On the Statistical Uncertainties of Time-domain-based Assessment of Stability Failures: Confidence Interval for the Mean and Variance of a Time Series

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1 Interntonl Shp Stblt Workshop 3 Proceedngs of the 3 th Interntonl Shp Stblt Workshop, Brest 3-6 September On the Sttstcl Uncertntes of Tme-domn-bsed Assessment of Stblt Flures: Confdence Intervl for the Men nd Vrnce of Tme Seres Vdm Belenk, Vlds Pprs, Chrstopher Kent, Mchel Hughes, Brdle Cmpbell, Tmoth Smth ABSTRACT Dvd Tlor Model Bsn SWCCD Unverst of orth Croln t Chpel Hll The pper ddresses one of the crtcl elements of sttstcl uncertnt of smulted or mesured roll motons confdence ntervl of the vrnce estmte. The pper revsts the dervton of the formul for the vrnce of the smple vrnce of sttonr stochstc process n order to reexmne the ssumptons, especll the one relted to the process hvng norml dstrbuton. The relton between the formul nd the confdence ntervl bsed on tretng the vrnce estmte of dfferent records s seprte dt ponts s lso consdered. KEYWORDS Roll motons, sttstcl uncertnt, tme-domn smultons ITRODUCTIO Wth the development of dvnced hdrodnmc codes cpble of predctng ver nonlner roll motons, there s n opportunt for the tme domn ssessment of dnmc stblt to become prt of the desgn process. Whle ddressng the ssue of the nonlnert of lrge-mpltude moton, tme domn smultons crete the ssue of sttstcl uncertnt. A tme domn smulton of shp motons n rregulr ses s Monte-Crlo method, so n result derved from them (such s the vrnce of mode of moton) s rndom number. The sme s true for the results of model tests n rregulr wves nd full scle sekeepng trls. Snce the rndom nture of these results s nherent nd cnnot be voded, t s essentl to chrcterze the uncertnt nd mke t prt of the desgn nlss. Chrcterzton of the sttstcl uncertnt of these results s the mn obectve of the pper. Whle evluton of the confdence ntervl of the smple vrnce s one of the most bsc sttstcl problems, there re severl detls tht tend to complcte ts evluton. Frst, the nert of the shp leds to sttstcl dependence between successve ponts of the moton tme seres. Second, the process of lrge-mpltude roll response s nonlner nd cnnot be ssumed to be Gussn. Unfortuntel, the stndrd formule for the confdence ntervl use ths ssumpton n one w or nother. Prmetrc roll s good exmple of such process; see e.g. Hshmoto et l (6). Thrd, the nonlnert of stblt-relted problems m led to the prctcl npplcblt of the ergodc ssumpton when multple records re requred to crr out the nlss. Ths problem becme prtculrl cler whle ttemptng to compre prmetrc roll results (Reed, ). Thus, the method of chrcterzton of sttstcl uncertnt of the results of the tme-

2 Interntonl Shp Stblt Workshop 3 Proceedngs of the 3 th Interntonl Shp Stblt Workshop, Brest 3-6 September domn numercl smultons (or model tests n rregulr wves) must be ble to tret these three fetures of lrge-mpltude roll moton: dependence, non-gussn dstrbuton nd prctcl non-ergodct. In n ttempt to ccount for dependence, Belenk & Weems (8) used stndrd formul (Prestle, 98) where the estmte of the utocorrelton functon ws ntroduced to hndle the dt dependence. Prctcl nonergodct s ddressed b consderng severl records of roll, s ws done n (Reed, ). The lst mle s the ssumpton of the Gussn dstrbuton used n stndrd formul n ll the cted works. The focus of ths pper s to understnd nfluence of ths ssumpton nd see f t cn be voded. THEORETICAL AALYSIS Mesure of Uncertnt The clculton of the confdence ntervl of sttstcl quntt requres n ssumpton of the dstrbuton of tht quntt. Wth few exceptons ( men vlue estmte of the norml vrble follows the Student t- dstrbuton, whle the estmte of the vrnce hs ch-squre dstrbuton), these dstrbutons re unknown. The ssumpton tht the estmte follows the norml dstrbuton s bsed on the centrl lmt theorem, snce estmtes nvolve summton of rndom numbers. The cvet s tht the smple sze should be lrge enough, s the centrl lmt theorem, strctl spekng, ddresses lmtng dstrbuton (s hnted b ts nme). One should be especll creful pplng the norml dstrbuton for the vrnce estmte s the vrnce s postve vlue b defnton, whle the norml dstrbuton lso supports negtve numbers. evertheless, f the smple sze s lrge enough, the confdence ntervl s expected to be reltvel smll nd the nfluence of smmetr of the rel dstrbuton of the estmte m be neglected. The smple sze s expected to be lrge, becuse severl records re needed to hndle prctcl nonergodct. Once the ssumpton of the norml dstrbuton of the vrnce estmte s ccepted, the vrnce of the vrnce s the onl vlue needed to clculte the confdence ntervl. Vrnce of Men Vlue Estmte Prestle (98) gves generl drecton on the dervton of the formule for the men vlue nd vrnce estmtes. Ths dervton s reproduced here, n order to understnd the necesst nd role of the Gussn ssumpton for the dstrbuton of the process. Consder the vrnce of the men vlue estmte mˆ (the smbol bove mens estmte ) of sttonr process x represented s record wth ponts wthout n further ssumptons. Vrmˆ Vr x Cov( x, x ) () where Vr(..) s the vrnce opertor nd Cov(..) s the covrnce opertor. Equton () s stndrd one; t expresses the vrnce of sum of dependent rndom vrbles. Snce the process x s ssumed sttonr, ts utocovrnce functon depends onl on the dfference n tme (tme lg) between the two ponts nd does not depend on prtculr tme nstnces: Cov( x, x ) R( t k,..., ) R( k ) () Consder sum of ll the elements of the covrnce mtrx tht re needed to compute the vrnce of the men estmte n Equton (): Cov( x, x ) (3) R R... R R R... R R R... R R R 3... R R R... R R

3 Interntonl Shp Stblt Workshop 3 Proceedngs of the 3 th Interntonl Shp Stblt Workshop, Brest 3-6 September 3 ote tht ll the elements of the mn dgonl of the covrnce mtrx re the sme nd equl to vrnce of the process V, snce the utocovrnce functon clculted for = s the vrnce: R( ) R() Vr ( x) V (4) In fct ll the elements on the lne prllel to the mn dgonl re lso the sme; the next element to the term R( )=V s lws R( ), then R( ) nd so forth. The mn dgonl of squre mtrx contns elements; the lne of elements prllel to the mn dgonl nd locted next to t, contns onl - elements. Ech next lne wll hve one element less, untl t comes to the low-left or upper-rght corner wth one element onl. Thus the sum n Equton () cn be presented s (hvng n mnd, tht the covrnce mtrx s smmetrc reltve to ts mn dgonl nd ll the lnes of elements except the mn dgonl re encountered twce): Cov( x, x ( ) R( ) ( ) R( )... R( V ) V ( ) R( ) (5) Substtuton of Equton (5) nto Equton () leds to the stndrd formul for the vrnce of the men vlue estmte (see e.g. Prestl, 98) V Vr mˆ R( ) (6) The frst term n Equton (6) s ctull vrnce of the men estmte of the rndom vrble, whle the second term ccounts for the dependence between the dt ponts of stochstc process. As expected, f the process x s uncorrelted whte nose (Wener process), the result s dentcl to one for the rndom vrble, becuse the uto-covrnce functon of the whte nose equls zero for ll non-zero tme lgs. ) Vrnce of Vrnce Estmte B defnton the vrnce s the verge of centered squres, thus process s ntroduced s: x m x ˆm (7) Then the estmte of the men vlue of the process s the estmte of the vrnce of the orgnl process x: ˆ Vˆ (8) m Then the vrnce of the men estmte of the process s the vrnce of the vrnce estmte of the process x: V ˆ V Vr R( ) (9) where V nd R re the vrnce nd the utocovrnce functon of the process of centered squres, respectvel. Ths s the plce when the ssumpton of the Gussn dstrbuton for the process x s mde n order to rrve t the stndrd formul of the vrnce of the vrnce estmte. If the process x hs norml dstrbuton: V V R ( ) R( ) () Substtuton of () nto (9) leds to the stndrd formul for vrnce of the vrnce estmte (see e.g. Prestl, 98): ˆ V 4 V R( ) Vr () Equton () cn lso be expressed n n lterntve form where the smmetrc propertes of the covrnce mtrx re not used. Ths form ws used, for exmple, n (Reed, ): ˆ V R( ( ) Vr ) () ote tht () does not hve n explct term tht ncludes the vrnce, but snce the ndex of the tme lg goes through zero, ths term s, ndeed, ncluded. It seems tht there s no pprent reson to use the Gussn ssumpton. The clculton of

4 Interntonl Shp Stblt Workshop 3 Proceedngs of the 3 th Interntonl Shp Stblt Workshop, Brest 3-6 September 4 the uto-covrnce functon of the centered squres requres lttle ddtonl computton effort n comprson wth strght utocovrnce functon. Vrnce of Ensemble Vrnce Consder n ensemble of r records, ech wth dt ponts. The tme ncrement s ssumed to be the sme for ll the records, whch s the usul prctce for both numercl smultons nd model tests. Then the sttstcl weght for ech record s expressed s follows W r totl (3) where totl s the totl number of ponts n the ensemble. The ensemble estmte for the men vlue s clculted for ll the ponts mˆ r r totl W x, x, r r totl W mˆ x, (4) where mˆ s the men vlue estmte for record. The dt pont x, n Equton (4) hs two ndexes for the record nd s the ndex wthn record. Snce the records cn be of dfferent lengths, the set of dt ponts x, do not consttute mtrx. The ensemble estmte for the vrnce s expressed nlogousl to the men vlue: r Vˆ WVˆ (5) where Vˆ s the vrnce estmte for record. The vrnce of the ensemble vrnce estmte cn be clculted s: Vr r V ˆ W Vr ( Vˆ ) (6) where the vrnce of the vrnce estmte for ech record s tken from Equton (9). Drect Estmte of Vrnce of the Vrnce Consder the vrnce estmte of ech record s relzton of rndom number. The verge vrnce of the record estmte s (ccountng for the fct tht ech vrnce estmte wth the ensemble hs sttstcl weght W,) r V ˆ W V ˆ Vˆ Vr ˆ (7) Equton (7) s not equvlent to Equton (6); t gves the verge vrnce of ech record, so t should be equvlent to Equton (9) verged through the ensemble. The vrnce of the ensemble estmte should be treted s the vrnce of the men of the record estmtes: r V ˆ W V ˆ Vˆ Vr ˆ (8) Substtutng Equton (8) nto (8): Vr ˆ r V ˆ W mˆ Vˆ r r W Vˆ, W, Vˆ (9) Usng the known formul for the squre of sum, one cn wrte: Vr ˆ r W V ˆ Vˆ Vˆ Vˆ,,, k () k The second term n equton () cn be consdered s n estmte for the utocovrnce functon of sngle record of centered squres tht uses populton men (5) nsted of record men. It hs to be dstngushed from the estmte bsed on the record dt onl: p k Rˆ ( ˆ ˆ (), V )(, k V ) The frst term n the formul () s the sme estmte uto-covrnce t zero tme lg. Thus

5 Vr V ˆ Interntonl Shp Stblt Workshop 3 Proceedngs of the 3 th Interntonl Shp Stblt Workshop, Brest 3-6 September r Rˆ p W Rˆ p () Equton () s smlr Equton (9). The dfference s tht not onl vergng over ll the records n the ensemble, but lso uses the populton men nsted of the record men for clculton of the uto-covrnce functon. Thus drect estmte of the vrnce of vrnce (7) s equvlent to populton verge of the record vrnce of vrnce, where populton men s used for evluton of uto-covrnce functon of the centered squres. UMERICAL AALYSIS Source of Shp Roll Dt A hbrd model (Weems & Wundrow, 3) ws used to reproduce roll moton s fst nd es w to reproduce roll motons wth the correct tpe of nonlnert. The model clcultes the Froude-Krlov nd hdrosttc forces on the ctul submerged volume for three degrees of freedom: heve, roll nd ptch. Clcultons were performed for the OR tumblehome topsde confgurton (Bshop, et l, 5); ths confgurton s representtve of n unconventonl hull desgn nd produces suffcentl nonlner motons brngng nto queston the Gussn ssumpton for roll motons whle ssessng sttstcl uncertnt. The motons were smulted for se stte descrbed b sgnfcnt wve heght of 7.5 m nd modl perod of 5s. Long-crested rregulr wves were modeled wth the Bretschneder spectrum. The speed ws 6 knots n stern-qurterng ses (45 degrees). The spectrum ws dscretzed wth unforml dstrbuted frequences tht fcltted modelng mnute long records. The ensemble (populton) conssted of 3 records totlng 5 hrs worth of dt. Estmton of Auto-Covrnce Strctl spekng, onl the uto-covrnce functon for centered squres s needed for Equton (9), however, t m be nstructve to look t the uto-covrnce of the orgnl process s well. The forml defnton of the uto-covrnce estmte s gven n Equton () nd rewrtten here for the process x Rˆ ( x mˆ )( x mˆ ) (3) When the tme lg I becomes lrge, the volume of the smple vlble for vergng decreses drmtcll. From Fgure, n ncrese n the mgntude of the utocovrnce functon for the lrge tme lgs cn be observed. Ths loss of ccurc cn be llevted b smple weghtng fctor: (-)/, re-wrtng Equton (3) s follows: Rˆ ( x mˆ )( x mˆ ) (4) The weghtng results n lttle chnge to the uto-covrnce functon for smll tme lgs s the dfference between nd - s not sgnfcnt for smll. When the ndex becomes lrge, the mount of vlble dt decreses nd therefore the nfluence of ts contrbuton lso decreses. The result of weghtng the estmte of the uto-covrnce functon s shown n Fg. Tme lg, s Fg. : Auto-covrnce functon estmted from sngle record usng Equton (3). Auto-covrnce, deg Auto-covrnce, deg - Tme lg, s Fg. : Auto-covrnce functon estmted from sngle record usng Equton (4) usng lner weghtng fctor. 5

6 Interntonl Shp Stblt Workshop 3 Proceedngs of the 3 th Interntonl Shp Stblt Workshop, Brest 3-6 September 6 Comprng Fgures nd, one cn see tht the ntl prt dd not chnge much, however the mount of numercl nose hs decresed sgnfcntl. Avergng the estmte cross the records further decreses ths nose nd ccounts for possble prctcl non-ergodct: r Rˆ W Rˆ (5) where r s the totl number of records, the number of ponts n -th record, W s weghng fctor of -th record. Fgure 3 shows the estmte of the uto-covrnce functon verged for 3 records. As expected, the nose s prctcll gone. - Fg. 3: Averged uto-covrnce functon, Equton (5). Estmton of Auto-Covrnce for the Centered Squres The estmton of the uto-covrnce functon for the centered squres process s smlr; frst the weghted record estmte s clculted, then the populton verge s evluted. Rˆ Rˆ r W Rˆ ( Vˆ)( Vˆ) (6) Fgure 4 shows the populton verge for the uto-covrnce of the centered squres, whle Fgure 5 contns the zoomed-n vew of the frst seconds of the estmte. 5 Auto-covrnce, deg Auto-covrnce of centered, squres deg Tme lg, s Tme lg, s Fg. 4: Averged uto-covrnce functon of the centered squres, Equton (7) The shpe of the uto-covrnce functon of the centered squres s drstcll dfferent compred wth the uto-covrnce functon of the orgnl process. The folds re locted mostl on the postve sde nd there s negtve tl slowl pprochng zero. The ppernce of the negtve tl s not result of numercl error, but consequence of mostl postve folds; t comes from the known propert of the uto-covrnce functon: R( ) R (7) If the folds re mostl postve, the rest of the uto-covrnce must be negtve to brng the sum (7) to zero Auto-covrnce of centered, squres deg Tme lg, s Fg. 5: Averged uto-covrnce functon of the centered squres, zoomed-n vew Possble Scheme of Clculton of the Vrnce of Vrnce Estmte Snce the lrge-mpltude roll response m be prctcll non-ergodc, t mkes sense to use the ensemble/populton estmte whenever possble. Thus the men vlue estmte (5) should be clculted frst to be used for further estmtes. Then the centered squres re clculted: x m (8) ˆ,, The men vlue estmte for centered squres s the vrnce estmte for the orgnl process r Vˆ mˆ W, (9)

7 Interntonl Shp Stblt Workshop 3 Proceedngs of the 3 th Interntonl Shp Stblt Workshop, Brest 3-6 September 7 The uto-covrnce of the centered squres s clculted wth Equton () nd verged over the populton: p r W Rˆ Rˆ (3) p Then the vrnce of the vrnce estmte for ech record needs to be clculted. To decrese vrblt for lrger tme lgs, t s proposed to remove the summnds bove the cutoff pont M: ˆ ˆ R M p V r ˆ V ˆ Rp (3) M It s proposed to set the cutoff pont M to hlf of the verge number of ponts of record. In n cse, for the correct defnton of the ensemble- verged uto-covrnce functon of centered squres (6): M mn (3) The fnl result s the vrnce of the vrnce estmte for the ensemble tht s clculted wth Equton (6). Once the vrnce of the vrnce estmte s clculted, the lst step s the ssessment of the confdence ntervl. Snce the estmte s ssumed to be dstrbuted normll, the hlf-wdth of the confdence ntervl s expressed s: Vˆ Vr ˆ ( ˆ ) (33) V Where s coeffcent dependent on the ccepted confdence probblt e.g:.95 ;.96 Fgure 6 shows comprson of three dfferent ws to compute the ensemble/ populton estmte of the vrnce wth confdence ntervl. The stndrd Gussn ssumpton overestmtes uncertnt compred to the two other methods. Equton (3) lso shows slght overestmton compre to the drect estmte (8). However, more clcultons re needed to conclude tht the observed dfferences re of generl nture Ensemble /populton estmte for vrnce of roll, 95% confdence probblt, deg Drect estmte (8) Wthout Gussn ssumpton (6 & 3) Wth Gussn ssumpton, (6 &) Fg. 6: Comprson of dfferent methods to compute confdence ntervl on the ensemble vrnce estmte. Equtons n prentheses. COCLUSIOS AD FUTURE WORK Contrr to populr opnon, the dervton of the formul for vrnce of the vrnce estmte s not bulk nd s qute strghtforwrd. The ssumpton of the Gussn dstrbuton of the process s ctull not necessr, f one cn estmte covrnce functon of centered squres of the process. Drect estmton when the vrnce of ech record s consdered s seprte dt pont s smlr to the formul of vrnce of the vrnce. The dfference ncludes use of the populton men nsted of the record men for the centered squres. Applng lner weghtng functon on the estmte of the utocovrnce functon helps to sgnfcntl reduce sttstcl nose cused b the decrese of vlble dt n lrge tme lgs. The next logcl step s to test these clcultons. Ths would nclude cretng lrge set of ensembles n order to see how well the computed confdence ntervl cptures the expected number of ensemble estmtes. The frcton of estmtes fllng wth the confdence ntervl should be close to the gven confdence probblt.

8 Interntonl Shp Stblt Workshop 3 Proceedngs of the 3 th Interntonl Shp Stblt Workshop, Brest 3-6 September 8 ACKOWLEGEMETS The work descrbed n ths pper hs been funded b the Offce of vl Reserch under Dr. Ptrck Purtell nd Dr. K-Hn Km. Ths support s grtefull cknowledged b the uthors. Prtcpton of Dr. Pprs ws supported b the summer fcult progrm supported b OR nd mnged b Dr. Jck Prce of Dvd Tlor Model Bsn. The uthors re grteful to Dr. D. Drzen nd Dr M. Levne for nternl support of ths work. Dscussons wth Dr. A. Reed hve been ver frutful. REFERECES Belenk, V. & K.M. Weems (8) Procedure for Probblstc Evluton of Lrge Ampltude Roll Motons n Proc. of the Osk Colloquum on Sekeepng nd Stblt of Shps, Osk, Jpn Hshmoto, H.,. Umed, A. Mtsud, & S. kmur, (6), Expermentl nd umercl Studes on Prmetrc Roll of Post-Pnmx Contner Shp n Irregulr Wves, Proc. 9th Intl. Conf. on the Stblt of Shps nd Ocen Vehcles, Ro de Jnero, Brzl, Vol., p 8-9. Prestle, M. B., (98), Spectrl Anlss nd Tme Seres, Vol., Acdemc Press, ew York Reed, A. () 6th ITTC Prmetrc Roll Benchmrk Stud, Proc. th Intl. Shp Stblt Workshop, Wshngton DC, USA pp Weems, K., & D. Wundrow (3) "Hbrd models for fst tme-domn smulton of stblt flures n rregulr wves wth volume-bsed clcultons for Froude-Krlov nd Hdrosttc Forces"; Proc. 3 th Intl. Shp Stblt Workshop, Brest, Frnce Bshop, R. C., W. Belknp, C. Turner, B. Smon, J. H. Km (5) Prmetrc Investgton on the Influence of GM, Roll dmpng, nd Above-Wter Form on the Roll Response of Model 563. Report SWCCD-5-TR-5/7.