Abstract: How to deal with information fusion problems on unreliable evidential sources? Because the kind of fusion problem is everywhere in information society, DSmT recently developed from DST and probability theory is much more powerful than the other method (DST, etc.). However, it needs to be improved for its limitation that its author didn`t mention of solving information fusion problem on unreliable evidence source. Therefore in this study we ourselves extend DSmT and solve the fusion problem on not only equal-reliable sources of evidence, as well as unreliable evidence source. At last, An example is illustrated to verify the benefit of this kind of method and a conclusion is reached that generalized DSmT method can fuse any class information and knowledge effectively whether handing static fusion problems or dynamic fusion problems in real time directly.
INTRODUCTION
Information fusion technology was a new subject that originated from martial application C4I system and IW system involving multi-target detection, recognizance, tracking and warfare field surveillance, evaluation of situation and menace, etc. With the transferring of information fusion technology from the martial application to civil one, presently the multi-sensors fusion technology won a chance to develop rapidly. In fact, since 1950, the development of new original theories dealing with uncertain and imprecise information has become very prolific. There are three major theories available now as alternative to the theory of probabilities for automatic plausible reasoning in expert systems as follows: Firstly, the fuzzy set theory developed by Zadeh (1965); Secondly, the Shafer’s theory of evidence in 1976 developed from his teacher (Dempster, 1967), known as DST (Shafer, 1976) and the theory of possibilities by Dubois and Prade (1988). Thirdly, recently, the unifying avant-gardise neutrosophy theory proposed by Smarandache (2000) especially, Dezert and Smarandache (2003a) proposed a new theory known as DSmT on which a panel discussion and a special session were also carried out at The 7th International Conference on Information Fusion 2004 (Dezert et al., 2004). Though obviously DSmT is a new and creative theory, which can deal with a wide range of evidence sources, it don’t distinctly propose how to deal with the unreliable sources.
THE D-S THEORY OF EVIDENCE
In this section we simply recall DST as follows:
The idea of using belief functions for representing someone’s subjective
feeling of uncertainty was first proposed by Shafer (1976), following the seminal
work of Dempster (1967) about upper and lower probabilities induced by a multi-valued
mappings. The use of belief functions as an alternative to subjective probabilities
for representing uncertainty was later justified axiomatically by Smets and
Kennes (1994), who introduced the Transferable Belief Model (TBM), providing
a clear and coherent interpretation of the various concepts underlying the theory.
Let θi (i = 1, 2, 3…n) be some exhaustive and exclusive elements (hypotheses) of interest taking on values in a finite discrete set Ω, called the frame of discernment. Let us assume that an agent entertains beliefs concerning the value of θi, given a certain evidential corpus. We postulate that these beliefs may be represented by a belief structure (or belief assignment), i.e., a function from 2Ω to [0,1] verifying ΣA⊆Ω m (A) = 1 and m (φ) = 0, here φ is empty set. For all A⊆Ω, the quantity m (A) represents the mass of belief allocated to proposition “θi ⊆ A” and that cannot be allocated to any strict sub-proposition because of lack of evidence. The subsets A of Ω such that m (A) are called the focal elements of m. The information contained in the belief structure may be equivalently represented as a belief function bel, or as a plausibility function pl, defined respectively as bel (A) = ΣB⊆A m (B) and pl (A) = ΣB∩A≠φ m (B). The quantity bel (A), called the credibility of A, is interpreted as the total degree of belief in A (i.e., in the proposition θi ⊆ A), whereas pl (A) denotes the amount of belief that could potentiality be transferred to A, taking into account the evidence that does not contradict that hypothesis.
Now we assume the simplest situation that two distinct pieces of evidence induce two belief structures m1 and m2. The orthogonal sum of m1 and m2, denoted as m = m1 ⊕ m2 is defined as:
| (1) |
Here,
The orthogonal sum (also called Dempster’s rule of combination) is commutative and associative. It plays a fundamental operation for combining different evidential sources in evidential sources in evidence theory. Decision-making is an important issue in any theory of uncertainty. In the TBM, a distinction is made between two levels of uncertainty representation: a credal level at which beliefs are entertained and represented using the formalism of belief functions and a decision level at which belief functions are converted to probability distributions to allow coherent betting behaviors (Smets and Kennes, 1994). Given a belief structure m, the Generalized insufficient reason principle leads to the definition of the pignistic probability distribution BetP as
where |A| denotes the cardinality of A. Of course, Jean Dezert and Florentin Smarandache have extended it and introduce a Generalized Pignistic Transformation (GPT) as a tool for decision-making (Dezert et al., 2004).
DEZERT-SMARANDACHE THEORY
Here we will simply introduce DSmT to the reader. If the reader want to know it in detail, please refer to the work by Smarandache and Dezert (2004). The practical limitations of the Dempster-Shafer Theory (DST) come essentially from its inherent following constraints, which are closely related with the acceptance of the third exclude principle.
(C1) The DST only considers a discrete and finite frame of discernment Ω based on a set of exhaustive and exclusive elementary elements θi (i = 1, 2, 3 … n).
(C2) The evidential sources are assumed independent and provide their own belief function on the power-set 2Ω but for same interpretation for Ω.
In most of practical fusion system, conflicts are unavoidable which can lead to the failure of making decision by using the DST. To solve it, some ad-hoc or heuristic techniques must always be added to the fusion process to manage or reduce the possibility of high degree of conflict between sources so that the complexity of reckoning increases. To overcome these major limitations and drawbacks relative to the Dempster’s rule of combination, The Dezert- Smarandache Theory (DSmT) of plausible and paradoxical reasoning emerges as the times require.
The foundations of the DSmT is to refute the principle of the third middle excluded and to allow the possibility for paradoxes (partial overlapping) between elements of the frame of discernment. The relaxation of the constraint C1 can be justified since the elements of Ω correspond generally only to imprecise/vague notions or concepts so that no refinement for satisfying C1 is actually possible (specially if natural language is used to describe elements of Ω). The DSmT refutes also the excessive requirement imposed by C2 since it seems clear that the frame is usually interpreted differently by the distinct sources of evidence (experts). Some subjectivity on the information provided by a source of information is almost unavoidable, otherwise, this would assume, as within the DST, that all corpora of evidence have an objective/universal (possibly uncertain) interpretation or measure of the phenomena under consideration, which unfortunately rarely (never) occurs in reality. Actually in most of cases, the sources of evidence provide their beliefs about some hypotheses only with respect to their own worlds of knowledge and experience without reference to the (inaccessible) absolute truth of the space of possibilities. The DSmT includes the possibility to deal with evidences arising from different sources of information, which don’t have access to absolute interpretation of the elements θi (i = 1,2,3…n) under consideration and can be interpreted as a general and direct extension of probability theory and the DST in the following sense. Let Ω = {θi, θ2} be the simplest frame of discernment involving only two elementary hypotheses (with no more additional assumptions on θ1 and θ2), then
| • | The probability theory solve basic probability assignments m (.) ε [0, 1] such that m (θ1) + m (θ2) = 1 |
| • | The DST deals with basic belief assignments (bba) m (.) ε [0, 1] such that m (θ1) + m (θ2) + m (θ1∪ θ2) = 1 |
| • | The DSmT theory deals with new bba m (.) ε [0, 1] such that m (θ1) + m (θ2) + m (θ1∪ θ2) + m (θ1∩ θ2) = 1 |
Next we continue introduce the combinational rule about free model and hybrid model of DsmT. Let Ω = {θ1, θ2, θ3 …θn} be a set of elements which can’t be precisely defined and separated so that no refinement of Ω in a new larger set Ωref of disjoint elementary hypotheses is possible. The hyper-power set DΩ is defined as the set of all compositions built from elements of Ω with ∪ and ∩ (Ω generates DΩ under operators ∪ and ∩) operators such that
| • | Φ, θ1,θ2,θ3…θn ε DΩ |
| • | If A,B ε DΩ, then A ∩ B ε DΩ and A ∪ B ε DΩ |
| • | No other elements belong to DΩ, except those obtained by using rules 1 or 2 |
The cardinality of DΩ is majored by
| (2) |
The DSm rule of combination is still commutative and associative.
While we also may give the presentation of DSm rule of combination for hybrid DSm models proposed by Smarandache and Dezert (2004), due to the limitation of length, we will write them directly here. (Smarandache and Dezert, 2004).
| (3) |
where
In fact, the hybrid DSm model is a genuine model, which has been extended to take into account all possible integrity constraints (if any) of the problem under consideration due to the true nature of elements/concepts involved into it. But whether the free DSmT or the hybrid DSmT just proposes the combinational role aimed at the equal-reliable evidential source. How to solve the unequal-reliable evidential sources with DSmT? We will extend DSmT in next section for this.
FURTHER EXTENSION FOR DSmT
Necessity of further extension for DSmT: With the rapid development of science and technology, the precision, correctness and real-time for acquiring and processing information and knowledge from the nature to serve for human are more and more required necessarily. Technique of information fusion experiences from the single source to multi-source and multi-sensor involving homogeneous and heterogeneous information. It is well known that the homogeneous sensors can’t give the same reliability sometimes for the sake of the difference of designing or operating, so that they can’t give the coherent result, let alone the heterogeneous sensors. To make the reader see it clearly, let us introduce three simple examples as follows:
Firstly, let us assume two homogeneous sensors (A and B), one (A) has a high occurrence of malfunction, however, the other (B) have a low one. If we use DST or DSmT (either the free model or the hybrid model) and then the equal quantity of m (•) is allocated respectively to each one (i.e., mA (•) = mB (•)). We all know, it is obvious to be unreasonable.
Secondly, let us assume someone goes to hospital, he sees two doctors (A, B), who have the same diploma from the same university, the same certification of qualification as a doctor, as well as the same age and length of service. But before giving a decision, one (A) answers for his diagnosis on the patient and asks some relative question of the symptom and examines carefully. While the other (B) don’t examine the patient throughout and then give a mistaken diagnosis result at once. Under this condition, can you assign the equal quantity of m (•) to two doctors?
Thirdly, supposed two experts (one (A) is younger, while the other is older) have a discussion about a plan to develop the company together. At first sight, the two experts all own abundant professional knowledge. We know that although the older is affluent in working experience, he has conservative thought, which might bring on blundering away. On the contrary, the younger has an open idea and powerful ability in assimilating exoteric knowledge and information, however, he is short of abundant working experience, which leads to rash advance (i.e., more haste, less speed). Regarding to this condition, when we allocate the quantity of m (•) to them, how will we solve this fusion problem?
Defining a generalized DSm theory: Let n evidential sources (S!,
S! … Sk), here we work out a uniform way in dealing
with the homogeneous and heterogeneous information sources. So we get the discernment
frame Ω = {θ1, θ2 … θn},
m (•) is the basic belief assignment, let mi (•) (i =
1, 2, 3 … k) be the evidential source Si’s observation and
let pi represent its corresponding estimation correctness rate in
history, considering
So we may generalize the combinational rule of DSmT as follows:
The classical DSm rule for k≥2 sources for free DSmT model is expressed as:
| (4) |
The hybrid DSm rule of combination for k≤2 sources is expressed as:
| (5) |
| Table 1: | Different in the BBA between DsmT and G-DSmT |
where
Supposed let pi = (i = 1, 2, 3, … k), in fact it becomes the free and hybrid DSmT model at once. So the combinational rules for free and hybrid DSmT model are a kind of special situation of G-DSmT. The G-DSmT pushes further the application of plausible and paradoxical reasoning. It not only may apply to any field where the DST or DSmT works, but also can deal with unreliable and independent evidential sources and even can solve the coupling ones (we will versify it in detail in next paper). We have recognized that the DSmT originated from the Probability theory and DST can settle a wider class of fusion problem directly and efficiently and owns a greater superiority than DST and the others, which can be found by Dezert and Smarandache (2003c) etc, let alone the G-DSmT.
ANALYSIS ON AN NUMBER EXAMPLE
In this section we give an example of the mobile robot, supposed where our new G-DSm theory is applied. To make the mobile robot located precisely, two different kinds of sensors (ultrasonic and laser) are used to work together. For the convenience of calculation, here we only assume one laser and one sonar rangefinder and the precision of location is scaled as precise and not precise, which is represented respectively as Ω = {θ1, θ2} in the discernment frame Ω. Let mu (•), ml (•) represent the basic belief assignment to ultrasonic and laser sensor and assume correctness rate pu of ultrasonic telemeter in history is 0.9, the laser one p1 = 0.8. We know the hyper-power set DΩ = {φ,θ1∩θ2,θ1,θ2,θ1∪θ2} and then may calculate it with DSmT and G-DSmT only with regards as the free model respectively in the following Table 2:
| Table 2: | Calculating result from DSmT and G-DSmt shown |
Results from the comparison in Table 2:
| • | For existing unreliability, the basic belief assignment of each source will decrease, for example, mu (θ1) transfers from 0.9 to 0.81, mu (θ2) transfers from 0.1 to 0.09; at the same time, the laser rangefinder also has the similar situation. However, the belief vacuous belief assignment VBA) will increase. It means that with the incensement of unreliability, the supporting measure to elements over the hyper-power set except the unknown hypotheses will decrease, which answers for the physical system and intuition of human. |
| • | The fusion result of two unreliable evidence sources is fused according
to the Eq. 4. Though here |
| • | It is shown clearly from the above Table 2 that the G-DSmT deal with the fusion problem more rational than DSmT, because it at least assigns the sensors’ occurrence of malfunction to the total ignorance of system and even increases the ability in fusing information. |
CONCLUSIONS
In this study, considering the limitation of DSm evidence theory in unreliable evidence sources, we generalize it to G-DSmT, in order to extend the application range of information fusion theory (DSmT) and improve the precision of fusion. We believe that G-DSmT will have more powerful vitality in society of information fusion presently or in the future.