Theory

RENORMALIZED CONJUNCTIVE FUSION

We assume here that two sources agree, even if there exists a small conflict between them. When there is a conflict, the result of conjunctive fusion $\pi_{\text{conj}}$ is sub-normalized. It is possible to find a normalized distribution by using the height h of the intersection between the two possibility distributions $\pi_1$ and $\pi_2$ :

$\begin{displaymath}\pi_{\text{renorm}}(\omega) = \frac{\pi_{\text{conj}}(\omega)}{h(\pi_1, \pi_2)}\end{displaymath}$

(53)

where $\pi_{\text{conj}}(\omega) = \min ( \pi_1(\omega), \pi_2(\omega) )$ and

$h(\pi_1, \pi_2) =\sup_{\omega \in \Omega} ( \min(\pi_1(\omega), \pi_2(\omega)) )$ .

Let h be the coherence degree between sources.

Naturally, renormalized conjunctive fusion is only used if two sources are in sufficient agreement (weak conflict). In cases of total conflict ( $h(\pi_1, \pi_2) = 0$ ), renormalized conjunctive fusion is not defined. So, by default, the result is null.

**Figure 25:** Renormalized conjunctive fusion
$\begin{figure}\begin{center}\fbox{\begin{tabular}{cc}\subfigure[Total confli......flict]{\epsfbox{c3-conj-renorm4.eps}} \\\end{tabular}}\end{center}\end{figure}$

Figure 25 shows how the renormalized conjunctive fusion behaves on applying operator ``minimum''.

(q): With total conflict, renormalized conjunctive fusion does not lead to any improvements compared to simple conjunctive fusion. This type of fusion has a conjunctive behaviour, so it requires a minimum amount of agreement between sources to provide a result. If disagreement is total (no intersection between possibility distributions given by the sources), then renormalized conjunctive fusion provides no result. In this case, the basic hypothesis - assuming reliability of sources to justify the use of conjunctive fusion - is completely false.
(r): When an intersection between possibility distributions occurs, it is possible to renormalize the distribution coming from conjunctive fusion. This renormalization allows the result set to reach a possibility degree of 1. But, this renormalization leads to an undesirable effect of discontinuity for a value near zero. Moreover, when conflict between sources is strong, renormalization assigns very high possibility degrees to values found possible by both sources (albeit to a low degree). So the use of conjunctive fusion is debatable when the sources are in strong conflict.
(s): It seems more reasonable to renormalize conjunctive fusion results in the event of weak conflict. This is close to a hypothesis of reliability between sources since conflict is weak. So renormalization hardly alters the result of conjunctive fusion.
(t): If there is no conflict, simple conjunctive fusion is again performed since the result is already normalized.

Let us note that renormalization removes conflict between sources. The degree of conflict can be assessed as being the distance between the maximum height of the intersection and degree 1, i.e. $1 - h(\pi_1, \pi_2)$ .

Renormalized conjunctive fusion is conceivable when conflict between sources is weak. But, as conflict becomes stronger, it is more and more improbable to assume that the sources are reliable. So, renormalization is less and less justifiable. Moreover, it is discontinuous around zero: therefore results obtained with strong conflict are not very reliable.

Renormalization creates two unwanted effects: instability around zero, and disappearance of information on conflict. However, we presented renormalized fusion so we could use it in the adaptive fusion formulae given in the next sections. The previously mentioned unwanted effects appear to decrease when renormalization is used jointly with other fusion methods.

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