IMPORTANCE MEASURES IN PRESENCE OF UNCERTAINTIES ABSTRACT

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1 P. ld, E. Zo, M. Comp MPORTNCE MESURES N PRESENCE OF UNCERTNTES R&RT # 13 pt 1 Vol. 009, Jun MPORTNCE MESURES N PRESENCE OF UNCERTNTES P. ld, E. Zo, M. Comp Engy Dptmnt, Polytchnc of Mln, Mln, tly -ml: nco.zo@polm.t STRCT Th pp pnt wok on th tudy of mpotnc mu n pnc of unctnt ogntng fom th lck of knowldg nd nfomton on th ytm ptmc unctnt. cton popod fo nkng th k contuto n pnc of unctnt dcd y polty dnty functon. 1 NTRODUCTON mpotnc Mu M ud to nk th contuton of componnt o c vnt to th ytm pfomnc, whch cn, fo xmpl, th ytm llty o k. M w ntlly ntoducd y num [1] n od to th contuton of th componnt to th ovll ytm llty; lt dffnt M hv n ntoducd to dd vou pct of llty, fty nd k gnfcnc Ful-Vly, Ctclty, Rk chvmnt Woth nd Rk Rducton Woth []. On th oth hnd, unctnt of two typ ffct th hvo of ytm [3]: ltoy nd ptmc; th fom typ lo fd to ducl o tochtc o ndom unctnty dc th nhnt vton octd wth th phycl ytm o th nvonmnt.g. vton n tmophc condton, n ftgu lf of compo nd tun ld; th ltt lo fd to ujctv o ducl unctnty, ntd, du to lck of knowldg of quntt o poc of th ytm o th nvonmnt.g. lck of xpmntl dt to chctz nw mtl nd poc, poo undtndng of coupld phyc phnomn, poo undtndng of ccdnt nttng vnt. n pctc M clcultd wthout du ccount of unctnt. Th ojctv of th wok thn to nvtgt how unctnt cn nflunc M nd how thy cn ccountd fo. Th unctnt condd of ptmc typ nd pntd y polty dnty functon. mthod popod fo nkng th contuto to th ytm pfomnc mu. Th pp ognzd follow. n Scton, th polm of compng th mpotnc mu of two componnt who llt unctn pntd to xpln th d yond th nkng mthod. n Scton 3, two c tud dcd: th ft ppl th compon mthod on th componnt of mpl ytm, whn unctnt ffct th llt; th cond vldt th mthod on lg ytm fo whch pocdu ntoducd fo ffcntly pfomng th nkng. Som concluon povdd wth gd to th compon twn th popod pocdu nd mthod pvouly pntd n th lttu [4]. COMPRNG THE MPORTNCE OF TWO COMPONENTS N PRESENCE OF UNCERTNTES 30

2 P. ld, E. Zo, M. Comp MPORTNCE MESURES N PRESENCE OF UNCERTNTES R&RT # 13 pt 1 Vol. 009, Jun Th m of th Scton to pnt pocdu fo compng th mpotnc of two componnt nd of n hypothtcl ytm n pnc of unctnty. n gnl, wth pct to th unctnty pntton, whn uffcntly nfomtv dt vll poltc dtuton ghtouly ud. Th unctnt octd wth th componnt pfomnc chcttc hould thn popgtd though th ytm modl, ldng to unctnt n th ytm pfomnc. Hnc, whn pfomng mpotnc mu clculton n pnc of unctnt ffctng th componnt pfomnc, th ult hould poply ccount fo uch unctnt nd o hould th nkng. Fo mplcty, unfomly dtutd unctnty umd to ffctng dctly th M of componnt nd. Tl 1 pot th ng of th M dtuton whl Fgu 1 nd how th copondng dtuton. Th M of componnt gnfcntly mo unctn thn tht of componnt. Tl 1: Unfom dtuton pmt. Unfom dtuton Low lmt Upp lmt Lookng t th dtuton of nd M nd on my ov tht E[ ] gt thn E[ ]; on th oth hnd th ng n whch th quntl lg thn th on. Fo xmpl, f on w to pfom th nkng d on th M 95 th quntl vlu, th concluon would tht componnt mo mpotnt thn. Fgu 1: Polty dnty functon pdf nd cumultv dtuton functon cdf of th ndom vl, nd nd - c nd d n c of M wth unfomly dtutd unctnt. Th dwck of compng th vlu copondng to pcfc quntl l n th lo of nfomton out th dtuton: wth fnc fo xmpl to Fgu 1, th fct tht th 95 th quntl of low thn th 95 th quntl of only mn tht th pont vlu whch low thn wth polty of 0.95 low thn th nlogou pont vlu fo. Th 31

3 P. ld, E. Zo, M. Comp MPORTNCE MESURES N PRESENCE OF UNCERTNTES R&RT # 13 pt 1 Vol. 009, Jun full nfomton on th ctul dffnc twn th dtuton of nd do not ply ny ol. n ovou wy to gv du ccount to th dffnc twn th dtuton to cond th ndom vl v - who pdf nd cdf hown n Fgu 1c nd 1d, pctvly. Th dtl on th nlytcl xpon gvn n ppndx 1. n od to tlh f componnt mo mpotnt thn, on cn cond th polty 1-F 0 tht gt thn ; fo xmpl, n th pnt c =P > =1-F 0=0.81, whch mn tht wth hgh polty componnt mo mpotnt thn. ccodng to th ov pocdu fo compng th mpotnc of two componnt nd, t ncy to fx thhold T on th vlu: f 1- F 0 lg thn T, thn mo mpotnt thn, othw no concluon cn gvn. Fo xmpl on my tk T=0.5 o T=0.7; th low th thhold th hgh th k octd wth th dcon. On th oth hnd, th choc of cp thhold poltc xcdnc mu h om ntnc lmtton ummzd n th followng pont: could fll jut on T: n th c gvn th nvtl ppoxmton nd unctnt ltd to th tmton of th M dtuton, no out concluon cn gvn on th componnt mpotnc Scton low. Condng th componnt,, nd C who M uch tht th cdf vlu n 0 of th M dffnc fll vy clo to T, t could hppn tht >, > C nd C >. Th lmtton cn ptlly ovcom y tkng not cp vlu of T ut ng [T l,t u ] fo xmpl o Gvn two componnt nd nd th dffnc - : f > T u, thn mo mpotnt thn. f < T l, thn mo mpotnt thn. f T l < < T u, thn qully mpotnt to. n th c, dffnt knd of ddtonl contnt/tgt cn gud th nkng od cot, tm, mpct on pulc opnon tc. t my of ntt to lt th ult povdd y th poltc xcdnc mu =P > to th tndd dvton of th M dtuton, σ nd σ. Fgu how th vton of fo ncng vlu of th tndd dvton σ nd σ, kpng fxd th mn vlu of nd nd th to k=σ /σ. n th xtm c of no unctnt on th knowldg of nd σ =0 nd σ =0, componnt mo mpotnt thn nd thu =1. ncng th tndd dvton, =1 hold long th pdf of nd do not ovlp,.. nd unctn quntt ut t not unctn tht >. Fnlly, th ovlppng twn pdf nc dc. Fom th condton on cn gu tht unctnt on th M cn ffct th nk od nd th ducton of th unctnt hould, n ctn c, condd fo dcng th k octd wth th dcon. 3

4 P. ld, E. Zo, M. Comp MPORTNCE MESURES N PRESENCE OF UNCERTNTES R&RT # 13 pt 1 Vol. 009, Jun Fgu : v σ nd σ, kpng σ /σ, E[ ] nd E[ ] contnt. 3 N EMPRCL PROCEDURE TO RNK COMPONENT MPORTNCE n th pvou Scton, poltc mu of xcdnc twn two v h n utlzd to comp componnt mpotnc mu n pnc of unctnt. To xtnd th mthod to lg num of componnt, pocdu fo uccv nkng mut ntoducd to vod th comntol xploon of pw compon. Th mthod popod n th pp n pplcton of on of th mot common otng lgothm, Quckot [5], whch pocd y choong n lmnt, clld pvot, nd movng ll mll lmnt fo th pvot nd ll lg lmnt ft t. n th pnt c th od lton twn th lmnt gudd y th mu of xcdnc ntoducd n Scton. To llutt th mthod, two c tud hv n xmnd: 1. flu t lognomlly dtutd fo th componnt ytm; th c llow u to tt th nkng cton;. flu t lognomlly dtutd fo mo complx ytm; th c tudy llow u to xpln n dtl th nkng pocdu, wth t dvntg nd lmtton. Fo mplcty only th num M condd, th onng mnng xctly th m fo th oth M. 3.1 Th componnt ytm Th ytm, ktchd n Fgu 3, md up of of nod: th ft conttutd y componnt n plll logc, th cond y ngl componnt. To ch of th componnt, cp llty vlu h n gnd o tht th M vlu potd n Tl, thd column, fo componnt nd th m tho condd n Scton. Dffnt M vlu of th th componnt gvn n Tl, column 3-6.Concluon hould tt concly th mot mpotnt popoton of th pp wll th utho vw of th pctcl mplcton of th ult. 33

5 P. ld, E. Zo, M. Comp MPORTNCE MESURES N PRESENCE OF UNCERTNTES R&RT # 13 pt 1 Vol. 009, Jun Fgu 3: Sytm Rllty lock Dgm. Tl : Componnt llty nd mpotnc mu. Rllty num F-V Ctclty RW RRW C Th componnt umd xponntl,.. wth contnt flu t λ, =,, C. Unctnt n th flu t dcd y lognoml dtuton, Fgu 4, lft, wth pmt gvn n Tl 3: [ ln λ μ ] σ f λ λ = 1 λ σ π t ch tm ntnt t th llty of componnt : λ, t = λt wth pdf fo 0<λ <1 Fgu 4, ght: ln λ ln μ t σ f λ, t = 3 λ ln λ σ π Th pmt of th dtuton of th flu t hv n chon uch tht th mn vlu of th llty t tm t=100 n ty unt of tm qul to th vlu n column 3 of Tl. n Fgu 5 nd, th pdf of th flu t nd llt t tm t=100 potd fo ll th componnt. 34

6 P. ld, E. Zo, M. Comp MPORTNCE MESURES N PRESENCE OF UNCERTNTES R&RT # 13 pt 1 Vol. 009, Jun Fgu 4: Lognoml dtuton of th flu t of componnt lft nd copondng pdf of th llty t dffnt tm ntnt ght. Tl 3: Pmt of th lognoml dtuton of th componnt flu t. Mn Vnc C n pt of th mplcty of th condd ytm, fndng th num M dtuton y n nlytcl ppoch mpctcl. To ovcom th dffcult Mont Clo mplng h n ppld. Th ultng dtuton plottd n Fgu 5 c nd d. t cn notd tht th dtuton of th M of componnt C dplcd to lg vlu thn nd, whch lv no dout tht th mot mpotnt componnt C, xpctd fom th tuctu of th ytm. fo th nkng of nd, on mut comput th polty P >.Th ult otnd y Mont Clo mplng = whch ld u to conclud tht =. On th conty, th num M vlu n Tl, column 3, nglctng unctnt, would ld to th concluon tht mo mpotnt thn. 3. mo complx ytm Whn th num of componnt n th ytm lg, th num of pw compon of th M nc dmtclly. Th cll fo ytmtc pocdu of nly to ffcntly pfom th mpotnc nkng. Lt u cond ytm md up of of th nod: th ft -out-of-3 uytm, th cond cont of ngl lmnt nd th thd plll ytm of two componnt. Th Rllty lock Dgm RD potd n Fgu 6. Tl 4 contn th dt ltv to th dtuton of th componnt flu t nd llt. Th copondng pdf plottd n Fgu 7. 35

7 P. ld, E. Zo, M. Comp MPORTNCE MESURES N PRESENCE OF UNCERTNTES R&RT # 13 pt 1 Vol. 009, Jun Fgu 5: pdf of th flu t, llty nd num M c nd d of th th componnt. 36

P. ld, E. Zo, M. Comp MPORTNCE MESURES N PRESENCE OF UNCERTNTES R&RT # 13 pt 1 Vol. 009, Jun oo3 Fgu 6: RD of th ytm. Tl 4: Pmt of th lognoml dtuton of th componnt flu t. Rllty Mn Vnc M 0.98 8.37-5 1.

8 P. ld, E. Zo, M. Comp MPORTNCE MESURES N PRESENCE OF UNCERTNTES R&RT # 13 pt 1 Vol. 009, Jun oo3 Fgu 6: RD of th ytm. Tl 4: Pmt of th lognoml dtuton of th componnt flu t. Rllty Mn Vnc M C D E F Fgu 7: Flu Rt dtuton. Th nkng pocdu popod n th Scton n pplcton of on of th mot fmou otng lgothm, Quckot [5], to th poltc mu of xcdnc j =P > j ntoducd n th pvou Scton. Quckot dvd nd conqu lgothm whch l on ptton opton: to ptton n y, on choo n lmnt, clld pvot, mov ll mll lmnt fo th pvot nd ll lg lmnt ft t. n th ttv pocdu, on thn cuvly ot ntnlly th ult of mll nd lg lmnt. n th c of ntt h, ttng fom th componnt nkng d on th llty mn vlu, on choo pvot lmnt th mot mpotnt componnt,.. th on wth lg mpotnc mu p clcultd d on th mn llt. Th pj =P p > j clcultd fo ch j p nd th p-dfnd thhold vlu T tt th lton od twn p nd j, wth pct to th F pj 0 cdf vlu. 37

9 P. ld, E. Zo, M. Comp MPORTNCE MESURES N PRESENCE OF UNCERTNTES R&RT # 13 pt 1 Vol. 009, Jun Tl 5: Column 1 pot th nkng otnd wthout condng unctnt n th M; column -6 th polty tht th componnt n th ow xcd th componnt n th column; th lt column how th nkng otnd y th popod mthod. Mn nkng C E F Fnl nk D D C C, C, E 0.71 E F F y pocdng th wy, th componnt odd n functon of th dtnc fom th pvot lmnt. Whn wppng occu twn th nk of two componnt, th chck on th xcdnc mu ptd fo ll th componnt downtm th ult. Dong o, t my hppn tht R >R j, R j =R k nd R k =R : n th c,, j nd k condd qully mpotnt. Th ult otnd y pplyng th pocdu on th ytm n Fgu 6 potd n Tl 5. xpctd fom th tuctu of th ytm, componnt D th mot mpotnt on, followd y th componnt n th -out-of-3 uytm. n ptcul, componnt th mot mpotnt of th uytm, thn, componnt nd C ult wth th m mpotnc; on th conty, fng to th mn nkng, componnt ult mo mpotnt thn componnt C. Fnlly, componnt E nd F n th plll uytm of Fgu 6 potzd ccodngly to th gnl ul tht th num M of componnt n plll ytm dc wth dcng llty of th componnt, oth nglctng Tl 4, column 7 nd ccountng fo unctnt Tl 5, column 7. tm of compon, th pocdu popod n [4] h n ppld fo th computton of n ltntv mu of xcdnc. t d on th followng two tp: j 1. Etmton, fo ch componnt, of th polty of occupyng pcfc od n th nkng polty m functon. Th chvd y pfomng Mont Clo mplng: t ch tl th componnt gvn nk od R =,,,F. Fgu 8 povd th dtuton of th otnd nk od.. Computton of th mu of xcdnc dfnd : j = P R > R = p R p R 4 j j n R = 1 R R j = 1 Th ult otnd whn pplyng th pocdu potd n Tl 6. j Rnk Tl 6: Rnk od otnd y pplyng th ltntv xcdnc mu popod n [4]. Mn nkng C E F 1 D D Fnl Rnkng ,C 4 C ,C 5 E E 6 F F 38

10 P. ld, E. Zo, M. Comp MPORTNCE MESURES N PRESENCE OF UNCERTNTES R&RT # 13 pt 1 Vol. 009, Jun Fo wht concn th fnl nkng, th two mthod condd povd th m ult. Notc, howv, tht f on cond dffnt ng [T l,t u ], fo xmpl [0.4,0.6] ntd of [0.3,0.7] th mthod popod n [4] ld to nkng componnt mo mpotnt thn C wh wth th mthod h popod, nd C hv th m mpotnc. Moov, f on cond vy mll ng, fo xmpl [T l,t u ]=[0.49,0.51], thn th mthod n [4] dtmn th nkng,, C fo th M wh th popod mthod tlh, C,. Fgu 8: Polty m functon of th nk od R, =,, F. Th dffnc n th nkng of th two mthod cud y th dffnt vlu of th xcdnc mu twn componnt nd C Tl 5 nd 6. n ptcul, th mthod popod n th wok ult n C =P C > =0.51, wh th mthod n [4] ult n =P C C > = Notc tht C dpnd only on th mpotnc mu of nd C wh dpnd on th polty tht componnt occup pcfc od nd thu lo on th C mpotnc mu of th oth componnt of th ytm. Moov, noth on to pf th pocdu h popod tht n th pocdu popod n [4], ft ch MC mplng th M of th componnt ud only to otn nk od, loong th nfomton on th ctul numcl dffnc of th M. Fnlly, wh fo j t hold tht j =1- j, th popty not vld fo, fo xmpl, wth fnc to Tl 6, j =0.665 nd C C = Th mn tht th fnl nkng od my dpnd on th choc of th pvot n th Quckot lgothm. 4 CONCLUSONS ND FUTURE WORKS n th wok, pocdu popod fo nkng ytm componnt n od of mpotnc whn n pnc of unctnt ffctng th componnt llt. Th pocdu d on th dfnton of poltc xcdnc mu tht pmt to comp th mpotnc of two componnt nd cn ummzd follow: 1. Rnk th componnt mpotnc ccodng to th M computd y condng th xpctd vlu of th llt, thu wthout condng unctnt. 39

11 P. ld, E. Zo, M. Comp MPORTNCE MESURES N PRESENCE OF UNCERTNTES R&RT # 13 pt 1 Vol. 009, Jun. Dfn th ng [T l,t u ] of vlu of th poltc xcdnc mu pj ; fo vlu pj n th ng t not pol to dcd f p > j o p < j nd th ld to cond componnt p nd j qully mpotnt. 3. pply th Quckot lgothm d on pj =P p > j : 3.1. Put th componnt n th nk od found n tp Choo th ft lmnt of th lt ult pvot lmnt, p Fo ch j n th ult comput th cdf, F pj, of p - j nd vlut pj =1- F pj 0: - f pj >T u, thn put j n th ult of lmnt l mpotnt thn p. - f pj <T l, thn put j n th ult of lmnt mo mpotnt thn p. - f pj fll n [T l,t u ], thn p qully mpotnt to j. 4. Fo ch ult go to 3. untl no ult wth mo thn on lmnt xt. Th pplcton of th popod pocdu to two c tud h hown th mpotnc of condng unctnt n th computton of M: th nkng of th componnt mpotnc otnd nglctng th unctnt ffctng th componnt llt cn dffnt fom th nkng otnd y condng unctnt ung th pocdu popod n th wok. Compd to noth ppoch popod n th lttu, th pocdu h pntd m to off om dvntg, fo xmpl, th ndpndnc of th fnl nk fom th choc of th pvot lmnt n th Quckot lgothm. Num fgu concutvly n th od n whch fnc md to thm n th txt, mkng no dtncton twn dgm nd photogph. Fgu, photogph, tc. cn n lck/wht o full colo, ut wll poducd n th ook n lck/wht only. Photogph hould wth good contt nd on gloy pp. Photogphc poducton cut fom ook o jounl, photocop of photogph nd cnd photogph unccptl. Th pocdng wll pntd n lck only. Fo th on vod th u of colou n fgu nd photogph. Colou lo nly lwy unncy fo cntfc wok. 5 CKNOWLEDGMENT Th wok fundd y CES RCERC S.p.. Th vw xpd n th pp olly tho of th utho. 6 REFERENCES [1] num, Z.W On th mpotnc of dffnt componnt n mult componnt ytm. Multvt nly, Nw Yok, cdmc P. [] Younglood, R.W Rk gnfcnc nd fty gnfcnc. Rllty Engnng nd Sytm Sfty; 73; pp [3] potolk, G.E Th concpt of polty n fty mnt of tchnologcl ytm, Scnc, 50, [4] Mod, M Rk nly n Engnng: Poltc Tchnqu, Tool nd Tnd, CRC P. [5] Ho, C Quckot, Comput Jounl, [6] Zo, E n ntoducton to th c of llty nd k nly, Sngpo, Wold Scntfc Pulhng. [7] Rund, M. & Hoylnd, Sytm Rllty Thoy, Wly. [8] oln, Rllty Engnng. Spng. [9] Schnw, W.G Rllty Modlng, LLoL-Vlg. 40

12 P. ld, E. Zo, M. Comp MPORTNCE MESURES N PRESENCE OF UNCERTNTES R&RT # 13 pt 1 Vol. 009, Jun 41 7 PPENDX 1 Gvn gnc unfom v x U,, t momnt gntng functon mgf gvn y: = φ Th v = - th convoluton of two unfomly dtutd ndom vl nd n ptcul: U, ; - U-,- ; Th mgf of gvn y: = φ + = φ Fo wht concn th nv tnfomton, t could notd tht mgf cn gdd th lgc um of functon lnly ncng/dcng wth th m lop. So th pdf nd cdf of gvn y: + = f = F 1 1

Exam 2 Solutions. Jonathan Turner 4/2/2012. CS 542 Advanced Data Structures and Algorithms

Exam 2 Solutions. Jonathan Turner 4/2/2012. CS 542 Advanced Data Structures and Algorithms CS 542 Avn Dt Stutu n Alotm Exm 2 Soluton Jontn Tun 4/2/202. (5 ont) Con n oton on t tton t tutu n w t n t 2 no. Wt t mllt num o no tt t tton t tutu oul ontn. Exln you nw. Sn n mut n you o u t n t, t n