Solution of the VBIED problem using Dezert-Smarandache Theory (DSmT)

Transcription

1 Solution of the VBIED problem using Dezert-Smarandache Theory (DSmT) Dr. Jean Dezert The French Aerospace Lab. - ONERA 29 Av. de la Division Leclerc Châtillon, France jean.dezert@onera.fr Dr. Florentin Smarandache Maths & Sciences Dept. University of New Mexico Gallup, NM 87301, USA smarand@unm.edu Web info : Fusion 2010, Edinburgh, UK, July 26-29th,

2 A very short introduction to DSmT Specificities: 1 - Integrity constraints of the static (or dynamic) fusion problem in the modeling. 2 - Fusion of uncertain, high conflicting and imprecise sources with PCR5 3 - New probabilistic transformation of mass of belief into probability (DSmP). July Vol.1 July Vol.2 June Vol p. 442 p. 758 p. Freely downladable at: 2

3 Frame and decision-support hypotheses Sub-frames: Θ 1 = {A = Suspicious person, Ā = not A} Θ 2 = {V =WhiteToyotaVehicle, V =notv } Θ 3 = {B =nearbuilding, B =notb} Joint frame: Θ = Θ 1 Θ 2 Θ 3 = {θ 1 (Ā, V, B), θ 2 (A, V, B), θ 3 (Ā, V, B), θ 4 (A, V, B), Decision-support hypotheses of threats «Optimistic» point of view «Cautious» point of view «Very cautious» point of view θ 5 (Ā, V,B), θ 6 (A, V,B), θ 7 (Ā, V, B), θ 8 (A, V, B)} θ 6 (A, V,B), θ 7 (Ā, V, B), and θ 8 (A, V, B) human bomb car bomb both Threat θ 8 =(A, V, B) Threat θ 7 θ 8 =(Ā, V, B) (A, V, B) Threat θ 6 θ 7 θ 8 =(A, V,B) (Ā, V, B) (A, V, B) Other possible threats Threat θ 6, θ 7, θ 6 θ 7 or θ 6 θ 8 3

4 Modeling the inputs (case 1) Source 0 (prior information): Suspect A drives a White Toyota m 0 (θ 4 θ 8 =(A, V, B) (A, V, B)) = 1 Source 1 (Analyst 1, 10 years experience): A is probably near the building B m 1 (θ 6 θ 8 =(A, V,B) (A, V, B)) = 0.75 m 1 (I t )=0.25 I t Total ignorance Source 2 (ANPR system): 30% probability that it is Aʼs white Toyota observed m 2 (θ 4 θ 8 =(A, V, B) (A, V, B)) = 0.30 m 2 (θ 4 θ 8 )=0.70 Source 3 (Analyst 2, no experience): It is improbable that A is near the building B m 3 (θ 6 θ 8 =(A, V,B) (A, V, B)) = 0.25 m 3 (I t )=0.75 Numerical values for m1 and m3 are ad-hoc but reflect the trend of what is expressed by sources 1 and 3. They can be changed (see next slides) in a finer analysis. Questions to answer: Q1: Should building B be evacuated? Q2: Is subsystem (Analyst 1,Analyst 2) better than ANPR system? 4

5 BetP DSmP Fusion results and answer to Q1 Assumption: All sources have same fiability and importance θ 4 θ θ 6 θ I t Inputs (case 1) PCR5 PCR6 focal element m PCR5 (.) m PCR6 (.) θ 1 θ 2 θ 3 θ 5 θ 6 θ θ θ 4 θ θ 6 θ I t Results of m 0 m 1 m 2 m 3 Singletons DSmP ɛ,p CR5 (.) DSmP ɛ,p CR6 (.) θ θ θ θ θ θ θ θ P(.) is in [Bel(.),Pl(.)] PCR5 PCR6 P (θ 6 θ 7 θ 8 ) [ , 1] P (θ 6 θ 7 θ 8 ) [ , 1] P (θ 6 θ 7 θ 8 ) [0, ] P (θ 6 θ 7 θ 8 ) [0, ] Singletons BetP PCR5 (.) BetP PCR6 (.) θ θ θ θ θ θ θ θ P (θ 7 θ 8 ) [ , 1] P (θ 7 θ 8 ) [0, ] P (θ 8 ) [ , ] P ( θ 8 ) [ , ] P (θ 7 θ 8 ) [ , 1] P (θ 7 θ 8 ) [0, ] P (θ 8 ) [ , ] P ( θ 8 ) [ , ] Answer to Q1 : Evacuation of the building 5

6 Fusion results and answer to Q2 Inputs (case 1) θ 4 θ θ 6 θ I t APNR system m 0 m 2 focal element m PCR5 (.) m PCR6 (.) θ 1 θ 2 θ 3 θ 5 θ 6 θ θ 4 θ Singletons DSmP ɛ,p CR5 (.) DSmP ɛ,p CR6 (.) θ θ θ θ θ θ θ θ (Analyst1, Analyst2) subsystem m 0 m 1 m 3 focal element m PCR5 (.) m PCR6 (.) θ θ 4 θ Singletons DSmP ɛ,p CR5 (.) DSmP ɛ,p CR6 (.) θ θ θ θ θ θ θ θ Answer to Q2 : (Analyst1,Analyst2) is more specific and informative than APNR 6

7 Fusion results with different inputs Case 2 : We change the inputs for sources 1 and 3 θ 4 θ θ 6 θ I t θ 4 θ θ 6 θ I t Fusion results focal element m PCR5 (.) m PCR6 (.) θ 1 θ 2 θ 3 θ 5 θ 6 θ θ θ 4 θ θ 6 θ I t The analysis yields same conclusions as for case 1 Answer to Q1 : Evacuation of the building Answer to Q2 : (Analyst1,Analyst2) is more specific and informative than APNR 7

8 Impact of the quality of prior information Very uncertain prior θ 4 θ θ 6 θ I t θ 4 θ θ 6 θ I t Fusion of all sources with PCR5 or PCR6 DSmP ɛ,p CR5 (θ 6 θ 7 θ 8 )= DSmP ɛ,p CR6 (θ 6 θ 7 θ 8 )= DSmP ɛ,p CR5 (θ 7 θ 8 )= DSmP ɛ,p CR6 (θ 7 θ 8 )= DSmP ɛ,p CR5 (θ 8 )= DSmP ɛ,p CR6 (θ 8 )= focal element m PCR5 (.) m PCR6 (.) θ θ 1 θ 2 θ 3 θ 5 θ 6 θ θ θ 4 θ θ 6 θ I t Singletons DSmP ɛ,p CR5 (.) DSmP ɛ,p CR6 (.) θ θ θ θ θ θ θ θ (A, V,B) (Ā, V, B) (A, V, B) Answer to Q1 : It depends on decision-support hypotheses one chooses!!! Using the very cautious point of view, Decision = Evacuate the building B Answer to Q2 : Using very cautious point of view, we will choose the most precise system which is (Analyst1, Analyst2). 8

9 Working with imprecise sources Operators for exact calculus with imprecision S 1 S 2 = {x x = s 1 + s 2,s 1 S 1,s 2 S 2 } S 1 S 2 = {x x = s 1 s 2,s 1 S 1,s 2 S 2 } S 1 S 2 = {x x = s 1 /s 2,s 1 S 1,s 2 S 2 } Imprecise inputs for sources 1 & 3 f 1 = θ 4 θ f 2 = θ 6 θ 8 0 [0.75,0.9] 0 [0.10,0.25] f 3 = I t 0 [0.1,0.25] 0 [0.75,0.9] PCR6 P (θ 6 θ 7 θ 8 ) [ , 1] P (θ 6 θ 7 θ 8 ) [0, ] P (θ 7 θ 8 ) [ , 1] P (θ 7 θ 8 ) [0, ] P (θ 8 ) [ , 1] P ( θ 8 ) [0, ] focal element m PCR6 (.) m approx PCR6 (.) θ 8 [ , ] [ , ] f 1 = θ 4 θ 8 [ , ] [ , ] f 2 = θ 6 θ 8 [ , ] [ , ] f 3 = θ 4 θ 8 [ , ] [ , ] I t [ , ] [ , ] Exact calculus Using bounds of cases 1 & 2 Answer to Q1: Evacuation of the building B Singletons DSmP ɛ,p CR6 (.) θ 1 [0.0181,0.0389] θ 2 [0.0181,0.0389] θ 3 [0.0181,0.0389] θ 4 [0.0008,0.0028] θ 5 [0.0181,0.0389] θ 6 [ ] θ 7 [0.0181,0.0389] θ 8 [0.2072,1] 9

10 Concluding remarks DmT allows to provide a solution to VBIED problem and to answer to question 1 & 2. Prior information and the choice of decision-support hypotheses have a strong impact on final decision. We can work with imprecise information as well. Reliability and importance of sources can be easily included in the fusion process. Paper available soon on HAL & ArXiv system: J. Dezert, F. Smarandache, Threat assessment of a possible Vehicle- Born Improvised Explosive Device using DSmT. 10

11 What we need for solving VBIED problem A good modeling of inputs, an efficient rule of fusion, and proper choice of decision-support hypotheses to evaluate. Fusion rule: Proportional Conflict Redistribution rule #5 (PCR5) m PCR5 ( ) =0 m PCR5 (X) =m 12 (X)+ Y G Θ \{X} X Y = [ m 1(X) 2 m 2 (Y ) m 1 (X)+m 2 (Y ) + m 2(X) 2 m 1 (Y ) m 2 (X)+m 1 (Y ) ] PCR5 transfers the partial conflicting masses only to the elements involved in the partial conflict and proportionally to mass m1(.) and m2(.) of elements involved in the partial conflict. It preserves the specificity of information in the fusion. A simpler rule (PCR6) was proposed by Martin & Osswald in [DSmT Book, Vol. 2]. DSmP versus BetP transformation BetP(X) = Y 2 Θ Y X Y m(y ) Bel(A) P {A} Pl(A) \{ }??? DSmP (X) = DSmP(.) reduces Shannon s entropy of P(.) w.r.t BetP(.) Y G Θ Z X Y C(Z)=1 Z Y C(Z)=1 m(z)+ C(X Y ) m(z)+ C(Y ) m(y ) 11