AXIOMATIC APPROACH IN THE ANALYSIS OF DATA FOR DESCRIBING COMPLEX SHAPES

Transcription

1 Proceedg of ICAD004 ICAD AXIOMAIC APPROACH I HE AALYSIS OF DAA FOR DESCRIBIG COMPLEX SHAPES Mchele Pappalardo mpappalardo@ua.t Departmet of Mechacal Egeerg Uverty of Salero Va Pote Do Melllo, 84084, Fcao (SA), Italy el ABSRAC I th paper the axom, of Axomatc Deg, are exteded to the o-probabltc ad repettve evet. he dea of formato, the theore of Fher ad Weer-Shao, a meaure oly o probabltc ad repettve evet. he dea of formato larger tha the probablty. It poble the formulato of a Exteded heory of Iformato for probabltc ad o-probabltc evet. A example tuded a applcato whch the umber of le the the umber of, ad the coupled deg caot be atfed. Keyword: deg, axome, formato, o-probabltc formato IRODUCIO he deg proce gve the tructure eceary for the traformato of the qualtatve eed, ofte tated o egeerg term, to the real product. h traformato acheved through the applcato of cetfc kowledge to the problem. Ug prevou deg databae the deg proce geerate everal alteratve to be evaluated frequetly. Uually the deg proce ubdvded to a ere of phae wth pecfc, evaluato to make betwee thee phae. Each evaluato determe whether the phae eed to be repeated, or f the deger eed to go back o oe or more phae. am P. Suh (990) propoe a axomatc method for hghly complex deg. he deg proce optmze elemet ug a et { } of fuctoal requremet ad of phycal parameter. a et { } He propoed two axom that could help to have a good deg. he relato of { } wth the { } mathematcally expreed a f he deg proce reduced to a ere of mappg from the deg' fuctoal requremet to the deg' parameter pace. he mappg proce betwee the doma repeated everal tme, wth the reult that prevou deg parameter determe the ext et of fuctoal requremet. he doma are defed by the vector: { } { } { PV} vector of fuctoal doma vector of phycal doma vector of proce doma he relato betwee thee two doma matrx otato wrtte a { } [ A]{ } () { } [ B]{ PV} () were [ A ] ad [ B ] are the deg matrce. I the problem whch the { } deped from o lear fucto, the equato () ca be wrtte dfferetal form { d } [ A]{ d} he elemet of deg matrx ca be wrtte a A ad the deg matrx Copyrght 004 by ICAD004 Page: /7

2 [ A].... matrx, but the phycal gfcace of the elemet A ca be lot. A deal deg matrx a quare dagoal.... matrx wth each related oe to oe to a gle. he ucoupled tolerace for a A. he propagato of tolerace for a decoupled deg wth a lower tragular matrx, expreed a A mall chage ay parameter may caue a devato the fuctoal requremet I lear deg A have are cotat. We A I (), the dagoal matrx a pecal cae. he deg matrx [ A ], geeral, a rectagular array of value. I the deg proce am P. Suh ha dcated two axom, o fuctoal requremet, order to exame the acto of plag. he axom are: Axom : he depedece Axom. Mata the depedece of fuctoal requremet. * A A * From equato () t evdet that. he coequece that decoupled deg ha le tolerace tha a ucoupled deg, ad the creae of the order of deg matrx make the lat tolerace maller. If the umber of greater tha the umber of, the the deg redudat. Whe the umber of le the the umber of, the the coupled deg caot be atfed. Suppoe that there a et of three{,, 3} ad a et of two {, }, the the equato matrx otato A A (3) A A 3 A3 A 3 I th equato 3 caot be alway atfed. he equato (3) ca be wrtte A + A A + A A + A 3 3 (4) Axom : he formato Axom. Mmze the formato cotet of the deg. he frt axom tate that the depedece of mut be alway mataed. he fuctoal et { } formato axom tate that the bet deg ha the mmum formato ad the mmum of fuctoal requremet. For comparg two deg, oe ca compare the formato cotet of the two deg whch ca atfy the fuctoally parameter. he formato cotet ca be decrbed by mea mlar to the Weer- Shao theory. he elemet of a deg matrx [ A ] ca be cotat or fucto, wth the coequece that the deg may be o-lear. Mathematcal techque ca traform a It ot poble to have a oluto of ytem (4) wthout to make chage to fuctoal requremet. DESIG WIH HE UMBER OF S LOWER HA HE UMBER OF I deal deg each fuctoal requremet mut be lked to oe deg parameter, ad vce vera each deg parameter ca atfy oe (or more) fuctoal parameter. From the ytem of equato (4) t tur out obvou that wth the umber of < t poble to have oly approxmate oluto. I th tuato the umber of uffcet to acheve all the exact mode. A good deg ha the mmum formato cotet. Aalyzg the formato of a deg t Copyrght 004 by ICAD004 Page: /7

3 A, B I J( B A) J( B) + J( A) (7) poble to udertad the phycal fluece of cotrat. he formato axom tate that the bet deg ha the mmum formato ad the mmum of fuctoal requremet. It ueful for to chooe betwee two or more deg. I abece of oluto we caot compare aythg: we eed at leat a oluto. Ug mathematcal traformato t poble to obta a approxmate oluto. If a et of tolerace mpoed to the doma of, the the tolerace are propagated from doma to doma ad the et of wll be modfed. If we ue the dea of formato metrc pace, ug the Laplace prcple of uffcet reao, accordg wth Maxmum Etropy Prcple of Jaye (MaxEt Prcple), we ca elect a oluto the dtrbuto that maxmze the Shao etropy meaure ad multaeouly cotet wth the value of cotrat. Wth MaxEt Prcple t poble to have a oluto whe the umber of le tha the umber of. ow t mportat to udertad the phycal gfcace of mathematcal traformato dervg from the ue of the Jaye prcple he dea of formato, the theore of Fher ad Weer-Shao, a meaure oly of probabltc ad repettve evet. he dea of formato larger tha the probablty ad the axom of Weer Shao ca be exteded to the o-probabltc ad repettve evet. Let Ω to be the feld of all evet ω (Fg. ), probabltc or o-probabltc, ad I a cla of part of Ω, I art ( Ω). Wth A I we ca aume the ext two axom: AXIOM I: he value of formato J(A) alway a umber o-egatve: J( A) + : I R (5) AXIOM II: he value of formato J(A) mootoou regard to cluo: A, B I, B A, J( B) J( A) (6) ow t poble the cotructo of ew algorthm term of formato, fouded oly o the frt ad ecod axom [3]. For depedet evet t opportue to aume a thrd axom: AXIOM III: If the evet A, B I are depedet for all the value of formato, we have: he thrd axom how that whe we are preece of depedet evet t poble to add up formato. If Ω a certa evet ad φ the mpoble evet tha, for a uveral valdty of J (A) ad J (φ ), for all Ω,I ad J mut be: J ( Ω) 0, J (φ) + (8) he expreo J ( Ω) 0 mea that Ω a certa evet wthout eed of formato. he expreo J (φ) + mea that f φ a mpoble evet wth the eed of fte formato. I a metrc pace Ω, f ω a evet I art ( Ω ), t meaure wll be alway correct. he kowledge of ω ot gve by t coordate Ω, but t poble oly to aert that ω lmted a ubet A I. If d( A ) the dameter of et A, tha, more the preco of meaure, le the meaure of dameter of evet A. If we aume that { P x,y } a et of deal data a cotuou cloed bouded ubet Ω [ D], gve ayε > 0, there a et { M x,y } of value of meaure wth uffcetly hgh preco uch that P M < ε for x, y (9) x,y x, y Ω But the probablty p of a exact meaure vere proporto to the preco, o the deal meaure of pot coordate ha ull probablty to be obtaed: t a mpoble evet. he mpoble evet φ ad the certa evet Ω are alway depedet from J ad A: they are uveral value. All three axom have correpodet axom Weer-Shao theory. Wth the axom J ( Ω) 0 ad J (φ) + t poble to cotruct model for formato very ueful applcato. I metrc pace Ω, for every evet A I we ca have a meaure of formato ug the mathematcal expreo : Copyrght 004 by ICAD004 Page: 3/7

4 J( A ) (0) Ay vector x ( x,x, L,x ) repreetg d( A ) proporto of ome whole ubect to the ut um cotrat x. Oe of mot uual dmlarte h defto of formato ha a atural applcato metrc pace [3]. I a metrc pace Ω, f ω d x, to meaure the dfferece betwee a evet I ( Ω ) art, better mut be the reult of t meaure tha maller the dameter of A, ad larger wll be J(A )[9]. If we aume that all the meaure are made wth equal care, ad for ay value of ω the data have a ormal dtrbuto, the probablty that the error d( A ) wll fall a mall terval δ gve for ω P ( ω ) exp ( d(a ) y [ σ ] δ y () σ π ) Smlar expreo ca be wrtte for all ω Ω. he tadard devatoσ a meaure of preco of the meauremet ad t a cotat for all the data. A the eparate meauremet are depedet for all evet, the probablty for all the product + P P( ω) exp ( d(a )) δy () k 0 σ π σ 0 Maxmum of P the g of the goode of the meaure of the dameter of ubet A. h wll occur whe 0 0 mmum d(a (3) ) I geeral, the crtero that the um of dameter d(a ) of et A mut be a mall a poble would have gve the better reult. We have that the ame formato ca be valued by the probablty ad by the o-probabltc meaure of dameter. So we ca have the meaure of formato from o-probabltc data. ow, for to acrbe ome formato to the realzed evet A, we ca aume a meaure of formato a o-probabltc fucto def [ d ( A ] J ( A ) Ψ ) J( A ) (4) d( A ) ad dtace x two compoto are Mkowk' dtace. I geeral, t poble to have the meaure of formato for probabltc ad o-probabltc evet ug emprcal or o-emprcal fucto o-attached to the probablty ad to the repettvee. It poble to defe a prcple for metrc pace o the groud of the heory of Iformato. O the aalogy of MaxEt prcple, the ame ca be Max Metrc Iformato Prcple (MaxMetrcIf). Itead of probablty, t poble to utlze a fte umber of approprate proporto ubect to a et of cotrat that add up to oe. I obervace of the Axom, let d,d,...,d be o-egatve real umber, let d d 0 ; ρ d + d d (5) ρ ; ; ρ 0 We ca ue a the meaure of formato the relato J(ρ) J( ρ, ρ,..., ρ ) ρ lρ (6) So that: J(ρ) maxmum whe ρ ρ... ρ J(ρ) mmum whe: oly oe umber zero I metrc pace, ug Eucld dtace, the formato maybe d ( x, ) * x ρ d ( x, x ) J ρ lo g ρ * * (7) he value of formato J(ρ) a meaure of equalty of umber amog themelve. Applg the ame formalm of MaxEt Prcple t eay to defe the MaxIf Prcple o the ba of o-called Laplace Prcple of uffcet reao. Max Metrc Iformato : Out of all kowledge, chooe the oluto earet to the uform dtrbuto of formato. Copyrght 004 by ICAD004 Page: 4/7

5 I the tuato whch we do ot have reao to prefer a oluto, t better chooe the oluto wth he etmator vector ha the dtrbuto uform dtrbuto, or the cloet to the uform dtrbuto of formato. d,,.., () 3 APPLICAIO O AXIOMAIC DESIG Oe applcato of the MaxMetrcIf prcple problem of approxmato a crtera to fd polyomal If the fucto f E x y,.. of emprcal for to repreet a gve et { }, data. Ideally, th proce hould take accout the relablty of the data, o the more relable data wll have grater weght o approxmatg fucto. I abece of kowledge, o ba of MaxIf prcple, we mut ue a polyomal, whch repreetg data, the devato from them chooe the oluto cloet the uform dtrbuto of formato. I axomatc method the deg obtaed from a et { } of fuctoal requremet ad a et { } of phycal parameter. he et fucto of the et f : he deg proce made wth a ere operato, wth whch the deg' fuctoal requremet are mapped to the deg' parameter pace. he mappg proce ha a reult that the prevou deg parameter determe the ext et of fuctoal requremet. Let be f the approxmatg fucto from whch we obta, from the data, the devato f. he etmator vector (,,.., ) d (8) From (5) we have f ρ f From (0) a fucto for to meaure the formato we ca ue the fucto J f (9) From MaxIf we have the max value for J whe J J... J... f f( ) (0) he max of formato obtaed whe the f ha the ame approxmatg fucto error. f... f ( ) () he devato of f (3) evaluated at certa abca ad the gve ordate correpodg to the ame abca: (4) From the oluto of the lear ytem A x b [ A] M M we have the value of from whch ca be valuated the approxmato wth the max formato. 4 EXAMPLE he example a cae tudy of the path of the pot P whch go from A to B (or to C), ad from B to C ug a lear path, wth the codto that the three pot, A,B ad C, are ot o oe traght. he are : Poblty of oe lear path from A to B : Poblty of oe lear path from A to C 3: Poblty of oe lear path from A to B ad from B to C. (5) he 3 caot be mapped phycal doma: t mpoble to go, ug a ler path, from A to C, cludg B. Deg parameter atfyg the fuctoal requremet ca be oly two: ad. he are Copyrght 004 by ICAD004 Page: 5/7

6 : Deg of path from A to B: vector AB (6) : Deg of path from A to B: vector AB : Deg of path from A to C: vector AC : Deg of path from A to C: vector AC * * * 3 : Deg of path from A to B to C Whe the umber of le the the umber of wth a tolerace, the the coupled deg caot be atfed. here are three{,, } ad two {, }. he equato 5 COCLUSIO 3 matrx otato {,, } [ A]{, } (7) 3 I th equato 3 caot be alway atfed. I lear depedece of from the A are cotat. We have A It poble a oluto wth the troducto of a tolerace o the preece of pot o the path. If t troduced a ukow tolerace the deg doma the a tolerace propagated to the fuctoal doma ad o. If, for the Max Metrc Iformato Prcple, mpoed that the the equal for all the elemet of { } relatohp betwee the two vector { } wrtte ad { } ca be A A A + A A + A he equato (5) may be wrtte matrx form A A A A 3 A3 A3 (8) (9) he equato (9) ca be ealy olved for obtag the vector of oluto uder tolerace {,, }.he oluto fgure 3. he value of Deg Parameter the tolerace. Ug the dea of formato, a larger way tha the dea of probablty, t poble the formulato of a Exted heory of Iformato for probabltc ad oprobabltc evet. o the queto o what happe whe the umber of le tha umber t poble to awer that ext a oluto whch fucto requremet ca be atfed approxmatg way. Whe umber of <, the the are uffcet to acheve all the. If mpoed to the doma of a et of tolerace, t poble to carry out a mathematcal traformato from whch t poble to obta all lackg value of.. he phycal gfcace of mathematcal traformato aalyzed wth the MaxEt Prcple of Jaye. he oluto, cotet wth the value of cotrat, obtaed electg the oluto that maxmze the Weer-Shao formato. I cocluo t poble to aert that: Whe umber of <, ug MaxEt Prcple, t poble to obta a approxmate oluto compatble wth boudary codto. 6 REFERECES [] J. Chg -.P. Suh, Computer aded geometrc topology ad hape deg wth axomatc deg framework, Proceedg of ICAD 00, Cambrdge, MA, Jue 0- [] Hubka, V. ad Eder, W.E. heory of echcal Sytem Sprger-Verlag, ew York [3] Jaye E.. Probablty heory: he logc of Scece. edu: 8008/ Jaye. [4] Kampe, J. de Feret, ote d eora dell'iformazoe, Ao Accademco 97-7 Aal dell'ittuto d Matematca Applcata dell'uvertà d Roma. 97. [5] Kapur,., Keava K., Etropy Optmzato Prcple, Acad. Pre Ic. Boto 99 Copyrght 004 by ICAD004 Page: 6/7

7 [6] Madhav V. Phadke Qualty egeerg Ug Robut Deg. 989 Pretce_Hall Iteratoal Edto, UK Lodo. [7] A. addeo, A. Doarumma,. Cappett, M. Pappalardo, Applcato of Fuzzy logc the tructural optmzato of a upport plate for electrcal accumulator a motor vehcle, Proceedg of the IMC 7, Galway, Irelad, 3-5/08/00, Page(): 47-56, alo publhed by ELSEVIER SCIECE o Joural of Materal Proceg echology, Iue: -3, Jauary 5, 00, pp [8] Pahl, G. Betz, W. Egeerg Deg. 988 he deg Coucl, Lodo. [9] Pappalardo, M. et al.- Meaure of depedece oft deg- Joural of Materal Proceg echology 4 (00)-ELSEVIER. [0] Pugh, S. Creatve Iovatve Product Ug otal Deg. 996 Addo-Weley Publhg Compay, Ic. ew York.. [] Sddal, J.. Probabltc Egeer Deg: Prcple ad applcato. Marcell Dekker, ew York, 983. [] Sddal, J.. Probabltc Modellg Deg- raacto of the ASME 330/Vol. 08, September 986. [3] Shao, C. E. he Bell Sytem echcal Joural Vol. 7, July, October, 948. [4] am P. Suh: he Prcple of Deg. Oxford Uverty Pre ew York 990. [5] am P. Suh Axomatc Deg-. MI-Pappalardo Sere Mechacal Egeerg Oxford Uverty Pre ew York 00. [6] Weer., Cyberetc, Par Herma (Act. Sc. 053) 948. Copyrght 004 by ICAD004 Page: 7/7