De los placeres mundanos de la sensualidad

Dempster-Shafer theory combines separate pieces of information (evidence) to calculate the belief in an event. Its key features are (1) its ability to specifically quantify and preserve ignorance, (2) its facility for assigning evidence to com- binations of choices - such as user in “’kitchen OR bedroom” as well as single- tons (unlike probability theory which must allocate probability to singletons), and (3) its use of domain knowledge as a method for belief distribution [37]. These features are relevant to situation recognition in context aware systems for several reasons:

• Sensors are unreliable; an ability to quantify this lack of reliability and preserve the resulting uncertainty will support the quantification of situ- ation uncertainty;

• Rules are uncertain, and this uncertainty can be used to contribute to situation uncertainty calculations;

• The theoretically sound basis for incorporating domain knowledge offers us a way to encode knowledge without relying on training data.

Each evidence source has a total available amount of belief to be allocated, to- talling to a value of 1. The mass function for each evidence source allocates a source’s belief across a set of choices. These choices are collectively called the Frame of Discernment. In this section, we describe the basic concepts of Dempster-Shafer theory: frames of discernment, mass functions, rule of evi- dence combination and discounting.

3.2.1

Frames of Discernment and mass functions

In a Dempster-Shafer theory reasoning scheme, the set of possible hypotheses are collectively called the frame of discernment. This frame Ω represents the set

of choices {h1,h2,...hn} available to the reasoning scheme, where sources (such

as sensors) assign belief or evidence across the hypotheses in the frame. Hy- potheses can be any subsets of the frame. i.e. to singletons in the frame or to combinations of elements in the frame. For example, a calendar sensor that monitors whether a user is scheduled to be in a meeting or not assigns belief across a frame of discernment that includes hypotheses {meeting, coffee break, busy at desk}. When the calendar indicates that the user does not have a meet- ing, belief is assigned by the calendar sensor to ’not meeting’ situations i.e. the combination of {coffee break, busy at desk}. It assigns zero belief to {meeting}.

Formally, 2Ω _{denote the set of all subsets of Ω to which a source of evidence}

can apply its belief. The function m : 2Ω _{→ [0, 1] is called a mass function that}

defines how belief is distributed across the frame, if the function satisfies the following conditions, for hypotheses A:

m(φ) = 0 (3.1)

X A⊆Θ

m(A) = 1 (3.2)

Based on these conditions, belief from an evidence source cannot be assigned to an empty or null hypothesis, and belief from the evidence source across the possible hypotheses (including combinations of hypotheses) must sum to 1, similar to probability theory. The least informative evidence (uncer- tainty) is the assignment of mass to a hypothesis containing all the elements

{h1, h2, ...hn}, because this evidence does not commit to any particular hypoth-

esis. This ’uncertainty’ is denoted by the symbolJ.

3.2.2

Combining evidence

A crucial part of the process of assessing evidence is the ability to fuse evi- dence from multiple sources. In Dempster-Shafer theory, the combination of evidence from two different independent sources is accomplished by Demp-

ster’s combination rule: m12(A) = P X∩Y =Am1(X).m2(Y ) 1 −P X∩Y =φm1(X).m2(Y ) (3.3)

where m12(A) is the combined belief for a given hypothesis A, and X and Y

represent all possible subsets of the frame. The numerator in equation 3.3 rep- resents evidence for hypotheses whose intersection is the exact hypothesis of interest, A. i.e. the agreement across the two sources about hypothesis A. The denominator, 1 − K is a normalisation factor, where K is a conflict factor repre- senting all combined evidence that does not match the hypothesis of interest, A. The value of conflict, K, when combining evidence is indicative of the level of disagreement amongst the sources of their belief in hypothesis A.

Dempster’s rule can be considered as a strict AND operation of the evidence sources [92]. An alternative will be required to cater for where sources are combined as OR scenarios.

3.2.3

Sensor discounting

Shafer defined an evidential operation for discounting sensor evidence [95]. When an evidence source is known to be less than 100% reliable, a discount- ing factor between 0 and 1 is applied to the source’s beliefs. Unused be- lief as a result of discounting is assigned to uncertainty. If a source is com- pletely reliable (r = 0) discounting has no effect. For example, a location sensor identifies a user’s location with belief distribution of 0.3 in Room A, and 0.7 in Room B or C. If a discount factor of 0.2 is applied (i.e. the sen- sor is 80% reliable), the belief assignments are discounted and the leftover discounted belief is assigned to uncertainty. The re-distributed belief is then

RoomA = 0.24, RoomB ∨ C = 0.56, Θ = 0.2. The impact of the discounting

factor on beliefs is represented formally by Lowrance as [71] as follows: For a discount factor, d, where (0 ≤ d ≤ 1), where Θ represents uncertainty:

md(A) =    (1 − d)m(A) if A 6= Θ d + (1 − d)md(Θ) if A = Θ (3.4)

3.2.4

Example of evidence combination

We use a worked example based on our own intelligent office data set to ex- plain the concepts of mass functions, frames of discernment and evidence com- bination.

Two sensors are used to detect user location in an office. The locations of interest to an application are the cafe, the user’s desk, the meeting room

and ’anywhere else’ in the building. Each of our sensors are capable of

discerning these locations. The frame of discernment for each of the sen- sors includes the singleton hypotheses {desk, caf e, meetingRoom, other}. It also, in theory, includes all possible combinations of the singletons, such as{desk ∧ caf e, desk ∧ caf e ∧ other} and the complete ignorance hypothesis {desk ∧ caf e ∧ meetingRoom ∧ other} which is represented as Θ. The first sen- sor detects the user’s location in the cafe. The sensor is 70% reliable, so its belief is assigned across the frame as {caf e 0.7, Θ 0.3}. The allocation of 0.3 to ignorance is generated by discounting the sensor evidence by 70%, as calcu- lated using equation 3.4. The second sensor has conflicting evidence, assigning its belief across the frame as: {meetingRoom 0.2, desk ∧ caf e ∧ other 0.6, Θ 0.2}. Combining those beliefs using Dempster’s rule of combination in equation 3.3, the calculations are shown in table 3.1. Prior to normalisation, the un- normalised masses as captured in the combined evidence from table 3.1 are: massCaf e = 0.42 + 0.14 = 0.56;

massmeeting = 0.06; massdesk,caf e,other = 0.18;

massΘ = 0.06;

conf lict = 0.14

The conflict of 0.14 represents the conflicting evidence from the sensors which has no overlap (i.e. desk versus meeting).

To normalise out conflict, the normalising factor is k = 1 − 0.14 = 0.86 The revised masses after normalisation, using equation 3.3 are:

masscaf e= 0.65; massmeeting = 0.07;

Mass Assignment from sources Cafe 0.7 Uncertainty 0.3 Desk, cafe, other

0.6 Cafe 0.42 Desk, Cafe, Other 0.18 Meeting 0.2 Conflict 0.14 Meeting 0.06 Uncertainty 0.2 Cafe 0.14 Uncertainty 0.06 Table 3.1: Evidence combination example massdesk,caf e,other = 0.21;

massuncertainty = 0.07