Dempster’s rule of combination, as described in equation 3.3, was the core fusion mechanism provided as part of the original Dempster-Shafer theory. This rule combines sources as an ’AND’ configuration. The rule has been en- hanced or changed by researchers in order to cater for specific applications of Dempster-Shafer theory because it is not suitable in all fusion scenarios. Gen- erally, these modifications are motivated by two problems. The first problem is how to treat conflict when sources disagree. Dempster’s rule can result in unexpected results when fusing conflicting evidence. This problem, termed Zadeh’s paradox [124] is described further in section 3.5.1. The second prob- lem is how to cater for evidence sources of varying trust, so that the credibility of sources is correctly reflected in the fusion result.
Various fusion rule modifications exist to cater for these problems. Yager’s [117] modified Dempster rule provides an alternative to the treatment of con- flicting evidence. Unlike Dempster’s rule, evidence in agreement is not nor- malised by a conflict factor, and conflict is stored separately. This gives a more transparent picture of conflict, so is useful in domains where conflict is critical, such as medical diagnosis. Dubois and Prade [33] defined a disjunctive fusion rule, where at least one source of information is telling the truth, but not all sources are trustworthy. The weakness of this rule is that if belief from a single source is 0, the fusion of all sources will be 0. This is not a credible outcome in a pervasive environment if a sensor breaks down. Murphy’s combination rule [76] eliminates the possibility of a single source dominating all other sources and is described further in section 3.6.1. Smet’s transferable belief model as- sumes that if sources conflict, such conflict can be assigned to other undefined options outside the frame of discernment - i.e. an open world assumption [99]. This is not suitable for pervasive systems if a closed situation group is under detection. Other modifications of Dempster’s rule that cater for vary- ing scenarios of conflict and trustworthiness of sources are Dubois and Prade’s exclusive and mixed disjunctive rules [24] and Inagaki’s Unified combination rule [92].
idence spread over time. Dempster’s rule assumes that all evidence is co- occurring. We wish to cater for evidence that may not be occurring at the same time, as described further in Chapter 4.
In the next section, we explain three particular problems for situation recogni- tion when we apply Dempster’s rule in our evidence decision network envi- ronment: Zadeh’s paradox, single sensor dominance and evidence spread over time. We then explain how we aim to address these in our evidence decision network by selecting two alternative fusion rules Murphys and averaging.
3.5.1
Zadeh’s paradox
Zadeh’s paradox is a well-documented problem with Dempster’s rule of com- bination [92]. Zadeh highlighted the fact that when sources in high conflict are combined using Dempster-Shafer rule, the results can be completely counter intuitive [124]. Using the example illustrated in figure 3.3, two location sensors assign belief across a frame {room a, room b, room c}. The first sensor assigns a belief of 0.99 that the correct location is {room a} and 0.01 belief to {room b}. The second sensor disagrees, assigning most of its belief 0.99 to {room c} and 0.01 to option {room b}. The sensors are almost completely conflicting, with a small degree of overlap. When the sensor masses are fused using Dempster rule of combination in equation 3.3, room b obtains all belief, with zero belief assigned to rooms a or c. This is because room b is the only option on which both sen- sors overlap, and all disagreeing evidence is normalised out. One reason to account for this is the possibility that both sensors are correct [42]. This possi- bility will work for domains such as medical diagnosis where two diseases can be detected together. But it does not work with context detection such as loca- tion, where a person can only be in one place at one time. Another possibility is that one source is not reliable [42, 99, 64], a possibility that we address by enabling rich sensor quality knowledge to be included as described in Chapter 4. A final possibility [99, 64] is the open world assumption in that both sources are wrong because choices outside the frame may exist. This is not a workable explanation in a pervasive environment where a closed world of possible val- ues (i.e. a known set of situations is being detected) is a reasonable assumption - unlike the medical domain where diseases outside the suggested possibilities may be possible. The most correct treatment in our opinion is to highlight an even distribution of belief for both room a and room c, with a small degree of belief for room b (and a large measure of conflict).
Room A B Room C
Sensor 1 belief: Majority to Room A
Sensor 2 belief: Majority to Room C Room B = least belief but overlapping belief = winner
Figure 3.3: Zadeh’s paradox: scenario of location sensors in conflict on room location
3.5.2
Single sensor dominance
A second problem that has gained far less attention in the literature is the potential dominance of a single sensor. Murphy [76] described how a single disagreeing sensor can overrule multiple other agreeing sensors in the fusion process. A categorical belief function is where all belief is assigned to one hy- pothesis in a frame [67]. For example, if five sensors are used to determine the location of a user in the house, a single categorical sensor that assigns all of its belief to a contradictory option will negate the evidence from the other four sensors. We suggest that this is particularly problematic for binary sen- sors which are increasingly being used in Smart Home deployments. Binary sensors have small frames of discernment, with just three states: {on, off , ig- norance}. Unless discounted, they will categorically assign all of their belief to the ’on’ or ’off’ states. A single malfunctioning binary sensor can in theory therefore overrule evidence from other correct binary sensors during the fu- sion process. A more intuitive result would be to allow the agreeing sensors to ’win’ but to represent the disagreeing sensors’ evidence as conflict.
3.5.3
Evidence spread over time
A third problem that we have observed occurs when sensor evidence of a higher level state is spread over time. For example, the detection of a break- fast activity by the triggering of a fridge sensor, then a kettle sensor, then the toaster sensor and so forth. At any point in time, only one of the sensors may be “on”, so fusion of all the sensor values at any point in time may result in the ’on’ sensor evidence being lost. The fusion rule should capture that some
evidence of the situation was observed even though it has been greatly contra- dicted by sensors that are off. It should not be wiped out by the overruling of the contradictory sensors, as will occur with Dempster’s rule of combination.