2 Complement Representation PIC. John J. Sudano Lockheed Martin Moorestown, NJ, 08057, USA

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1 The yste Probablty nforaton ontent P Relatonshp to ontrbutng oponents obnng ndependent Mult-ource elefs Hybrd and Pedgree Pgnstc Probabltes ohn. udano Lockheed Martn Moorestown U john.j.sudano@lco.co bstract - n the desgn of nforaton fuson systes the reducton of coputatonal coplexty s a key desgn paraeter for real-te pleentatons. One way to splfy the coputatons s to decopose the syste nto subsystes of non-correlated nforatonal coponents such as a qualtatve nforatonal coponent a quanttatve nforatonal coponent and a copleent nforatonal coponent. probablty nforaton content P varable assgns an nforaton content value to any set of syste or sub-syste probablty dstrbutons. The P varable s the noralzed entropy coputed fro the probablty dstrbuton. Ths artcle derves a P varable for a subsyste represented by the copleent probabltes. Ths artcle also derves a relatonshp between the P varable of sub-syste coponents and the syste nforatonal P varable. hybrd pgnstc probablty s ntroduced that s robust n estatng a probablty for any aturty of the ncoplete data set. new ethodology of cobnng ndependent ultsource belefs s presented. pedgree pgnstc probablty s ntroduced that uses soe nforaton of the orgnal fused data sets to copute a better pgnstc probablty. ntroducton probablty nforaton content P varable assgns an nforaton content value to a probablty easure. The P varable s the noralzed entropy coputed fro the probablty dstrbuton. P value of one ndcates total knowledge s avalable and there s no abguty n the decson akng.e. one of the hypotheses has a probablty value of one and the rest have zero. P value of zero ndcates that all the hypotheses have an equal probablty of occurrng and t s not possble to ake a good decson. Gven a probablty easure P on a set { a a a } wth respectve probabltes { P a P a P a } the probablty nforaton content s: P Α + P a P a n the desgn of nforaton fuson systes the reducton of coputatonal coplexty s a key desgn paraeter for real-te pleentatons. One way to splfy the coputatons s to decopose the ncoplete syste nforaton set nto subsystes of uncorrelated nforatonal coponents. The subsystes of uncorrelated nforatonal coponents can be labeled as a ature or non-ature data set. The Pgnstc probabltes proportonal to belefs Prl and self-consstent pgnstc probablty PrcP are vald only for a ature data set. ets pgnstc probablty etp the pgnstc probablty proportonal to all the plausbltes PraPl the pgnstc probablty proportonal to the plausbltes PrPl and a new ntroduced hybrd pgnstc probablty PrHyb are robust for all type of data sets. oe subsystes of uncorrelated nforatonal coponents can be represented by sets { } of basc belef assgnents s a new ethodology s presented for cobnng or fusng the s to produce one set of { }. Pedgree Pgnstc Probablty PrPed s ntroduced that uses the fused wth the pedgree nforaton of each subsystes to copute a better pgnstc probablty. opleent Representaton P The copleent c of an event s the set of all outcoes n the saple space that are not ncluded n the outcoes of event. f the probablty of event 77

2 occurrng s P then the probablty of ts copleent s: P P. nce the su of the sngleton probabltes s one P 3 t follows that the su of the probablty of the sngleton copleent s -: P P. 4 The probablty nforaton content for a syste represented by copleent probabltes s: P P P Α + 5 noralzed copleent probablty s defned as: 6 P ˆ P P The probablty nforaton content for a syste represented by noralzed copleent probabltes s: P ˆ P ˆ P ˆ Mutually Exclusve ubsyste Dsjont Decoposton n the desgn of nforaton fuson systes the reducton of coputatonal coplexty s a key desgn paraeter for real-te pleentatons. One way to splfy the coputatons s to decopose the syste nto subsystes of utually exclusve dsjoned coponents. Let Ω be the set of possble outcoes where the outcoes are utually exclusve and exhaustve sngleton eleents of the decson envronent. n soe systes the Ω set s decoposed nto utually exclusve subsystes Ω Ω. For systes wth a coplex nput e.g. real-te sensor easureents ultdensonal fltered feature extractons real-te data base and a pror data base nforaton content real-te natural language text and sybols parsng evdence quanttatve and qualtatve councaton clues and nconsstent errors a power set of the outcoes s a better representaton of the ncoplete nforaton set. Power - set Ω Ω Ω Ω 8 ote that ths power set s uch saller than the full syste power set for two or ore subsystes. Ω Ω Ω Ω 9 For the subsystes supportng the nuber of possble hypothess wth respectve coponents j k the sub-syste P values are coputed fro the ndvdual probabltes: P P + P P + P P j P P j + P P j 0 P P The syste probablty can be coputed fro the uncorrelated subsyste probablty dstrbutons as: P s P P j P k ote that the syste probablty s noralzed to one provded that the subsyste probablty dstrbutons are noralzed to one. s P s s P s s P s j P k The syste P varable s: j. P P j P k P j P s k. P k P s s P P + 3 ubsttutng the syste probablty n ters of the subsyste probabltes: P P + j k. P P j P k P P j P k L 78

3 4 Expandng the logarthc ter:. P P j P k P + P j + P k + j k P P ung over the soltary probablty ters: P P + P P + P j P j + P k j k ubsttutng the subsyste P varables: P P + P k P P + P P + P P plfyng the logarthc coeffcents shows that the syste P s a weghted average of the subsyste Ps: P P + P P + P P + P P ote that f all the subsyste nforaton sets support the sae nuber of hypotheses.e. then the syste P equaton splfes to the average of the subsyste Ps. P P P P + P P + P P plfyng otaton 9 n order to splfy the dervatons the copoundto-su of sngletons operator s used. The copoundto-su operator on sngleton eleents has the followng propertes: el{ j k z el j + el k + + el z Pl{ j k z} Pl j + Pl k + + Pl z { j k z} j + k + + z 0 5 Hybrd Pgnstc Probablty seres of Pgnstc probabltes were ntroduced n that use soe Posteror nforaton to proportonally decopose the copound basc belef assgnents nto Pgnstc probabltes. The Pgnstc probablty proportonal to all plausbltes PraPl s coputed usng the followng equaton. wth PraPl el + ε Pl Ω Ω - el ε Pl The hybrd pgnstc probablty transforaton dstrbutes the basc belef assgnents proportonally to PraPl aong each sngleton eleent of Ω wth Ω for all. PrHyb PraPl 3 PraPl M M M The pgnstc probablty s noralzed to one. PrHyb Ω For each sngleton eleent the pgnstc probablty s bound between the elef and the Plausblty. el 6 Dscusson PrHyb Pl 4 5 n coplex decson akng the ncoplete nforaton set can be dvded nto ndependent coponents and each coponent can be labeled as a ature or non-ature data set. The Pgnstc probabltes proportonal to belefs Prl and self-consstent pgnstc probablty PrcP are vald only for a ature data set. ets pgnstc probablty etp the pgnstc probablty proportonal to all the plausbltes PraPl the pgnstc probablty proportonal to the plausbltes PrPl and hybrd pgnstc probablty PrHyb are robust for all type of data sets. oe of these pgnstc probabltes are descrbed by the probablty proportonally functon { Pl PraPl el} 6 wth the correspondng pgnstc probabltes. Π etp PrPl PrHyb Prl } 7 { 79

4 7 ew Methodology of obnng ndependent Mult-ource elefs n soe systes the ncoplete nforaton set can be dvded nto ndependent coponent sets } { of basc belef assgnents s and M sets of probabltes. Ths secton ntroduces a new ethodology of cobnng the s. f f 8 s the noralzng factor of sung the to one. Ω 9 The cobned s can be used wth an approprate pgnstc probablty transfor to ake good decsons. ote that n the cobnaton or fuson process of the s the pedgree nforaton of each set of s s lost. avng the probablty proportonally functon for each eber of the sngleton set can be used to copute the pgnstc probablty transfor. 8 Pedgree Pgnstc Probablty The Pedgree Pgnstc Probablty PrPed s ntroduced that uses the fused s wth the probablty proportonally functons to copute a better pgnstc probablty estate a ayesan equvalent. The Pedgree Pgnstc Probablty s coputed for each sngleton eleent of wth Ω for all Ω. PrPed g g 30 s the noralzng factor of sung the Pgnstc probablty to one. 9 ardnalty plfcaton The ethodology of belef cobnaton and the pedgree pgnstc probablty splfes sgnfcantly when used n conjuncton wth probablty proportonally functon of the cardnalty value of the power set eber. f cardnalty f 3 ubsttutng one equaton nto the other: cardnalty 3 The Pedgree Pgnstc Probablty s: PrPed g cardnalty cardnalty g 33 ubsttutng one equaton nto the other: cardnalty cardnalty PrPed 34 nce s a sngleton eber ts cardnalty s one..e. cardnalty cardnalty PrPed 35 Defnng a new set of the s as: cardnalty The Pedgree Pgnstc Probablty for the new set of s becoes: cardnalty cardnalty PrPed 38 alculatng ets pgnstc probablty for the new set of. 80

5 etp 39 For ths new set of redefned s the Pedgree pgnstc probablty and ets pgnstc probablty are equvalent. PrPed cardnalt y etp 40 n lterature Fster and Mtchell 3 address the cobnaton of two belef subsystes: 4 s the noralzng factor of sung the to one and s the cardnalty of. 0 Exaple: The followng exaple llustrates soe desgn concepts presented n ths artcle. set of the s s gven as: The belefs are calculated as: 4 el _ For these s the Plausbltes are calculated to be: Pl _ The ets pgnstc probabltes are coputed fro the above s as: etp _ The pgnstc probablty proportonal to all Plausbltes PraPl s coputed as: PraPl_ wth ε Let another set of the s be: The belefs are calculated as: 47 el _ For these s the Plausbltes are calculated to be: Pl _ oputng the pgnstc probablty proportonal to all plausbltes. - el_ wth Ω 50 ε Pl_ PraPl Ω el + ε Pl 5 Pr apl _ set of the s s gven as: 8

6 The belefs are calculated as: 53 el _ For these s the Plausbltes are calculated to be: Pl _ ardnalty Weghtng 0 3 cardnalty 3 D D cardnalty cardnalty cardnalty cardnalty cardnalty cardnalty cardnalty cardnalty cardnalty cardnalty cardnalty cardnalty cardnalty cardnalty cardnalty The Pedgree Pgnstc Probablty s coputed wth the followng equaton: PrPed cardnalty cardnalty 58 PrPed cardnalt y Defnng a new set of the s as: alculatng ets pgnstc probablty for the new set of. etp 6 etp The Pedgree pgnstc probablty and ets pgnstc probablty are the sae. PrPed cardnalt y etp 64 ote that by usng a pror nforaton nherent n the cardnalty probablty proportonally functon the saller probabltes tend to be overestated whle the larger probabltes tend to be underestated. Mxed Weghtng 8

7 For ths exaple the xed weghtng descrbed by the followng belef hybrd belef H probablty proportonally functon: {el_0 PraPl_ el_} 65 f D H 0 D D 0 f D 66 el _ 0 Pr apl _ el 0 el Pr apl _ el _ D 000 H H H H H H H H H H H H H H H The Pedgree Pgnstc Probablty s coputed wth the followng equaton: g PrPed H 68 el _ 0 Pr apl _ el _ g el _ 0 Pr apl _ el _ 69 PrPedH ote that by usng soe posteror nforaton nherent n the probablty proportonally functon better estates of the pgnstc probabltes are obtaned. n the desgn of nforaton fuson systes the reducton of coputatonal coplexty s a key desgn paraeter for real-te pleentatons. One ethod of obtanng soe splfcaton of the coputatons s to decopose the syste nto subsystes of utually exclusve dsjoned nforatonal coponents. Ths artcle derved a sple relatonshp between the total syste nforatonal P varable and the weghted average of dsjont decoposton sub-syste coponents Ps. The probablty nforaton content for a syste represented by copleent probabltes was deonstrated. hybrd pgnstc probablty was ntroduced that s robust n estatng a probablty for any aturty of the ncoplete data set. new ethodology of cobnng ndependent ultsource belefs was presented. pedgree pgnstc probablty was ntroduced that use soe nforaton of the orgnal fused data sets to copute a better pgnstc probablty. References: udano ohn. Pgnstc Probablty Transfors for Mxes of Low- and Hgh- Probablty Events 4 th nternatonal onference on nforaton Fuson 00 Montreal Q anada ugust 00 pages TU ets P. ennes R. The Transferable elef Model rtfcal ntellgence vol. 66 pages Fster T. Mtchell R. Modfed Depster-hafer wth Entropy ased elef ody opresson Proc. 994 ont ervce obat dentfcaton ystes onference aval Postgraduate chool ugust 994 pp PrPed cardnalty oncluson: 83