Optimal design of an accelerated degradation experiment with reciprocal Weibull degradation rate
|
|
- Cuthbert Cross
- 5 years ago
- Views:
Transcription
1 Unversty of South Florda Scholar Coons Graduate Theses and Dssertatons Graduate School 4 Optal desgn of an accelerated degradaton experent wth recprocal Weull degradaton rate Indra Polavarapu Unversty of South Florda Follow ths and addtonal works at: Part of the Aercan Studes Coons Scholar Coons Ctaton Polavarapu Indra "Optal desgn of an accelerated degradaton experent wth recprocal Weull degradaton rate" (4). Graduate Theses and Dssertatons. Ths Thess s rought to you for free and open access y the Graduate School at Scholar Coons. It has een accepted for ncluson n Graduate Theses and Dssertatons y an authorzed adnstrator of Scholar Coons. For ore nforaton please contact scholarcoons@usf.edu.
2 Optal Desgn of An Accelerated Degradaton Experent wth Recprocal Weull Degradaton Rate y Indra Polavarapu A thess sutted n partal fulfllent of the requreents for the degree of Master of Scence n Industral Engneerng Departent of Industral and Manageent Systes Engneerng College of Engneerng Unversty of South Florda Major Professor: Okogaa Geoffrey Ph.D. Rao A.N.V Ph.D. Qang Huang Ph.D. Date of Approval: Septeer 4 Keywords: hghly relale products lfe testng degradaton testng recprocal weull dstruton degradaton falure Copyrght 4 Indra Polavarapu
3 ACKNOWLEDGMENT I would lke to express y grattude to all those who gave e the posslty to coplete ths thess. I a deeply ndeted to y advsor Dr. O. Geoffrey Okogaa whose help stulatng suggestons and encourageent helped e n all the te of research and splfed y task n copleton of ths thess. Hs understandng encouragng and personal gudance have provded a good ass for ths work. Specal thanks to Dr.A.N.V.Rao for hs wde knowledge and expertse and hs portant support as a cottee eer throughout ths work. I warly thank y other cottee eer Dr.Qang Huang for hs valuale advce and frendly support. Durng ths work I have collaorated wth any frends and colleagues for who I have great regard and I wsh to extend y warest thanks to all those who have helped e wth y work n the Departent of Industral & Manageent Systes Engneerng. I would lke to thank all the staff at The Insttute on Black Lfe for ther help support and frendshp. Grateful thanks to y faly for eng there fro the very start standng the whole way through stors and sunshne. Wthout ther encourageent and understandng t would have een possle for e to fnsh ths work.
4 TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES ABSTRACT v v CHAPTER INTRODUCTION. Background. ADTs Versus Other Testng Methods.3 Relalty Analyss 3.4 Applcatons 4.4. Lght Ettng Dodes 5.4. Logc Integrated Crcuts Power Crcuts 5.5 Motvaton for Ths Research 6.6 Organzaton of Ths Thess 7 CHAPTER LITERATURE REVIEW 8. Introducton 8. Degradaton Models 9.. Lnear Degradaton 9.. Convex Degradaton 9..3 Concave Degradaton.3 General Degradaton Path Model.4 Degradaton and Falure Types 3.4. Soft Falures: Specfed Degradaton Level 3.4. Hard Falures: Jont Dstruton of Degradaton and Falure Level 3.5 Constant Stress Degradaton Models 4.5. The Arrhenus Rate Degradaton Model 4.5. Inverse Power Relatonshp Eyrng Relatonshp 5.6 Acceleraton Model 5.6. Elevated Teperature Acceleraton 6.6. Non-lnear Degradaton Path Reacton-rate Acceleraton 6
5 .6.3 Lnear Degradaton Path Reacton-rate Acceleraton Degradaton wth Parallel Reactons 8.7 Estaton of Accelerated Degradaton Model Paraeters 8 CHAPTER 3 PROBLEM STATEMENT The Optzaton Prole 3 CHAPTER 4 METHODOLOGY 5 4. Degradaton Model wth Rando Coeffcent 5 4. Assuptons The Mean-Te-To-Falure The Coputaton of [ φ ] 4.5 The Coputaton of [ φ ] φ 7 E 3 φ ( n) 4.6 The Cost Functon TC { f l } = The Optzaton Model Algorth 35 CHAPTER 5 EXAMPLE Sulaton Experent Optal Test Plan Based on the LED Data Optal Paraeters Based on the ADT Experent Senstvty Analyss Test Plans under a Varety of C Test Plans for Dfferent Values of and a Varety of Conatons of { S } = Senstvty Analyss of Ms-specfyng u andσ 53 CHAPTER 6 CONCLUSIONS AND FUTURE RESEARCH Future Research Drectons 57 REFERENCES 59
6 LIST OF TABLES Tale 5. The Standardzed Saple Degradaton Paths at Stress Level S 4 Tale 5. The Standardzed Saple Degradaton Paths at Stress Level S 4 Tale 5.3 The Standardzed Saple Degradaton Paths at Stress Level S 3 4 Tale 5.4 Optal Degradaton Test Plans under Varous Cost Condtons (C s C p C C d C ) 48 Tale 5.5 The Optal Test Plans for Soe Values of C 5 Tale 5.6 The Optal Test Plans for Varous Values of and Conatons of Stress Levels 5 3 { n } * * * Tale 5.7 The Optal Soluton ( f l ) for the Case that { ε } 3 = are = Changed over the Ranges ±.5% 53 3 { n } * for the Case that { ε } 3 = are = Changed over the Ranges ± 5% 54 * * Tale 5.8 The Optal Soluton ( f l ) 3 { n } * * * Tale 5.9 The Optal Soluton ( f l ) for the Case that { ε } 3 = are = Changed over the Ranges ± % 54
7 LIST OF FIGURES Fgure. Possle Shapes of Degradaton Curves 8 Fgure. Propagaton Delay Growth Curve wth a Plateau (Maxu Degradaton) 9 Fgure.3 Equal Log-Spacng Plan for Measureent Fgure 5. The Standardzed Saple Degradaton Paths under (a) S () S and (c) S 3 43 Fgure 5. The Plots of ω j (t) versus t.5 under (a) S () S and (c) S 3 44 Fgure 5.3 The Weull Proalty Plot for { } 3 5 ˆ β 46 Fgure 5.4 The Noral Proalty Plots for Resduals under (a) S () S (c) S 3 47 j j= v
8 OPTIMAL DESIGN OF AN ACCELERATED DEGRADATION EXPERIMENT WITH RECIPROCAL WEIBULL DEGRADATION RATE INDIRA POLAVARAPU ABSTRACT To eet ncreasng copetton get products to arket n the shortest possle te and satsfy heghtened custoer expectatons products ust e ade ore roust and fewer falures ust e oserved n a short developent perod. In ths crcustance assessng product relalty ased on degradaton data at hgh stress levels ecoes necessary. Ths assessent s accoplshed through accelerated degradaton tests (ADT). These tests nvolve over stress testng n whch nstead of lfe product perforance s easured as t degrades over te. Due to the role these tests play n deternng proper relalty estates for the product t s necessary to scentfcally desgn these test plans so as to save te and expense and provde ore accurate estates of relalty for a gven nuer of test unts and test te. In ADTs several decson varales such as nspecton frequency the saple sze and the ternaton te at each stress level are portant. In ths research an optal plan s developed for the desgn of accelerated degradaton test wth a recprocal Weull degradaton data usng the ean te to falure (MTTF) as the nzng crtera. A non lnear nteger prograng prole s developed under the constrant that the total experental cost does not exceed a pre- v
9 deterned udget. The optal conaton of saple sze nspecton frequency and the ternaton te at each stress level s found. A case exaple ased on Lght Ettng Dode (LED) exaple s used to llustrate the proposed ethod. Senstvty analyses on the cost paraeters and the paraeters of the underlyng proalty dstruton are perfored to assess the roustness of the proposed ethod. v
10 CHAPTER INTRODUCTION. Background Today s anufacturers are facng new pressures to develop hghly sophstcated products to atch rapd advances n technology ntense gloal copetton and ncreasng custoer expectatons. As a result anufacturers ust produce coponents n record te whle provng productvty relalty and overall qualty of the coponent. It s a sgnfcant challenge to desgn develop test and anufacture hghly relale products wthn short turn around tes and rean wthn the strngent condtons posed y oth nternal and external crcustances. Estatng the te-to-falure dstruton or long-ter perforance of coponents of hgh relalty products s partcularly dffcult. Most odern products are desgned to operate wthout falure for several years. Thus few of such unts wll fal or degrade to a sgnfcant aount n a test of any practcal length ased on noral use condtons. For exaple durng the desgn and constructon of a councaton satellte there ay e only 6 onths avalale to test the coponents whch are eant to e n servce for 5 to years. The coponents used n suarne cales are often requred to operate for 5 years under the sea. Very few test unts are avalale that wll actually reflect the lfe profles of these coponents. For these reasons Accelerated tests (ATs) are used wdely
11 n anufacturng ndustres partcularly to otan tely nforaton on the relalty of products.. ADTs Versus Other Testng Methods Tradtonally relalty assessent of new products has een ased on accelerated lfe tests (ALTs) that record falure and censorng tes of products sujected to elevated stress. However ths approach ay offer lttle help for hghly relale products whch are not lkely to fal durng an experent of reasonale length. An alternatve approach s to assess the relalty fro the changes n perforance (degradaton) oserved durng the experent f there exsts a qualty characterstc of the product whose degradaton over te can e related to relalty. Accelerated degradaton tests copared to other tests have the advantage of analyzng perforance efore the ateral or the coponent fals. Degradaton tests deterne how uch lfe there s left n a ateral or n coponents and such knowledge enales lfe extenson. Extrapolatng perforance degradaton to estate when t reaches falure level enales analyss of degradaton data. However such analyss s correct only f a good odel for extrapolaton of perforance degradaton and a sutale perforance falure have een estalshed. Soe of the general assuptons of accelerated degradaton odels are Degradaton s not reversle. A odel apples to a sngle degradaton process echans or falure ode. If there are sultaneous degradaton processes and falure odes each requres ts own odel. Degradaton of specen perforance efore the test starts s neglgle. Perforance s easured wth neglgle rando error.
12 The falure process at hgh stress levels are the sae as at the desgn or use stress levels. Accelerated degradaton tests are usually very expensve and thus t s essental to plan the carefully. Good test plans yeld etter results for a gven cost and te paraeters on the other hand poor test plans waste te and resources and ay not even yeld the desred nforaton. When conductng an ADT the followng ssues are usually of nterest. How long should the test e run? or How any unts should e tested at each stress level? Thus to address the ssues a scentfc plan s needed to ake the ost effcent use of test resources and ultately to otan an accurate estate of the lfe profle of an entty under the noral use condtons..3 Relalty Analyss Most thngs have a lfe span defned n soe for or another. These lfe tes when easured present us wth data sets that are used for scentfc or other purposes. It s natural to study the lfe te dstruton of an entty through a set of easured data. An area of research whch s stll vary uch actve s the theory of relalty. A generc defnton of relalty s: Relalty s the proalty that a product or a syste wll perfor ts ntended functon wthout falure for a specfed perod of te under specfc operatng condtons. To express ths relatonshp atheatcally we defne the contnuous rando varale to e the te to falure of the coponent or syste. Thus relalty at te t can e expressed as: ( ) = Pr{ T t} ; R t T Where : R () t R( ) = L R() t = t 3
13 In relalty analyss the ajor ssue s the proalty dstruton of the lfe tes of the entty under study. For ths purpose the standard ethod s to take a set of oserved lfe tes T T... T n or censored soetes where we assue that the oserved lfe te s a functon of an unknown paraeter θ whch can e expressed as T ~ F (.; θ). Where F (.; θ) s the proalty densty functon of an unknown paraeter θ. Fro the lkelhood functon constructed fro ths saple we can ake an nference wth respect to the unknown paraeter θ. When the for of F (.; θ) s known and the coplete dstruton of F s deterned y a fnte densonal paraeter θ then we have a classcal paraetrc odel; f F s copletely unknown except for soe qualtatve descrptons such as contnuty or soothness then we have a non-paraetrc odel; fnally f F s unknown ut the paraeter θ has soe structure to explore then we have a se-paraetrc odel. When the paraeter θ exhts soe structure we wll naturally eed our nference prole nto tradtonal and te-tested odels for statstcal analyss. These nclude technques such as experental desgn regresson logstc regresson accelerated lfe testng etc. These ethods ncorporate varous stuatons that ay e encountered n practce. There s no need however to restrct the nference to the classcal frequentsts paraetrc setup. We can f the stuaton requres use the Bayesan ethod or even the eprcal Bayes technques..4 Applcatons Applcatons of accelerated degradaton tests nclude lght ettng dodes logc ntegrated crcuts power supples etc. 4
14 .4. Lght Ettng Dodes (LED) Lght ettng dodes are wdely used n any felds rangng fro consuer electroncs to optcal fer transsson systes. The LED has any features such as less power consupton sall volue good vsual effects and long lfe. Nowadays they are used as electronc oards on hghways and streets and as soke censors on celngs etc. Because of ther hgh relalty t s dffcult to otan the product lfe te nforaton under noral stress levels n a relatvely short te. Thus ADTs are used to otan the relalty nforaton of LED products [6 7]..4. Logc Integrated Crcuts Soe logc ntegrated crcuts are used as coponents of suarne cales. The portant paraeter n deternng the relalty nforaton of logc ntegrated crcuts s propagaton delay [8]. The logc ntegrated crcuts ght not functon f the propagaton delay of a logc gate ncreases (degrades). For exaple a logc crcut whch s desgned to have a axu propagaton delay of nsec fro nput to the output requres that the coned propagaton delay of the ndvdual logc gates n the crtcal path does not exceed nsec. These logc ntegrated crcuts are requred to operate for at least 5 years under the sea wthout falure. So accelerated degradaton tests are eployed to predct the lfe of logc ntegrated crcut and to study the assocated propagaton delay..4.3 Power Crcuts For power supples falures are coon due to low DC output. Power supply unts show downward drft n ther DC output. Accelerated degradaton tests are used to easure the DC output and to ontor the devce for relalty nforaton. 5
15 Nelson [3 pp 5-548] lsts other applcatons of accelerated degradaton tests whch nclude: etals se conductors and cro electroncs delectrcs and nsulatons food and drugs and plastcs and polyers..5 Motvaton for Ths Research An Accelerated Degradaton test s a echans desgned to shorten the lfe of products y sujectng the test unts to hgher levels of stresses that are ore severe than the noral use stress levels. The nforaton fro hgh stress levels s extrapolated through a reasonale statstcal ethod to otan estates of lfe or long-ter perforance at the noral use stress level. Tradtonal approaches are ased on lfe tests that record only te-to-falure. Such analyses have een extensvely studed and developed over the past few decades and any artcles have een pulshed n ths area. Due to the fact that tradtonal lfe testng of hghly relale products does not gve good relalty estates relalty assessent usng degradaton data has ecoe ncreasngly portant. In the lterature ost degradaton odels assues that the degradaton paths or transfored degradaton paths are lnear and are developed for only the noral use stress level [3-8]. Most of the lterature focuses on estatng the paraeters n the lnear degradaton odel and the lfe dstruton. Research aout accelerated test plannng s also reported. But when carryng out the accelerated degradaton tests several decson varales such as nspecton frequency the saple sze and the ternaton te at each stress level are portant. The prary ojectve for ths research s to deterne the optal paraeters of an ADT wth respect to products whose degradaton rates follow the recprocal weull 6
16 dstruton. Ths s accoplshed y takng the MLE s of the varance of the confdence nterval of the MTTF under the constrant that the total experental cost does not exceed a pre-deterned udget. A non lnear prograng prole s developed to deterne the optal value of the decson varales such as saple sze nspecton frequency and the ternaton te at each stress level..6 Organzaton of the Thess In Chapter a revew of the lterature s dscussed. Chapter 3 dscusses the prole stateent assuptons ade and notatons used. An optzaton prole s proposed. In Chapter 4 an optal plan for solvng the optzaton prole s presented. To llustrate the optal plan an exaple of LED degradaton s presented n chapter 5.Fnally the conclusons of the study and suggestons for further research are presented n chapter 6. 7
17 CHAPTER LITERATURE REVIEW. Introducton Durng the 99 s Nelson[3] (chapter ) provded a farly thorough survey on ADT whch ncluded areas of applcatons statstcal odels descres asc deas on ADT odels and usng a specfc exaple shows how to analyze a type of degradaton data. Carey and Koeng[4] (99) have descred a data-analyss strategy and a odelfttng ethod to extract relalty nforaton fro oservatons on the degradaton of ntegrated logc devces that are coponents n a new generaton of suarne cales. Most falures can e traced to an underlyng degradaton process. Meeker and Escoar (998) gave soe exaples of three general shapes for degradaton curves n artrary unts of degradaton and te: lnear convex and concave whch are shown n fg... The dashed horzontal lne represents the degradaton level at whch falure would occur. Fgure. Possle Shapes of Degradaton Curves 8
18 . Degradaton Models.. Lnear Degradaton Meeker and Haada [7] uses lnear degradaton n soe sple wear processes lke autoole tre wear. Let D (t) represents the aount of autoole tre tread at te t and dd ( t) wear rate dt = C then D (t) = D () + C t. The paraeters D () and C could e taken as constant for ndvdual unts ut rando fro unt-to-unt... Convex Degradaton The convex degradaton approach s used n odels for whch the degradaton rate ncreases wth the level of degradaton such as n odelng the growth of fatgue cracks. Let a (t) denote the sze of a crack at te t. The Pars odel[8] s gven as da( t) dt = [ ( )] (.) C k a Where a = sze of the crack C and = ateral propertes and k(a) = stress-ntensty factor 9
19 s often used to descre the growth of fatgue cracks. Lu and Meeker [8] use ths odel n whch k ( a) = a for descrng the growth rate of fatgue cracks wthn a certan sze range. Then ( () ) = a a t (.) /( { [ a()] C( ) t} ) where a () =.9 nches s the ntal crack length at t =. Dvdng oth sdes of Eq. (.) y a () yelds /( ) a (t)/a () = / { [ a ()] C( ) t} (. )..3 Concave Degradaton Meeker and LuValle [9] descre odels for growth of falure-causng conductng flaents of chlorne-copper copounds n prnted-crcut oards. They consder degradaton fro a frst-order checal reacton. These flaents cause falure when they reach fro one copper-plated through-hole to another. Let A (t) e the aount of chlorne avalale for reacton at te t and A (t) e the aount of falure- causng chlorne-copper copound at te t. Under approprate condtons copper cones wth chlorne A to produce the chlorne-copper copound A wth a constant rate k. The equatons for the rate for ths process are da dt = ka and da dt = ka Let c and A () e the ntal aounts. Assung A () = we get
20 A (t) = A ( )[-exp(-kt)]. (.3) Ths functon s llustrated y the concave curve n Fg... The asyptote at A ( ) = A () reflects the aount of chlorne avalale for the reacton producng copounds. LuValle and Meeker [9] also suggest other ore elaorate odels for ths falure process. Carey and Koeng [4] use slar odels to descre degradaton of electronc coponents..3 General Degradaton Path Model Lu & Meeker (993) use the followng odel [Eq.3] for the analyss of degradaton data at a fxed level of stress (.e. no acceleraton) and to estate a te-tofalure dstruton. They denote the true degradaton path of a partcular unt(a functon of te) y D(t) t >. In applcatons values of D(t) are sapled at dscrete ponts n tet t.the oserved saple degradaton path for unt at te t j s a unt s actual degradaton path plus error and s gven y y = + =.. n j = (.3) j D j j Where D j = D (t j β ) s the actual path of the th unt at te t j (the tes need not e the sae for all unts) ( σ ) j ~ N s the devaton fro the assued odel for unt at te t j β = ( β. β k ) s a vector of k unknown paraeters for unt. The devatons are used to descre the easureent error. The total nuer of nspectons on unt s denoted y. Te t could e real-te operatng te or soe surrogate lke les for autoole tres or loadng cycles n fatgue tests. Typcally sall paths are descred wth a odel that has up to four ponts (.e. k= 3 4). Soe of the
21 paraeters n β could e rando fro unt-to-unt and soe of the could e odeled as constant across all unts. The scales of y and t can e chosen to splfy the for of D(t β ). The choce of a degradaton odel requres not only the specfcaton of the for of D(t β ) functon ut also the specfcaton of whch of the paraeters n β are rando and whch are fxed as well as the jont dstruton of the rando coponents n β. Lu & Meeker (993) descre the use of a general faly of transforatons to a ultvarate noral dstruton wth ean vector µ β and covarance atrx β. It s generally reasonale to assue that the rando coponents of the vector β are ndependent of the j. We also assue that j are ndependent and dentcally dstruted. Because the y j are taken serally on a unt however there s potental for autocorrelaton aong the j. Especally f there are any closely spaced readngs. In any practcal applcatons nvolvng nference on the degradaton of unts fro a populaton or process however f the odel ft s adequate and f the testng and easureent processes are n control then the autocorrelaton s typcally weak and oreover t s donated y the unt-to-unt varalty n the β values and thus can e gnored. Also t s well known that pont estates of regresson odels are not serously affected y autocorrelaton ut gnorng autocorrelaton can result n standard errors that are serously ased. Ths however s not a prole when confdence ntervals are constructed y usng an approprate sulaton-ased ootstrap ethod. In ore coplcated stuatons t ay also happen that σ wll depend on the level of the acceleraton varale. Often however approprate odelng (for varance stratfcaton
22 e.g. transforaton of the degradaton response) wll allow the use of a spler odel ased on constantσ..4 Degradaton and Falure Types.4. Soft Falures: Specfed Degradaton Level For soe products there s a gradual loss of perforance (e.g. decreasng lght output fro a fluorescent lght ul). Then falure would e defned n an artrary anner at a specfed level of degradaton such as 6 % of ntal output. Tseng Haada and Chao (995) explan ths wth an exaple and defned ths as soft falure. A fxed value of D f s used to denote the crtcal level for the degradaton path aove (or elow) whch falure s assued to have occurred. The falure te T s defned as the te when the actual path D(t) crosses the crtcal degradaton level c and t c s used to denote the planned stoppng te n the experent. Inferences are ade on the falure-te dstruton of a partcular product or ateral. For soft falures t ay e possle to contnue oservaton eyond D f..4. Hard Falures: Jont Dstruton of Degradaton and Falure Level For soe products a falure event s defned as when the product stops workng (e.g. when the resstance of a resstor devates too uch fro ts nonal value causng the oscllator n an electronc crcut to stop oscllatng or when an ncandescent lght ul urns out). These are called hard falures. In general wth hard falures falure tes correspond to a partcular level of degradaton. But the level of degradaton at whch falure occurs s rando fro unt to unt or fro te to te. Ths could e odeled y usng a dstruton to descre unt-to-unt varalty n D f or ore generally the jont dstruton of β and the stochastc ehavor of D f. 3
23 .5 Constant Stress Degradaton Models Nelson (99) refly descres soe asc degradaton odels for constant stress. The followng are the ost wdely used constant stress degradaton odels..5. The Arrhenus Rate Model The Arrhenus rate relatonshp s wdely used for teperature-accelerated degradaton. Ths odel s ostly used n pharaceutcals nsulatons delectrcs plastcs polyers Adhesves attery cells and ncandescent lap flaents. Arrhenus law: Accordng to the Arrhenus rate law the rate of a sple (frst-order) checal reacton depends on teperature as follows rate = A exp[ E /( kt)] (.5) where: E s the actvaton energy of the reacton usually n electron-volts. k s Boltzann s constant electron-volts per C. T s the asolute Kelvn teperature; t equals the teperature n Centgrade plus 73.6 degrees; the asolute Rankne teperature equals the Fahrenhet teperature plus Fahrenhet degrees. A s a constant that s characterstc of the product falure echans and the test condtons. The product s assued to fal when soe crtcal aount of the checal has reacted (or dffused); Crtcal aount = (rate) (te to falure) or Te to falure = (crtcal aount) / (rate) 4
24 Therefore the nonal te τ to falure ( lfe ) s nversely proportonal to the rate. Ths yelds the Arrhenus lfe relatonshp τ = Aexp[ E /( kt)].5. Inverse Power Relatonshp The nverse power relatonshp s wdely used to odel product lfe as a functon of an acceleratng stress. Ths s ostly used n electrcal nsulatons delectrcs n voltage-endurance tests all and roller earngs ncandescent laps and flash laps etc. The relatonshp s soetes called the nverse power law or sply the power law. Suppose that the acceleratng stress varale V s postve. The nverse power relatonshp etween nonal lfe τ of a product and V s τ γ ( V ) A/ V = ; (.5) Here A and γ are paraeters characterstc of the product specen geoetry and farcaton the test ethod etc. The paraeter γ s called the power or exponent..5.3 Eyrng Relatonshp An alteratve to the Arrhenus relatonshp for teperature acceleraton s the Eyrng relatonshp. The Eyrng relatonshp for nonal lfe τ as a functon of asolute teperature T s τ = ( A/ T )exp[ B /( kt)] ; (.53) here A and B are constants that are characterstc of the product ad test ethod and k s the Boltzann s constant..6 Acceleraton Model To otan tely nforaton fro laoratory tests soetes t s requred to use soe for of acceleraton. In soe falure echanss such as the weakenng of an adhesve echancal ond or the growth of a conductng flaent through an nsulator 5
25 the checal or other degradaton process can e accelerated y ncreasng the level of acceleraton varales lke teperature hudty voltage or pressure. If an adequate physcally-ased statstcal odel s avalale to relate falure te to levels of acceleratng varales the odel can e used to estate lfete or degradaton rates at product use condtons. Lu Meeker & Escoar [5] entoned the followng acceleraton odels..6. Elevated Teperature Acceleraton The Arrhenus odel whch descres the effect of teperature on the rate of a sple frst-order checal reacton s Ea Ea 65 R( tep) = γ exp = γ exp (.6) k B ( tep ) tep Where tep s teperature n C and k B = /65 s the Boltzann s constant n unts of electron volts per C. The pre-exponental factor γ and the reacton actvaton energy E a n unts of electron volts are characterstcs of the partcular checal reacton. Takng the rato of the reacton rates at teperatures tep and Acceleraton Factor tep U cancels γ gvng an AF(tep tep U E a ) = R( tep) R( tep U ).6. Non-lnear Degradaton Path and Reacton-rate Acceleraton The sple checal degradaton path odel wth a teperature acceleraton factor affectng the rate of reacton s gven y D( t; tep) = D { exp[ RU AF( tep) t]} (.6) 6
26 Here R U s the rate reacton at use teperature tep U R U AF(tep) s the rate reacton at teperature tep and D s the asyptote. When degradaton s easured on a scale decreasng fro zero D < then the falure occurs at the sallest t such that D(t) D f. Equatng D(T ; tep) to D f and solvng for T gves the falure te at teperature tep as T ( tep) = D log R U D AF( tep) f = T ( tep U ) AF( tep) Where T ( tep ) = - (/R U ) log (-D f /D ) s falure te at use condtons. U Here the lfe/teperature odel nduced y the sple degradaton process and the Arrhenus-acceleraton odel results n a Scale Accelerated Falure Te (SAFT) odel. Under the SAFT odel the degradaton path of a unt at any teperature can e used to deterne the degradaton path that the sae unt would have had at any other specfed teperature y scalng the te axs y the acceleraton factor AF (tep).6.3 Lnear Degradaton Path Reacton-rate Acceleraton Consder the odel wth nonlnear degradaton path and reacton rate acceleraton along wth the crtcal level D f. When D(t) s sall relatve to D D( t; tep) D { exp[ RU AF ( tep) t]} = (.63) D + RU AF( tep) t = RU AF( tep t ) If falure occurs when D(T) D f then D(T;tep)= D f and the falure te s gven as D f T ( tepu ) + T ( tep) = = where T(tep + U ) = D f / RU s falure te at use R AF( tep) AF( tep) U condtons. Ths s also an SAFT odel. 7
27 .6.4 Degradaton wth Parallel Reactons A ore coplcated degradaton path odel wth two parallel one-step falure causng checal reactons s gven y : D( t; tep) = D + D { exp[ R { exp[ R U U AF ( tep) t]} AF ( tep) t]} (.64) Where R U and R U are the use-condton rates of the two parallel reactons contrutng to falure. Ths degradaton odel does not lead to an SAFT odel ecause the teperature affects the degradaton processes dfferently nducng nonlnearty nto the acceleraton functon relatng tes at two dfferent teperatures..7 Estaton of Accelerated Degradaton Model Paraeters Lu and Meeker (993) use a two-stage ethod to estate the paraeters of the xed-effects accelerated degradaton odel. Stage - For each unt ft the degradaton odel to the saple path and otan the estate of the odel paraeters of each unt. Stage - Cone the estate of the odel paraeters of each unt n the frst stage to produce estates of the populaton paraeters. In another research Lu Meeker & Escoar [5] suggest that n soe cases an approxate axu lkelhood (ML) s faster than n nonlnear least squares estatons requred for the two-stage ethod. ML estaton also has the advantages of desrale large-sple propertes and easy to use saple paths for whch all of the paraeters cannot e estated. The two-stage estaton s useful for otanng startng values for the ML approach for odelng especally when another dstruton other than a jont noral dstruton for the rando effects s eng consdered. 8
28 Meeker and Escoar (993) have gven an updated lterature survey on varous approaches used to assess relalty nforaton fro degradaton data. Boulanger and Escoar[7] address the prole of desgnng a class of ADTs. They assue that each unt s sujected to an elevated constant stress level over the duraton of the experent. The perforance degradaton of each test unt at a stress level could e descred y a growth curve whch levels off to a plateau (axu degradaton) after a perod of te. Fgure. shows the degradaton aount over te. The odel s gve y: y(t) = α[-exp(-(βt) γ )] + є(t) Where y(t) = oserved change of propagaton delay up to te t α β = plateau where the degradaton wll level off = rando coeffcent γ = a constant whch s equal to.5 and є (t) = easureent error. Fgure. Propagaton Delay Growth Curve wth a Plateau (Maxu Degradaton) 9
29 The authors consder α as lognorally dstruted and stress dependent. The desgn prole they consder s to nze the varance of the estate of the ean of the logarth of the plateau ln(α) at use condton. Ther ojectve here s to provde soe gudelnes n desgnng a useful plan for a specal class of accelerated degradaton tests. They frst deterne the optal stress levels and the proporton of unts allocated to each stress level and then deterne optal tes to easure the perforance degradaton of unts at each stress level. The test stress levels are chosen to e 448 K and 373 K whch are the hghest teperatures the plastc package can wthstand and the nu teperature the easureent equpent can detect any degradaton at the end of the experent respectvely. Equal log-spacng plan shown n Fg..3. s used for easureent ecause the process shows a great deal of degradaton at the early stage and then stalzes. Fgure.3 Equal Log-Spacng Plan for Measureent Although the result s nterestng t s not practcal snce an approprate ternaton te for an experent s usually not known n advance. In the lterature ost of the degradaton odels are lnear or can e transfored to lnear odels. Also ost of the lterature focuses on estatng the paraeters n the degradaton odel and the lfe dstruton. Yu and Tseng[8] proposed an ntutvely onlne and real-te rule to deterne an approprate ternaton te for an ADT. Park and Yu[9] develop optal accelerated degradaton test plans under the assuptons of
30 destructve testng and the sple constant rate relatonshp etween stress and product perforance. The authors deterne stress levels the proporton of test unts allocated to each stress level and easureent tes such that the asyptotc varance of the axu lkelhood estator of the MTTF at the use condton s nzed. Yu and Tseng [ -4] descre a ethod for conductng a degradaton experent effcently consderng several factors such as the nspecton frequency ternaton te and the saple sze. They consder a typcal degradaton path of an LED whch s ln(-ln(y(t))) = ln(α) + β ln(t) + є(t) Where y (t) = standardzed lght ntensty of an LED devce at te t α β = paraeters of the degradaton path and є (t) = easureent error of the devce at te t. Based on data (t k y j (t k )) s the for stress level k = where k s the ndex that represents the th easureent for unt j the degradaton paraeters α β for unt j can e otaned. The falure te of unt j can e found usng ln( D) τ = α β assung D s the crtcal value for the standardzed lght ntensty when an LED fals. In these papers they deal wth the optal desgn for a degradaton experent under the constrant that the total experental cost does not exceed a predeterned udget. The optal decson varales are otaned y nzng the varance of the estated pth percentle of the lfete dstruton. But these three decson varales have a
31 great deal of nfluence upon the experental cost and the precson of selectng the ost relale product usng degradaton data. A nonlnear nteger prograng prole s developed to deterne the optal conaton of saple sze nspecton frequency and ternaton te. As s evdent fro the aove revew an extensve lterature on the desgn of degradaton tests and accelerated degradaton tests exsts. But when desgnng a degradaton test the dstruton of the degradaton rate of the product/coponent at whch t degrades s very portant. The Weull and lognoral dstrutons are two ost popular lfete odels n relalty analyss that have een used for ths purpose. An ncorrect choce of the dstruton ay lead to serous as.
32 CHAPTER 3 PROBLEM STATEMENT Wth regard to hghly relale products t s portant to consder the ssues of how to plan tests that provde the ost effcent use of resources especally as t relates to conductng an accelerated degradaton test. The prole s to nze the expected value of the Mean-Te-To-Falure of a product suject to the constrant that the total cost of the test does not exceed a predeterned test udget. 3. The Optzaton Prole The followng decson varales are portant n conductng an ADT effcently (Yu & Chao [ ]). These varales not only affect the experental cost ut also affect the precson of estatng the MTTF ( φ ) whch can e defned as the expected or the ean value of the falure te. The pertnent questons regardng the test plan are How s an approprate nspecton frequency (f ) deterned? How any tes (l ) should the product s perforance e nspected at each stress level? How any devces (saple sze n) should e taken for testng at each stress level? In order to easure the precson of estatng the MTTF the expected wdth of the confdence nterval values of the Mean-Te-To-Falure( φ ) s coputed. The expected value of a real-valued rando varale gves the ean or central tendency of the 3
33 dstruton of the varale. The unased axu lkelhood estator of the MTTF can e acheved y estatng the asyptotc varance. The asyptotc varance can e otaned y nzng ether the ean square error or the expected value of the range of MTTF (MTTF ax -MTTF n ). Snce we do not have a pror estate of the ean square error we wll estate the asyptotc varance y nzng the MTTF. Thus the optal decson prole ased on the expected range of the MTTF s forulated as follows: Where as TC ({ f l} n) ({ f l } n) = Mnze [ φ ] E (3.) φ Suject to TC { f l } ( n) C (3.) = f l n є N = { 3 } (3.3) = 3. = denote the total cost of conductng an ADT. ˆ φ denote an estator of [ φ φ ] denote a (-p) % CI of [ φ ] φ ( n) = ( f l n) φ ased on a test plan { f l } φ fro the test plan { } = E denotes the expected wdth of the ( p)% CI of φ C s the total cost of the udget. p s the percentle of the lfe te dstruton of the product at noral use condton. 4
34 CHAPTER 4 METHODOLOGY 4. Degradaton Model wth Rando Coeffcent Let η() t denote the qualty characterstc (degradaton path) of the product at te t. Assue that there exsts a sutale functon ω (.) such that α ω ( η( t)) = βt t (4.) Where α > s a fxed and known constant; β > s a rando coeffcent that vares fro unt to unt. 4. Assuptons The ADT uses stress levels S S S S satsfyng (S ) S S S where S s the use condton. Due to the easureent errors the actual degradaton path cannot e oserved drectly. Let y j (t k ) denote the saple degradaton path of the j th devce at te t k under the stress level S.The path can e expressed as follows: α ω y ( t )) = β t + ( t ) (4.) ( j k j k j k The unts put nto test are randoly selected fro the saples and are randoly assgned to test stress levels. At each stress level n devces are randoly selected for testng. 5
35 Suppose that under stress level S the nspectons are ade l tes for every f unts of te (e.g. f hours or f days) untl te t l = f l t u where l s a postve nteger and t u s a unt of te. Assue that β j follows a recprocal weull- dstruton then β - follows a weull dstruton wth scale paraeter θ and shape paraeter δ (whch s denoted y β - ~ Weull (θ δ ) The shape paraeter δ does not depend on the stress level and the scale paraeter or the characterstc lfe θ s a functon of transfored levels of stress: ln ( θ ) γ + γ X = (4.) Where γ and γ are unknown paraeters to e estated fro the data X =X (S ) and X (.) s a sutale transforaton. Two falar exaples for X (.) are as follows: X (S ) = / S f an Arrhenus odel s assued = ln S f an nverse-power odel s assued Soe other relatonshps whch are coonly used are entoned n the lterature revew n chapter. In order to solve the optzaton prole the MTTF has to e coputed frst. 4.3 The Mean-Te-To-Falure The product lfe te (τ ) s sutaly defned as the te when η crosses the crtcal level D. Fro Eq (3.) τ can e expressed as ω( D) τ = β / α Takng natural logarth on oth sdes ln( τ ) = [ ln( ω( D )) ln β ] (4.3) α 6
36 Snce β follows recprocal weull dstruton -ln β follows the extree value dstruton wth scale paraeter u and locaton paraeter (whch s denoted y -ln β ~ Extree (u ) and n equaton (4.) wth α fxed t can e shown that τ follows the Weull dstruton wth scale paraeter ( θ *( D / α ω( ))) and shape paraeterαδ. Let τ denote the product s lfete under S. Thus we have τ / ( θ * ( ω( ))) αδ ) α = Weull D The MTTF φ of the product s lfete dstruton under S s Where u = γ + γ X ) ( S / ( θ ( ω( ))) Γ + α φ = D αδ / α u φ ( ( )) Γ = ω D exp + (4.3) α α Here the prole s to desgn an effcent ADT such that φ can e estated precsely. The optzaton prole can e solved y usng the followng steps. 4.4 The Coputaton of [ φ φ ] l For j n and ased on the oservatons {( t y ( t )} k j k ) k = the least-squares estator (LSE) βˆ of β condtonal on j j β j can e coputed y nzng l = LS( β j ) k = { ( ( )) } α ω y t + β t j k j k Thus we otan l ω( yj ( t k )) t l α t k α k = k ˆ β = (4.4) j k = 7
37 and σ can e estated y σ = j= ( ˆ β ) n = LS j (4.4) n( l ) By consderng the frst-order Taylor seres expanson aout of followng approxate forula for ln βˆ j : βˆ j ln β j we can otan the ˆ β ˆ j ln β ln + j β j (4.43) β j where condtonal on ˆ β j β j E = β j Var ˆ β β l α / = t k k and [ j j ] σ =.Hence t s β j seen that ˆ β j β j as l k = t α k (4.44) Fro equaton (4.44) t s seen that the asyptotc dstruton of un-condtonal follows an extree value dstruton wth ( ˆ ˆ ). ln βˆj u Thus ( uˆ ˆ ) axu lkelhood estators (MLEs) of ( u ) can e otaned drectly (Lawless98) y: the conventonal e uˆ n x j = exp n j ˆ = ˆ and 8
38 9 ˆ exp exp = = = = = = n j j n j j n j j j x n x x x where. ˆ ln n j x j j = β Here û and ˆ can e solved y usng soe nuercal ethods( e.g. Newton s ethod) wth an teratve procedure. Based on the asyptotcally effcent property of axu lkelhood estate (Lawless 98) the jont densty of û and ˆ follows an asyptotcally varate noral dstruton as follows. ~ ˆ ˆ u N u (4.45) where [ ] [ ] [ ] [ ] = Var Cov u Cov u u Var ˆ ˆ ˆ ˆ ˆ ˆ = I - (u ) denotes the covarance atrx of û and ˆ. The fsher nforaton atrx I(u ) can e expressed as follows: ( ) ( ) ( ) ( ) ( ) ln ln ln ln = u L E u u L E u u L E u u L E u I where ( ) = = j j n j u x u x u L exp exp
39 By usng the technque of ntegraton y parts Var [ û ] Cov [ û ˆ ] and Var [ ˆ ] can e otaned as follows. Var 6 π [ uˆ ] = + ( γ ) (4.46) n π 6 [ ˆ 6 ] = ( γ ) ˆ Cov u (4.47) n π Var [ ˆ 6 ] = (4.48) n π Where Γ (x) s the gaa functon and γ =.577 s the Euler s constant. In a real stuaton the experent s only conducted up to te t l.thus the paraeters u ) can e slghtly calrated y the condtonal expectaton technque. ( Assung ( u l l ) denotes the paraeters after refned calraton the approxate relatons etween ( ) l l u and u ) can e expressed (Hong Fwu Yu []) as follows: ( u l u + γ ( l ) where: l + 6σ θ Γ + δ α α l α tu f k = t k To assure that k = l t α k s suffcently large t s reasonale to set 6σ θ Γ + δ α α l t f u = t k α k / s < s < (4.49) 3
40 Ths equaton ndcates that the slower the qualty characterstc of a product degrades the longer the degradaton test should last. Thus ˆ can e further estated as follows: ˆ = (4.4) ˆ ˆ = Based on these estators { } can e otaned as follows (Lawless [6]): u = the LSEs ˆ γ ˆ ) of γ ) n Equaton (4.) ˆ γ = ˆ γ ( γ T T ( X X ) X Y ( γ where X T = X ( S )... X ( S )... X ( S ) and Y T ( u uˆ... uˆ ) ˆ =.Thus u can e estated y = ˆ γ + ˆ γ X ( ) (4.4) uˆ S The approxate dstruton of û and ˆ s as follows 6 Π + H u ˆ ~ N u * γ ( γ ) (4.4) n Π 6 G = S = where H = X ( S ) + X ( S ) X ( S ) X ( ) ( = S = and G = X S ) ( X ( )) ˆ 6 ~ N (4.43) n Π The approxate ( p) % and ( p ) % confdence nterval (CI) of u and can e otaned as follows: 3
41 [ u u ] = Ζ exp uˆ uˆ ) Ζ exp p / ˆ / var( ˆ u p u uˆ var( uˆ ) and [ ] = Ζ exp ˆ p / var ˆ () ˆ Ζ ˆexp p / var ˆ () ˆ Where: Ζ s the( p ) th percentle of standard noral dstruton and Ζ / s p / p the( p ) th percentle of standard noral dstruton. p and p are the percentle values for scale(u ) and shape() paraeters respectvely. Now susttutng ( u ) and ( u ) ( p)( p )% CI for φ as follows: [ φ φ ] = ω( ) nto Equaton (4.3) we otan an approxate u u ( ) Γ + α ( ( )) Γ ω α D exp D exp + (4.44) α α α α 4.5 The Coputaton of E [ φ ] φ By takng the natural logarth of oth sdes of Eq (4.3) we have ln ˆ φ ( ) = f l n [ ln( ω( D) ) + u ] + ln Γ + α α (4.5) 3
42 33 The asyptotc dstruton of ( ) n l f ˆ ln φ follows a noral dstruton ( ) ν N where ( ) ( ) ( ) [ ] ( ) ( ) 6 ln ln ln ˆ var ln ˆ ln Γ + + Γ + Γ + + = Γ = γ π α α α α α ν α ω α n E E u and E u D Hence the asyptotc ean of { } ( ) n l f ˆ = φ can e expressed as follows: { } ( ) [ ] + = exp ˆ ν φ n l f E (4.5) Therefore [ ] ( ) + exp ν ν φ φ E where ( ) ( ) ( ) ( ) [ ] ( ) ( ) [ ] ( ) Γ + Γ + + Γ + + Γ + Γ + Γ + + = Γ + Γ = n n E E E E u u and E E u u D ln ln * 6 ln ln ln ln ˆ var ˆ var ln ln ˆ ˆ ln α α γ π α α α α α α ν ν α α ω α
43 The Cost Functon { } ( ) n l f TC = The total cost { } ( ) n l f TC = of conductng an ADT s dvded nto three parts (Yu and Tseng [ ]). The cost of conductng an ADT s { } = + p s l f C l f C ax where C s denotes the operator s salary per unt of te and C p denotes the unt cost of power of the testng equpent. The easureent cost s = l n C where C denotes the unt cost of easureent. The cost to test the devces s n C d where C d denotes the unt cost per devce. Therefore the total cost of the experent s { } ( ) { } n C l n C l f l f n l f TC d + + = = = = p s + C ax C (4.6) 4.7 The Optzaton Model Fro the foregong results the optzaton prole can e expressed as follows ( ) ( ) { } ( )( ) n N n l f p p p where p p p s k f t C n C l n C l f C l f C s t Mn l k u d p s 3... : (4.73) * 6 (4.7) ax.. (4.7) exp = < < = Γ = = = δ θ σ ν ν ε α α α
44 4.8 Algorth Due to the coplexty of the ojectve functon t s dffcult to fnd an effcent ethod to solve the optzaton odel n equaton (4.7). The ojectve functon can e ( n) expressed as a functon of p and { f l } =. Hence wth splcty structure of the constrant and the nteger restrcton on the decson varales an approxate soluton can e otaned y the followng steps. ( ( n ) = [ φ φ ] Let p { f l } = E. Partton the nterval ( ) Set p equally nto l (say l =) suntervals. p p ( k) = k k =.( l -). For each ( ) l p k the correspondng ( ) optal conaton { f k) l ( k) } n( k) Gven { f }. = ( Deterne the correspondng { } Deterne the correspondng n. ( ) [ ] = can e otaned as follows. l = y Equaton (4.49). Copute V { f} = φ fro the test plan { f l} E φ = ({ } n ( k) ) The optal soluton f ( k) l ( k) { } f = Where = satsfy V { f } = n n () ( ) ( n) = can e deterned f ( ) V { f} =... ( ) (4.8) = f f f f f 6σ θ Γ + δ = α tu π s G 35
45 and x G denotes the largest nteger not greater than x. In the nzaton ( n) process of (4.6) any { f l } would not e taken nto consderaton. * * * Fnally an approxate optal soluton p { f l } = that does not satsfy the cost constrant * ( ( n ) = can e deterned f * * * * * * * ( p ({ f l } n ) = n ( p( k) ){ f ( k) l ( k) } n ( k ( )) = = k l 36
46 CHAPTER 5 EXAMPLE The relalty of electronc devces s of a crtcal concern especally for ltary aerospace and councaton applcatons. LEDs (lght ettng dodes) are consdered a good lght source for optcal lnks wth good teperature dependence sall power consupton and hgh relalty. Snce LEDs are desgned to e n servce for several years wthout falure t s hard to oserve falures under noral operatng condtons n a short te. The relalty perforance of LEDs (Lght Ettng Dodes) has nearly always een superor to that of ncandescent neon and other type laps. In addton today s LED s have uch hgher relaltes than early LED devces. Iproved assely growth ethods and structures along wth new aterals have allowed for the developent and ass producton of extreely relale hgh rghtness LEDs n all colors ncludng whte. The expected useale lfete of an LED s usually estated y the extrapolaton of easured data or y estatng the value fro accelerated testng. Accelerated testng nvolves sujectng the LED to ore extree condtons (.e.: hgher teperature and/or hgher currents) than would e expected under noral operatng condtons. Ths s necessary snce t s often dffcult and practcal to actually test an LED for hours or over years. The an concern wth accelerated testng of LEDs s understandng how to accurately translate these results to noral operatng condtons. 37
47 The lfete of an LED s defned as the te t takes for the lght output to reach 5% of ts ntal value. The average lfete specfed y LED anufacturers s hours. Ths does not ean that the LED wll cease to operate after K hours; n fact ost LEDs wll functon for thousands of hours eyond the specfed lfete value. It eans that after hours the LED wll e half as rght as ts ntal lunosty level. In ths chapter the applcalty of the proposed odel s deonstrated y a nuercal exaple. 5. Sulaton Experent The purpose of the sulaton experent s to generate the data that would e used to estate the relalty of LEDs (type GaAlAs) at noral operatng condton wth teperature S = 78 K (5 C) y usng the degradaton data otaned at the three accelerated stress levels S = 98 K (5 C) S = 338 K (65 C) S 3 = 378 K (5 C). The data for twenty fve LEDs were sulated at each of these three teperatures. The duraton of each cycle (Sulaton run) s 336 hours and the total nuer of cycles s 9. Each cycle represents an nspecton nterval. Let ω (t) denote the oserved standardzed lght ntensty of the jth LED at te t j under S. The data s sulated n Matla y assung the rando varale β follows a recprocal Weull dstruton. By usng the Arrhenus relatonshp etween teperature and te the degradaton data was generated at the three stress levels S S and S 3. The data represents the standardzed lght ntensty of each coponent at a partcular te.65 t.the resultng data s gven n tales Fgure 5. shows the sulated saple j 38
48 degradaton paths of 5 LEDs. Fgure 5. s the plots of ω (t) versus j.65 t under S S and S 3. It s seen fro the fgure that there exsts a lnear relatonshp etween ω (t) andt. 65 s gven y: j.65 ω ( t) = β t + ( t) (5.) j j j Where (t) s the error ter. j 39
49 4 Tale 5.: The Sulated Standardzed Saple Degradaton Paths at Stress Level S Te (hr) w (t) w (t) w 3(t) w 4(t) w 5(t) w 6(t) w 7(t) w 8(t) w 9(t) w (t)
50 4 Tale 5.: The Sulated Standardzed Saple Degradaton Paths at Stress Level S Te (hr) w (t) w (t) w 3(t) w 4(t) w 5(t) w 6(t) w 7(t) w 8(t) w 9(t) w (t)
51 4 Tale 5.3: The Sulated Standardzed Saple Degradaton Paths at Stress Level S 3 Te (hr) w 3(t) w 3(t) w 33(t) w 34(t) w 35(t) w 36(t) w 37(t) w 38(t) w 39(t) w 3(t)
52 wj(t) -E-4-4E-4-6E-4-8E-4 -E-3 -.E-3 -.4E te(hour) -E- -4E- -6E- -8E- -E- -.E- -.4E (a) te(hour) () -5E E -8 w3j(t) -.5E -8 -E E -8-3E -8 t e (h o u r) (c) Fgure 5.. The Standardzed Saple Degradaton Paths under (a) S () S and (c) S 3 43
53 -E E wj(t) -6E-4-8E-4 -E-3 -.E-3 -.4E-3 t^.65 (a) wj(t) -E- -4E E- -8E- -E- -.E- -.4E t^ () -5E-9 -E w3j(t) -.5E-8 -E-8 -.5E-8-3E-8 t^.65 (c) Fgure 5.. The Plots of ω j (t) versus t.65 under (a) S () S and (c) S 3 44
54 Based on the oservatons {( ( ))} 3 k j t k t ω the LSEs βˆ j and σˆ k = j can e coputed. To ake sure of the approprateness of the Weull-dstruton Weull proalty plots were constructed for each hgher stress level (Fgure 5.3). All of the trends appear lnear aout the reference lne. Fgure 5.4 shows the noral proalty plots for the resduals under S S and S 3. The plots ndcate that the dstruton assuptons for β and (t) are reasonale. Fro Equatons and 4. we have ˆ σ = ˆ =.68 and the Arrhenus relatonshp s gven y: ˆ γ = ˆ γ + (5.) S uˆ Where ˆ γ =.9977 and ˆ γ = To otan the optal test plan for the ADT of LED we need the actual values ofσ and ( γ γ ). For convenence these estates are treated as the true values to evaluate the optal test plan of LED data. 45
55 (a) () (c) Fgure 5.3 The Weull Proalty Plot for { β } 3 ˆ j 5 j= 46
56 (a) () (c) Fgure 5.4 The Noral Proalty Plots for Resduals under (a) S () S (c) S 3 47
System in Weibull Distribution
Internatonal Matheatcal Foru 4 9 no. 9 94-95 Relablty Equvalence Factors of a Seres-Parallel Syste n Webull Dstrbuton M. A. El-Dacese Matheatcs Departent Faculty of Scence Tanta Unversty Tanta Egypt eldacese@yahoo.co
More informationExcess Error, Approximation Error, and Estimation Error
E0 370 Statstcal Learnng Theory Lecture 10 Sep 15, 011 Excess Error, Approxaton Error, and Estaton Error Lecturer: Shvan Agarwal Scrbe: Shvan Agarwal 1 Introducton So far, we have consdered the fnte saple
More informationPROBABILITY AND STATISTICS Vol. III - Analysis of Variance and Analysis of Covariance - V. Nollau ANALYSIS OF VARIANCE AND ANALYSIS OF COVARIANCE
ANALYSIS OF VARIANCE AND ANALYSIS OF COVARIANCE V. Nollau Insttute of Matheatcal Stochastcs, Techncal Unversty of Dresden, Gerany Keywords: Analyss of varance, least squares ethod, odels wth fxed effects,
More informationXII.3 The EM (Expectation-Maximization) Algorithm
XII.3 The EM (Expectaton-Maxzaton) Algorth Toshnor Munaata 3/7/06 The EM algorth s a technque to deal wth varous types of ncoplete data or hdden varables. It can be appled to a wde range of learnng probles
More informationBAYESIAN CURVE FITTING USING PIECEWISE POLYNOMIALS. Dariusz Biskup
BAYESIAN CURVE FITTING USING PIECEWISE POLYNOMIALS Darusz Bskup 1. Introducton The paper presents a nonparaetrc procedure for estaton of an unknown functon f n the regresson odel y = f x + ε = N. (1) (
More informationStatistical analysis of Accelerated life testing under Weibull distribution based on fuzzy theory
Statstcal analyss of Accelerated lfe testng under Webull dstrbuton based on fuzzy theory Han Xu, Scence & Technology on Relablty & Envronental Engneerng Laboratory, School of Relablty and Syste Engneerng,
More informationChapter 12 Lyes KADEM [Thermodynamics II] 2007
Chapter 2 Lyes KDEM [Therodynacs II] 2007 Gas Mxtures In ths chapter we wll develop ethods for deternng therodynac propertes of a xture n order to apply the frst law to systes nvolvng xtures. Ths wll be
More informationLeast Squares Fitting of Data
Least Squares Fttng of Data Davd Eberly Geoetrc Tools, LLC http://www.geoetrctools.co/ Copyrght c 1998-2014. All Rghts Reserved. Created: July 15, 1999 Last Modfed: February 9, 2008 Contents 1 Lnear Fttng
More informationx yi In chapter 14, we want to perform inference (i.e. calculate confidence intervals and perform tests of significance) in this setting.
The Practce of Statstcs, nd ed. Chapter 14 Inference for Regresson Introducton In chapter 3 we used a least-squares regresson lne (LSRL) to represent a lnear relatonshp etween two quanttatve explanator
More informationITERATIVE ESTIMATION PROCEDURE FOR GEOSTATISTICAL REGRESSION AND GEOSTATISTICAL KRIGING
ESE 5 ITERATIVE ESTIMATION PROCEDURE FOR GEOSTATISTICAL REGRESSION AND GEOSTATISTICAL KRIGING Gven a geostatstcal regresson odel: k Y () s x () s () s x () s () s, s R wth () unknown () E[ ( s)], s R ()
More informationwhere I = (n x n) diagonal identity matrix with diagonal elements = 1 and off-diagonal elements = 0; and σ 2 e = variance of (Y X).
11.4.1 Estmaton of Multple Regresson Coeffcents In multple lnear regresson, we essentally solve n equatons for the p unnown parameters. hus n must e equal to or greater than p and n practce n should e
More informationLeast Squares Fitting of Data
Least Squares Fttng of Data Davd Eberly Geoetrc Tools, LLC http://www.geoetrctools.co/ Copyrght c 1998-2015. All Rghts Reserved. Created: July 15, 1999 Last Modfed: January 5, 2015 Contents 1 Lnear Fttng
More informationMultipoint Analysis for Sibling Pairs. Biostatistics 666 Lecture 18
Multpont Analyss for Sblng ars Bostatstcs 666 Lecture 8 revously Lnkage analyss wth pars of ndvduals Non-paraetrc BS Methods Maxu Lkelhood BD Based Method ossble Trangle Constrant AS Methods Covered So
More informationWhat is LP? LP is an optimization technique that allocates limited resources among competing activities in the best possible manner.
(C) 998 Gerald B Sheblé, all rghts reserved Lnear Prograng Introducton Contents I. What s LP? II. LP Theor III. The Splex Method IV. Refneents to the Splex Method What s LP? LP s an optzaton technque that
More informationApplied Mathematics Letters
Appled Matheatcs Letters 2 (2) 46 5 Contents lsts avalable at ScenceDrect Appled Matheatcs Letters journal hoepage: wwwelseverco/locate/al Calculaton of coeffcents of a cardnal B-splne Gradr V Mlovanovć
More information1 Definition of Rademacher Complexity
COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture #9 Scrbe: Josh Chen March 5, 2013 We ve spent the past few classes provng bounds on the generalzaton error of PAClearnng algorths for the
More informationEstimation of Reliability in Multicomponent Stress-Strength Based on Generalized Rayleigh Distribution
Journal of Modern Appled Statstcal Methods Volue 13 Issue 1 Artcle 4 5-1-014 Estaton of Relablty n Multcoponent Stress-Strength Based on Generalzed Raylegh Dstrbuton Gadde Srnvasa Rao Unversty of Dodoa,
More informationReliability estimation in Pareto-I distribution based on progressively type II censored sample with binomial removals
Journal of Scentfc esearch Developent (): 08-3 05 Avalable onlne at wwwjsradorg ISSN 5-7569 05 JSAD elablty estaton n Pareto-I dstrbuton based on progressvely type II censored saple wth bnoal reovals Ilhan
More information1.3 Hence, calculate a formula for the force required to break the bond (i.e. the maximum value of F)
EN40: Dynacs and Vbratons Hoework 4: Work, Energy and Lnear Moentu Due Frday March 6 th School of Engneerng Brown Unversty 1. The Rydberg potental s a sple odel of atoc nteractons. It specfes the potental
More informationEstimation in Step-stress Partially Accelerated Life Test for Exponentiated Pareto Distribution under Progressive Censoring with Random Removal
Journal of Advances n Matheatcs and Coputer Scence 5(): -6, 07; Artcle no.jamcs.3469 Prevously known as Brtsh Journal of Matheatcs & Coputer Scence ISSN: 3-085 Estaton n Step-stress Partally Accelerated
More informationDescription of the Force Method Procedure. Indeterminate Analysis Force Method 1. Force Method con t. Force Method con t
Indeternate Analyss Force Method The force (flexblty) ethod expresses the relatonshps between dsplaceents and forces that exst n a structure. Prary objectve of the force ethod s to deterne the chosen set
More informationElastic Collisions. Definition: two point masses on which no external forces act collide without losing any energy.
Elastc Collsons Defnton: to pont asses on hch no external forces act collde thout losng any energy v Prerequstes: θ θ collsons n one denson conservaton of oentu and energy occurs frequently n everyday
More informationDenote the function derivatives f(x) in given points. x a b. Using relationships (1.2), polynomials (1.1) are written in the form
SET OF METHODS FO SOUTION THE AUHY POBEM FO STIFF SYSTEMS OF ODINAY DIFFEENTIA EUATIONS AF atypov and YuV Nulchev Insttute of Theoretcal and Appled Mechancs SB AS 639 Novosbrs ussa Introducton A constructon
More informationPARAMETER ESTIMATION IN WEIBULL DISTRIBUTION ON PROGRESSIVELY TYPE- II CENSORED SAMPLE WITH BETA-BINOMIAL REMOVALS
Econoy & Busness ISSN 1314-7242, Volue 10, 2016 PARAMETER ESTIMATION IN WEIBULL DISTRIBUTION ON PROGRESSIVELY TYPE- II CENSORED SAMPLE WITH BETA-BINOMIAL REMOVALS Ilhan Usta, Hanef Gezer Departent of Statstcs,
More informationFermi-Dirac statistics
UCC/Physcs/MK/EM/October 8, 205 Fer-Drac statstcs Fer-Drac dstrbuton Matter partcles that are eleentary ostly have a type of angular oentu called spn. hese partcles are known to have a agnetc oent whch
More informationPsychology 282 Lecture #24 Outline Regression Diagnostics: Outliers
Psychology 282 Lecture #24 Outlne Regresson Dagnostcs: Outlers In an earler lecture we studed the statstcal assumptons underlyng the regresson model, ncludng the followng ponts: Formal statement of assumptons.
More informationEcon107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4)
I. Classcal Assumptons Econ7 Appled Econometrcs Topc 3: Classcal Model (Studenmund, Chapter 4) We have defned OLS and studed some algebrac propertes of OLS. In ths topc we wll study statstcal propertes
More informationChapter 11: Simple Linear Regression and Correlation
Chapter 11: Smple Lnear Regresson and Correlaton 11-1 Emprcal Models 11-2 Smple Lnear Regresson 11-3 Propertes of the Least Squares Estmators 11-4 Hypothess Test n Smple Lnear Regresson 11-4.1 Use of t-tests
More information1 Review From Last Time
COS 5: Foundatons of Machne Learnng Rob Schapre Lecture #8 Scrbe: Monrul I Sharf Aprl 0, 2003 Revew Fro Last Te Last te, we were talkng about how to odel dstrbutons, and we had ths setup: Gven - exaples
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More informationHandling Overload (G. Buttazzo, Hard Real-Time Systems, Ch. 9) Causes for Overload
PS-663: Real-Te Systes Handlng Overloads Handlng Overload (G Buttazzo, Hard Real-Te Systes, h 9) auses for Overload Bad syste desgn eg poor estaton of worst-case executon tes Sultaneous arrval of unexpected
More informationKernel Methods and SVMs Extension
Kernel Methods and SVMs Extenson The purpose of ths document s to revew materal covered n Machne Learnng 1 Supervsed Learnng regardng support vector machnes (SVMs). Ths document also provdes a general
More informationFall 2012 Analysis of Experimental Measurements B. Eisenstein/rev. S. Errede. ) with a symmetric Pcovariance matrix of the y( x ) measurements V
Fall Analyss o Experental Measureents B Esensten/rev S Errede General Least Squares wth General Constrants: Suppose we have easureents y( x ( y( x, y( x,, y( x wth a syetrc covarance atrx o the y( x easureents
More informationOur focus will be on linear systems. A system is linear if it obeys the principle of superposition and homogenity, i.e.
SSTEM MODELLIN In order to solve a control syste proble, the descrptons of the syste and ts coponents ust be put nto a for sutable for analyss and evaluaton. The followng ethods can be used to odel physcal
More informationComputational and Statistical Learning theory Assignment 4
Coputatonal and Statstcal Learnng theory Assgnent 4 Due: March 2nd Eal solutons to : karthk at ttc dot edu Notatons/Defntons Recall the defnton of saple based Radeacher coplexty : [ ] R S F) := E ɛ {±}
More informationIntroducing Entropy Distributions
Graubner, Schdt & Proske: Proceedngs of the 6 th Internatonal Probablstc Workshop, Darstadt 8 Introducng Entropy Dstrbutons Noel van Erp & Peter van Gelder Structural Hydraulc Engneerng and Probablstc
More informationRichard Socher, Henning Peters Elements of Statistical Learning I E[X] = arg min. E[(X b) 2 ]
1 Prolem (10P) Show that f X s a random varale, then E[X] = arg mn E[(X ) 2 ] Thus a good predcton for X s E[X] f the squared dfference s used as the metrc. The followng rules are used n the proof: 1.
More informationBAYESIAN AND NON BAYESIAN ESTIMATION OF ERLANG DISTRIBUTION UNDER PROGRESSIVE CENSORING
www.arpapress.co/volues/volissue3/ijrras 3_8.pdf BAYESIAN AND NON BAYESIAN ESTIMATION OF ERLANG DISTRIBUTION UNDER PROGRESSIVE CENSORING R.A. Bakoban Departent of Statstcs, Scences Faculty for Grls, Kng
More informationOn Pfaff s solution of the Pfaff problem
Zur Pfaff scen Lösung des Pfaff scen Probles Mat. Ann. 7 (880) 53-530. On Pfaff s soluton of te Pfaff proble By A. MAYER n Lepzg Translated by D. H. Delpenc Te way tat Pfaff adopted for te ntegraton of
More information3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
More informationChapter One Mixture of Ideal Gases
herodynacs II AA Chapter One Mxture of Ideal Gases. Coposton of a Gas Mxture: Mass and Mole Fractons o deterne the propertes of a xture, we need to now the coposton of the xture as well as the propertes
More informationChapter 13. Gas Mixtures. Study Guide in PowerPoint. Thermodynamics: An Engineering Approach, 5th edition by Yunus A. Çengel and Michael A.
Chapter 3 Gas Mxtures Study Gude n PowerPont to accopany Therodynacs: An Engneerng Approach, 5th edton by Yunus A. Çengel and Mchael A. Boles The dscussons n ths chapter are restrcted to nonreactve deal-gas
More informationIntegral Transforms and Dual Integral Equations to Solve Heat Equation with Mixed Conditions
Int J Open Probles Copt Math, Vol 7, No 4, Deceber 214 ISSN 1998-6262; Copyrght ICSS Publcaton, 214 www-csrsorg Integral Transfors and Dual Integral Equatons to Solve Heat Equaton wth Mxed Condtons Naser
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analyss of Varance and Desgn of Experment-I MODULE VII LECTURE - 3 ANALYSIS OF COVARIANCE Dr Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur Any scentfc experment s performed
More informationDetermination of the Confidence Level of PSD Estimation with Given D.O.F. Based on WELCH Algorithm
Internatonal Conference on Inforaton Technology and Manageent Innovaton (ICITMI 05) Deternaton of the Confdence Level of PSD Estaton wth Gven D.O.F. Based on WELCH Algorth Xue-wang Zhu, *, S-jan Zhang
More informationLECTURE :FACTOR ANALYSIS
LCUR :FACOR ANALYSIS Rta Osadchy Based on Lecture Notes by A. Ng Motvaton Dstrbuton coes fro MoG Have suffcent aount of data: >>n denson Use M to ft Mture of Gaussans nu. of tranng ponts If
More informationCollaborative Filtering Recommendation Algorithm
Vol.141 (GST 2016), pp.199-203 http://dx.do.org/10.14257/astl.2016.141.43 Collaboratve Flterng Recoendaton Algorth Dong Lang Qongta Teachers College, Haou 570100, Chna, 18689851015@163.co Abstract. Ths
More informationDesigning Fuzzy Time Series Model Using Generalized Wang s Method and Its application to Forecasting Interest Rate of Bank Indonesia Certificate
The Frst Internatonal Senar on Scence and Technology, Islac Unversty of Indonesa, 4-5 January 009. Desgnng Fuzzy Te Seres odel Usng Generalzed Wang s ethod and Its applcaton to Forecastng Interest Rate
More information2016 Wiley. Study Session 2: Ethical and Professional Standards Application
6 Wley Study Sesson : Ethcal and Professonal Standards Applcaton LESSON : CORRECTION ANALYSIS Readng 9: Correlaton and Regresson LOS 9a: Calculate and nterpret a sample covarance and a sample correlaton
More informationGlobal Sensitivity. Tuesday 20 th February, 2018
Global Senstvty Tuesday 2 th February, 28 ) Local Senstvty Most senstvty analyses [] are based on local estmates of senstvty, typcally by expandng the response n a Taylor seres about some specfc values
More informationhalftoning Journal of Electronic Imaging, vol. 11, no. 4, Oct Je-Ho Lee and Jan P. Allebach
olorant-based drect bnary search» halftonng Journal of Electronc Iagng, vol., no. 4, Oct. 22 Je-Ho Lee and Jan P. Allebach School of Electrcal Engneerng & oputer Scence Kyungpook Natonal Unversty Abstract
More informationLecture Notes on Linear Regression
Lecture Notes on Lnear Regresson Feng L fl@sdueducn Shandong Unversty, Chna Lnear Regresson Problem In regresson problem, we am at predct a contnuous target value gven an nput feature vector We assume
More informationStatistics for Economics & Business
Statstcs for Economcs & Busness Smple Lnear Regresson Learnng Objectves In ths chapter, you learn: How to use regresson analyss to predct the value of a dependent varable based on an ndependent varable
More informationCOS 511: Theoretical Machine Learning
COS 5: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture #0 Scrbe: José Sões Ferrera March 06, 203 In the last lecture the concept of Radeacher coplexty was ntroduced, wth the goal of showng that
More informationAN ANALYSIS OF A FRACTAL KINETICS CURVE OF SAVAGEAU
AN ANALYI OF A FRACTAL KINETIC CURE OF AAGEAU by John Maloney and Jack Hedel Departent of Matheatcs Unversty of Nebraska at Oaha Oaha, Nebraska 688 Eal addresses: aloney@unoaha.edu, jhedel@unoaha.edu Runnng
More informationStatistics II Final Exam 26/6/18
Statstcs II Fnal Exam 26/6/18 Academc Year 2017/18 Solutons Exam duraton: 2 h 30 mn 1. (3 ponts) A town hall s conductng a study to determne the amount of leftover food produced by the restaurants n the
More informationPolynomial Regression Models
LINEAR REGRESSION ANALYSIS MODULE XII Lecture - 6 Polynomal Regresson Models Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur Test of sgnfcance To test the sgnfcance
More informationAn Optimal Bound for Sum of Square Roots of Special Type of Integers
The Sxth Internatonal Syposu on Operatons Research and Its Applcatons ISORA 06 Xnang, Chna, August 8 12, 2006 Copyrght 2006 ORSC & APORC pp. 206 211 An Optal Bound for Su of Square Roots of Specal Type
More informationBayesian Planning of Hit-Miss Inspection Tests
Bayesan Plannng of Ht-Mss Inspecton Tests Yew-Meng Koh a and Wllam Q Meeker a a Center for Nondestructve Evaluaton, Department of Statstcs, Iowa State Unversty, Ames, Iowa 5000 Abstract Although some useful
More informationScattering by a perfectly conducting infinite cylinder
Scatterng by a perfectly conductng nfnte cylnder Reeber that ths s the full soluton everywhere. We are actually nterested n the scatterng n the far feld lt. We agan use the asyptotc relatonshp exp exp
More informationThe Parity of the Number of Irreducible Factors for Some Pentanomials
The Party of the Nuber of Irreducble Factors for Soe Pentanoals Wolfra Koepf 1, Ryul K 1 Departent of Matheatcs Unversty of Kassel, Kassel, F. R. Gerany Faculty of Matheatcs and Mechancs K Il Sung Unversty,
More informationSmall-Sample Equating With Prior Information
Research Report Sall-Saple Equatng Wth Pror Inforaton Sauel A Lvngston Charles Lews June 009 ETS RR-09-5 Lstenng Learnng Leadng Sall-Saple Equatng Wth Pror Inforaton Sauel A Lvngston and Charles Lews ETS,
More informationx i1 =1 for all i (the constant ).
Chapter 5 The Multple Regresson Model Consder an economc model where the dependent varable s a functon of K explanatory varables. The economc model has the form: y = f ( x,x,..., ) xk Approxmate ths by
More informationRevision: December 13, E Main Suite D Pullman, WA (509) Voice and Fax
.9.1: AC power analyss Reson: Deceber 13, 010 15 E Man Sute D Pullan, WA 99163 (509 334 6306 Voce and Fax Oerew n chapter.9.0, we ntroduced soe basc quanttes relate to delery of power usng snusodal sgnals.
More informationChapter 8 Indicator Variables
Chapter 8 Indcator Varables In general, e explanatory varables n any regresson analyss are assumed to be quanttatve n nature. For example, e varables lke temperature, dstance, age etc. are quanttatve n
More informationStructure and Drive Paul A. Jensen Copyright July 20, 2003
Structure and Drve Paul A. Jensen Copyrght July 20, 2003 A system s made up of several operatons wth flow passng between them. The structure of the system descrbes the flow paths from nputs to outputs.
More information4 Analysis of Variance (ANOVA) 5 ANOVA. 5.1 Introduction. 5.2 Fixed Effects ANOVA
4 Analyss of Varance (ANOVA) 5 ANOVA 51 Introducton ANOVA ANOVA s a way to estmate and test the means of multple populatons We wll start wth one-way ANOVA If the populatons ncluded n the study are selected
More informationPHYS 1443 Section 002 Lecture #20
PHYS 1443 Secton 002 Lecture #20 Dr. Jae Condtons for Equlbru & Mechancal Equlbru How to Solve Equlbru Probles? A ew Exaples of Mechancal Equlbru Elastc Propertes of Solds Densty and Specfc Gravty lud
More informationLINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity
LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 30 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 2 Remedes for multcollnearty Varous technques have
More informationComparison of the Population Variance Estimators. of 2-Parameter Exponential Distribution Based on. Multiple Criteria Decision Making Method
Appled Mathematcal Scences, Vol. 7, 0, no. 47, 07-0 HIARI Ltd, www.m-hkar.com Comparson of the Populaton Varance Estmators of -Parameter Exponental Dstrbuton Based on Multple Crtera Decson Makng Method
More informationComparison of Regression Lines
STATGRAPHICS Rev. 9/13/2013 Comparson of Regresson Lnes Summary... 1 Data Input... 3 Analyss Summary... 4 Plot of Ftted Model... 6 Condtonal Sums of Squares... 6 Analyss Optons... 7 Forecasts... 8 Confdence
More informationOutline. Prior Information and Subjective Probability. Subjective Probability. The Histogram Approach. Subjective Determination of the Prior Density
Outlne Pror Inforaton and Subjectve Probablty u89603 1 Subjectve Probablty Subjectve Deternaton of the Pror Densty Nonnforatve Prors Maxu Entropy Prors Usng the Margnal Dstrbuton to Deterne the Pror Herarchcal
More informationStudy of the possibility of eliminating the Gibbs paradox within the framework of classical thermodynamics *
tudy of the possblty of elnatng the Gbbs paradox wthn the fraework of classcal therodynacs * V. Ihnatovych Departent of Phlosophy, Natonal echncal Unversty of Ukrane Kyv Polytechnc Insttute, Kyv, Ukrane
More informationRELIABILITY ASSESSMENT
CHAPTER Rsk Analyss n Engneerng and Economcs RELIABILITY ASSESSMENT A. J. Clark School of Engneerng Department of Cvl and Envronmental Engneerng 4a CHAPMAN HALL/CRC Rsk Analyss for Engneerng Department
More informationLinear Multiple Regression Model of High Performance Liquid Chromatography
Lnear Multple Regresson Model of Hgh Perforance Lqud Chroatography STANISLAVA LABÁTOVÁ Insttute of Inforatcs Departent of dscrete processes odelng and control Slovak Acadey of Scences Dúbravská 9, 845
More informationCOMP th April, 2007 Clement Pang
COMP 540 12 th Aprl, 2007 Cleent Pang Boostng Cobnng weak classers Fts an Addtve Model Is essentally Forward Stagewse Addtve Modelng wth Exponental Loss Loss Functons Classcaton: Msclasscaton, Exponental,
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationOptimal Marketing Strategies for a Customer Data Intermediary. Technical Appendix
Optal Marketng Strateges for a Custoer Data Interedary Techncal Appendx oseph Pancras Unversty of Connectcut School of Busness Marketng Departent 00 Hllsde Road, Unt 04 Storrs, CT 0669-04 oseph.pancras@busness.uconn.edu
More informationELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More informationBayesian estimation using MCMC approach based on progressive first-failure censoring from generalized Pareto distribution
Aercan Journal of Theoretcal and Appled Statstcs 03; (5): 8-4 Publshed onlne August 30 03 (http://www.scencepublshnggroup.co/j/ajtas) do: 0.648/j.ajtas.03005.3 Bayesan estaton usng MCMC approach based
More informationXiangwen Li. March 8th and March 13th, 2001
CS49I Approxaton Algorths The Vertex-Cover Proble Lecture Notes Xangwen L March 8th and March 3th, 00 Absolute Approxaton Gven an optzaton proble P, an algorth A s an approxaton algorth for P f, for an
More informationDepartment of Quantitative Methods & Information Systems. Time Series and Their Components QMIS 320. Chapter 6
Department of Quanttatve Methods & Informaton Systems Tme Seres and Ther Components QMIS 30 Chapter 6 Fall 00 Dr. Mohammad Zanal These sldes were modfed from ther orgnal source for educatonal purpose only.
More informationNegative Binomial Regression
STATGRAPHICS Rev. 9/16/2013 Negatve Bnomal Regresson Summary... 1 Data Input... 3 Statstcal Model... 3 Analyss Summary... 4 Analyss Optons... 7 Plot of Ftted Model... 8 Observed Versus Predcted... 10 Predctons...
More informationLinear Regression Analysis: Terminology and Notation
ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page ) Lnear Regresson Analyss: Termnology and Notaton Consder the generc verson of the smple (two-varable) lnear regresson model. It s represented
More informationThe Impact of the Earth s Movement through the Space on Measuring the Velocity of Light
Journal of Appled Matheatcs and Physcs, 6, 4, 68-78 Publshed Onlne June 6 n ScRes http://wwwscrporg/journal/jap http://dxdoorg/436/jap646 The Ipact of the Earth s Moeent through the Space on Measurng the
More informationWeek3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity
Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle
More informationEconomics 130. Lecture 4 Simple Linear Regression Continued
Economcs 130 Lecture 4 Contnued Readngs for Week 4 Text, Chapter and 3. We contnue wth addressng our second ssue + add n how we evaluate these relatonshps: Where do we get data to do ths analyss? How do
More informationCHAPTER 6 CONSTRAINED OPTIMIZATION 1: K-T CONDITIONS
Chapter 6: Constraned Optzaton CHAPER 6 CONSRAINED OPIMIZAION : K- CONDIIONS Introducton We now begn our dscusson of gradent-based constraned optzaton. Recall that n Chapter 3 we looked at gradent-based
More informationPredictive Analytics : QM901.1x Prof U Dinesh Kumar, IIMB. All Rights Reserved, Indian Institute of Management Bangalore
Sesson Outlne Introducton to classfcaton problems and dscrete choce models. Introducton to Logstcs Regresson. Logstc functon and Logt functon. Maxmum Lkelhood Estmator (MLE) for estmaton of LR parameters.
More informationOn the correction of the h-index for career length
1 On the correcton of the h-ndex for career length by L. Egghe Unverstet Hasselt (UHasselt), Campus Depenbeek, Agoralaan, B-3590 Depenbeek, Belgum 1 and Unverstet Antwerpen (UA), IBW, Stadscampus, Venusstraat
More informationQuantum Particle Motion in Physical Space
Adv. Studes Theor. Phys., Vol. 8, 014, no. 1, 7-34 HIKARI Ltd, www.-hkar.co http://dx.do.org/10.1988/astp.014.311136 Quantu Partcle Moton n Physcal Space A. Yu. Saarn Dept. of Physcs, Saara State Techncal
More informationCHAPT II : Prob-stats, estimation
CHAPT II : Prob-stats, estaton Randoness, probablty Probablty densty functons and cuulatve densty functons. Jont, argnal and condtonal dstrbutons. The Bayes forula. Saplng and statstcs Descrptve and nferental
More information,..., k N. , k 2. ,..., k i. The derivative with respect to temperature T is calculated by using the chain rule: & ( (5) dj j dt = "J j. k i.
Suppleentary Materal Dervaton of Eq. 1a. Assue j s a functon of the rate constants for the N coponent reactons: j j (k 1,,..., k,..., k N ( The dervatve wth respect to teperature T s calculated by usng
More informationLimited Dependent Variables
Lmted Dependent Varables. What f the left-hand sde varable s not a contnuous thng spread from mnus nfnty to plus nfnty? That s, gven a model = f (, β, ε, where a. s bounded below at zero, such as wages
More informationLINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity
LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 31 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 6. Rdge regresson The OLSE s the best lnear unbased
More informationHere is the rationale: If X and y have a strong positive relationship to one another, then ( x x) will tend to be positive when ( y y)
Secton 1.5 Correlaton In the prevous sectons, we looked at regresson and the value r was a measurement of how much of the varaton n y can be attrbuted to the lnear relatonshp between y and x. In ths secton,
More informationClass: Life-Science Subject: Physics
Class: Lfe-Scence Subject: Physcs Frst year (6 pts): Graphc desgn of an energy exchange A partcle (B) of ass =g oves on an nclned plane of an nclned angle α = 3 relatve to the horzontal. We want to study
More informationSignal-noise Ratio Recognition Algorithm Based on Singular Value Decomposition
4th Internatonal Conference on Machnery, Materals and Coputng Technology (ICMMCT 06) Sgnal-nose Rato Recognton Algorth Based on Sngular Value Decoposton Qao Y, a, Cu Qan, b, Zhang We, c and Lu Yan, d Bejng
More informationy new = M x old Feature Selection: Linear Transformations Constraint Optimization (insertion)
Feature Selecton: Lnear ransforatons new = M x old Constrant Optzaton (nserton) 3 Proble: Gven an objectve functon f(x) to be optzed and let constrants be gven b h k (x)=c k, ovng constants to the left,
More information, are assumed to fluctuate around zero, with E( i) 0. Now imagine that this overall random effect, , is composed of many independent factors,
Part II. Contnuous Spatal Data Analyss 3. Spatally-Dependent Rando Effects Observe that all regressons n the llustratons above [startng wth expresson (..3) n the Sudan ranfall exaple] have reled on an
More information