Formalisms For Fusion Belief in Design

Transcription

1 XII ADM Internatonal Conference - Grand Hotel - Rmn Italy - Sept. 5 th -7 th, 200 Formalsms For Fuson Belef n Desgn Mchele Pappalardo DIMEC-Department of Mechancal Engneerng Unversty of Salerno, Italy Abstract The nature and scope of fuson of mprecse data n the desgn s explored and dscussed wth subectve vew of de Fnett. Every stage n the desgn process ncludes a level of approxmaton and s typcally hgh n the prelmnary phase. Engneerng proects often are modfed n several desgn teratons before beng completed. Informaton receved from many groups or experts workng on a proect wll often necesstate changes n a desgn. The nteracton between dfferent groups assocated wth a desgn proect often takes the form of an nformal fuson of data. Ths form of nteracton commonly arses when engneerng nformaton s mprecse. For the fuson of belef the theory of Dempster and Shafer s used. DS theory generalses the Bayesan rule and provdes the mechansm for the fuson of nformaton.. INTRODUCTION The analyss of belef s a method for manpulatng nformaton on prelmnarly desgns. It s possble to combne preference usng the aggregaton of two or more desgn functon. In desgn the subectve nterpretaton of Carnap, probablty as measure degree of belef, s very often supported. The probablty of a statement s the degree of confrmaton on one event A. If the posteror probablty P new (A) s greater or less of the pror P old (A) probablty than we have the measure of degree of confrmaton of Carnap new old Measure of Carnap = P ( A) P ( A) It s a measure for confrmng or nfrmng a hypothess. In Bayesan approach, let A be a proposton the set of possble truths s { A, A}, wth the axom of probablty we have P ( A) + P( A) = Denotng varous propostons A, B, C, etc., f the propostons A B s true and A s false than we have the Laplace s mathematcal representaton of process of learnng (Bayes Theorem): P ( A B, C) = P( B A, C) P( AC) P( B C) The Bayes Theorem s an algorthm for update results and for acqurng new nformaton. P ( AC) Is the pror probablty for A when we know only C and p ( A B, C) s the posteror nformaton and updated as a results acqurng new nformaton B. A represents the hypothess under analyss C represents what we know about A (table of truth) before gettng B A2-50

2 XII ADM Internatonal Conference - Grand Hotel - Rmn Italy - Sept. 5 th -7 th, 200 (new data). The orgnal degree of belef s replaced by a new degree of belef when new evdence s obtaned. Bayes theorem s smple a rule for manpulatng probabltes but t cannot by tself help us to assgn the probabltes. When the detals of a desgn are unknown and there s not suffcent nformaton, the prelmnary decsons are the most mportant. The Lap lace s Prncple of Insuffcent Reason asserts that, when one has not suffcent nformaton to dstngush between the belefs of a lot of events, the best strategy s to consder all cases equally lkely. The Laplace s prncple, mathematcal translaton of an ancent prncple of wsdom, s a decson-makng prncple snce assgn values of belef. Fg. One basc problem n makng decson s how to evaluate the belef of an uncertan event. Snce dfferent hypothess assgn dfferent dstrbutons of belef, the problem s haw to take n account dfferent hypothess and to assgn only one fnal set of proper values whch can came from probablty, possblty or theory of evdence or from more varety of nterpretatons of probablty have been proposed. In every desgn we can n stuatons where the nformaton s ncomplete and the truth-value of proposton s ndetermnate. We must decde what s possble true and possble false and to do next on bass of a plausble reasonng wthout to lack nformaton. Accordng to N. Wener n the learnng the nformaton results fundamental on the acton n progress snce t results so possble the acton of feed-back. In the choce of the followng acton the prncple of the feedback means that the behavor s compared perodcally wth the result from to acheve, and that the success or the falure of ths behavor t modfes the future result. The comparson s founded on the measure of the nformaton that s founded n turn on the value of the probabltes. For havng the control of uncertanty, t s necessary to measure nformaton on t. We can use as weght of evdence the Turng s defnton of nformaton. If x s a fnte partton of a probablty space, than the nformaton functon of x s a step functon whose value on an element of x s the negatve of the logarthm of the probablty of ths element (entropy). If we assume that the entropy of a system, n a gven state, s drectly proportonal to the logarthm of the probablty of fndng t n that state, than entropy s the measure of the nformaton of the system. The Shannon s defnton of entropy s = = = n under the constrant p ( x ) n ( p( x ),..., p( x )) p( x ) p( x ) H n = = = log. When the probablty s equally lkely the value of entropy s max. The entropy s the measure of uncertanty. When we have new results wth new nformaton, the new effect can be evaluated measurng the dstance D(p:q) between the dstrbuton a pror dstrbuton q from the new dstrbuton p obtaned usng new data. The measure of dstance can be calculated usng the Kulback s formula of cross-entropy A2-5

3 XII ADM Internatonal Conference - Grand Hotel - Rmn Italy - Sept. 5 th -7 th, 200 ( p : q) = p ln( p q ). If our a pror dstrbuton B ( A) = B( A) = 0. 5 s of max entropy than D values of cross-entropy are equal to Shannon s values of entropy. Vald soluton has mnmum entropy. A system can have nfnte soluton consstent wth gven constrant, but only wth maxmum entropy, or mnmum cross-entropy, probablstc enttes adust ther values as to gve the optmum soluton (MaxEnt prncple of E.T.Jaynes). The Maxmum Entropy prncple assgns the probabltes. For to have an deal soluton, wth mnmum entropy, a system must depend from a fnte and lmted number of parameter. In a desgn we can have far solutons f t s observed the fundamental axom of Soft Desgn: Vald desgn has mnmum values of nformaton and depends on a fnte and lmted number of ndependent, or soft dependent, parameters. The restrctons of the axom nvolve reproducblty and stablty of the same system. For to select a fnal vald proper selecton, among possble solutons, t s necessary to fx a weghtng parameter. The real art s choosng an approprate space of possbltes. MaxEnt assgn a pror probablty dstrbuton over ths space. For mnmzng expected loss, a Desgner starts selectng a subset from the pror plausble data so that effcent processng s possble, and for to get a posteror plausble subset. The crtcal moment s the fnal selecton from solutons consstent wth gven constrant. 2. INFERENCE Probablstc models use the probablty dstrbutons for to descrbe uncertantes. It s a good practce to construct probablstc model of uncertantes usng subectve nformaton because desgners clam that probablty does not only express the frequency of an event but t also descrbes a subectve strength of belef that the event can occur. One can use probablstc models when has suffcent numercal data to estmate dstrbuton of random varable. In absence of suffcent data the probablstc model for desgn can be constructed on belef of experts. Bayesans beleve that t s possble to make probablty assgnment even n presence of subectve degree of belef and n absence of frequency nformaton. The probablty s central to the modelng of engneerng phenomena and at the same tme there s much dsagreement about nterpretaton that should be attached to concept. We use two best-known nterpretatons: the classcal of Bernoull and Laplace and the subectve of de Fnett. In the classcal nterpretaton an uncertan event can be decompose nto equally lkely cases and the probablty s the rato of favorable to total cases. The prncple of Laplace asserts that, when, on an uncertan event one has not suffcent nformaton to dstngush between the belefs of the cases, the best strategy s to consder all equally lkely. Wth subectve vew of de Fnett, the probablty s the subectve strength of belef that can be assgned to any event repeatable or not. In subectve belef f p s the probablty of an event E f S s bettng amount (postve or negatve) than the gan wll be (-p)s. The condton of coherence of de Fnett s that gan (-p)s and bet ps wll be always of opposte sgn. The mathematcal translaton s p ( p) S S s always postve than wll be p ( p) 0. Ths condton of coherence s always vald when s 0 p. In the analyss of a desgn, f G s the gan and C s the loss (or cost), on the bass of MaxEnt prncple, the relatons between gan and loss gve method for to fnd values of belef useful for to get vald solutons. Indcatng wth ( C / G) = γ > 0, the expected utlty relaton of von Neumann-Morgenstern, between gan and cost on event A, s B A G B A C B A B A γ = ( ) ( ( )) = ( ) ( ( )) k A2-52

4 XII ADM Internatonal Conference - Grand Hotel - Rmn Italy - Sept. 5 th -7 th, 200 Analyzng the coherence of expected utlty relaton usng the subectve defnton of probablty of de Fnett we must to have T = [ B( A)( B( A)) γ ] 0 The coherence s possble only f t s 0 B ( A) Fg.2 In total absence of nformaton the Laplacan strategy s to consder equally lkely G, C, B(A) and B( A). Then f s G=C than s γ = and MaxEnt prncple gves the Laplacan dstrbuton B ( A) = B( A) = 0. 5 and k=0. The Laplacan soluton of s coherent wth prncple of de Fnett T = [ B( A)( B( A)) γ ] = Wth constant expected utlty k, for each modfcaton of γ, we get a new dstrbuton. B(A) represents learnng and tells us how update belef when our state of knowledge change trough acquston of new values of γ. B ( A) = ( γ + k) ( + γ ) The new effect can be evaluated measurng the measure of Carnap as dstance D ( p q) between the Laplacan dstrbuton q ( a pror dstrbuton) from the new dstrbuton p obtaned usng new data. The condton of coherence on the expected utlty relaton of von Neumann-Morgenstern can be obtaned applyng the condton of Laplace. The prncple of Laplace asserts that when n a utlty relaton, has not suffcent nformaton to dstngush between the belefs of the cases, the best strategy s to consder all equally lkely the gan and the bet, than B ( A) = ( B( A) ) γ ; k = 0 ; B ( A) = γ ( + γ ) The value of γ s always 0 and the range of belef s 0 B ( A). The condton of coherence gves the range of belef. The values of belef based on a pror dstrbuton wth k=0 s the best procedure for to obtan nference. Wth expected utlty k=0, for each modfcaton of γ, we get a new dstrbuton and B(A) s the Break Value of belef. We defne the mnmum measure of belef for havng expected utlty non negatve as the break values BV s. The value of k=0 n the expected utlty relaton of von Neumann-Morgenstern s the same condton of coherence of de Fnett on the gan and bet. The break value tells us how update belef when our state of knowledge change trough acquston of new values of γ. A2-53

5 XII ADM Internatonal Conference - Grand Hotel - Rmn Italy - Sept. 5 th -7 th, 200 Break Value = B( A) = Bel( A) = m( A) It s possble to fnd from γ a convex dstrbutons of mnmum measures of the belef. The degree of truth, n hs conventonal meanng of degree of plausblty and truth (but also of satsfacton) can be evaluated usng the DS rules. For the mass m s m Fg.3 ) + m ({ A} ) + m A} ) = The ntersecton of nformaton s sets (assumng that the nformaton s, for defnton, true) tends to ncrease the degree of truth for whch an hypothess s defntely or confrmed or dened. The lack of convergence of the analyss means that t s not verfed the premse of the process. The DS theory gves a method for to have the selecton, among desgn alternatves, wth greater belef. From the sets of break values the nference s Bel ( A) = {[ Bel ( A) Bel ( A) ] Bel ( A) } BELIEF In nference problems the belef can be represented by mathematcal enttes Bel(.) [ 0,] wth the sum to ts negaton equal to. In Bayesan approach on an probablstc event A the belef s Bel(A) and Bel ( A) f uncertanty s equal to zero than Bel( A) = Bel( A) Usng the belef we wants to allow to more general approach to representng uncertanty than Bayesan approach, thus f Bel(A) s the mass of A does not mean that Bel ( A) n the weght of belef n A. In our framework the measures correspond to the belef functon. Usng belef we have more general measure then probablty and every event A can le n an nterval. We can take belef and plausblty as lower and upper bounds of the probablty assgned to an event. A method for handlng data n presence of uncertanty wth qualtatve values s theory of Dempster-Shafer. The DS theory of s a method for reasonng under uncertanty, nclude Bayesan probablty as specal case, and ntroduce the belef functon as lower probabltes A2-54

6 XII ADM Internatonal Conference - Grand Hotel - Rmn Italy - Sept. 5 th -7 th, 200 and the plausblty functon as upper probabltes. Here we are nterested n applyng theory because the numercal nformaton requred by Bayesan methods are not avalable. Numercal measure n presence of uncertanty may be assgned to set of propostons as well as sngle proposton. The probabltes are apportoned to subsets and the mass v can move Θ = x,.. be the frame of dscernment over each element. Let the fnte non empty set { x n } whch s the set of all hypothess. The basc probablty s assgned n the range [,] 0 to the n 2 subset of Θ consstng of a sngleton or conuncton of sngleton of n elements x. The basc probablty s a functon whch assgn the wegh to the subset such that m( A ) = m( Φ ) 0 A Θ = The lower probablty P ( A ) s defned as P ( A ) = A A m( A ) And the upper probablty P ( A ) s defned as A A P ( A ) = m( A ) The m( A ) values are the ndependent basc values of probablty nferred on each subset A. The belef functon of set M f s gven by Bel ( M ) = A M m( A ) Pl ( M ) = A M Θ m( A ). The evdental nterval that provdes a lower and upper bound s Evdental Interval = Bl( M ),Pl( M ) If m and m 2 are the ndependent basc probabltes from ndependent evdence, and { A } A the sets of focal ponts, then the theorem of Shafer gves the rule of combnaton. and { } 2 Let m and m 2 two ndependent basc probabltes from ndependent evdence. If A ( ) 2 ( 2 ) > 0 A2 Φ m A m A then def m( A ) m2 ( A ) A A2 = Ak 2 Θ m( A) = ( m m2 )( A) =, A Φ, ( m m2 ) = 2 [ 0,] ( ) 2( 2 ) A A2 =Φ m A m A gve the rule for combnng two or more probablty gven from ndependent evdence. [ ] Fg.4 A2-55

7 XII ADM Internatonal Conference - Grand Hotel - Rmn Italy - Sept. 5 th -7 th, 200 If we assume that belef functon Bl, Bl2, Bl3,..., Bln are assgned n the same frame of dscernment, than accordng wth the Shafer s rule of combnaton, the new belef functon may be yelded va Bel( A) = { [( Bel( A) Bel2( A) ) Bel3( A) ]...} 4. APPLICATION The example s a prelmnary desgn task consstng n a selecton between two soluton proposed from experts. The result of experts consst n two set of break values of belef. If the belef on two desgn s Expert E Expert E2 m E ( A) m E 2 ( A) Soluton Soluton The solutons of experts s subectve and gves the mass on m ({ A}) and m A}). For Soluton the mass of belef are Expert E ({ A A} ) = m A} )+ m ) + m ) = = Expert E 2 ({ A A} ) = m A} )+ m ) + m ) = = Soluton m E ) 0.0 m E ) m E ( { A, A } ) =0.8 m E 2 ) =0.0 { A } =0.0 { A } =0.0 m E 2 ) =0.3 { A} =0.06 { A} =0.4 m E 2 A} ) =0.7 { A } =0.0 { A} =0.24 { A, A} =0.56 m, 2 ) =0, m,2 ) = =0.44, m,2 A} ) =0.56, Pl Soluton ) =- m,2 ) =0.56 For Soluton 2 the mass of belef are Expert E ({ A A} ) = m A} )+ m ) + m ) = = Expert E 2 ({ A A} ) = m A} )+ m ) + m ) = = Soluton 2 m E ) 0.0 m E ) m E ( { A, A } ) =0.6 m E 2 ) =0.0 { A } =0.0 { A } =0.0 m E 2 ) =0. { A} =0.04 { A} =0.06 m E 2 A} ) =0.9 { A } =0.0 { A} =0.36 { A, A} =0.54 m, 2 ) =0, m,2 ) = =0.46, m,2 A} ) =0.54 Pl Sluton 2 ) =- m,2 ) =0.54 We have Pl Soluton ) > Pl Sluton 2 ) Soluton s more plausble than the Soluton 2. A2-56

8 XII ADM Internatonal Conference - Grand Hotel - Rmn Italy - Sept. 5 th -7 th, CONCLUSIONS The nature and scope of fuson of mprecse data n engneerng s explored and dscussed wth subectve vew of de Fnett, and applcaton of the method s llustrated. The ntersecton of nformaton sets tends to ncrease the degree of truth for whch an nformaton s defntely ether confrmed or dened. Preferences are expressed on an absolute scale, where the mass of belefs ndcates a completely acceptable value or completely unacceptable value. The fuson of desgn decson combnes many ndvdual preferences nto a sngle, overall preference. The lack of convergence of the analyss means that has not verfed the premse of the desgn. Prof. MICHELE PAPPALARDO DIMEC-Department of Mechancal Engneerng Unversty of Salerno Fscano (SA) - Italy tel fax e-mal: pappalar@brdge.dma.unsa.t REFERENCES [] Lukasewcz J. Modal Logc. Polsh Scentfc Publsher. Warzawa. 970 [2] B. de Fnett, Theory of Probablty, 2 vols. (New York, 974). [3] Jaynes E.T. Baysan Methods: General Background. In Proceedngs Volume. Maxmum Entropy and Methods n Appled Statstcs. J.H. Justce, Edtor, Cambrdge Unversty Press (985); pp.-25. [4] Zmmerman H.J.: Fuzzy Sets Theory and ts Applcaton. Boston 985. [5] Shafer G. A Mathematcal Theory of Evdence -Prnceton Unversty Press76 [6] A. Donnarumma, M. Pappalardo - Desgnng n Many-Valued Logc IPMM99. 2 nd Internatonal Processng and Manufacturng of Materals, Hawa, 999. [7] A. Donnarumma M. Pappalardo Uncertanty n Soft Desgn - 3 nd Internatonal Conference on Qualty - Oxford, England 30 nd -3 rd March [8] A. Donnarumma M. Pappalardo Maxmum Entropy n Soft Desgn MaxEnt th Internatonal workng on Byesan Inference and Maxmum entropy Methods n Scence and Engneerng, Yvette,France July [9] A. Donnarumma, M. Pappalardo, A. Pellegrno Measure of Independence n Soft Desgn, IMC 7 7 th annual Conference of the Irsh Manufacturng Commttee Galway, Ere, August 23-25, [0] A. Donnarumma, M. Pappalardo Selecton of a synchromesh wth the evdence CIM 200 Zakopane 7-9 March, 200 Poland. A2-57