Optimal Maintenance Modeling and Data Analysis for HM-60 Machining Center. Michael Jong Kim

Transcription

1 Optmal Mantenance Modelng and Data Analyss for HM-6 Machnng Center Mchael Jong Km A thess submtted n partal fulfllment of the requrements for the degree of BACHELOR OF APPLIED SCIENCE Supervsor: V. Maks Department of Mechancal and Industral Engneerng Unversty of Toronto March, 27

2 Abstract The thess conssts of two parts. Part I ams to formulate a mathematcal model, whch wll be used to determne the optmal mantenance polcy for a stochastcally deteroratng system. A modfed polcy-teraton algorthm usng the embedded technque s developed as the computatonal algorthm and s used to determne the optmal mantenance polcy. Part II ams to provde a comprehensve revew of varous relablty data analyss methodologes and apply these methodologes to real data sets obtaned from the HM-6 Machnng Centers at the Comtech Company. Partcularly, the adequacy of the two and three-parameter Webull dstrbutons are tested on falure data sets correspondng to machnes 29, 215, and 246 at the Comtech Company.

3 Acknowledgments My deepest grattude and apprecaton goes out frst and foremost to my dearest famly: Shaddy, Hobby, Sammy, L, Jona, Mom, and Dad. Wthout your love and support ths thess, and all else, would not be possble. Of course, I would also lke to thank my supervsor Professor Vlam Maks. Workng wth you has been the academc hghlght of my undergraduate career and has nspred me to acheve much more n the years to come.

4 Table of Contents 1. Part I Introducton The Model The Embedded Algorthm Numercal Analyss Concluson Part II Introducton Data Analyss Methodologes The Three-Parameter Webull Dstrbuton Censored Data Models wth Covarates Parameter Estmatons Goodness-of-Ft Tests Applcaton to HM-6 Machnng Center Falure Data Prelmnary Data Analyss Parameter Estmatons Goodness-of-Ft Tests Concluson References Fgures and Tables Appendces.. 68 Appendx A Appendx B Appendx C Appendx D Appendx E Appendx F... 8 Appendx G... 82

5 Lst of Symbols O Set of operatonal states M Set of mnor falure states F Set of major falure states r (a) Repared state: startng n state, after acton a s carred out Expected tme the system wll reman n operatonal state when no repar s carred out p j (a) Transton probabltes d Operatng cost per unt tme b (a) Mean cost to carry out acton a t (a) Mean tme to carry out acton a c (a) Mean cost ncurred untl next decson epoch (a) Mean sojourn tme untl next decson epoch Z Set of all statonary polces E (z) Set of states n whch repars take place ( ) p E z j ( a) Probablty that the frst state the system wll enter n set E(z) s state j gven that the current state s ( ) c E z ( a) Expected cost ncurred untl the system enters some state n E(z) gven the current state s E( z) ( a) Expected tme untl the system enters some state n E(z) gven the current state s Ft () Cumulatve dstrbuton f() t Probablty densty functon () t Falure rate functon Rt () Relablty functon Scale parameter for Webull dstrbuton Shape parameter for Webull dstrbuton Locaton parameter for Webull dstrbuton t MTTF n r t * Mean tme to falure Lnk functon for covarate models Sample sze Number of observed falures (uncensored data ponts) Test termnaton tme (for Type I sngle censorng)

6 v t () th data pont n the ordered data set k smallest nteger greater than or equal to k MLE Maxmum lkelhood estmator L( t1,, tn 1,, k ) L( 1,, k ) Lkelhood functon wth respect to parameters 1,, k gven data t 1,, t k H H 1 Null hypothess Alternatve hypothess

7 v Lst of Fgures 1. Fgure 1.1 Natural Evoluton of the Process Fgure 2.1 Flexblty of the Webull Dstrbuton Fgure 2.2 Type I Sngle Censorng Fgure 2.3 Type II Sngle Censorng

8 v Lst of Tables 1. Table 1.1 Expected Costs and Tmes Table 1.2 Summary of Embedded Components Table 1.3 Summary of Polces After Each Iteraton Table 1.4 Expected Costs and Tmes (Second Numercal Example) Table 1.5 Summary of Embedded Components (Second Numercal Example) Ez ( ) 6. Table 1.6 Summary of embedded components: p ( z ) E( z1 ) 7. Table 1.7 Summary of embedded components: p ( z ) Table 1.8 Summary of Polces After Each Iteraton (Second Numercal Example) Table 2.1 Flexblty of the Webull Dstrbuton Table 2.2 Ch-Square Crtcal Values j j 1

9 Part I: Optmal Mantenance Modelng 1

10 Introducton Part I of the thess ams to extend and generalze the mathematcal formulaton presented n Optmal mantenance polcy for a mult-state deteroratng system wth two types of falure under general repar whch provded a general repar model for a multstate deteroratng system subject to a fnte number of mnor falures and a sngle major falure. Snce t s often very costly to repar or replace a faled system, preventve mantenance s usually carred out followng mantenance polces, whch specfy when repars should take place. The theory of Sem-Markov Decson Processes was used to model such deteroratng systems. A modfed polcy-teraton algorthm usng the embedded technque was developed as the computatonal approach used to calculate the optmal mantenance polcy. Part I of the thess s structured as follows. Secton 1.2 presents the mathematcal model, whch represents the deteroratng system. Secton 1.3 presents the modfed polcy-teraton algorthm usng the embedded technque. Secton 1.4 provdes numercal examples, whch llustrate the mplementaton of the computatonal approach. Secton 1.5 contans concludng remarks.

11 Model Formulaton In ths secton we formulate a mathematcal model of a mult-state deteroratng system that s subject to major and mnor falures. If a system experences a major falure t must be replaced mmedately. That s, upon a major system falure, dong nothng or reparng the system s not an opton. If the system experences a mnor falure t must be repared to the operatonal state t was functonng at just pror to falure. At each operatonal state of the system a decson s made whether to perform a general repar. The process s modeled as a SMDP. We denote the set of operatonal states as O {1,2,, N}. State 1 represents the state of a new system. The degree of deteroraton ncreases wth each subsequent operatonal state. We suppose that there are K types of mnor falures that can occur from any operatonal state. In real systems ths may correspond to the falure of any of K mnor components of a system as opposed to a major system falure. If a mnor falure of type occurs from operatonal state j, then the state of the system s represented by M. We denote the set of all mnor falure states as M { M 1,2,, K, j O}. There are also L major falure states denoted as F 1, F 2,, F L. The set of all major falure states s denoted as F { F1, F2,, F L }. The system can enter any falure state F l, l 1,2,, L, from any operatonal state. Thus, the state space s S OM F. Fgure 1.1 llustrates the relatonshp between operatonal and falure states. At each decson epoch, f the system s n some operatonal state and a general repar a takes place, the system s repared from state to a state a. Choosng a corresponds to performng no repar on the system. The acton space for each operatonal state s therefore gve as A( ) { a a 1}. If the system s n some mnor falure j j

12 4 state M, a repar R takes place whch repars the system back to operatonal state j. If j j the system s n the major falure state F l, l 1,2,, L, a repar Rl takes place whch repars the system back to operatonal state 1. Thus, A( F ) { R } and A( M ) { R }. For eachsafter repar a A() s carred out the system wll be n repared state l l j j a, f O r ( a) j, f M kj M 1, f F (1.1) It should be noted that for any operatonal state f we restrct A( ) {,1, 1}, the acton space reduces to the acton space defned by Moustafa et al [15]. If the system s n some state and general repar a A() s performed, the system wll be repared to state r ( a ). For eachsand a A() we defne pj ( a) as the probablty that the system wll be n state j at the next decson epoch gven that ts current state s and a repar a s performed. Thus, we have pj ( a) pr ( a), j (). Furthermore, for each par of operatonal states and j, pj () whenever j. We let represent the expected tme the system wll reman n operatonal state when no repar s carred out. When the system s n operatonal state the operatng cost per unt tme s denoted as d. For each statessthe mean cost and tme to carry out repar a A() are denoted as b( a) and t ( a ), respectvely. It s assumed that b() t (). We let m represent the dle cost per unt tme whle the system s undergong general repar. Thus, for each Sand a A() the mean cost ncurred untl the next decson epoch s c ( a) b ( a) mt ( a) d (1.2) r( a) r( a)

13 5 and the mean sojourn tme untl the next decson epoch s ( n) t ( a) (1.3) r ( a)

14 The Computatonal Approach Our optmalty crteron s to mnmze the long-run expected average cost per unt tme. We denote the set of all statonary polces by Z. Every statonary polcy zzs a functon that assgns to each state San acton z( ) A( ). To determne an optmal mantenance polcy for the formulated sem-markov decson process we use a modfed verson of the polcy-teraton algorthm n whch an embedded subset of the state space s defned whch attempts to decrease the sze of the lnear system durng the value-determnaton step. Ths can sgnfcantly reduce computatonal efforts for large state spaces. We frst revew the steps of the polcyteraton algorthm wthout the use of the embedded technque. The Polcy-Iteraton Algorthm Step : Intalzaton. Choose any statonary polcy z Z. Step 1: Value-determnaton step. Compute g and v, S, as the soluton to the followng system of lnear equatons v c ( z( )) g ( z( )) p ( z( )) v, S, j j js v, for some s S s (1.4) Step 2: Polcy-mprovement step. For each state S, usng the values g and v, S, obtaned n the prevous step, determne the acton a A() that mnmzes the expresson c ( a ) g ( a ) p ( a ) v (1.5) j j js The new polcy zzs obtaned by takng z() a for each S.

15 7 Step 3: Convergence test. If z z, the teraton stops and the optmal statonary polcy s z. Otherwse, return to Step 1 replacng z wth z. Under polcy zzdefne the embedded set of states as E( z) { S z( ) }. Thus Ezs () the set of all states n whch repars take place. To use the embedded technque we requre that E() z S and that () Ezbe accessble from every state S. Both of these clearly hold snce state1 Ez ( ) and M F E() z. We defne p Ez ( ) j () z as the probablty that the frst state the system wll enter n set Ezs () state j gven that the current state s and polcy z s used. If the ntal state s an element of Ez, () we take the frst state entered n the set Ezto () be the frst state entered n Ezupon () the next return to Ez. () To calculate p z, Ezs () thought of as a set of absorbng states and the Ez ( ) j () followng system of lnear equatons s solved p ( z) p () p () p ( z), E( z), j E( z) (1.6) E( z) E( z) j j k kj ke ( z) For the remanng states E() z we have p ( z) q ( z) p ( z),, j E( z) (1.7) E( z) E( z) j k kj ke ( z) where q ( z), E( z), j E( z) s the soluton to the followng system of equatons j q ( z) p ( z( )) p ( z( )) q ( z), E( z), j E( z) (1.8) j j k kj ke ( z) E( z Next we defne ) E( z c ( z) and ) ( z) as the expected cost ncurred and expected tme untl the system enters some state n Ezgven () the current state s and polcy z s used. Then

16 8 E( z for each E() z we compute ) E( z c ( z) and ) ( z) by solvng the followng system of equatons E( z) E( z) j j je ( z) c ( z) a p () c ( z) (1.9) ( z) p () ( z) (1.1) E( z) E( z) j j je ( z) For the remanng states E() z f r ( z( )) E( z) c ( z) c ( z) ( b ( z( )) mt ( z( ))) (1.11) E( z) E( z) r ( z) ( z) ( z) t ( z( )) (1.12) E( z) E( z) r ( z) and f r ( z( )) E( z) we solve the system of equatons c ( z) c ( z( )) p () c ( z) (1.13) E( z) E( z) r ( z( )), j j js ( z) ( z( )) p () ( z) (1.14) E( z) E( z) r ( z( )), j j js Tjms [2] showed that once Ez, () p z, Ez ( ) j () ( ) E( z c ( z) and ) ( z) were defned, system of E z lnear equatons defned n (1.4) can be replaced by v c ( z) g ( z) p ( z) v, E( z) E( z) E( z) E( z) j j je ( z) v, for some s E( z) s (1.15) To calculated v for the remanng states E() z requres smply substtutng values n the equatons v c ( z) g ( z) p ( z) v, E( z) (1.16) E( z) E( z) E( z) j j je ( z)

17 Numercal Examples The frst example s the same as the one presented by Moustafa et al. [15], whch s solved n ths paper by usng the embedded technque. The model presented n ths paper s an extenson of the model presented by Moustafa et al. [15] where only one major falure was consdered and only three types of mantenance decsons were allowed at each state of the system partcularly: do-nothng, mnmal mantenance or replacement. Thus, the use of the embedded technque to solve the numercal example presented by Moustafa et al. [15] should yeld dentcal results. Ths numercal example consders a state space wth 9 operatonal states (states 1 to 9), no mnor falure states, and one major falure state (state 1). Table 1.1 provdes the mean transton costs and tmes of the process. The natural evoluton of the system s descrbed by the one-step probablty matrx We start wth an ntal polcy z {,,,,,,,,, R}, whch performs no (1.17) repars n operatonal states. Thus, the embedded set of states s Ez ( ) {1}. Usng Ez ( ) E( z ) E z equatons (1.6)-(1.14) we obtan values for p ( z ), c ( z ) and ( z ) as j ( )

18 1 summarzed n Table 1.2. Carryng out the value-determnaton step as descrbed n subsecton 3.3, we solve the system of equatons n (15)-(16) and obtan v , v , v , v , v , v , v , v8 17.3, v , v1, g After carryng out the polcymprovement step usng equaton (1.5) we get a new polcy z 1 {,1,1,1,1, R, R, R, R, R} z. Thus, we return to Step 1 takng the ntal polcy to be z 1. The polcy obtaned after every teraton s summarzed n the Table 1.3. We obtan a fnal optmal polcy of z* {,1,1,1,1,1,1, R, R, R} and a long run expected average cost per unt tme of g Ths s dentcal to the optmal polcy and optmal expected cost rate obtaned by Moustafa et al. [15]. In the next numercal example we determne the optmal mantenance polcy usng the embedded technque for a system wth state space S {1,2,3, M, M, M, M, M, M, F}. In ths example, F denotes a sngle major falure state as opposed to the set of all major falure states. We also determne the optmal polcy usng the standard polcy-teraton algorthm (wthout the use of the embedded technque) to valdate the results obtaned when usng embedded technque. The computatonal tmes both algorthms are provded at the end of ths secton. Table 1.4 provdes the mean transton costs and tmes of the process. The evoluton of the system s descrbed by the one-step transton probablty matrx

19 We start wth an ntal polcy z {,,, R 11, R21, R12, R22, R13, R23, R}, whch (1.18) performs no repars n operatonal states. Thus, the embedded set of states s E( z ) { M, M, M, M, M, M, F}. Usng equatons (1.6)-(1.14) we obtan Ez ( ) E( z ) E z values for p ( z ), c ( z ) and ( z ) as summarzed n Tables Carryng j ( ) out the value-determnaton step as descrbed n subsecton 3.3, we solve the system of equatons n (1.15)-(1.16). Choosng vf we obtan v , v2 5.52, v3 46.8, vm 448.1, v M, v M, v M, 22 v , vm , g After carryng out the polcy-mprovement step 23 M13 usng equaton (1.5) we get a new polcy z 1 {,1,2, R 11, R 21, R 12, R 22, R 13, R 23, R} z. Thus, we return to Step 1 takng the ntal polcy to be z 1. The polces obtaned after each teraton are summarzed n Table 1.8. We obtan a fnal optmal long-run expected average cost per unt tme of g The computatonal tme for runnng the modfed polcy-teraton algorthm (wth the use of the embedded technque) was.3 seconds usng MATLAB verson 7.2.

20 12 Next we compute the optmal polcy usng the standard polcy-teraton algorthm wthout the use of the embedded technque. We start wth the same ntal polcy z {,,, R, R, R, R, R, R, R}, whch performs no repars n operatonal states. Table 1.9 summarzes the values of v and g obtaned durng each teraton as well as the new polcy obtaned at the end of the teraton. We see that the results obtaned when usng the polcy-teraton algorthm wthout the use of the embedded technque, are consstent wth the results obtaned when the embedded technque s used. In both cases, the optmal mantenance polcy s gven as z 1 {,1,2, R 11, R 21, R 12, R 22, R 13, R 23, R} z wth an optmal cost rate of g The computatonal tme for runnng the standard polcy-teraton algorthm (wthout the use of the embedded technque) was also.3 seconds usng MATLAB verson 7.2.

21 Concluson Part I of the thess presented a general repar model for a mult-state deteroratng system subject to a fnte number of major and mnor falures. The problem was formulated as a sem-markov decson process wth the optmalty crteron beng the mnmzaton of the long-run expected average cost per unt tme. A modfed polcyteraton algorthm usng the embedded technque was developed as the computatonal approach used to fnd the optmal mantenance polcy. Two numercal examples were presented. The frst numercal example appled the embedded technque to the numercal example presented by Moustafa et al. [15] and dentcal results were obtaned. The second numercal example compared the results and computatonal tmes when usng the embedded technque and polcy-teraton algorthm. It was found that dentcal mantenance polces and computatonal tmes were obtaned from the two algorthms.

22 Part II: Relablty Data Analyss Methodologes and Applcaton to HM-6 Machnng Center Falure Data 14

23 Introducton Part II of the thess provdes both a revew (Secton 2.2) and applcaton (Secton 2.3) of varous relablty data analyss methodologes. The methodologes are appled to the Comtech Company HM-6 Machnng Center falure data. In partcular, the falure data (.e. tmes between falures) correspondng to three dfferent machnes: M29, M215, and M246 wll be analyzed. Secton 2.2, the revew of the data analyss methodologes, s structured as follows. Secton ntroduces the three-parameter Webull dstrbuton. The data analyss n ths thess wll focus, for the most part, on ths probablty dstrbuton. Secton summarzes sngly rght censored (Type I and Type II) and complete data sets. Secton dscusses models wth covarates, specfcally the proportonal hazards model (PHM). Secton revews the method of maxmum lkelhood (MML) for the Webull dstrbuton and PHM. Secton presents varous goodness-of-ft tests, whch can be used to valdate the use of the Webull dstrbuton when censored and complete data s used. Secton 2.3, the applcaton of the methodologes to the HM-6 falure data, s structured as follows. Secton contans prelmnary data analyss technques usng computer-based software (BestFt and Mntab). In ths secton statstcal summares, hstograms, probablty plots, gve reasonable suggestons as to the types of dstrbutons that wll best ft the data. Secton apples the theory of maxmum lkelhood estmates on the three data sets. The estmates are compared to the prelmnary estmates found n Secton Secton performs goodness-of-ft tests (Ch-square, Mann s, and K-S) are to valdate the adequacy of the ftted dstrbutons for the three data sets.

24 Data Analyss Methodologes The Three-Parameter Webull Dstrbuton The Webull dstrbuton s one of the most useful dstrbutons n lfetme data analyss and relablty engneerng. Its usefulness comes from the flexblty of ts falure rate functon defned by 1 t ( t),, t (2.1) The parameters and are referred to as the shape and scale parameters, respectvely. Many mportant probablty dstrbutons such as the exponental dstrbuton or Raylegh dstrbuton are characterzed by the fact that ther correspondng falure rate functons are constant (CFR) or lnearly ncreasng (LFR), respectvely. As we see from equaton (2.1), dependng on the choce of the shape parameter, the falure rate functon of the Webull dstrbuton can be constant, lnear, ncreasng, and even decreasng 1. It follows, by defnton, that the relablty, cumulatve dstrbuton and probablty densty functons are gven as t R( t) exp ( s) ds exp t (2.2) F( t) 1 R( t) 1 exp t 1 t f ( t) ( t) R( t) exp t (2.3) (2.4) Sometmes, when a lower bound for falure data s known to exst, a thrd parametert s ntroduced whch effectvely shfts the dstrbuton byt unts. Ths parameter s aptly 1 See Table 2.1 and Fgure 2.1, whch llustrate the flexblty of the Webull dstrbuton.

25 17 referred to as the locaton parameter. Introducng ths parameter modfes equatons (2.1) (2.4), the Two Parameter Webull equatons, nto the Three Parameter Webull equatons () t 1 tt (2.1a) t t Rt ( ) exp (2.2a) t t Ft ( ) 1 exp (2.3a) 1 t t t t f( t) exp (2.4a) where,, and t t. The ntroducton of the locaton parameter assumes that no falures wll occur before tme t. Two mportant summary measures of relablty for any gven probablty dstrbuton are the mean tme to falure (MTTF) and the varance. The MTTF s defned by MTTF E( T) tf ( t) dt (2.5) The MTTF can also be equvalently defned as (See e.g. Ebelng [7]) MTTF R() t dt (2.5a) whch s often easer to compute 2. It was also shown [7] that the MTTF for the Webull dstrbuton s gven by the equaton 2 See Appendx B for the dervaton of equaton (2.5a).

26 18 1 MTTF t 1 (2.6) y x y e dy, s known as the gamma functon 3. The varance, whch s x1 where another measure used to descrbe a falure dstrbuton, s defned as 2 2 V[ T] ( t MTTF ) f ( t) dt (2.7) Lke the case of the MTTF t can be shown [7] that the varance can be equvalently defned as 2 2 t f () t dtmttf 2 (2.7a) for ease of computaton 4. The varance for the Webull dstrbuton was shown [7] to be gven by the equaton (2.8) 3 See Appendx C for the dervaton of equaton (2.6). 4 See Appendx D for the dervaton of equaton (2.7a). 5 See Appendx E for the dervaton of equaton (2.8).

27 Censored Data: Type I and Type II sngle rght censorng Gven a data set of sample sze n, f the falure tmes of all n tems are known, the data set s referred to as a complete data set. Sometmes t s not always practcal (or even possble) to record the exact falure tmes of all tems. When the exact tme to falure of only a strct subset of all the tems s known (say only r of n tems had an observed falure tme) the data set s referred to as a censored data set. Ths thess consders two types of censorng, Type I rght censorng (tme censorng) and Type II rght censorng (order statstc censorng). Type I censorng termnates the collecton of falure tmes after a predetermned amount of tme t * has elapsed. Usng ths type of censorng, the number of falure data values that wll be observed s a random varable. Type II censorng termnates the collecton of falure tmes after a predetermned number r of falure tmes are observed. That s to say the test s termnated at tme t ( r ), the tme of the rth falure. Usng ths type of censorng, the tme untl the test s termnated s a random varable rather than the number of observed data ponts. Fgure 2.2 and Fgure 2.3 llustrate the mplementaton of Type I and Type II censorng on a data set of eght tems.

28 Models wth Covarates: The Proportonal Hazards Model A covarate s any varable that affects the probablty characterstcs of a random q varablet representng falure tme of a gven tem. The vector zr s commonly used to denote a tuple of q covarates. Covarates nteract and affect probablty dstrbutons va a lnk functon : R q R. The lnk functon typcally q satsfes ( ) 1and ( z) for each zr. A proportonal hazards model (PHM) s a covarate model n whch the lnk functon s ncorporated nto the falure rate functon defned by the equaton ( t) ( z ) ( t), t (2.9) In ths model () t represents the baselne falure rate functon (.e. f no covarates were present). From equaton (2.9) t s clear that covarates n a PHM nteract wth the probablstc characterstcs of the populaton falure tmes by ncreasng the falure rate functon when ( z) 1and decreasng t when ( z ) 1. The lnk functon commonly takes the log lnear form defned by the equaton ( z) expβz (2.1) where β ( 1,, q )' s a vector of regresson coeffcents correspondng to the q covarates of vector z. It s clear from the equaton that ths partcular lnk functon satsfes ( ) 1and ( z ). If the covarates z are tme ndependent and baselne falure rate functon s Webull, the falure rate functon s gven by 1 tt ( ) ( ),, t z t t (2.11)

29 21 It follows drectly that the relablty, cumulatve dstrbuton and probablty densty functons are gven as t t R( t) exp ( ) t t z (2.12) t t F( t) 1 exp ( ) t t z (2.13) 1 t t t t f ( t) ( z) exp ( z ) t t (2.14) In equatons (2.11) (2.14) t s mportant to menton that the lnk functon () z, havng the log lnear form, depends not only on the values of the covarates z, but also on the unknown regresson parameters β ( 1,, q )' that typcally need to be estmated (dscussed n the followng secton).

30 Parameter Estmatons In ths secton we frst revew the method of maxmum lkelhood (MML) and then use the method to estmate the parameters of the Webull dstrbuton and the proportonal hazards model. Supposet 1, t 2,, t n s a sample of sze n. We want to fnd estmators for parameters for a supposed populaton dstrbuton, whch maxmze the lkelhood of observng ths sample. These maxmum lkelhood estmators, for the supposed probablty dstrbuton, are found by maxmzng the lkelhood functon L( t,, t,, ) f ( t,, ) (2.15) 1 n 1 k 1 k 1 n wth respect to parameters 1,, k. For short we represent L( t1,, tn 1,, k) as L( 1,, k ). It s usually easer to maxmze the natural logarthm of the lkelhood functon. Thus, solvng for the values of 1,, k whch maxmze the natural logarthm of the lkelhood functon requres solvng the followng system of equatons ln L( 1,, k ) 1,2,, k (2.16) Snce the natural logarthm s an ncreasng monotonc functon, the parameter values whch maxmze the natural logarthm of the lkelhood functon wll necessarly maxmze the lkelhood functon tself 6. For sngly censored (Type I and Type II) data, the lkelhood functon s easly modfed to account for the fact that only a lower bound wll be known for n r data ponts. We let U represents the set of uncensored 6 See Appendx F for a mathematcal argument of ths clam.

31 23 tems, C represents the set of censored tems, and θ ( 1,, k ) s the vector of dstrbuton parameters. The lkelhood functon for Type I data can be represented compactly as L( θ) f ( t θ) R( t* θ ) (2.17) U C and Cohen [3] showed that for Type II data the lkelhood functon of ordered data t (1),, t ( r ) can be represented as n! L( θ) f ( t( ) θ) R( t( r) θ ) (2.18) ( n r)! U C for t (1) t (2) t ( r). Ebelng [7] showed that the MLE for the shape parameter of the Webull dstrbuton s computed by solvng the equaton r ˆ ˆ t 1 t n r t s t s r ˆ ˆ t ( ) 1 n r t s ln ( ) ln r ˆ 1 1 g( ) ln t ˆ r 1 (2.19) In general equaton (2.19) cannot be solved drectly. Rather, a soluton to the equaton must be obtaned numercally. Ths can be acheved usng the Newton-Raphson method, whch teratvely solves for ˆ usng the followng recursve equaton ˆ ˆ ˆ g( j ) dg( x) j 1 j where g'( x) g '( ˆ ) dx (2.2) j Ebelng [7] then went on to show that once the estmator for the shape parameter s determned, the MLE for the scale parameter s solved through the equaton 1 for complete data (2.21) for Type IIdata 1 ˆ r ˆ ˆ ˆ 1 t ( n r) ts wherets t* for Type I data r 1 t(r)

32 24 To ntalze the Newton-Raphson method, Neter et al. [17] used the least squares estmaton method and found the ntal shape parameter estmator to be ˆ r 1 ( X X )( Y Y ) r 1 ( X X) 2 (2.22) where X ln x, Y n 1 ln ln, and X andy are the respectve means. The method nx for solvng the MLE for the locaton parameter s not well defned as s the case for the shape and scale parameters. Muraldhar et al. [16] developed an estmaton technque for the Webull locaton parameter and determned the best estmator (for complete data sets) to be gven by tˆ 2 t1t n t j t t 2t 1 n j (2.23) where j np (the smallest nteger greater than or equal to np ) and.3437 p.8829n. As was mentoned n secton 2.2.3, the proportonal hazards model s characterzed by the falure rate functon havng the form ( t) ( z ) ( t), t (2.24) where s a lnk functon whose arguments are covarates and () t s the baselne falure rate functon. For defnteness (and smplcty), n ths subsecton, the lnk functon s assumed to have the commonly used log lnear form ( z) expβz (2.25) where β s a vector of regresson coeffcents correspondng to the q covarates of vector z. In general however, the lnk functon need not be of the form n equaton (2.25).

33 25 In fact, lnk functon can be any functon whch satsfes ( ) 1and ( z) for each z. The purpose of ths subsecton s to estmate the values of the regresson coeffcent vector β from a data set consstng of n tems and r observed (uncensored) falure tmes. Let the vector z ( z 1, z2, zq )' be the covarates collected for each tem 1,2,, n. For complete data the lkelhood functon wth covarates can be represented as n L( θ, β) f ( t, z, θ, β ) (2.26) 1 for Type I censorng the lkelhood functon s L( θ, β) f ( t, z, θ, β) R( t, z, θ, β ) (2.27) U * C and for Type II censorng the lkelhood functon s n! L( θ, β) f ( t( ), z, θ, β) R( t( r), z, θ, β ) (2.28) ( n r)! U C As before, the natural logarthm form of the lkelhood functon s usually easer to work wth. For example, for Type I censored data, equaton (2.27), we have ln L( θ, β ) ln f ( t, z, θ, β ) ln R( t, z, θ, β ) (2.29) U * C Or equvalently ln L( θ, β ) ln h( t, z, θ, β ) F( t, z, θ, β ) (2.3) U * 1 n Substtutng for the Webull dstrbuton and log lnear lnk functon we get 1 n 1 t t ' t * t ln L(,, t, β) β' z ln e U βz (2.31) 1

34 26 Thus, solvng for the values of θ ( 1,, k ) andβ ( 1,, q )' whch maxmze the natural logarthm of the lkelhood functon requres solvng the followng system of nonlnear equatons ln L(,, t, β),, t, 1,, q (2.32).e. takng the frst partal dervatve of the natural logarthm of the lkelhood functon wth respect to,, t, andβ ( 1,, q )' and settng each partal dervatve equal to zero. As n the prevous subsecton, there s no easy way to solve equatons (2.32) drectly. Thus, numercal methods must be employed to solve equatons (2.32) or equvalently maxmze equaton (2.31). One approach to maxmzng equaton (2.31) s to use nonlnear programmng (NLP). We see that nonlnear program 1 n 1 t t ' t * t max ln L(,, t, β) β' z ln e U βz 1 s.t. t mn( t ) (2.33) s a specal case of the general NLP max f ( x) s.t. g ( x) b 1 1 (2.34) g m ( x) b m Wnston [23] gave the necessary and suffcent condtons (Kuhn-Tucker Condtons) that are requred to solve the nonlnear program defned n (2.33). However, Hormaza and Smth [9] found that emprcal studes have shown that the suffcent Kuhn-Tucker condtons requred to solve the nonlnear program defned n (2.33) are almost never satsfed n the three parameter Webull case. Thus, they [9] suggest usng software

35 27 packages such as GAMS to numercally solve the system of nonlnear equatons n the orgnal form defned n (2.32) (rather than (2.33)).

36 Goodness-of-Ft Tests In ths secton we dscuss varous goodness-of-ft tests that can be used to check the goodness of a dstrbuton s ft to emprcal data. The tests dscussed n ths secton are applcable to the Three Parameter Webull dstrbuton; the probablty dstrbuton ths thess focuses on. We start wth the ch-square goodness-of-ft test, whch can be used for both dscrete and contnuous dstrbutons. Ch-Square Goodness-of-Ft Test The ch-square goodness-of-ft test can be used when parameters for contnuous dstrbutons are estmated from the MLEs. Thus, ths test s applcable n determnng the valdty of fttng a Webull dstrbuton to a data set. The draw back of ths test s that s only useful for large sample szes. The null and alternatve hypotheses are H : The falure tmes came from the supposed dstrbuton H : 1 The falure tmes dd not come from the supposed dstrbuton The ch-square goodness-of-ft test statstc s k 2 2 ( O E) (2.35) E 1 where k represents the number of classes, O represents the observed number of falures n the th class, E nprepresents the expected number of falures n the th class, n represents the sample sze, and p F( a ) F( a 1) represents the probablty of falure occurrng n the th class gven the falure tmes came from the supposed dstrbuton. The th class s defned by ( a 1, a] wth a. For censored data (Type I and Type II), t * and t ( r ) should be ncluded n the last class wth the class upper bound. It s

37 29 usually recommended that p be the same value p for each nterval (equal probablty). It should be noted that for complete data sets, ths approach s reasonable but for censored data fndng ntervals wth equal probablty can be dffcult n general. To ensure that the ch-square test s unbased p s chosen to be greater than or equal to 5. The number of n ntervals k s taken to be 1. The test statstc n equaton (2.35) has a ch-squared p dstrbuton wth k 1degrees of freedom. Table 2.2 dsplays the crtcal values whch, when compared to the test statstc, determne f the supposed dstrbuton s a good ft. 2 2 If crt, then H s rejected. Mann s Goodness-of-Ft Test Ths test s a goodness-of-ft test desgned specfcally for the Webull dstrbuton. The null and alternatve hypotheses are The test statstc s H : The falure tmes are Webull dstrbuted. H : 1 The falure tmes are not Webull dstrbuted. r1 k (ln t ln t ) M M k t t M 1 k 1 1 k1 2 (ln 1 1 ln ) (2.36) r where k 1 2, k2 r 1 2, M Z 1 Z,and Z.5 ln ln 1 n.25. The test statstc n equaton (2.36) has an F-dstrbuton wth 2k2 degrees of freedom for the numerator and 2k1 degrees of freedom for the denomnator. The crtcal values of the F

38 3 dstrbuton, when compared to the test statstc, determne f the supposed dstrbuton s a good ft. If M Fcrt, then H s rejected. Kolmogorov-Smrnov Goodness-of-Ft Test Ths test s a goodness-of-ft test desgned for contnuous dstrbutons. Ft ˆ () denotes the emprcal cumulatve dstrbuton functon, Ft () denotes the actual underlyng populaton cumulatve dstrbuton functon, and F () t denotes the ftted cumulatve dstrbuton functon. The null and alternatve hypotheses for ths goodnessof-ft test are H : F( t) F ( t) H : F( t) F ( t) 1 The test statstc s D sup Fˆ ( t) F ( t) (2.37) n t Larger values of the test statstc D suggest a larger dscrepancy between Ft ˆ () and F () t n whch n turn suggests a poor ft. A computatonal method for determnng D s to defne the followng n D n max F ( t( ) ) n 1,2,, n (2.38) D n 1 max F ( t( ) ) n 1,2,, n (2.39) and let D max{ D, D }. If the test statstc exceeds a chosen crtcal value, the null n n n hypothess H s rejected. For Type I censorng equaton (2.37) s modfed to D sup Fˆ ( t) F ( t) (2.4) np, t c

39 31 where c s the tme that the test s termnated and p F () c s the expected proporton censored under H. For Type II censorng equaton (2.37) s modfed to D sup Fˆ ( t) F ( t) (2.41) nr, t t ( r ) The computatonal formulas for censored data are smlarly defned. For Type I censorng D max F ( t ) n n, p ( ) c (2.42) D 1 max F ( t ) n n, p ( ) c (2.43) where Dn, p max{ Dn, p, Dn, p}. for Type II censorng D max F ( t ) n n, r ( ) 1,2,, r (2.44) D 1 max F ( t ) n n, r ( ) 1,2,, r (2.45) where Dn, r max{ Dn, r, Dn, r}.

40 Applcaton to HM-6 Machnng Center Falure Data Prelmnary Data Analyss The Two-Parameter Webull Dstrbuton For a prelmnary analyss of the data the BestFt and Mntab software were used to gve reasonable suggestons as to the types of dstrbutons that wll best ft the data. The table below shows the top three dstrbutons BestFt found for the three data sets. The rankng was based on the K-S test. From the table t s clear that the two-parameter Webull dstrbuton s the best choce for all three data sets. Machne Rank: 1 Rank: 2 Rank: 3 29 Webull (.89,272) Lognorm (353,792) Lognorm2 (4.97,1.34) 215 Webull (.88,444) Pearson VI (1.21,3.4,829) Lognorm (567,123) 246 Webull (.65,32) Lognorm (439,13) Lognorm2 (4.94,1.51) BestFt also produced graphs (below) that llustrate the two-parameter Webull dstrbuton ftted over the actual data for the three sets of data. Vsually, the Webull dstrbuton seems to be a reasonable choce. Machne 29 Fttng Webull Dstrbuton Comparson of Input Dstrbuton and Webull(.89,2.72e+2).3.2 Input Webull Values n 1^2 Machne 215 Fttng Webull Dstrbuton

41 Percent 33 Comparson of Input Dstrbuton and Webull(.88,4.44e+2).16.8 Input Webull Values n 1^3 Machne 246 Fttng Webull Dstrbuton Comparson of Input Dstrbuton and Webull(.65,3.2e+2).16.8 Input Webull Values n 1^3 Usng Mntab, probablty plots were obtaned to gve an ndcaton of how well the two-parameter Webull dstrbuton fts the data. From the graphs, both the shape and scale parameters were estmated. Probablty Plot of TTF M29 Webull - 95% CI Shape.8895 Scale N 16 AD.739 P-Value TTF M

42 Percent Percent 34 Probablty Plot of TTF Machne 215 Webull - 95% CI Shape.876 Scale N 75 AD.388 P-Value > TTF Machne Probablty Plot of TTF M246 Webull - 95% CI Shape.656 Scale 38.5 N 8 AD P-Value < TTF M The probablty plots ndcate that the two-parameter Webull dstrbuton s a poor ft for the falure data of machnes 29 and 246. The p-values obtaned from each of the plots ndcate the level of adequacy of the ftted dstrbutons. Lower p-values ndcate a worse ft. It s clear that the two-parameter Webull dstrbuton s a poor ft for the falure tmes of machnes 29 and 246 as shown below. Machne p-values <.1 The probablty plots show that the frst group of ordered data ponts form a concave shape and fall outsde the confdence nterval bounds. Ths suggested that,

43 Percent Percent Percent 35 nstead of the 2-parameter Webull dstrbuton, a 3-parameter Webull dstrbuton should be used to mprove the ft. The Three-Parameter Webull Dstrbuton Usng the Mntab software, probablty plots were agan obtaned for the 3 data sets, ths tme wth the 3-parameter Webull as the ftted dstrbuton. Probablty Plot of TTF M29 3-Parameter Webull - 95% CI Shape.892 Scale Thresh N 16 AD.741 P-Value TTF M29 - Threshold 1 1 Probablty Plot of TTF Machne Parameter Webull - 95% CI Shape.8512 Scale Thresh N 75 AD.367 P-Value TTF Machne Threshold 1 Probablty Plot of TTF M246 3-Parameter Webull - 95% CI Shape.5845 Scale Thresh N 8 AD 1.2 P-Value TTF M246 - Threshold 1.

44 Percent 36 Machne 2-p Webull 3-p Webull <.1.14 It was found that n each of the data sets, the 3-parameter Webull was a better ft that the 2-parameter Webull. However, for machnes 29 and 246, the p-values were stll only.57 and.14, whch ndcated that the 3-parameter Webull stll dd not provde an adequate dstrbuton ft. Lookng closely at the probablty plots for machnes 29 and 246, t was decded that the frst ordered data pont for each of the plots would be treated as outlers. Thus, the frst of the ordered data ponts n the machne 29 and 246 falure data sets were removed. The probablty plots of the 3-parameter Webull dstrbuton excludng the outlyng data values are shown below. Probablty Plot of TTF Machne 29 3-Parameter Webull - 95% CI Shape.873 Scale Thresh 13.2 N 15 AD.524 P-Value TTF Machne 29 - Threshold 1

45 Percent 37 Probablty Plot of TTF Machne Parameter Webull - 95% CI Shape.5418 Scale 249. Thresh N 79 AD.513 P-Value TTF Machne Threshold 1. The p-values (below) show a drastc ncrease when the outlyng data ponts were removed. It s clear that the 3-parameter Webull dstrbuton s a reasonable ft for the falure data. Machne 2-p Webull 3-p Webull (wth Outlers) 3-p Webull (w/o Outlers) <

46 Parameter Estmatons Two-Parameter Webull Dstrbuton The MLE for the shape parameter of the Webull dstrbuton s computed by solvng the equaton (2.19). For complete data sets, the equaton becomes n β t lnt n ( ) lnt β t β n 1 g β (2.46) In general t cannot be solved drectly. Rather, a soluton to the equaton must be generated numercally. Ths can be acheved usng the Newton-Raphson method that teratvely solves for ˆ usng recursve equaton (2.2). Dfferentatng equaton (2.46) gves g '( β) 2 β t β β β t t lnt t lnt 1 β (2.47) Usng ths teratve method, the shape parameters obtaned for each of the three data sets are gven n the tables below. The value of.5 s chosen to ntate the teratons. Machne 29 Machne 215 Machne 246 Shape Value Shape Value Shape Value B1.5 B B B B B B1.5 B B B B B B1.5 B B B B

47 39 Once the estmator for the shape parameter s determned, the MLE for the scale parameter s solved through equaton (2.21). For complete data sets the equaton becomes ˆ 1 n n ˆ t 1 1 ˆ (2.48) For the three sets of data we obtan the scale parameters below. Machne 29 Machne 215 Machne Comparng these values to the estmates obtaned n Secton 2.3.1, we see that for all three data sets they the estmates are dentcal. Three-Parameter Webull Dstrbuton For complete data sets the lkelhood functon s gven by n n β1 β β t μ t μ L( μ, θ, β) exp (2.49) θ 1 θ θ and the natural logarthm of the lkelhood functon gven by n n t μ ln L( μ, θ, β) nln β nβ ln θ ( β 1) ln( t μ) 1 1 θ β (2.5) For the three-parameter Webull dstrbuton, f the locaton parameter s unknown, the method used above to estmate the parameters of the two-parameter Webull dstrbuton can not be used. Lawless [11] proposed the followng method for estmatng the parameters of a three-parameter Webull dstrbuton. Fx dfferent values of the locaton parameter μ takng values between zero and the frst ordered data value. For these fxed values of the locaton parameter the valuest μ can be consdered as observatons from a

48 4 two-parameter Webull dstrbuton and the Newton-Raphson method can be used to estmate the shape and scale parameters. The fxed value of the locaton parameter, along wth ts correspondng shape and scale parameters, whch maxmzes the equaton (2.5) are taken to be the estmates for the parameters of the three-parameter Webull dstrbuton. The tables below gve the estmates for the parameters for dfferent values of the locaton parameter. The bolded values ndcate the fnal estmates. Machne 29 Locaton Shape Scale LogLkelhood Machne 215 Locaton Shape Scale LogLkelhood

49 Machne 246 Locaton Shape Scale LogLkelhood A summary of the parameter estmates for all three machnes are gven n the table below Machne Locaton Shape Scale

50 Goodness-of-Ft Tests Two-Parameter Webull Dstrbuton Three goodness-of-ft tests were carred out on the three falure data sets. For all three tests an alpha value of.5 was used. It was found that the Webull dstrbuton for the falure data correspondng to machnes 29 and 215 was not rejected n any of the three tests. However, the Webull dstrbuton for the falure data correspondng to machnes 246 was rejected n both the Ch-square test and Mann s test. Ch-square Goodness-of-Ft Test The ch-square goodness-of-ft test and ts correspondng test statstc are gven by equaton (2.35). A summary of all three data sets s gven below. Machne 29 H : The falure tmes came from Webull (.89,272) H : 1 The falure tmes dd not come from Webull (.89,272) n = 16 P =.5 k = 2 df = 19 alpha =.5 Interval Actual Frequency Expected Frequency

51 Ch Stat Ch Crt Thus, the null hypothess s not rejected. Machne 215 H : The falure tmes came from Webull (.88,444) H : 1 The falure tmes dd not come from Webull (.88,444) n = 75 p =.6667 k = 15 df = 14 alpha =.5 Interval Actual Frequency Expected Frequency

52 Ch Stat 14 Ch Crt Thus, the null hypothess s not rejected. Machne 246 H : The falure tmes came from Webull (.65,32) H : 1 The falure tmes dd not come from Webull (.65,32) n = 8 p =.1 k = 1 df = 9 alpha =.5

53 45 Interval Actual Frequency Expected Frequency Ch Stat Ch Crt Thus, the null hypothess s rejected. Mann s Goodness-of-Ft Test Mann s goodness-of-ft test and ts correspondng test statstc are gven by equaton (2.36). A summary of all three data sets s gven below. Machne 29 H : The falure tmes came from Webull (.89,272) H : 1 The falure tmes dd not come from Webull (.89,272) M

54 46 M Crt 1.35 Thus, the null hypothess s not rejected. Machne 215 H : The falure tmes came from Webull (.88,444) H : 1 The falure tmes dd not come from Webull (.88,444) M M Crt 1.48 Thus, the null hypothess s not rejected. Machne 246 H : The falure tmes came from Webull (.65,32) H : 1 The falure tmes dd not come from Webull (.65,32) M M Crt 1.45 Thus, the null hypothess s rejected. K-S Goodness-of-Ft Test The K-S goodness-of-ft test and ts correspondng test statstcs are gven by equaton (2.37). A summary of all three data sets s gven below. Machne 29 H : The falure tmes came from Webull (.89,272) H : 1 The falure tmes dd not come from Webull (.89,272) Dn+ Dn- D D Crt

55 47 Thus, the null hypothess s not rejected. Machne 215 H : The falure tmes came from Webull (.88,444) H : 1 The falure tmes dd not come from Webull (.88,444) Dn+ Dn- D D Crt Thus, the null hypothess s not rejected. Machne 246 H : The falure tmes came from Webull (.65,32) H : 1 The falure tmes dd not come from Webull (.65,32) Dn+ Dn- D D Crt Thus, the null hypothess s not rejected. Three-Parameter Webull Dstrbuton As was the case wth the 2-parameter Webull dstrbuton, the 3-parameter Webull dstrbuton (ncludng the data outlers) for the falure data correspondng to machne 246 was rejected n both the Ch-square test and Mann s test. Ch-square Goodness-of-Ft Test A summary of all three data sets s gven below. Machne 29 n = 16 p =.5 k = 2 df = 19 alpha =.5 Interval Actual Expected

56 48 Frequency Frequency Ch Stat

57 49 Ch Crt Thus, the null hypothess s not rejected. Machne 215 n = 75 p =.6667 k = 15 df = 14 alpha =.5 Interval Actual Frequency Expected Frequency Ch Stat 14 Ch Crt Thus, the null hypothess s not rejected.

58 5 Machne 246 n = 8 p =.1 k = 1 df = 9 alpha =.5 Interval Actual Frequency Expected Frequency Ch Stat Ch Crt Thus, the null hypothess s rejected. Mann s Goodness-of-ft Test A summary of all three data sets s gven below. Machne 29 M M Crt 1.35 Thus, the null hypothess s not rejected.

59 51 Machne 215 M M Crt 1.48 Thus, the null hypothess s not rejected. Machne 246 M M Crt 1.45 Thus, the null hypothess s rejected. K-S Goodness-of-Ft Test A summary of all three data sets s gven below. Machne 29 Dn+ Dn- D D Crt Thus, the null hypothess s not rejected. Machne 215 Dn+ Dn- D D Crt Thus, the null hypothess s not rejected. Machne 246 Dn+ Dn- D D Crt Thus, the null hypothess s not rejected.

60 52 When the outlyng data values were removed, the three ftted dstrbutons passed all three (Ch-squared, Mann s, and K-S) of the goodness-of-ft tests. Ch-square Goodness-of-Ft Test A summary of all three data sets s gven below. Machne 29 n = 15 p =.5 k = 2 df = 19 alpha =.5 Interval Actual Frequency Expected Frequency

61 Ch Stat Ch Crt Thus, the null hypothess s not rejected. Machne 215 n = 75 p =.6667 k = 15 df = 14 alpha =.5 Interval Actual Frequency Expected Frequency

62 Ch Stat 14 Ch Crt Thus, the null hypothess s not rejected. Machne 246 n = 79 p =.1 k = 1 df = 9 alpha =.5 Interval Actual Frequency Expected Frequency