Address: Vul Voloska 10, 04070
More informationKriging Models Predicting Atrazine Concentrations in Surface Water Draining Agricultural Watersheds
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Kriging Models Predicing Arazine Concenraions in Surface Waer Draining Agriculural Waersheds Paul L. Mosquin, Jeremy Aldworh, Wenlin Chen Supplemenal Maerial Number
More informationPhysics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle
Chaper 2 Newonian Mechanics Single Paricle In his Chaper we will review wha Newon s laws of mechanics ell us abou he moion of a single paricle. Newon s laws are only valid in suiable reference frames,
More informationA Robust Exponentially Weighted Moving Average Control Chart for the Process Mean
Journal of Modern Applied Saisical Mehods Volume 5 Issue Aricle --005 A Robus Exponenially Weighed Moving Average Conrol Char for he Process Mean Michael B. C. Khoo Universii Sains, Malaysia, mkbc@usm.my
More informationRobust estimation based on the first- and third-moment restrictions of the power transformation model
h Inernaional Congress on Modelling and Simulaion, Adelaide, Ausralia, 6 December 3 www.mssanz.org.au/modsim3 Robus esimaion based on he firs- and hird-momen resricions of he power ransformaion Nawaa,
More information12: AUTOREGRESSIVE AND MOVING AVERAGE PROCESSES IN DISCRETE TIME. Σ j =
1: AUTOREGRESSIVE AND MOVING AVERAGE PROCESSES IN DISCRETE TIME Moving Averages Recall ha a whie noise process is a series { } = having variance σ. The whie noise process has specral densiy f (λ) = of
More informationStationary Time Series
3-Jul-3 Time Series Analysis Assoc. Prof. Dr. Sevap Kesel July 03 Saionary Time Series Sricly saionary process: If he oin dis. of is he same as he oin dis. of ( X,... X n) ( X h,... X nh) Weakly Saionary
More informationACE 562 Fall Lecture 5: The Simple Linear Regression Model: Sampling Properties of the Least Squares Estimators. by Professor Scott H.
ACE 56 Fall 005 Lecure 5: he Simple Linear Regression Model: Sampling Properies of he Leas Squares Esimaors by Professor Sco H. Irwin Required Reading: Griffihs, Hill and Judge. "Inference in he Simple
More informationNature Neuroscience: doi: /nn Supplementary Figure 1. Spike-count autocorrelations in time.
Supplemenary Figure 1 Spike-coun auocorrelaions in ime. Normalized auocorrelaion marices are shown for each area in a daase. The marix shows he mean correlaion of he spike coun in each ime bin wih he spike
More informationA Study of the Allan Variance for Constant-Mean Non-Stationary Processes
A Sudy of he Allan Variance for Consan-Mean Non-Saionary Processes Haoian Xu Séphane Guerrier Robero Molinari & Yuming Zhang arxiv:70207795v3 mahst] 22 Jun 207 Absrac The Allan Variance AV is a widely
More informationSTRUCTURAL CHANGE IN TIME SERIES OF THE EXCHANGE RATES BETWEEN YEN-DOLLAR AND YEN-EURO IN
Inernaional Journal of Applied Economerics and Quaniaive Sudies. Vol.1-3(004) STRUCTURAL CHANGE IN TIME SERIES OF THE EXCHANGE RATES BETWEEN YEN-DOLLAR AND YEN-EURO IN 001-004 OBARA, Takashi * Absrac The
More informationLecture Notes 2. The Hilbert Space Approach to Time Series
Time Series Seven N. Durlauf Universiy of Wisconsin. Basic ideas Lecure Noes. The Hilber Space Approach o Time Series The Hilber space framework provides a very powerful language for discussing he relaionship
More informationInstitute for Mathematical Methods in Economics. University of Technology Vienna. Singapore, May Manfred Deistler
MULTIVARIATE TIME SERIES ANALYSIS AND FORECASTING Manfred Deisler E O S Economerics and Sysems Theory Insiue for Mahemaical Mehods in Economics Universiy of Technology Vienna Singapore, May 2004 Inroducion
More informationFinancial Econometrics Jeffrey R. Russell Midterm Winter 2009 SOLUTIONS
Name SOLUTIONS Financial Economerics Jeffrey R. Russell Miderm Winer 009 SOLUTIONS You have 80 minues o complee he exam. Use can use a calculaor and noes. Try o fi all your work in he space provided. If
More informationSensors, Signals and Noise
Sensors, Signals and Noise COURSE OUTLINE Inroducion Signals and Noise: 1) Descripion Filering Sensors and associaed elecronics rv 2017/02/08 1 Noise Descripion Noise Waveforms and Samples Saisics of Noise
More informationThe Simple Linear Regression Model: Reporting the Results and Choosing the Functional Form
Chaper 6 The Simple Linear Regression Model: Reporing he Resuls and Choosing he Funcional Form To complee he analysis of he simple linear regression model, in his chaper we will consider how o measure
More informationACE 562 Fall Lecture 8: The Simple Linear Regression Model: R 2, Reporting the Results and Prediction. by Professor Scott H.
ACE 56 Fall 5 Lecure 8: The Simple Linear Regression Model: R, Reporing he Resuls and Predicion by Professor Sco H. Irwin Required Readings: Griffihs, Hill and Judge. "Explaining Variaion in he Dependen
More informationChapter 16. Regression with Time Series Data
Chaper 16 Regression wih Time Series Daa The analysis of ime series daa is of vial ineres o many groups, such as macroeconomiss sudying he behavior of naional and inernaional economies, finance economiss
More informationProperties of Autocorrelated Processes Economics 30331
Properies of Auocorrelaed Processes Economics 3033 Bill Evans Fall 05 Suppose we have ime series daa series labeled as where =,,3, T (he final period) Some examples are he dail closing price of he S&500,
More informationEcon107 Applied Econometrics Topic 7: Multicollinearity (Studenmund, Chapter 8)
I. Definiions and Problems A. Perfec Mulicollineariy Econ7 Applied Economerics Topic 7: Mulicollineariy (Sudenmund, Chaper 8) Definiion: Perfec mulicollineariy exiss in a following K-variable regression
More informationCONFIDENCE LIMITS AND THEIR ROBUSTNESS
CONFIDENCE LIMITS AND THEIR ROBUSTNESS Rajendran Raja Fermi Naional Acceleraor laboraory Baavia, IL 60510 Absrac Confidence limis are common place in physics analysis. Grea care mus be aken in heir calculaion
More informationOn Measuring Pro-Poor Growth. 1. On Various Ways of Measuring Pro-Poor Growth: A Short Review of the Literature
On Measuring Pro-Poor Growh 1. On Various Ways of Measuring Pro-Poor Growh: A Shor eview of he Lieraure During he pas en years or so here have been various suggesions concerning he way one should check
More informationExcel-Based Solution Method For The Optimal Policy Of The Hadley And Whittin s Exact Model With Arma Demand
Excel-Based Soluion Mehod For The Opimal Policy Of The Hadley And Whiin s Exac Model Wih Arma Demand Kal Nami School of Business and Economics Winson Salem Sae Universiy Winson Salem, NC 27110 Phone: (336)750-2338
More informationDistribution of Least Squares
Disribuion of Leas Squares In classic regression, if he errors are iid normal, and independen of he regressors, hen he leas squares esimaes have an exac normal disribuion, no jus asympoic his is no rue
More informationUnit Root Time Series. Univariate random walk
Uni Roo ime Series Univariae random walk Consider he regression y y where ~ iid N 0, he leas squares esimae of is: ˆ yy y y yy Now wha if = If y y hen le y 0 =0 so ha y j j If ~ iid N 0, hen y ~ N 0, he
More informationExponentially Weighted Moving Average (EWMA) Chart Based on Six Delta Initiatives
hps://doi.org/0.545/mjis.08.600 Exponenially Weighed Moving Average (EWMA) Char Based on Six Dela Iniiaives KALPESH S. TAILOR Deparmen of Saisics, M. K. Bhavnagar Universiy, Bhavnagar-36400 E-mail: kalpesh_lr@yahoo.co.in
More informationSPC Procedures for Monitoring Autocorrelated Processes
Qualiy Technology & Quaniaive Managemen Vol. 4, No. 4, pp. 501-540, 007 QTQM ICAQM 007 SPC Procedures for Monioring Auocorrelaed Processes S. Psarakis and G. E. A. Papaleonida Ahens Universiy of Economics
More informationChapter 5. Heterocedastic Models. Introduction to time series (2008) 1
Chaper 5 Heerocedasic Models Inroducion o ime series (2008) 1 Chaper 5. Conens. 5.1. The ARCH model. 5.2. The GARCH model. 5.3. The exponenial GARCH model. 5.4. The CHARMA model. 5.5. Random coefficien
More informationRegression with Time Series Data
Regression wih Time Series Daa y = β 0 + β 1 x 1 +...+ β k x k + u Serial Correlaion and Heeroskedasiciy Time Series - Serial Correlaion and Heeroskedasiciy 1 Serially Correlaed Errors: Consequences Wih
More informationGMM - Generalized Method of Moments
GMM - Generalized Mehod of Momens Conens GMM esimaion, shor inroducion 2 GMM inuiion: Maching momens 2 3 General overview of GMM esimaion. 3 3. Weighing marix...........................................
More informationChapter 11. Heteroskedasticity The Nature of Heteroskedasticity. In Chapter 3 we introduced the linear model (11.1.1)
Chaper 11 Heeroskedasiciy 11.1 The Naure of Heeroskedasiciy In Chaper 3 we inroduced he linear model y = β+β x (11.1.1) 1 o explain household expendiure on food (y) as a funcion of household income (x).
More informationLecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still.
Lecure - Kinemaics in One Dimension Displacemen, Velociy and Acceleraion Everyhing in he world is moving. Nohing says sill. Moion occurs a all scales of he universe, saring from he moion of elecrons in
More informationChapter 15. Time Series: Descriptive Analyses, Models, and Forecasting
Chaper 15 Time Series: Descripive Analyses, Models, and Forecasing Descripive Analysis: Index Numbers Index Number a number ha measures he change in a variable over ime relaive o he value of he variable
More informationForecasting optimally
I) ile: Forecas Evaluaion II) Conens: Evaluaing forecass, properies of opimal forecass, esing properies of opimal forecass, saisical comparison of forecas accuracy III) Documenaion: - Diebold, Francis
More informationI. Return Calculations (20 pts, 4 points each)
Universiy of Washingon Spring 015 Deparmen of Economics Eric Zivo Econ 44 Miderm Exam Soluions This is a closed book and closed noe exam. However, you are allowed one page of noes (8.5 by 11 or A4 double-sided)
More informationReferences are appeared in the last slide. Last update: (1393/08/19)
SYSEM IDEIFICAIO Ali Karimpour Associae Professor Ferdowsi Universi of Mashhad References are appeared in he las slide. Las updae: 0..204 393/08/9 Lecure 5 lecure 5 Parameer Esimaion Mehods opics o be
More informationMeasurement Error 1: Consequences Page 1. Definitions. For two variables, X and Y, the following hold: Expectation, or Mean, of X.
Measuremen Error 1: Consequences of Measuremen Error Richard Williams, Universiy of Nore Dame, hps://www3.nd.edu/~rwilliam/ Las revised January 1, 015 Definiions. For wo variables, X and Y, he following
More informationGeneralized Least Squares
Generalized Leas Squares Augus 006 1 Modified Model Original assumpions: 1 Specificaion: y = Xβ + ε (1) Eε =0 3 EX 0 ε =0 4 Eεε 0 = σ I In his secion, we consider relaxing assumpion (4) Insead, assume
More informationRichard A. Davis Colorado State University Bojan Basrak Eurandom Thomas Mikosch University of Groningen
Mulivariae Regular Variaion wih Applicaion o Financial Time Series Models Richard A. Davis Colorado Sae Universiy Bojan Basrak Eurandom Thomas Mikosch Universiy of Groningen Ouline + Characerisics of some
More informationWisconsin Unemployment Rate Forecast Revisited
Wisconsin Unemploymen Rae Forecas Revisied Forecas in Lecure Wisconsin unemploymen November 06 was 4.% Forecass Poin Forecas 50% Inerval 80% Inerval Forecas Forecas December 06 4.0% (4.0%, 4.0%) (3.95%,
More informationThe electromagnetic interference in case of onboard navy ships computers - a new approach
The elecromagneic inerference in case of onboard navy ships compuers - a new approach Prof. dr. ing. Alexandru SOTIR Naval Academy Mircea cel Bărân, Fulgerului Sree, Consanţa, soiralexandru@yahoo.com Absrac.
More informationACE 562 Fall Lecture 4: Simple Linear Regression Model: Specification and Estimation. by Professor Scott H. Irwin
ACE 56 Fall 005 Lecure 4: Simple Linear Regression Model: Specificaion and Esimaion by Professor Sco H. Irwin Required Reading: Griffihs, Hill and Judge. "Simple Regression: Economic and Saisical Model
More informationStability and Bifurcation in a Neural Network Model with Two Delays
Inernaional Mahemaical Forum, Vol. 6, 11, no. 35, 175-1731 Sabiliy and Bifurcaion in a Neural Nework Model wih Two Delays GuangPing Hu and XiaoLing Li School of Mahemaics and Physics, Nanjing Universiy
More informationRandom Processes 1/24
Random Processes 1/24 Random Process Oher Names : Random Signal Sochasic Process A Random Process is an exension of he concep of a Random variable (RV) Simples View : A Random Process is a RV ha is a Funcion
More informationECON 482 / WH Hong Time Series Data Analysis 1. The Nature of Time Series Data. Example of time series data (inflation and unemployment rates)
ECON 48 / WH Hong Time Series Daa Analysis. The Naure of Time Series Daa Example of ime series daa (inflaion and unemploymen raes) ECON 48 / WH Hong Time Series Daa Analysis The naure of ime series daa
More informationIntroduction D P. r = constant discount rate, g = Gordon Model (1962): constant dividend growth rate.
Inroducion Gordon Model (1962): D P = r g r = consan discoun rae, g = consan dividend growh rae. If raional expecaions of fuure discoun raes and dividend growh vary over ime, so should he D/P raio. Since
More information- The whole joint distribution is independent of the date at which it is measured and depends only on the lag.
Saionary Processes Sricly saionary - The whole join disribuion is indeenden of he dae a which i is measured and deends only on he lag. - E y ) is a finie consan. ( - V y ) is a finie consan. ( ( y, y s
More informationStochastic Structural Dynamics. Lecture-6
Sochasic Srucural Dynamics Lecure-6 Random processes- Dr C S Manohar Deparmen of Civil Engineering Professor of Srucural Engineering Indian Insiue of Science Bangalore 560 0 India manohar@civil.iisc.erne.in
More informationTesting the Random Walk Model. i.i.d. ( ) r
he random walk heory saes: esing he Random Walk Model µ ε () np = + np + Momen Condiions where where ε ~ i.i.d he idea here is o es direcly he resricions imposed by momen condiions. lnp lnp µ ( lnp lnp
More informationTime series Decomposition method
Time series Decomposiion mehod A ime series is described using a mulifacor model such as = f (rend, cyclical, seasonal, error) = f (T, C, S, e) Long- Iner-mediaed Seasonal Irregular erm erm effec, effec,
More informationDynamic Models, Autocorrelation and Forecasting
ECON 4551 Economerics II Memorial Universiy of Newfoundland Dynamic Models, Auocorrelaion and Forecasing Adaped from Vera Tabakova s noes 9.1 Inroducion 9.2 Lags in he Error Term: Auocorrelaion 9.3 Esimaing
More informationFailure of the work-hamiltonian connection for free energy calculations. Abstract
Failure of he work-hamilonian connecion for free energy calculaions Jose M. G. Vilar 1 and J. Miguel Rubi 1 Compuaional Biology Program, Memorial Sloan-Keering Cancer Cener, 175 York Avenue, New York,
More informationSTATE-SPACE MODELLING. A mass balance across the tank gives:
B. Lennox and N.F. Thornhill, 9, Sae Space Modelling, IChemE Process Managemen and Conrol Subjec Group Newsleer STE-SPACE MODELLING Inroducion: Over he pas decade or so here has been an ever increasing
More informationSample Autocorrelations for Financial Time Series Models. Richard A. Davis Colorado State University Thomas Mikosch University of Copenhagen
Sample Auocorrelaions for Financial Time Series Models Richard A. Davis Colorado Sae Universiy Thomas Mikosch Universiy of Copenhagen Ouline Characerisics of some financial ime series IBM reurns NZ-USA
More informationNotes on Kalman Filtering
Noes on Kalman Filering Brian Borchers and Rick Aser November 7, Inroducion Daa Assimilaion is he problem of merging model predicions wih acual measuremens of a sysem o produce an opimal esimae of he curren
More informationA Specification Test for Linear Dynamic Stochastic General Equilibrium Models
Journal of Saisical and Economeric Mehods, vol.1, no.2, 2012, 65-70 ISSN: 2241-0384 (prin), 2241-0376 (online) Scienpress Ld, 2012 A Specificaion Tes for Linear Dynamic Sochasic General Equilibrium Models
More informationHow to Deal with Structural Breaks in Practical Cointegration Analysis
How o Deal wih Srucural Breaks in Pracical Coinegraion Analysis Roselyne Joyeux * School of Economic and Financial Sudies Macquarie Universiy December 00 ABSTRACT In his noe we consider he reamen of srucural
More information18 Biological models with discrete time
8 Biological models wih discree ime The mos imporan applicaions, however, may be pedagogical. The elegan body of mahemaical heory peraining o linear sysems (Fourier analysis, orhogonal funcions, and so
More information4 Sequences of measurable functions
4 Sequences of measurable funcions 1. Le (Ω, A, µ) be a measure space (complee, afer a possible applicaion of he compleion heorem). In his chaper we invesigae relaions beween various (nonequivalen) convergences
More informationL07. KALMAN FILTERING FOR NON-LINEAR SYSTEMS. NA568 Mobile Robotics: Methods & Algorithms
L07. KALMAN FILTERING FOR NON-LINEAR SYSTEMS NA568 Mobile Roboics: Mehods & Algorihms Today s Topic Quick review on (Linear) Kalman Filer Kalman Filering for Non-Linear Sysems Exended Kalman Filer (EKF)
More informationOn Multicomponent System Reliability with Microshocks - Microdamages Type of Components Interaction
On Mulicomponen Sysem Reliabiliy wih Microshocks - Microdamages Type of Componens Ineracion Jerzy K. Filus, and Lidia Z. Filus Absrac Consider a wo componen parallel sysem. The defined new sochasic dependences
More informationTwo Coupled Oscillators / Normal Modes
Lecure 3 Phys 3750 Two Coupled Oscillaors / Normal Modes Overview and Moivaion: Today we ake a small, bu significan, sep owards wave moion. We will no ye observe waves, bu his sep is imporan in is own
More informationDEPARTMENT OF STATISTICS
A Tes for Mulivariae ARCH Effecs R. Sco Hacker and Abdulnasser Haemi-J 004: DEPARTMENT OF STATISTICS S-0 07 LUND SWEDEN A Tes for Mulivariae ARCH Effecs R. Sco Hacker Jönköping Inernaional Business School
More informationTesting for a Single Factor Model in the Multivariate State Space Framework
esing for a Single Facor Model in he Mulivariae Sae Space Framework Chen C.-Y. M. Chiba and M. Kobayashi Inernaional Graduae School of Social Sciences Yokohama Naional Universiy Japan Faculy of Economics
More informationEmpirical Process Theory
Empirical Process heory 4.384 ime Series Analysis, Fall 27 Reciaion by Paul Schrimpf Supplemenary o lecures given by Anna Mikusheva Ocober 7, 28 Reciaion 7 Empirical Process heory Le x be a real-valued
More informationInnova Junior College H2 Mathematics JC2 Preliminary Examinations Paper 2 Solutions 0 (*)
Soluion 3 x 4x3 x 3 x 0 4x3 x 4x3 x 4x3 x 4x3 x x 3x 3 4x3 x Innova Junior College H Mahemaics JC Preliminary Examinaions Paper Soluions 3x 3 4x 3x 0 4x 3 4x 3 0 (*) 0 0 + + + - 3 3 4 3 3 3 3 Hence x or
More informationSmoothing. Backward smoother: At any give T, replace the observation yt by a combination of observations at & before T
Smoohing Consan process Separae signal & noise Smooh he daa: Backward smooher: A an give, replace he observaion b a combinaion of observaions a & before Simple smooher : replace he curren observaion wih
More informationQuarterly ice cream sales are high each summer, and the series tends to repeat itself each year, so that the seasonal period is 4.
Seasonal models Many business and economic ime series conain a seasonal componen ha repeas iself afer a regular period of ime. The smalles ime period for his repeiion is called he seasonal period, and
More informationRandom Walk with Anti-Correlated Steps
Random Walk wih Ani-Correlaed Seps John Noga Dirk Wagner 2 Absrac We conjecure he expeced value of random walks wih ani-correlaed seps o be exacly. We suppor his conjecure wih 2 plausibiliy argumens and
More informationBifurcation Analysis of a Stage-Structured Prey-Predator System with Discrete and Continuous Delays
Applied Mahemaics 4 59-64 hp://dx.doi.org/.46/am..4744 Published Online July (hp://www.scirp.org/ournal/am) Bifurcaion Analysis of a Sage-Srucured Prey-Predaor Sysem wih Discree and Coninuous Delays Shunyi
More informationLecture 5. Time series: ECM. Bernardina Algieri Department Economics, Statistics and Finance
Lecure 5 Time series: ECM Bernardina Algieri Deparmen Economics, Saisics and Finance Conens Time Series Modelling Coinegraion Error Correcion Model Two Seps, Engle-Granger procedure Error Correcion Model
More informationExplaining Total Factor Productivity. Ulrich Kohli University of Geneva December 2015
Explaining Toal Facor Produciviy Ulrich Kohli Universiy of Geneva December 2015 Needed: A Theory of Toal Facor Produciviy Edward C. Presco (1998) 2 1. Inroducion Toal Facor Produciviy (TFP) has become
More informationBasic definitions and relations
Basic definiions and relaions Lecurer: Dmiri A. Molchanov E-mail: molchan@cs.u.fi hp://www.cs.u.fi/kurssi/tlt-2716/ Kendall s noaion for queuing sysems: Arrival processes; Service ime disribuions; Examples.
More informationVectorautoregressive Model and Cointegration Analysis. Time Series Analysis Dr. Sevtap Kestel 1
Vecorauoregressive Model and Coinegraion Analysis Par V Time Series Analysis Dr. Sevap Kesel 1 Vecorauoregression Vecor auoregression (VAR) is an economeric model used o capure he evoluion and he inerdependencies
More informationChapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
More informationCash Flow Valuation Mode Lin Discrete Time
IOSR Journal of Mahemaics (IOSR-JM) e-issn: 2278-5728,p-ISSN: 2319-765X, 6, Issue 6 (May. - Jun. 2013), PP 35-41 Cash Flow Valuaion Mode Lin Discree Time Olayiwola. M. A. and Oni, N. O. Deparmen of Mahemaics
More informationEchocardiography Project and Finite Fourier Series
Echocardiography Projec and Finie Fourier Series 1 U M An echocardiagram is a plo of how a porion of he hear moves as he funcion of ime over he one or more hearbea cycles If he hearbea repeas iself every
More informationChristos Papadimitriou & Luca Trevisan November 22, 2016
U.C. Bereley CS170: Algorihms Handou LN-11-22 Chrisos Papadimiriou & Luca Trevisan November 22, 2016 Sreaming algorihms In his lecure and he nex one we sudy memory-efficien algorihms ha process a sream
More informationChoice of Spectral Density Estimator in Ng-Perron Test: A Comparative Analysis
Inernaional Economeric Review (IER) Choice of Specral Densiy Esimaor in Ng-Perron Tes: A Comparaive Analysis Muhammad Irfan Malik and Aiq-ur-Rehman Inernaional Islamic Universiy Islamabad and Inernaional
More informationV AK (t) I T (t) I TRM. V AK( full area) (t) t t 1 Axial turn-on. Switching losses for Phase Control and Bi- Directionally Controlled Thyristors
Applicaion Noe Swiching losses for Phase Conrol and Bi- Direcionally Conrolled Thyrisors V AK () I T () Causing W on I TRM V AK( full area) () 1 Axial urn-on Plasma spread 2 Swiching losses for Phase Conrol
More informationLesson 2, page 1. Outline of lesson 2
Lesson 2, page Ouline of lesson 2 Inroduce he Auocorrelaion Coefficien Undersand and define saionariy Discuss ransformaion Discuss rend and rend removal C:\Kyrre\sudier\drgrad\Kurs\series\lecure 02 03022.doc,
More informationInventory Control of Perishable Items in a Two-Echelon Supply Chain
Journal of Indusrial Engineering, Universiy of ehran, Special Issue,, PP. 69-77 69 Invenory Conrol of Perishable Iems in a wo-echelon Supply Chain Fariborz Jolai *, Elmira Gheisariha and Farnaz Nojavan
More informationPhysics 127b: Statistical Mechanics. Fokker-Planck Equation. Time Evolution
Physics 7b: Saisical Mechanics Fokker-Planck Equaion The Langevin equaion approach o he evoluion of he velociy disribuion for he Brownian paricle migh leave you uncomforable. A more formal reamen of his
More informationBox-Jenkins Modelling of Nigerian Stock Prices Data
Greener Journal of Science Engineering and Technological Research ISSN: 76-7835 Vol. (), pp. 03-038, Sepember 0. Research Aricle Box-Jenkins Modelling of Nigerian Sock Prices Daa Ee Harrison Euk*, Barholomew
More informationState-Space Models. Initialization, Estimation and Smoothing of the Kalman Filter
Sae-Space Models Iniializaion, Esimaion and Smoohing of he Kalman Filer Iniializaion of he Kalman Filer The Kalman filer shows how o updae pas predicors and he corresponding predicion error variances when
More informationModeling and Forecasting Volatility Autoregressive Conditional Heteroskedasticity Models. Economic Forecasting Anthony Tay Slide 1
Modeling and Forecasing Volailiy Auoregressive Condiional Heeroskedasiciy Models Anhony Tay Slide 1 smpl @all line(m) sii dl_sii S TII D L _ S TII 4,000. 3,000.1.0,000 -.1 1,000 -. 0 86 88 90 9 94 96 98
More informationLecture 33: November 29
36-705: Inermediae Saisics Fall 2017 Lecurer: Siva Balakrishnan Lecure 33: November 29 Today we will coninue discussing he boosrap, and hen ry o undersand why i works in a simple case. In he las lecure
More information( ) ( ) if t = t. It must satisfy the identity. So, bulkiness of the unit impulse (hyper)function is equal to 1. The defining characteristic is
UNIT IMPULSE RESPONSE, UNIT STEP RESPONSE, STABILITY. Uni impulse funcion (Dirac dela funcion, dela funcion) rigorously defined is no sricly a funcion, bu disribuion (or measure), precise reamen requires
More information