Theory

Theory

It is possible that, among the sources to be fused, some are considered more reliable than others.

For example, the size of an individual can be evaluated either

using indirect information, a comparison with other people around for instance, or
using direct information like using a gauge.

It is obvious that direct information is more precise than indirect information. In the presence of conflict between direct and indirect sources, confidence will be given rather to the direct sources and information provided by the indirect sources will be discarded.

A set of information sources K can be classified according to their reliability.

Let $\{K_1, K_2, \dots, K_n\}$ be a K-partition such that the sources brought together in K_i are considered to have the same reliability, and to be more reliable than those of sets K_j with j > i. Information fusion provided by sets $K_1, K_2, \dots, K_n$ will be performed as follows (figure 37):

1): information of set K₁ are merged by means of quantified adaptive method for instance;
2): the preceding result is refined with information of set K₂ only if they do not contradict those of set K₁;
3): the process is similar with sets K₃ to K_n.

**Figure 37:** Fusion by priority of 4 sets of information sources.
$\begin{figure}\begin{center}\epsfbox{c3-fusion-prio.eps}\end{center}\end{figure}$

Let

$\pi_i$ the possibility distribution built after fusion of set K_i,
$h(\pi_1, \pi_2)$ the degree of consensus between distributions $\pi_1$ and $\pi_2$ when the number n of sets of sources equals 2.

We can define a rule for fusion by priority with a conjunctive behaviour or a disjunctive behaviour.

Conjunctive behaviour:

The rule for fusion by priority is then the following:

$\begin{displaymath}\forall \omega \in \Omega, \quad\pi_{\text{prio}_{\text{conj}}}^{1>2}(\omega) =\min(\pi_1, \max(\pi_2, 1-h(\pi_1, \pi_2)))\end{displaymath}$

(73)

This rule expresses that the sources of set K₁ provide reliable information, whereas information resulting from the set K₂ is regarded as reliable only with a degree of certainty equal to the degree of consensus h between sets K₁ and K₂.

If $h(\pi_1, \pi_2) = 0$ , then $\pi_{\text{prio}_{\text{conj}}}^{1>2}(\omega) = \pi_1$ . This means that information coming from set K₂ are not taken into account because they are in total conflict with those of set K₁.
On the contrary, if $h(\pi_1, \pi_2) = 1$ , then $\pi_{\text{prio}_{\text{conj}}}^{1>2}(\omega) = \min(\pi_1, \pi_2)$ . That means that information of set K₂ agree with those of set K₁. One then uses the first to refine the seconds. The agreement of the sets of sources strengthens the credibility which can be given to the intersection of their possibility distributions.

This rule corresponds to an operation of weighted conjunction whose weighting

equal 1 for distribution $\pi_1$ ,
equal $h(\pi_1, \pi_2)$ for distribution $\pi_2$ .

Let us notice that the result of this fusion by priority is under-normalized as soon as $h(\pi_1, \pi_2) < 1$ .

Disjunctive behaviour

Rule 73 has a conjunctive behaviour. Symmetrically can be defined a rule with disjunctive behaviour:

$\begin{displaymath}\forall \omega \in \Omega, \quad\pi_{\text{prio}_{\text{disj}}}^{1>2}(\omega) =\max(\pi_1, \min(\pi_2, h(\pi_1, \pi_2)))\end{displaymath}$

(74)

This rule expresses that if one of the two sets is supposed mistaken, it is surely the least reliable group K₂. So, information brought by K₂ is stored only if they are in agreement with K₁. The more the conflict between K₁ and K₂ increases, and the more is truncated the information of K₂ to which one makes less confidence. When the conflict is total, only information of K₁ will be stored.

Let us notice that the result of this fusion by priority is under-normalized as soon as $h(\pi_1, \pi_2) < 1$ .

**Figure 38:** Fusion by priority.
$\begin{figure}\begin{center}\epsfbox{c3-prio1a4.eps}\end{center}\end{figure}$

IRIT-UPS