In this section, an application of proposed discounted methods to temporal discounting is studied.
e principal idea behind this discounting is the fact that a piece of information becomes partially obsolete with time. is can happen because the entity described by this particular information is dynamic, changes or is not observed any more. It is important to underline that different pieces of information become obsolete at possibly different rates. is example motivates why there is a need for introducing new contextual discounting schemes and why the existing one is not sufficient. e first part demonstrates some postulates about temporal discounting itself. Next, the existing contex- tual discounting scheme is applied to temporal discounting. Finally, the application of the proposed methods is demonstrated.
.. Temporal discounting using contextual discounting
is section will present an aempt to use contextual discounting as presented by Mercier, ost, and Denœux () and Mercier, ost, and Denœux () and a counter-example demonstrating that this discounting scheme is not adapted for this aim.
Computing discounting mass function Instead of calculating discounting mass function mΘby applying the disjunctive operator, one can compute it directly using (Mercier, ost, and Denœux, Proposition ):
mΘ(A) =
θ∈Θ θ⊆A
αθ·
θ∈Θ θ⊆A
(1−αθ) (.)
Computing discounted mass function Once again, direct computation is possible to obtain dis- counted mass functionα∪,Θm using the results from Equations.and., which yields¹:
α∪,Θm(A) = (m∪ mΘ) (A) (.)
=
B∪C=A
m(B)· mΘ(C)
=
B⊆A
m(B)·
C⊆A, C⊇A\B
mΘ(C)
¹It is supposed that no discount rate has been defined for the empty set.
. Case study: temporal discounting using proposed methods Simplified computation Let us suppose that m is a normal mass function, i.e. the mass aributed to the empty set is null. is enables us to simplify Equation.for singletons to:
α∪,Θm({θ}) =
B∪C=θ
m(B)· mΘ(C) (.)
=
C⊆θ
m({θ})· mΘ(C)
= m({θ})·
C⊆θ
mΘ(C)
= m({θ})· belΘ({θ})
Use for temporal discounting
In order to calculate discount ratesα of contextual discounting from parametersκ of temporal dis- counting, let us compare side by side temporal discounting (Equation.) as obtained thanks to the above stated postulates:
αm({θ}) = m({θ})·e−λθt (.)
= m({θ})·κθ ∀θ∈Θ, 0< κθ≤1 with the simplified expression of contextually discounted mass (Equation.):
αm({θ}) = m({θ})· belΘ(θ) (.) which, given that m({θ})= 0, ∀θ∈Θ, yields:
m({θ})·κθ ≡ m({θ})· belΘ(θ) / : m({θ}) (.)
κθ ≡ belΘ(θ) (.)
κθ ≡
B∈Θ B⊆θ
(1−αB) (.)
LetK=|Ω|=|Θ|. Creating a system of equations for allθ∈Θusing Equation.issues:
κθ1 =
B∈Θ B⊆θ1
(1−αB) ...
κθK =
B∈Θ B⊆θK
(1−αB)
(.)
By solving the above equation it, with the convention that
i∈∅
xi = 1, one obtains:
αi = 1− K−1
j=i
κθj
κKθi−2 (.)
From Equations.and., we obtain:
κθ(t) =e−λθt (.)
αi(t) = 1− K−1
j=i
e−λjt
(e−λit)K−2 (.)
Example and counterexample
Let us consider sourceS that provides mass functions mΩS defined on the frame of discernmentΩ = {ω1, ω2, ω3}. Relying on the information that we possess about this source, we can examine two cases C1andC2 for which the half-life times differ. For eachω ∈Ω, half-life timest1/2are known to be:
C1 C2
t1/2, C1 = [1,4,15]s t1/2, C2 = [5,4,15]s
CaseC1can be interpreted as follows. Additional knowledge about sourceSis available and it states that classesω1,ω2andω3become obsolete with different rates. Namely,ω1is known to be worth a half of its initial value¹aer second,ω2andω3— aer s and s respectively. Analogical interpretation should be given to caseC2 with the sole difference that the half-life period of class ω1 is longer and equal to s.
Using Equation., decay parametersλare computed:
λC1 ≈[0.6931,0.1733,0.0462] (.) λC2 ≈[0.1386,0.1733,0.0462] (.)
en, thanks to Equations .and., let compute parametersκand discount factor vectorα for instantt= s:
κC1(t)≈[0.0625,0.5000,0.8312] (.) κC2(t)≈[0.5743,0.5000,0.8312] (.)
αC1 ≈[−1.5787,0.6777,0.8061] (.)
αC2 ≈[0.1493,0.0228,0.4122] (.)
Discounting for caseC1 e above steps demonstrate that the desired temporal discounting cannot be expressed in terms of contextual discounting as proposed in (Mercier, ost, and Denœux).
Indeed, αC1 contains a negative value, which is incompatible with this method and the outcome of such a discounting would not satisfy the condition of a mass function as required in Equation..
¹e wordvaluecorresponds to some subjective value of a piece of information from the point of view of the fusion system.
. Case study: temporal discounting using proposed methods
Discounting for caseC2
For caseC2, we can compute discounting mass function mΘusing Equation.:
mΘ(∅) = 0.4886 mΘ({ω1}) = 0.0858 mΘ({ω2}) = 0.0114 mΘ({ω1, ω2}) = 0.0020 mΘ({ω3}) = 0.3427 mΘ({ω1, ω3}) = 0.0602 mΘ({ω2, ω3}) = 0.0080 mΘ({ω1, ω2, ω3}) = 0.0014
Given function m with masses aributed as follows:
m(∅) = 0 m({ω1}) = 0.3
m({ω2}) = 0.4 m({ω1, ω2}) = 0 m({ω3}) = 0.2 m({ω1, ω3}) = 0 m({ω2, ω3}) = 0 m({ω1, ω2, ω3}) = 0.1
using the contextual discounting operation as expressed by Equation., we obtain discounted mass functiontm=α∪m:
tm(∅) = 0 tm({ω1}) = 0.1723
tm({ω2}) = 0.2 tm({ω1, ω2}) = 0.0391
tm({ω3}) = 0.1662 tm({ω1, ω3}) = 0.15
tm({ω2, ω3}) = 0.1442 tm({ω1, ω2, ω3}) = 0.1281
.. Contextual temporal discounting inconveniences
Contextual temporal discounting such as presented presents some undesirable properties. Let us reuse the example of caseC2from the section... Lett= s. We would expect that the masses aributed to all classes will diminish , and times respectively¹. e resulting discounted mass function is equal to:
tm(∅) = 0 (.)
tm({ω1}) = 0.0000732 (.)
tm({ω2}) = 0.0000122 (.)
tm({ω1, ω2}) = 0.000156 (.)
tm({ω3}) (.)
tm({ω1, ω3}) = 0.3410 (.)
tm({ω2, ω3}) = 0.0405 (.)
tm({ω1, ω2, ω3}) = 0.6058 (.)
As expected, we observe thattm({ω1}) = .×− ≈ m(4096{ω1}). Similarly, our expectations are satisfied fortm({ω2})andtm({ω3}). However, it is observed that e.g. tm({ω1, ω2})has a relatively
¹ese numbers come from a simple computation2
t t1/2.
Table . – Temporal discounting using the proposed discount schemes. Case .
t1/2, C1 = [1,4,15]s
A m(A) αom(A) αpm(A) αcm(A)
∅
{ω1} . 0.281 0.281 0.281
{ω2} . 0.1 0.1 0.1
{ω1, ω2} . 0.2 0.1453125 0.09375 {ω3} . 0.034 0.034 0.034
{ω1, ω3}
{ω2, ω3}
Ω . . . .
Table . – Temporal discounting using the proposed discount schemes. Case .
Result of Mercier’s contextual discounting in the rightmost column.
t1/2, C2 = [5,4,15]s
A m(A) αom(A) αpm(A) αcm(A) α∪m(A)
∅
{ω1} . 0.12771 0.12771 0.12771 0.1723
{ω2} . 0.1 0.1 0.1 .
{ω1, ω2} . 0.2 0.1069275 0.04257 .
{ω3} . 0.03376 0.03376 0.03376 .
{ω1, ω3} .
{ω2, ω3} .
Ω . . . . .
large value, only slightly smaller than m({ω1})+m({ω40962})+tm({ω1, ω2}) = 40960.7 =.×−, but bigger than 327680.7 =.×−.
.. Temporal discounting using proposed discounting schemes
On the contrary to contextual discounting, the proposed methods are expressive enough to reflect the desired behaviour of temporal discounting. Let us reuse the same two cases evoked in Section...
e computation of decay parametersλandκis common to both methods. Moreover, discount rate vector valuesα correspond directly to values ofκas shown by:
α1 = [ω1 →0.0625, ω2 →0.5, ω3 →0.8312] (.)
α2 = [ω1 →0.5743, ω2 →0.5, ω3 →0.8312] (.) Tables.and.show different discounting methods for the two analysed cases.
. Conclusion
In this chapter, we have proposed and defined three types of contextual discounting: conservative, pro- portional and optimistic. ese methods allow fine-grained modelling of the reliability of the sources.
Moreover, the introduced techniques can be applied to temporal discounting which has been described as well. It has been demonstrated that the existing contextual discounting introduced by Mercier, ost, and Denœux () is not strong enough to model temporal discounting.
In addition to the already given applications, the authors consider the use of temporal discounting in
. Conclusion the context of intelligent transportation perception. Various object classes seen by a vehicle should not be forgoen at the same rate. For instance, information about objects recognised as buildings shall be kept longer than static but possibly mobile objects. In turn, mobile static objects would persist longer than moving objects.
As a practical advantage, one can mention that for a given discount rate vector, factors by which masses are multiplied to obtain discounted mass function can be precomputed and stored for later use. e computational complexity of such an algorithm grows linearly with the size of the power set2Ωequally for time and space.
It would be interesting to automatically or semi-automatically define which type of discounting has to be used in particular situation. Moreover, a profound study of the properties of the proposed discounting rules seems to be significantly important. ese tasks are le for future research.