Data fusion
Evidential theory
Modeling
Mass function
Plausibility function
Affaiblissement
αj=0 : total ignorance
Belief function
Estimation
Simple support functions
Fonction de croyance complémentaire
Fonction sur singletons
Combinaison
Mass modification based
Conflict redistribution based
Dempster Shafer rule
Focality
Synopsis
+Certainty convergence
---Total certainty to minor opinion
--- Loss of majority opinion
---unearned mass
Corrective strategy of body evidence
Total/partial conflict redistribution strategy
Décision
Max de plausibilité
Max de croyance
Can represent imprecisions
Vote decisional method
Combination
MEk(x) =Σmj=1Mjk(x)
is associative
is commutative
Problems
if sources are pair
total incertitude
Decision
majority voting
absolute majority voting
if $$M^{E}_{k} (x) > m/2$$
Source ponderation
$$\mathcal{M}_{k}^{E}(x)\ = \Sigma_{j=1}^{m} \alpha _{j}M^{j}_k (x) \ $$
non idempotent
needs learning
success criterions : accuracy...
$$\mathcal{M}_{k}^{E}(x)\ = \Sigma_{j=1}^{m} \alpha _{jk}M^{j}_k (x) \ $$
Simple, naturlisch
no prior knowledge
Modèle probabiliste
Approche objectiviste
Approche subjective (Bayésienne)
Representing ignorance
Principe de raison insuffiante
if information is absent, take uniform law
ignorance : uniform law => maximum entropy
Combinaisons
Bayesian
other operators
$$ p(d_{i}|S_1,...,S_m) = \frac{p(S_1,S_2,...,S_m|d_i)p(d_i)}{p(S_1,...,S_m)} $$
$$ p(d_{i}|S_1,...,S_m) = \frac{p(S_1|d_i)p(S_2|S_1,d_i) ... p(S_m|S_1,...,S_{m-1} ,d_i ) p(d_i) }{ p(S_1) p(S_2|S_1) ... p(S_m|S_1,...,S_{m-1}) } $$
$$Using\ Independence\ hypothesis\ : \frac{\Pi ^m_{j=1} p(S_j|d_i) p(d_i) }{\Pi ^m_{j=1} p(S_j) } $$
Posterior
$$ p(H_i /x) = \frac{v_x (H_i) p(H_i) }{ \Sigma_{H_j \in \Omega} v_x (H_j) p(H_j) } $$
Likelihood
$$ v_{x_{1,2} }(H_i) = \frac{v_{x_1} (H_i) v_{x_2} (H_i) }{ \Sigma_{H_j \in \Omega} v_{x_1} (H_j) v_{x2} (H_j) } $$
Conflict
No available conflict notion
total conflict, no possible likelihood calculation
Decision
posterior max : MAP
Max likelihood : MV
confusion between ignorance and equiprobability
good for rich knowledges
no coflict modelisation
closed world exclusivity, exhaustivity
continuous distribution
Fuzzy logic
permits representing imprecisions and incertitudes
introduce semantics
combine variations of info types
Fuzzy sets theory
$$\mathcal{X} (x)$$
$$1\ if\ x\ \in\ A$$
$$0\ if\ x\ \notin\ A$$
Subjective neighboring notion
Noyau
$$ Noy(A)\ ={x \in A : \mu_A (x) = 1 } $$
$$Normalised\ sub-set : Noy(A) != \emptyset $$
Support
$$ {x \in A : \mu_A (x) > 0 } $$
Height
$$sup_{x \in A}(x) $$
Cardinal
$$|A| = \Sigma_{s \in S} \mu_A (x)$$
$$Coupe\ \alpha$$
$$A_{\alpha} = { X\in S : \mu _A (x) \geq \alpha }$$
for defuzzifying
Fuzzy number
approximately tolerable interval
Possibility theory
fuzzifying
needs
univers de discours
partition classifying of that universe
belonging functions
Goal
get a more sure and precise decision
$$m_{DS}(X)=\frac{m_{12}(H)}{1-m_{12}(\emptyset)}$$ $$m_{12}(H)= \Sigma m_1(H_1) m_2 (H_2)$$
Calcul du consensus
+Disappearing ignorance
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