Diagnosing Problems of Distribution-Free Multivariate Control Chart
|
|
- Ralf Dennis
- 5 years ago
- Views:
Transcription
1 Advaced Materals Research Ole: ISSN: , Vols , 6-66 do:.48/ 4 ras ech Publcatos, Swtzerlad Dagosg Problems of Dstrbuto-Free Multvarate Cotrol Chart Wel Sh, a ad Xuem Z,b, a Uversty of echology ad Educato, a, P. R. Cha a shwel43@63.com.c, b z_xuem@alyu.com.c Keywords: Dstrbuto-Free; Multvarate Statstcal Process Cotrol; Fault Dagoss; LASSO Abstract. I order to solve the roblem of oly have a few hstorcal data that ca be used multvarate rocess motorg, a ew dstrbuto-free multvarate cotrol chart has bee roosed. Ad the cotrol chart structure the cotrol lmts are determed o-le wth the future observatos ad the hstorcal data. herefore, the roosed cotrol chart has very mortat alcato ractce. However, the research does t study the roblem of the fault dagoss after the cotrol chart alarms. So we use LASSO-based dagostc framewor to detfy whe a detected shft has occurred ad to solate the shfted comoets. Itroducto I moder qualty cotrol, t s very commo to motor several qualty characterstcs a rocess smultaeously []. hs motorg ad dagostc about multvarate rocess observatos s ofte called multvarate statstcal rocess cotrol (referred to as MSPC). Because the basc tas of MSPC cludes: Frstly, determg whether statstcal rocess has chaged, that s, whether a sgal occurred cotrol chart. Secod, detfyg the locato of shft occurred ad solatg the comoets of shfted. I the recet research, Che, Z & Zou reseted a dstrbuto-free multvarate cotrol chart []. herefore, the urose of ths aer based o the ther research s comletg the secod tas, amely the fault dagoss or fault detfcato of the dstrbuto-free multvarate cotrol chart. I the recet research, for the dagosg roblems MSPC, Zou, Jag & sug have rased a ufed LASSO-based dagostc framewor [3]. he ma urose of the dagoss s to determe the ma chaged arameters, uder the stuato that other MSPC methods have used to be detected ad have estmated the chage-ot correctly. I ths case, the chage-ot arttos the observatos to two subsets wth dfferet arameter values. Hece the fault solato as two samle selecto roblem to combe Bayesa formato crtero (referred to as BIC)wth a ealzed techque to facltate the fault tracg rocess ad suggest a ractcal LASSO-based dagostc rocedure [4,5]. Wth the method above we ca fd the arameters that are resosble for the chage. herefore, the cotet of ths aer s that usg based o LASSO dagostc framewor to comlete the ost-sgal dagostc aalyss, whch s based o dstrbuto-free multvarate cotrol chart, so that ths cotrol chart has a better alcato ractce. he rest cotet of ths aer s as followed: Secto descrbes dstrbuto-free multvarate cotrol chart. Secto 3 rovdes LASSO-based dagostc method. Secto 4 develos the ost-sgal dagostc aalyss of dstrbuto-free multvarate cotrol chart, whch s the focus of ths aer. Secto 5 gves the coclusos of the aer. Dstrbuto-free Multvarate Cotrol Chart Suose that there are m deedet ad detcally dstrbuted hstorcal observatos, x m,... + x R,where are some teger.ad suose the th future observato x = ( x,..., x ) s collected from the follow multvarate locato chage-ot model All rghts reserved. No art of cotets of ths aer may be reroduced or trasmtted ay form or by ay meas wthout the wrtte ermsso of ras ech Publcatos, (#69868, Pesylvaa State Uversty, Uversty Par, USA-8/9/6,:54:8)
2 Advaced Materals Research Vols x F ( x, µ ), = m +,...,,,..., τ F ( x, µ ), = τ +,... () whereτ s the shft ot, F ad F are the -cotrol ad out-of-cotrol dstrbuto fuctos. I ractcal alcatos, they may be the same or may be dfferet, but for ther locato arameters µ = ( µ,... µ ) ad µ = ( µ,... µ ), we ofte assume that they are uequal. Cosdered the motorg roblem () s close to the locato of the two samle arameter hyothess test of the roblem. herefore, we use oarametrc two-samle test methods to cosder + ad { x,..., x} as F ( x ; µ ) ad (, ) ths roblem. Suose { x m,..., x τ } τ + are two deedet samles, ad dstrbuted F x µ. he cosder the ull hyothess test, H : µ µ = versus H : µ µ. Because the ull hyothess µ = µ s equvalet to the µ = µ,where =,.... So, t s straghtforward to cosder the Wlcoxo ra-sum test for each comoet, = = τ +, () τ r ( τ )( + ) ( + )( τ ) Where r s the ra of x amog the samle. hrough the above dscusso to costruct a EWMA cotrol chart ad use ths cotrol chart to motor the model (). Let X, = ( x,..., x ),ad X ( X,,... X, ) =. Choose a smoothg arameter λ ad a wdow sze ω.at each ot, costruct a chartg statstcs ω, λ = ω, λ, where = ( ω, λ) = ( = ω+ r ( m + + ) ( + m ω )( m + + ), (3) ω Smlarly, r s the ra of x amog the samle X +. Suose τ s the locato of chage ot, m, the ( ω, λ) would become a large value for > τ, ad corresodgly would become a large value, ad the caused a alarm. I order to costruct a comlete cotrol chart after the costructo of the statstc, aother mortat tas s to determe the cotrol lmts. hs ew method roose to determe the cotrol lmts C( α ) by solvg the followg equatos, ( ω λ > C α F) = α ( ) Pr,, Pr ω, λ > C ( α) ( ω, λ) < C α, <, F = α, >, where α s the re-secfed false alarm rate, F = τ τ I( x ) t, ad = F ( t ) ( m = + ) I ( x t ). So ths cotrol chart costtuted by the ad C( α ), termed as = m + dstrbuto-free multvarate EWMA chart(referred to as DFEWMA). he LASSO-based EBIC Dagoss Method I ths secto, we wll troduce the method we use ths aer, amely the LASSO-based EBIC dagoss method [3]. I may alcatos, all varables shft at the same tme s very rare, ad the umber of smultaeously chaged varables s relatvely small. herefore, we suose µ = µ + δ, (4)
3 64 New echologes for Egeerg Research ad Desg Idustry whereδ = ( δ,..., δ ). We geerally suose that the maorty comoets are zero, whch s called sarse features [6]. So the fault dagoss s essetally smlar to the model or varable selecto roblem, that s, oe wshes to choose those values that devate sgfcatly from δ. Because of Schwarz s Bayesa formato crtero teds to better detfy the true sarse model, so we wll cosder the dagostc ad wth BIC [4]. However, whe the model sace s large but the samle sze s moderate, the ordary BIC s somewhat lberal for model selecto. herefore, we wll cosder tae advatage of the exteded famly of BIC (referred to as EBIC) [7]. Accordg to a heurstc dervato rovded by Zou & Qu [3], the defto of EBIC s EBICs = f ( δ s ) + l + l s, (5) + Where f ( δ s ) = ( µ µ δ s ) ( Σ + Σ) ( µ µ δ s ), ad s s a caddate model, whch cotas the corresodg dex of chaged arameters. I ractcal alcatos, whe the dmeso s large, usg the (5) to calculate all the values of the EBIC s mossble. herefore, ths method combed some ealzed techques wth the EBIC. Next, we wll cosder the ealzed loss fucto, ( δ ; ) ( δ ) ( δ ) PL = f + g, (6) = Where (,..., ) g s the ealty fucto. By usg the adatve LASSO [8], the ealzed loss fucto coverted to APL = are the ealty arameters, ( ; ) f = δ = δ + δ, (7) herefore, for a gve, model δ ca be obtaed by mmzg ( ; ) s = { : δ }. By substtutg δ to (5) we wll get APL δ. hus we get a caddate l EBIC = f δ + + l. (8) + Where reresets the umber of o-zero values r δ. Let = δ, where r > s some gve costat, e.g..5 or, ad δ = µ µ, the the ealzed loss fucto (7) coverted to ( ; ) = PL α = δ α Γ δ α + α. (9) Where r r r α = δ δ, = dag( δ,..., δ ),ad Γ = Ω + Ω. herefore, wth the hel of LARS algorthm[9], the dagostc results ca be easly obtaed. hs dagostc method s dvded to three stes: ().Fd corresodg estmates of µ ad covarace matrx Σ, where =, determe f ( δ ). ().Use the LARS algorthm to solve (9), the obta ALASSO solutos. (3).Substtute these solutos to (8), ad fd a δ m dagostc corresodg result s s = { : δ m }. m. hus we so that EBIC value s the smallest. he the
4 Advaced Materals Research Vols he Post-Sgal Dagoss of DFEWMA Chart Che, Z & Zou (4) roosed the method of dstrbuto-free multvarate cotrol chart, but ther study dd ot clude the ost-sgal dagoss of the DFEWMA chart [], so ths art of aer wll study the fault dagoss of DFEWMA based o ther research. Frstly, we suose that the cotrol chart sgals at th observato the we wll have m hstorcal IC observatos ad ew observatos. Now, we assume that a shft occurs after the τ th samle ( τ < ), the gve a estmate of shft locato v, we ca calculate the oarametrc test statstc ( v, arg max τ < (,.herefore, we ca get the estmator of the shft locato,τ, s τ = v. () After the shft locato s estmated, the chage-otτ wll dvded the observatos to two X τ m + X τ + samles, ad. herefore, we ca use the above-metoed LASSO-based EBIC dagoss rocedure [3], by dagostc aalyss of chage-ot to determe whch comoets have chaged. Suose that the mea vector are µ, the covarace matrx are Σ, where =,. he f δ x x δ x x δ = ( ) Σ + Σ ( ), () Where x ad Σ are the estmates of the mea vector ad covarace matrx for the th samle. herefore, we ca use the above-metoed basc stes to dagose the locatoτ of the shft, ad obta dagostc results s = { : δ }. Next we show the results of our research through data smulato. Here we use a multvarate ormal dstrbuto, the mea vector µ s set to zero vector ad the covarace matrx s ( σ ) Σ =,where σ = ad.5 σ =. Now we are gog to cosder the followg case: x =, 3 + δ s used as a out of cotrol model whe x m =, =., α =. 5 > τ, where δ s shft vector. I our smulato, λ ad let chaged locato at τ = 55. So we frst get a DFEWMA cotrol chart show Fgure. 6 DFEWMA sgal 统统量 Samle Idex Fgure.DFEWMA chart. he blac dot reresets the ( ω,, the red dot reresets the C ( α ), ad the blue sgal ot s = v Fgure. he values of ( v, v <. for
5 66 New echologes for Egeerg Research ad Desg Idustry After the cotrol chart sgals at = 6, the we go to comlete the ost-sgal dagoss. Frst, we v, λ reaches the maxmum use () to fd the chage-otτ. From Fgure we ca see, 57. whe v = 55. So usg () we ca get a very accurate estmate of the chage-ot τ, that s τ = 55. Now, we have = 8 ad = 6. he, we use the LASSO-based EBIC dagoss method to detfy the chaged arameters. Our results are showed able. hese values of EBIC dcates the shft may have occurred the four comoets. herefore, we ca get the dagostc result of DFEWMA cotrol chart s s = {, 7, 9,}. able. Dagostc results of the LASSO-based EBIC dagoss method about DFEWMA chart EBIC δ Cocluso hs aer based o the study of dstrbuto-free multvarate cotrol chart, usg the LASSO-based EBIC dagoss method to comlete the fault dagostc aalyss of DFEWMA cotrol chart. Whe the DFEWMA cotrol chart sgals, we through the dagostc aalyss ca accurately determe the locato of the shft, ad detfy the ma elemets of the shft. I ractcal alcatos, ths wll be able to hel busess maagers ad egeers to detfy ad elmate the root causes of fault qucly ad accurately, so that ths cotrol chart has better alcato. Acowledgemet hs research was suorted by the NNSF of Cha Grats 36. Refereces [] Woodall, W. H. ad Motgomery, D. Some Curret Drectos the heory ad Alcato of Statstcal Process Motorg, Joural of Qualty echology, to aear. (3) [] Na Che,Xuem Z ad Chaglag Zou.A Dstrbuto-free Multvarate Cotrol Chart(4) [3] Zou, C., Jag, W., ad sug, F. A Lasso-Based Dagostc Framewor for Multvarate Statstcal Process Cotrol, echometrcs, 53, () [4] Schwarz, G. Estmatg the Dmeso of a Model, he Aals of Statstcs, 6, (978) [5] bshra, R. J. Regresso Shrage ad Selecto va the LASSO, Joural of the Royal Statstcal Socety: Seres B, 58, (996) [6] Zou, C., ad Qu, P. Multvarate Statstcal Process Cotrol Usg LASSO, Joural of the Amerca Statstcal Assocato, 4, (9) [7] Che, J., ad Che, Z. Exteded Bayesa Iformato Crtero for Model Selecto wth Large Model Saces, Bometra, 95, (8) [8] Zou, H. he Adatve Lasso ad Its Oracle Proertes, Joural of the Amerca Statstcal Assocato,, (6) [9] Efro, B., Haste,., Johstoe, I., ad bshra, R. Least Agle Regresso, he Aals of Statstcs, 3, (4)
STRONG CONSISTENCY FOR SIMPLE LINEAR EV MODEL WITH v/ -MIXING
Joural of tatstcs: Advaces Theory ad Alcatos Volume 5, Number, 6, Pages 3- Avalable at htt://scetfcadvaces.co. DOI: htt://d.do.org/.864/jsata_7678 TRONG CONITENCY FOR IMPLE LINEAR EV MODEL WITH v/ -MIXING
More informationFunctions of Random Variables
Fuctos of Radom Varables Chapter Fve Fuctos of Radom Varables 5. Itroducto A geeral egeerg aalyss model s show Fg. 5.. The model output (respose) cotas the performaces of a system or product, such as weght,
More informationSTK3100 and STK4100 Autumn 2018
SK3 ad SK4 Autum 8 Geeralzed lear models Part III Covers the followg materal from chaters 4 ad 5: Cofdece tervals by vertg tests Cosder a model wth a sgle arameter β We may obta a ( α% cofdece terval for
More informationSTK3100 and STK4100 Autumn 2017
SK3 ad SK4 Autum 7 Geeralzed lear models Part III Covers the followg materal from chaters 4 ad 5: Sectos 4..5, 4.3.5, 4.3.6, 4.4., 4.4., ad 4.4.3 Sectos 5.., 5.., ad 5.5. Ørulf Borga Deartmet of Mathematcs
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Aalyss of Varace ad Desg of Exermets-I MODULE II LECTURE - GENERAL LINEAR HYPOTHESIS AND ANALYSIS OF VARIANCE Dr Shalabh Deartmet of Mathematcs ad Statstcs Ida Isttute of Techology Kaur Tukey s rocedure
More informationApplication of Calibration Approach for Regression Coefficient Estimation under Two-stage Sampling Design
Authors: Pradp Basak, Kaustav Adtya, Hukum Chadra ad U.C. Sud Applcato of Calbrato Approach for Regresso Coeffcet Estmato uder Two-stage Samplg Desg Pradp Basak, Kaustav Adtya, Hukum Chadra ad U.C. Sud
More information2. Independence and Bernoulli Trials
. Ideedece ad Beroull Trals Ideedece: Evets ad B are deedet f B B. - It s easy to show that, B deedet mles, B;, B are all deedet ars. For examle, ad so that B or B B B B B φ,.e., ad B are deedet evets.,
More informationAnalysis of Variance with Weibull Data
Aalyss of Varace wth Webull Data Lahaa Watthaacheewaul Abstract I statstcal data aalyss by aalyss of varace, the usual basc assumptos are that the model s addtve ad the errors are radomly, depedetly, ad
More informationCS 2750 Machine Learning Lecture 5. Density estimation. Density estimation
CS 750 Mache Learg Lecture 5 esty estmato Mlos Hausrecht mlos@tt.edu 539 Seott Square esty estmato esty estmato: s a usuervsed learg roblem Goal: Lear a model that rereset the relatos amog attrbutes the
More informationRecursive linear estimation for discrete time systems in the presence of different multiplicative observation noises
Recursve lear estmato for dscrete tme systems the resece of dfferet multlcatve observato oses C. Sáchez Gozález,*,.M. García Muñoz Deartameto de Métodos Cuattatvos ara la Ecoomía y la Emresa, Facultad
More information2SLS Estimates ECON In this case, begin with the assumption that E[ i
SLS Estmates ECON 3033 Bll Evas Fall 05 Two-Stage Least Squares (SLS Cosder a stadard lear bvarate regresso model y 0 x. I ths case, beg wth the assumto that E[ x] 0 whch meas that OLS estmates of wll
More informationESS Line Fitting
ESS 5 014 17. Le Fttg A very commo problem data aalyss s lookg for relatoshpetwee dfferet parameters ad fttg les or surfaces to data. The smplest example s fttg a straght le ad we wll dscuss that here
More informationOrdinary Least Squares Regression. Simple Regression. Algebra and Assumptions.
Ordary Least Squares egresso. Smple egresso. Algebra ad Assumptos. I ths part of the course we are gog to study a techque for aalysg the lear relatoshp betwee two varables Y ad X. We have pars of observatos
More informationCHAPTER VI Statistical Analysis of Experimental Data
Chapter VI Statstcal Aalyss of Expermetal Data CHAPTER VI Statstcal Aalyss of Expermetal Data Measuremets do ot lead to a uque value. Ths s a result of the multtude of errors (maly radom errors) that ca
More informationSummary of the lecture in Biostatistics
Summary of the lecture Bostatstcs Probablty Desty Fucto For a cotuos radom varable, a probablty desty fucto s a fucto such that: 0 dx a b) b a dx A probablty desty fucto provdes a smple descrpto of the
More informationD KL (P Q) := p i ln p i q i
Cheroff-Bouds 1 The Geeral Boud Let P 1,, m ) ad Q q 1,, q m ) be two dstrbutos o m elemets, e,, q 0, for 1,, m, ad m 1 m 1 q 1 The Kullback-Lebler dvergece or relatve etroy of P ad Q s defed as m D KL
More informationA New Family of Transformations for Lifetime Data
Proceedgs of the World Cogress o Egeerg 4 Vol I, WCE 4, July - 4, 4, Lodo, U.K. A New Famly of Trasformatos for Lfetme Data Lakhaa Watthaacheewakul Abstract A famly of trasformatos s the oe of several
More informationUNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS
UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Postpoed exam: ECON430 Statstcs Date of exam: Jauary 0, 0 Tme for exam: 09:00 a.m. :00 oo The problem set covers 5 pages Resources allowed: All wrtte ad prted
More informationIntroduction to local (nonparametric) density estimation. methods
Itroducto to local (oparametrc) desty estmato methods A slecture by Yu Lu for ECE 66 Sprg 014 1. Itroducto Ths slecture troduces two local desty estmato methods whch are Parze desty estmato ad k-earest
More informationNon-uniform Turán-type problems
Joural of Combatoral Theory, Seres A 111 2005 106 110 wwwelsevercomlocatecta No-uform Turá-type problems DhruvMubay 1, Y Zhao 2 Departmet of Mathematcs, Statstcs, ad Computer Scece, Uversty of Illos at
More informationPTAS for Bin-Packing
CS 663: Patter Matchg Algorthms Scrbe: Che Jag /9/00. Itroducto PTAS for B-Packg The B-Packg problem s NP-hard. If we use approxmato algorthms, the B-Packg problem could be solved polyomal tme. For example,
More informationThe number of observed cases The number of parameters. ith case of the dichotomous dependent variable. the ith case of the jth parameter
LOGISTIC REGRESSION Notato Model Logstc regresso regresses a dchotomous depedet varable o a set of depedet varables. Several methods are mplemeted for selectg the depedet varables. The followg otato s
More informationFeature Selection: Part 2. 1 Greedy Algorithms (continued from the last lecture)
CSE 546: Mache Learg Lecture 6 Feature Selecto: Part 2 Istructor: Sham Kakade Greedy Algorthms (cotued from the last lecture) There are varety of greedy algorthms ad umerous amg covetos for these algorthms.
More informationON BIVARIATE GEOMETRIC DISTRIBUTION. K. Jayakumar, D.A. Mundassery 1. INTRODUCTION
STATISTICA, ao LXVII, 4, 007 O BIVARIATE GEOMETRIC DISTRIBUTIO ITRODUCTIO Probablty dstrbutos of radom sums of deedetly ad detcally dstrbuted radom varables are maly aled modelg ractcal roblems that deal
More informationA Combination of Adaptive and Line Intercept Sampling Applicable in Agricultural and Environmental Studies
ISSN 1684-8403 Joural of Statstcs Volume 15, 008, pp. 44-53 Abstract A Combato of Adaptve ad Le Itercept Samplg Applcable Agrcultural ad Evrometal Studes Azmer Kha 1 A adaptve procedure s descrbed for
More informationAnalysis of Lagrange Interpolation Formula
P IJISET - Iteratoal Joural of Iovatve Scece, Egeerg & Techology, Vol. Issue, December 4. www.jset.com ISS 348 7968 Aalyss of Lagrage Iterpolato Formula Vjay Dahya PDepartmet of MathematcsMaharaja Surajmal
More informationOn the characteristics of partial differential equations
Sur les caractérstques des équatos au dérvées artelles Bull Soc Math Frace 5 (897) 8- O the characterstcs of artal dfferetal equatos By JULES BEUDON Traslated by D H Delhech I a ote that was reseted to
More informationSTK4011 and STK9011 Autumn 2016
STK4 ad STK9 Autum 6 Pot estmato Covers (most of the followg materal from chapter 7: Secto 7.: pages 3-3 Secto 7..: pages 3-33 Secto 7..: pages 35-3 Secto 7..3: pages 34-35 Secto 7.3.: pages 33-33 Secto
More informationSimple Linear Regression
Statstcal Methods I (EST 75) Page 139 Smple Lear Regresso Smple regresso applcatos are used to ft a model descrbg a lear relatoshp betwee two varables. The aspects of least squares regresso ad correlato
More informationLecture 9. Some Useful Discrete Distributions. Some Useful Discrete Distributions. The observations generated by different experiments have
NM 7 Lecture 9 Some Useful Dscrete Dstrbutos Some Useful Dscrete Dstrbutos The observatos geerated by dfferet eermets have the same geeral tye of behavor. Cosequetly, radom varables assocated wth these
More informationMedian as a Weighted Arithmetic Mean of All Sample Observations
Meda as a Weghted Arthmetc Mea of All Sample Observatos SK Mshra Dept. of Ecoomcs NEHU, Shllog (Ida). Itroducto: Iumerably may textbooks Statstcs explctly meto that oe of the weakesses (or propertes) of
More informationBayes Estimator for Exponential Distribution with Extension of Jeffery Prior Information
Malaysa Joural of Mathematcal Sceces (): 97- (9) Bayes Estmator for Expoetal Dstrbuto wth Exteso of Jeffery Pror Iformato Hadeel Salm Al-Kutub ad Noor Akma Ibrahm Isttute for Mathematcal Research, Uverst
More informationCHAPTER 6. d. With success = observation greater than 10, x = # of successes = 4, and
CHAPTR 6 Secto 6.. a. We use the samle mea, to estmate the oulato mea µ. Σ 9.80 µ 8.407 7 ~ 7. b. We use the samle meda, 7 (the mddle observato whe arraged ascedg order. c. We use the samle stadard devato,
More informationLINEAR REGRESSION ANALYSIS
LINEAR REGRESSION ANALYSIS MODULE V Lecture - Correctg Model Iadequaces Through Trasformato ad Weghtg Dr. Shalabh Departmet of Mathematcs ad Statstcs Ida Isttute of Techology Kapur Aalytcal methods for
More informationComparing Different Estimators of three Parameters for Transmuted Weibull Distribution
Global Joural of Pure ad Appled Mathematcs. ISSN 0973-768 Volume 3, Number 9 (207), pp. 55-528 Research Ida Publcatos http://www.rpublcato.com Comparg Dfferet Estmators of three Parameters for Trasmuted
More informationDerivation of 3-Point Block Method Formula for Solving First Order Stiff Ordinary Differential Equations
Dervato of -Pot Block Method Formula for Solvg Frst Order Stff Ordary Dfferetal Equatos Kharul Hamd Kharul Auar, Kharl Iskadar Othma, Zara Bb Ibrahm Abstract Dervato of pot block method formula wth costat
More informationSolving Constrained Flow-Shop Scheduling. Problems with Three Machines
It J Cotemp Math Sceces, Vol 5, 2010, o 19, 921-929 Solvg Costraed Flow-Shop Schedulg Problems wth Three Maches P Pada ad P Rajedra Departmet of Mathematcs, School of Advaced Sceces, VIT Uversty, Vellore-632
More informationPoint Estimation: definition of estimators
Pot Estmato: defto of estmators Pot estmator: ay fucto W (X,..., X ) of a data sample. The exercse of pot estmato s to use partcular fuctos of the data order to estmate certa ukow populato parameters.
More informationbest estimate (mean) for X uncertainty or error in the measurement (systematic, random or statistical) best
Error Aalyss Preamble Wheever a measuremet s made, the result followg from that measuremet s always subject to ucertaty The ucertaty ca be reduced by makg several measuremets of the same quatty or by mprovg
More informationTESTS BASED ON MAXIMUM LIKELIHOOD
ESE 5 Toy E. Smth. The Basc Example. TESTS BASED ON MAXIMUM LIKELIHOOD To llustrate the propertes of maxmum lkelhood estmates ad tests, we cosder the smplest possble case of estmatg the mea of the ormal
More informationå 1 13 Practice Final Examination Solutions - = CS109 Dec 5, 2018
Chrs Pech Fal Practce CS09 Dec 5, 08 Practce Fal Examato Solutos. Aswer: 4/5 8/7. There are multle ways to obta ths aswer; here are two: The frst commo method s to sum over all ossbltes for the rak of
More informationTHE ROYAL STATISTICAL SOCIETY 2016 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE MODULE 5
THE ROYAL STATISTICAL SOCIETY 06 EAMINATIONS SOLUTIONS HIGHER CERTIFICATE MODULE 5 The Socety s provdg these solutos to assst cadtes preparg for the examatos 07. The solutos are teded as learg ads ad should
More informationA tighter lower bound on the circuit size of the hardest Boolean functions
Electroc Colloquum o Computatoal Complexty, Report No. 86 2011) A tghter lower boud o the crcut sze of the hardest Boolea fuctos Masak Yamamoto Abstract I [IPL2005], Fradse ad Mlterse mproved bouds o the
More information{ }{ ( )} (, ) = ( ) ( ) ( ) Chapter 14 Exercises in Sampling Theory. Exercise 1 (Simple random sampling): Solution:
Chapter 4 Exercses Samplg Theory Exercse (Smple radom samplg: Let there be two correlated radom varables X ad A sample of sze s draw from a populato by smple radom samplg wthout replacemet The observed
More informationLogistic regression (continued)
STAT562 page 138 Logstc regresso (cotued) Suppose we ow cosder more complex models to descrbe the relatoshp betwee a categorcal respose varable (Y) that takes o two (2) possble outcomes ad a set of p explaatory
More informationChannel Models with Memory. Channel Models with Memory. Channel Models with Memory. Channel Models with Memory
Chael Models wth Memory Chael Models wth Memory Hayder radha Electrcal ad Comuter Egeerg Mchga State Uversty I may ractcal etworkg scearos (cludg the Iteret ad wreless etworks), the uderlyg chaels are
More informationLecture Note to Rice Chapter 8
ECON 430 HG revsed Nov 06 Lecture Note to Rce Chapter 8 Radom matrces Let Y, =,,, m, =,,, be radom varables (r.v. s). The matrx Y Y Y Y Y Y Y Y Y Y = m m m s called a radom matrx ( wth a ot m-dmesoal dstrbuto,
More informationLecture 07: Poles and Zeros
Lecture 07: Poles ad Zeros Defto of poles ad zeros The trasfer fucto provdes a bass for determg mportat system respose characterstcs wthout solvg the complete dfferetal equato. As defed, the trasfer fucto
More informationComparison of Parameters of Lognormal Distribution Based On the Classical and Posterior Estimates
Joural of Moder Appled Statstcal Methods Volume Issue Artcle 8 --03 Comparso of Parameters of Logormal Dstrbuto Based O the Classcal ad Posteror Estmates Raja Sulta Uversty of Kashmr, Sragar, Ida, hamzasulta8@yahoo.com
More informationChapter 5 Properties of a Random Sample
Lecture 3 o BST 63: Statstcal Theory I Ku Zhag, /6/006 Revew for the revous lecture Cocets: radom samle, samle mea, samle varace Theorems: roertes of a radom samle, samle mea, samle varace Examles: how
More informationFault Diagnosis Using Feature Vectors and Fuzzy Fault Pattern Rulebase
Fault Dagoss Usg Feature Vectors ad Fuzzy Fault Patter Rulebase Prepared by: FL Lews Updated: Wedesday, ovember 03, 004 Feature Vectors The requred puts for the dagostc models are termed the feature vectors
More informationA New Method for Solving Fuzzy Linear. Programming by Solving Linear Programming
ppled Matheatcal Sceces Vol 008 o 50 7-80 New Method for Solvg Fuzzy Lear Prograg by Solvg Lear Prograg S H Nasser a Departet of Matheatcs Faculty of Basc Sceces Mazadara Uversty Babolsar Ira b The Research
More informationELEC 6041 LECTURE NOTES WEEK 1 Dr. Amir G. Aghdam Concordia University
ELEC 604 LECTURE NOTES WEEK Dr mr G ghdam Cocorda Uverst Itroducto - Large-scale sstems are the mult-ut mult-outut (MIMO) sstems cosstg of geograhcall searated comoets - Eamles of large-scale sstems clude
More informationBeam Warming Second-Order Upwind Method
Beam Warmg Secod-Order Upwd Method Petr Valeta Jauary 6, 015 Ths documet s a part of the assessmet work for the subject 1DRP Dfferetal Equatos o Computer lectured o FNSPE CTU Prague. Abstract Ths documet
More information9.1 Introduction to the probit and logit models
EC3000 Ecoometrcs Lecture 9 Probt & Logt Aalss 9. Itroducto to the probt ad logt models 9. The logt model 9.3 The probt model Appedx 9. Itroducto to the probt ad logt models These models are used regressos
More informationUNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS
UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Exam: ECON430 Statstcs Date of exam: Frday, December 8, 07 Grades are gve: Jauary 4, 08 Tme for exam: 0900 am 00 oo The problem set covers 5 pages Resources allowed:
More informationVOL. 3, NO. 11, November 2013 ISSN ARPN Journal of Science and Technology All rights reserved.
VOL., NO., November 0 ISSN 5-77 ARPN Joural of Scece ad Techology 0-0. All rghts reserved. http://www.ejouralofscece.org Usg Square-Root Iverted Gamma Dstrbuto as Pror to Draw Iferece o the Raylegh Dstrbuto
More informationLecture 7. Confidence Intervals and Hypothesis Tests in the Simple CLR Model
Lecture 7. Cofdece Itervals ad Hypothess Tests the Smple CLR Model I lecture 6 we troduced the Classcal Lear Regresso (CLR) model that s the radom expermet of whch the data Y,,, K, are the outcomes. The
More informationBlock-Based Compact Thermal Modeling of Semiconductor Integrated Circuits
Block-Based Compact hermal Modelg of Semcoductor Itegrated Crcuts Master s hess Defese Caddate: Jg Ba Commttee Members: Dr. Mg-Cheg Cheg Dr. Daqg Hou Dr. Robert Schllg July 27, 2009 Outle Itroducto Backgroud
More informationComparison of Dual to Ratio-Cum-Product Estimators of Population Mean
Research Joural of Mathematcal ad Statstcal Sceces ISS 30 6047 Vol. 1(), 5-1, ovember (013) Res. J. Mathematcal ad Statstcal Sc. Comparso of Dual to Rato-Cum-Product Estmators of Populato Mea Abstract
More informationLecture 3 Probability review (cont d)
STATS 00: Itroducto to Statstcal Iferece Autum 06 Lecture 3 Probablty revew (cot d) 3. Jot dstrbutos If radom varables X,..., X k are depedet, the ther dstrbuto may be specfed by specfyg the dvdual dstrbuto
More informationUnimodality Tests for Global Optimization of Single Variable Functions Using Statistical Methods
Malaysa Umodalty Joural Tests of Mathematcal for Global Optmzato Sceces (): of 05 Sgle - 5 Varable (007) Fuctos Usg Statstcal Methods Umodalty Tests for Global Optmzato of Sgle Varable Fuctos Usg Statstcal
More information2006 Jamie Trahan, Autar Kaw, Kevin Martin University of South Florida United States of America
SOLUTION OF SYSTEMS OF SIMULTANEOUS LINEAR EQUATIONS Gauss-Sedel Method 006 Jame Traha, Autar Kaw, Kev Mart Uversty of South Florda Uted States of Amerca kaw@eg.usf.edu Itroducto Ths worksheet demostrates
More informationThe Mathematical Appendix
The Mathematcal Appedx Defto A: If ( Λ, Ω, where ( λ λ λ whch the probablty dstrbutos,,..., Defto A. uppose that ( Λ,,..., s a expermet type, the σ-algebra o λ λ λ are defed s deoted by ( (,,...,, σ Ω.
More informationON THE USE OF OBSERVED FISHER INFORMATION IN WALD AND SCORE TEST
N THE USE F BSERVED FISHER INFRMATIN IN WALD AND SCRE TEST Vasudeva Guddattu 1 & Arua Rao Abstract I the recet years there s a large alcato of large samle tests may scetfc vestgatos. The commoly used large
More informationEconometric Methods. Review of Estimation
Ecoometrc Methods Revew of Estmato Estmatg the populato mea Radom samplg Pot ad terval estmators Lear estmators Ubased estmators Lear Ubased Estmators (LUEs) Effcecy (mmum varace) ad Best Lear Ubased Estmators
More informationUNIT 4 SOME OTHER SAMPLING SCHEMES
UIT 4 SOE OTHER SAPLIG SCHEES Some Other Samplg Schemes Structure 4. Itroducto Objectves 4. Itroducto to Systematc Samplg 4.3 ethods of Systematc Samplg Lear Systematc Samplg Crcular Systematc Samplg Advatages
More informationApplication of Generating Functions to the Theory of Success Runs
Aled Mathematcal Sceces, Vol. 10, 2016, o. 50, 2491-2495 HIKARI Ltd, www.m-hkar.com htt://dx.do.org/10.12988/ams.2016.66197 Alcato of Geeratg Fuctos to the Theory of Success Rus B.M. Bekker, O.A. Ivaov
More informationChapter 13, Part A Analysis of Variance and Experimental Design. Introduction to Analysis of Variance. Introduction to Analysis of Variance
Chapter, Part A Aalyss of Varace ad Epermetal Desg Itroducto to Aalyss of Varace Aalyss of Varace: Testg for the Equalty of Populato Meas Multple Comparso Procedures Itroducto to Aalyss of Varace Aalyss
More informationStudy of Correlation using Bayes Approach under bivariate Distributions
Iteratoal Joural of Scece Egeerg ad Techolog Research IJSETR Volume Issue Februar 4 Stud of Correlato usg Baes Approach uder bvarate Dstrbutos N.S.Padharkar* ad. M.N.Deshpade** *Govt.Vdarbha Isttute of
More informationBayesian Inferences for Two Parameter Weibull Distribution Kipkoech W. Cheruiyot 1, Abel Ouko 2, Emily Kirimi 3
IOSR Joural of Mathematcs IOSR-JM e-issn: 78-578, p-issn: 9-765X. Volume, Issue Ver. II Ja - Feb. 05, PP 4- www.osrjourals.org Bayesa Ifereces for Two Parameter Webull Dstrbuto Kpkoech W. Cheruyot, Abel
More informationEntropy ISSN by MDPI
Etropy 2003, 5, 233-238 Etropy ISSN 1099-4300 2003 by MDPI www.mdp.org/etropy O the Measure Etropy of Addtve Cellular Automata Hasa Aı Arts ad Sceces Faculty, Departmet of Mathematcs, Harra Uversty; 63100,
More informationGENERALIZED METHOD OF MOMENTS CHARACTERISTICS AND ITS APPLICATION ON PANELDATA
Sc.It.(Lahore),26(3),985-990,2014 ISSN 1013-5316; CODEN: SINTE 8 GENERALIZED METHOD OF MOMENTS CHARACTERISTICS AND ITS APPLICATION ON PANELDATA Beradhta H. S. Utam 1, Warsoo 1, Da Kurasar 1, Mustofa Usma
More informationBounds on the expected entropy and KL-divergence of sampled multinomial distributions. Brandon C. Roy
Bouds o the expected etropy ad KL-dvergece of sampled multomal dstrbutos Brado C. Roy bcroy@meda.mt.edu Orgal: May 18, 2011 Revsed: Jue 6, 2011 Abstract Iformato theoretc quattes calculated from a sampled
More informationTwo Fuzzy Probability Measures
Two Fuzzy robablty Measures Zdeěk Karíšek Isttute of Mathematcs Faculty of Mechacal Egeerg Bro Uversty of Techology Techcká 2 66 69 Bro Czech Reublc e-mal: karsek@umfmevutbrcz Karel Slavíček System dmstrato
More informationPermutation Tests for More Than Two Samples
Permutato Tests for ore Tha Two Samples Ferry Butar Butar, Ph.D. Abstract A F statstc s a classcal test for the aalyss of varace where the uderlyg dstrbuto s a ormal. For uspecfed dstrbutos, the permutato
More informationSome Statistical Inferences on the Records Weibull Distribution Using Shannon Entropy and Renyi Entropy
OPEN ACCESS Coferece Proceedgs Paper Etropy www.scforum.et/coferece/ecea- Some Statstcal Ifereces o the Records Webull Dstrbuto Usg Shao Etropy ad Rey Etropy Gholamhosse Yar, Rezva Rezae * School of Mathematcs,
More informationProbability and Statistics. What is probability? What is statistics?
robablt ad Statstcs What s robablt? What s statstcs? robablt ad Statstcs robablt Formall defed usg a set of aoms Seeks to determe the lkelhood that a gve evet or observato or measuremet wll or has haeed
More informationANALYSIS ON THE NATURE OF THE BASIC EQUATIONS IN SYNERGETIC INTER-REPRESENTATION NETWORK
Far East Joural of Appled Mathematcs Volume, Number, 2008, Pages Ths paper s avalable ole at http://www.pphm.com 2008 Pushpa Publshg House ANALYSIS ON THE NATURE OF THE ASI EQUATIONS IN SYNERGETI INTER-REPRESENTATION
More informationUnit 9. The Tangent Bundle
Ut 9. The Taget Budle ========================================================================================== ---------- The taget sace of a submafold of R, detfcato of taget vectors wth dervatos at
More informationChapter 3 Experimental Design Models
Chater 3 Exermetal Desg Models We cosder the models whch are used desgg a exermet. The exermetal codtos, exermetal setu ad the obectve of the study essetally determe that what tye of desg s to be used
More information1 Lyapunov Stability Theory
Lyapuov Stablty heory I ths secto we cosder proofs of stablty of equlbra of autoomous systems. hs s stadard theory for olear systems, ad oe of the most mportat tools the aalyss of olear systems. It may
More informationAnalysis of a Repairable (n-1)-out-of-n: G System with Failure and Repair Times Arbitrarily Distributed
Amerca Joural of Mathematcs ad Statstcs. ; (: -8 DOI:.593/j.ajms.. Aalyss of a Reparable (--out-of-: G System wth Falure ad Repar Tmes Arbtrarly Dstrbuted M. Gherda, M. Boushaba, Departmet of Mathematcs,
More informationMulti Objective Fuzzy Inventory Model with. Demand Dependent Unit Cost and Lead Time. Constraints A Karush Kuhn Tucker Conditions.
It. Joural of Math. Aalyss, Vol. 8, 204, o. 4, 87-93 HIKARI Ltd, www.m-hkar.com http://dx.do.org/0.2988/jma.204.30252 Mult Objectve Fuzzy Ivetory Model wth Demad Depedet Ut Cost ad Lead Tme Costrats A
More informationBootstrap Method for Testing of Equality of Several Coefficients of Variation
Cloud Publcatos Iteratoal Joural of Advaced Mathematcs ad Statstcs Volume, pp. -6, Artcle ID Sc- Research Artcle Ope Access Bootstrap Method for Testg of Equalty of Several Coeffcets of Varato Dr. Navee
More informationChapter 5 Properties of a Random Sample
Lecture 6 o BST 63: Statstcal Theory I Ku Zhag, /0/008 Revew for the prevous lecture Cocepts: t-dstrbuto, F-dstrbuto Theorems: Dstrbutos of sample mea ad sample varace, relatoshp betwee sample mea ad sample
More informationChapter 3 Sampling For Proportions and Percentages
Chapter 3 Samplg For Proportos ad Percetages I may stuatos, the characterstc uder study o whch the observatos are collected are qualtatve ature For example, the resposes of customers may marketg surveys
More informationρ < 1 be five real numbers. The
Lecture o BST 63: Statstcal Theory I Ku Zhag, /0/006 Revew for the prevous lecture Deftos: covarace, correlato Examples: How to calculate covarace ad correlato Theorems: propertes of correlato ad covarace
More informationLecture Notes Forecasting the process of estimating or predicting unknown situations
Lecture Notes. Ecoomc Forecastg. Forecastg the process of estmatg or predctg ukow stuatos Eample usuall ecoomsts predct future ecoomc varables Forecastg apples to a varet of data () tme seres data predctg
More information1 Solution to Problem 6.40
1 Soluto to Problem 6.40 (a We wll wrte T τ (X 1,...,X where the X s are..d. wth PDF f(x µ, σ 1 ( x µ σ g, σ where the locato parameter µ s ay real umber ad the scale parameter σ s > 0. Lettg Z X µ σ we
More informationLecture 3. Sampling, sampling distributions, and parameter estimation
Lecture 3 Samplg, samplg dstrbutos, ad parameter estmato Samplg Defto Populato s defed as the collecto of all the possble observatos of terest. The collecto of observatos we take from the populato s called
More informationA Robust Total Least Mean Square Algorithm For Nonlinear Adaptive Filter
A Robust otal east Mea Square Algorthm For Nolear Adaptve Flter Ruxua We School of Electroc ad Iformato Egeerg X'a Jaotog Uversty X'a 70049, P.R. Cha rxwe@chare.com Chogzhao Ha, azhe u School of Electroc
More informationThe Empirical Performances of the Selection Criteria for Nonparametric Regression Using Smoothing Spline
Proceedgs of the 5th WSEAS It. Cof. o COMPUTATIONAL INTELLIGENCE, MAN-MACHINE SYSTEMS AND CYBERNETICS, Vece, Italy, November 0-, 006 8 The Emrcal Performaces of the Selecto Crtera for Noarametrc Regresso
More informationECON 5360 Class Notes GMM
ECON 560 Class Notes GMM Geeralzed Method of Momets (GMM) I beg by outlg the classcal method of momets techque (Fsher, 95) ad the proceed to geeralzed method of momets (Hase, 98).. radtoal Method of Momets
More informationNP!= P. By Liu Ran. Table of Contents. The P versus NP problem is a major unsolved problem in computer
NP!= P By Lu Ra Table of Cotets. Itroduce 2. Prelmary theorem 3. Proof 4. Expla 5. Cocluso. Itroduce The P versus NP problem s a major usolved problem computer scece. Iformally, t asks whether a computer
More informationA New Method for Decision Making Based on Soft Matrix Theory
Joural of Scetfc esearch & eports 3(5): 0-7, 04; rtcle o. JS.04.5.00 SCIENCEDOMIN teratoal www.scecedoma.org New Method for Decso Mag Based o Soft Matrx Theory Zhmg Zhag * College of Mathematcs ad Computer
More informationLecture Notes 2. The ability to manipulate matrices is critical in economics.
Lecture Notes. Revew of Matrces he ablt to mapulate matrces s crtcal ecoomcs.. Matr a rectagular arra of umbers, parameters, or varables placed rows ad colums. Matrces are assocated wth lear equatos. lemets
More informationEstimation of Stress- Strength Reliability model using finite mixture of exponential distributions
Iteratoal Joural of Computatoal Egeerg Research Vol, 0 Issue, Estmato of Stress- Stregth Relablty model usg fte mxture of expoetal dstrbutos K.Sadhya, T.S.Umamaheswar Departmet of Mathematcs, Lal Bhadur
More informationTo use adaptive cluster sampling we must first make some definitions of the sampling universe:
8.3 ADAPTIVE SAMPLING Most of the methods dscussed samplg theory are lmted to samplg desgs hch the selecto of the samples ca be doe before the survey, so that oe of the decsos about samplg deped ay ay
More informationGenerating Multivariate Nonnormal Distribution Random Numbers Based on Copula Function
7659, Eglad, UK Joural of Iformato ad Computg Scece Vol. 2, No. 3, 2007, pp. 9-96 Geeratg Multvarate Noormal Dstrbuto Radom Numbers Based o Copula Fucto Xaopg Hu +, Jam He ad Hogsheg Ly School of Ecoomcs
More information