Continuous Belief Functions: Focal Intervals Properties.

Transcription

1 Contnuous Belef Functons: Focal Intervals Propertes. Jean-Marc Vannobel To cte ths verson: Jean-Marc Vannobel. Contnuous Belef Functons: Focal Intervals Propertes.. BELIEF 212, May 212, Compègne, France. pp hal-7668 HAL Id: hal Submtted on 17 Dec 212 HAL s a mult-dscplnary open access archve for the depost and dssemnaton of scentfc research documents, whether they are publshed or not. The documents may come from teachng and research nsttutons n France or abroad, or from publc or prvate research centers. L archve ouverte plurdscplnare HAL, est destnée au dépôt et à la dffuson de documents scentfques de nveau recherche, publés ou non, émanant des établssements d ensegnement et de recherche franças ou étrangers, des laboratores publcs ou prvés.

2 Contnuous belef functons: focal ntervals propertes Jean-Marc Vannobel Abstract The set of focal elements resultng from a conunctve or dsunctve combnaton of consonant belef functons s regretfully not consonant and s thus very dffcult to represent. In ths paper, we propose a graphcal representaton of the cross product of two focal sets orgnatng from unvarate Gaussan pdfs. Ths representaton allows to represent ntal focal ntervals as well as focal ntervals resultng from a combnaton operaton. We show n case of conunctve or dsunctve combnaton operatons, that the whole doman can be separated n four subsets of ntervals havng same propertes. At last, we focus on dentcal length focal ntervals resultng from a combnaton. We show that such ntervals are organed n connected lne segments on our graphcal representaton. 1 Introducton 1.1 Sources of nformaton Consder a source of nformaton S wth knowledge modeled by a unvarate convex (unmodal and consonant) probablty densty functon betf of a contnuous random varable X. The support of betf s called Ω = [Ω,Ω+ ] wth Ω,Ω+ R [4]. The mode and the varance of betf are respectvely noted µ and σ 2. Suppose now E = (F,m ), the pece of evdence deduced from betf. E s totally descrbed by the par composed of m, the Least Commted sopgnstc basc belef densty (bbd) deduced from betf [1] [4] and F = {I Ω m (I) > }, the focal set of ntervals wth elements n Ω. Jean-Marc Vannobel LAGIS, Unversté Llle1, e-mal: ean-marc.vannobel@unv-llle1.fr 1

3 2 Jean-Marc Vannobel 1.2 Focal ntervals An nterval A = [A,A + ] wth A,A + R such as m (A) s called focal nterval of E thus A F. All elements of F are nested ntervals n case of a consonant pdf betf and correspond to horontal cuts of betf as shown n fgure 1(a). It s convenent to label the elements of F accordng to ther ncluson order by a contnuous ndex. Ths can be done for nstance wrt the pdf value at focal nterval bounds [2] or wrt the half-length of the focal nterval [3]. Ths last opton allows n general to defne a sngle bbd s expresson for a whole famly of pdfs [6]. In case of symmetrcal pdfs lke Gaussan ones as well as Laplace ones, focal ntervals can be labeled by an ndex such as A = [A,A + ] wth: = x µ, R +, (1) σ A = µ σ,a [Ω,µ], (2) A + = µ + σ,a + [µ,ω + ]. (3) 2 Focal sets graphcal representatons 2.1 General consderatons We consder n what follows two peces of evdence E = (F,m ) and E = (F,m ) deduced respectvely from the Gaussan pdfs betf (x;µ,σ 2) and betf (x;µ,σ 2) wth µ µ. Focal ntervals are denoted by A k wth k the value taken by the label obtaned usng relaton (1). We assume the use of Gaussan pdfs snce many sensors model uncertanty by such pdfs but any other symmetrcal bell shaped pdfs lke Laplace or Cauchy ones would also serve the purpose. F, s the focal set resultng from a conunctve (resp. dsunctve) combnaton of E and E. Elements of F, correspond to the non empty ntersecton (resp. unon) of pars n F F. The content of F, and the length dependences between ts elements depend of course on the chosen combnaton rule. 2.2 Bell shaped probablty densty functons The graphcal representaton proposed n fgure 1(a) shows the focal set F obtaned from a Gaussan pdf Betf. Elements of F are ordered wrt label, dfferng n that pont from the graphcal representaton proposed by Strat [5].

4 Contnuous belef functons: focal ntervals propertes 3 When labelng focal ntervals wrt ther length as defned n (1), the focal set F s encompassed by two symmetrcal half-lnes defnng an sosceles trangle. Equatons of these half-lnes are deduced from relatons (2) and (3) and correspond to: = A µ σ, (4) = A+ µ σ. As shown n fgure 1(b), ths s a convenent way to graphcally compare focal ntervals comng from dfferent focal sets. One can see n ths fgure the result of the ntersecton or unon of two ntervals A k F and B l F whch are ndexed resp. by k and l. For nstance, t also allows to show the doman of ntervals B F that do not ntersect wth A k (f any). It s obvous from relaton (1) that the label value of these B ntervals s n [, A k + µ ). 2.3 Bounds relatons of F F elements Fgure 2(a) shows the pars of ntervals (A x,b x ) F F havng a common bound x Ω. The ndex pars ( x,x ) R+2 correspondng to (A x,b x ) draw the lnes 1, 2 and 3 as shown n Fgure 2(a). These lnes are defned by 1 : 1 : x = µ µ 2 : x = µ µ 3 : x = µ µ + σ σ x, x [,µ ], x, x [µ,µ ], + σ x, x [µ,+ ]. Pars ( x,x ) on the half lne called 1 lead to x µ such as: { A x = B x = x, A x + B x +. Pars ( x,x ) on lne segment 2 correspond to x [µ,µ ] such as: (5) (6) A x + = B x = x. (7) At last, pars ( x,x ) on the half lne 3 correspond to x µ 2 such as: { A x + = B x + = x, A x B x. (8) 1 proof s not gven here due to lack of space

5 4 Jean-Marc Vannobel To outlne exstng partal conflct k, between the agents E and E [6], the label values of the modes are denoted by K and K : { K = µ µ σ, K = µ µ. (9) Relatons (5) show that the absolute value of lne drectons of 1, 2 and 3 are equal to arctg( σ ) and correspond to angles α 1,α 2 and α 3 n fgure 2(a). 2.4 Focal ntervals ntersecton or unon overvew As shown n fgure 2(b), focal ntervals A k F and B l F can drectly be drawn on the chart by respectvely vertcal and horontal lne segments snce 1, 2 and 3 correspond to the focal ntervals bounds. The path defned by half-lnes 1, 2 and 3 covers Ω. It becomes thus easy to analye pars n F F to deduce ther ntersecton or unon. For nstance, A k ntervals shown n fgure 2(b) are such as A k B l = [ B l,a ] k+ and B l = [ A k,b ] l+. When consderng k and l respectvely as horontal and vertcal cursors t s also possble to fnd the lmts of ntervals ntersectng or ncludng one another or not. 2.5 Partcular domans Fgure 2(b) shows also that the 1, 2 and 3 lnes separate F F n four domans called D 1, D 2, D 3 ans D 2 4. For µ < µ,,, defnng pars (A,B ) F F, we have: > K + σ A < K + σ A < K + σ, > K σ A B < K σ, A B B (D 1 ), B (D 2 ),, > K + σ / { },A,B (D 3 ), = (D 4 ). (1) When µ 1 = µ 2 only D 1 and D 2 exst and are separated by the half-lne 2 = σ1 σ proof s not gven here due to lack of space

6 Contnuous belef functons: focal ntervals propertes Consonant subsets of F, The focal set F, obtaned after a conunctve or a dsunctve combnaton operaton of E and E s composed of an nfnte number of nested focal ntervals subsets. These consonant subsets appear both n D 1, D 2 and D 3 shown n fgure 2(b) and are partally represented n fgure 1(b) wth the dark gray area. These subsets are not dsont and consequently f the same nterval belongs to several of them, t s necessary to ntegrate to get ts total weght. The doman D 4 dffers from the other ones as t s empty n case of a conunctve combnaton operaton and composed of nested non convex ntervals n case of a dsunctve one. 3 Same length ntervals resultng from ntersecton and unon operatons of focal ones 3.1 Intersecton of focal ntervals The length l of the ntersecton of the pars (A,B ) F F can be deduced from the characterstcs of D 1, D 2, D 3 and D 4 gven by relatons (1). l only depends on the pdfs betf and betf parameters and the focal ntervals ndexes and : D 1 : A B = A, l(a B ) = 2σ, D 2 : A B = B, l(a B ) = 2, D 3 : A B = [B,A + ], l(a B ) = µ 2 µ 1 + σ +, D 4 : A B =, l(a B ) =. (11) We can fnd the elements of F F havng an dentcal ntersecton length L = l(a B ) wth the help of relatons (11). For nstance, the ndexes of these elements n D 3 are: thus: = L + µ 2 µ 1 σ (12) = A σ wth A = L + K. (13) Ths s graphcally represented n fgure 2(c) by the lne segment 6 bounded by ponts (F,C) and (D,E) and crossng the pont (, ) = (,A) wth A as defned n relaton (13). Pars (A,B ) n D 1 havng also an ntersecton length equal to L correspond to the ponts of the half-lne 5 crossng (F,C) snce A B = A n D 1. Pars of ntervals on the half lne 7 crossng (D,E) n fgure 2(c) have an ntersecton length equal to L too snce A

7 6 Jean-Marc Vannobel B = B n D 2. At last, values A to F are qute nterdependent 3 dependng on K and K values. Ths s due to the symmetry propertes of straght lnes 1 to 7 n fgures 2(b) and 2(c). Many relatons dependng on these parameters lead to the value of L such as L = 2 (C K ) = 2σ (D K ) for nstance. 3.2 Unon of focal ntervals One can deduce from the relatons (1) related to the domans D 1, D 2, D 3, D 4 that for the unon operaton of pars (A,B ) F F, we have: D 1 : A B = B, l(a B ) = 2, D 2 : A B = A, l(a B ) = 2σ, D 3 : A B = [B,A + ], l(a B ) = µ 2 µ 1 + σ +, D 4 : A B =, l(a B ) = 2(σ + ). (14) As expressed n (14), concernng D 3, one can wrte: l(a B ) µ 2 µ 1 = σ +. (15) From (15), pars (A,B ) D 3 havng a constant unon length L = l(a ) satsfy thus: B = A σ wth A = L K. (16) Ths relaton corresponds n fgure 2(d) to the lne segment 6 bounded by ponts (F,C) and (D,E) and crossng the pont (, ) = (,A) wth A as defned n relaton (16). Pars n D 1 havng a unon length equal to L correspond to the ponts of the lne segment 8. Ths s also the case n D 2 for the ponts of the lne segment 9. As n the case of ntersecton operaton, many relatons 4 lnk the value of L to the parameters of pdfs betf and betf. For nstance we have L = 2 C = 2σ D. 4 Concluson and Acknowledgment As we have seen, the focal set F, obtaned n case of a conunctve (resp. dsunctve) combnaton of two peces of evdence E and E wth consonant 3 proof s not gven here due to lack of space 4 proof s not gven here due to lack of space

8 Contnuous belef functons: focal ntervals propertes 7 Betf - A k k = k k k Betf ( A ) Betf ( A ) k k A A x k A = [ k A, k A ] F k l k A A k U l B k A U l B l B k A U B = x (a) Focal ntervals doman F resultng from a Gaussan pdf. (b) Intersecton and unon of focal ntervals resultng from two Gaussan pdfs. Fg. 1 Focal ntervals graphcal representaton relatvely to the label value. focal domans s not as heterogeneous as t seems to be. Intervals belongng to F, are sorted nto only four domans. In each of these domans, pars (A F,B F ) of focal ntervals share common propertes regardng ntersecton A B and unon A B. These four domans can be graphcally represented n a lnear space where they are separated by straght lnes when the focal sets F and F are composed of centered and consonant ntervals. At last, elements of F, havng a same length are lnked by lnear relatons. Ths can be useful n problems where nterval lengths have to be taken nto account. Authors are ndebted to J. Klen for the revew of ths work. References 1. Caron, F., Rstc, B., Duflos, E., Vanheeghe, P.: Least commted basc belef densty nduced by a multvarate gaussan: formulaton wth applcatons. Internatonal Journal on Approxmate Reasonng, vol. 48, no. 2, (28) 2. Doré P.-E., Martn A., Khenchaf A.: Constructng of least commted basc belef densty lnked to a multmodal probablty densty, n COGIS, Pars (Fr), Rstc, B., Smets,P.: Belef functon theory on the contnuous space wth an applcaton to model based classfcaton. In IPMU 4, Informaton Processng and Management of Uncertanty n Knowledge Based Systems, pp , Pars (Fr) (24) 4. Smets, P.: Belef functons on real numbers. Internatonal Journal of Approxmate Reasonng, vol. 4, no. 3, (25)

9 8 Jean-Marc Vannobel 5. Strat, T.H.: Contnuous belef functons for evdental reasonng, n AAAI-84 (Ed., Natonal Conference on Artfcal Intellgence, , (1984) 6. Vannobel, J.-M.: Contnuous belef functons: sngletons plausblty functon n conunctve and dsunctve combnaton operatons of consonant bbds. Proceedngs of Workshop on the theory of belef functons, CDROM, 6 pages. Brest(Fr) (21) Z Z 1 D 1 k A 1 X 1 3 l l B D 3 3 l B K 2 x K 2 x 2 X K 3 Z D 4 K k A k D 2 Z (a) Pars of ntervals havng a common lower or upper bound (b) Domans of dentcal propertes Z Z 5 A 4 A 4 C 6 C 8 6 K E 7 K E 9 F K D B Z F K D B Z (c) Same length ntervals resultng from a conunctve combnaton (d) Same length ntervals resultng from a dsunctve combnaton Fg. 2 Graphcal representatons of focal ntervals propertes.