International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 3
Number 4
December 2009
423
1 INTRODUCTION
There are stochastic and epistemic uncertainties dis-
tinguished. Stochastic also called aleatory uncertain-
ty reflects unknown, usually unpredictable behav-
iour of a system. The system behaves in stochastic
way when its states are random ones. They can be
identified based on traditional probability theory. In
maritime traffic engineering attempt to find devia-
tion from intended track is related to the aleatory un-
certainty.
Shortage of knowledge or incomplete evidence
creates another kind of uncertainty. Epistemic or
subjective uncertainty results from insufficient or
vague evidence. Question of identity of new spotted
object refers to this sort of uncertainty. It is quite of-
ten when observer at monitoring station spots new
radar mark and tries to find out what vessel this
could be. Usually there is some evidence available,
for example radar echo signature and estimate of
speed can be helpful. Modern AIS technology trans-
fers data useful in identification. Problem was dis-
cussed by the author in his previous papers (Filip-
owicz 2007 & Filipowicz 2008).). Navigational aids
deliver plenty of data used for position fixing. The
quality of data is different and depends on many fac-
tors. Such imprecise and sometimes incomplete data
are further combined for position refinement. Quan-
tifying navigational status regarding an obstacle is
crucial from safety standards point of view.
In classical probability theory the knowledge of
probability of an event can be used to calculate like-
lihood of the contrary statement. In this approach if
one navigational aid indicates position within certain
area with probability of 0.6, that mean that navigator
believes that he is outside the area with the probabil-
ity of 0.4. The theory also requires that data regard-
ing probability of all considered events is at dispos-
al. The theory is limited in its ability when dealing
with epistemic uncertainty.
Mathematical Theory of Evidence (MTE for
short) is more flexible in this respect. MTE is a theo-
ry (initiated by Dempster & Shafer) based on belief
and plausibility functions and scheme of reasoning
in order to combine separate pieces of evidence to
calculate the probability of an event. Contrary to
probability theory it enables modelling knowledge
and ignorance. Evidence can be combined therefore
even partial knowledge associated with less mean-
ingful facts may end up in valuable conclusions.
Combining evidence leads to data enrichment and
improved probability judgments can be obtained for
each considered hypothesis. Fundamental for MTE
is Dempster-Shafer scheme of reasoning initially in-
tended for crisp values. New extensions to cope with
imprecision are also available since it is often that to
obtain precise figures is infeasible. Imprecision is
expressed as interval values or fuzzy figures. In the
paper and elsewhere fuzzy values are considered as
a set of intervals given for selected possibility levels.
Problem of position refinement that involves ep-
istemic uncertainty could be defined as below.
Given:
− navigation aids indicating different positions, dif-
ferent distances from an obstacle
An Application of Mathematical Theory of
Evidence in Navigation
W. Filipowicz
Gdynia Maritime University, Gdynia, Poland
ABSTRACT: Plenty of various quality data are available to the officer of watch. The data of various qualities
comes from different navigational aids. This kind of data creates new challenge regarding information associ-
ation. The challenge is met by Mathematical Theory of Evidence. The theory delivers methods enabling com-
bination of various sources of data. Results of association have informative context increased. Associated data
enable the navigator to refine his position and his status regarding dangerous places. The procedure involves
uncertainty, ambiguity and vague evidence. Imprecise and incomplete evidence can be combined using ex-
tended Dempster-Shafer reasoning scheme.
424
− each aid has reliability and accuracy characteris-
tic assigned to it
− linguistic terms referring to close, sufficient and
safe distances are available as membership func-
tions
Question:
− what is credibility that the real distance to the ob-
stacle is safe one?
First part of the paper is devoted to basic proba-
bility assignment. Then necessity to deal with im-
precision is depicted. Further on interval values are
introduced and belief structure defined. Short de-
scription of Dempster-Shafer method is also includ-
ed. Last part of the paper deals with identification of
navigational status referring to an obstacle. Two
navigational aids are considered. Their indications
are combined in order to quantify distance from cer-
tain shallow water area.
2 PROBABILITY ASSIGNMENT
Frame of discernment in Mathematical Theory of
Evidence consists of possible events. Events are un-
derstood very widely. Examples of events that are of
interest in navigation could be: route taken by a
spotted vessel, position fixing based on an electronic
aid, attempt to refine unidentified object etc. Events
are considered as atomic or structured ones. Consid-
ering limited set of objects as a single entity means
dealing with molecular or structured event. For ex-
ample new spotted object must be large container
carrier or medium bulk vessel because no other traf-
fic is expected within the area. It is assumed that in
case of structured event all constituents are equally
possible.
Figure 1. Intended route forecast problem involves three
events: taking route r
1
, taking r
2
and joint r
1
or r
2
Let us consider example on reasoning which of
possible and treated as equivalent routes r1 or r2 will
be taken by the vessel shown at figure 1. The frame
of discernment embraces three events Ω = ({r
1
},
{r
2
}, {r
1
, r
2
}). First event is related to route r1 as
possibly taken by the vessel, selecting route r2
means occurrence of the second event. Third molec-
ular event expresses uncertainty, it constituents r
1
or
r
2
are assumed to be equally possible.
Some evidence supporting reasoning on intended
ship’s itinerary is assumed. Recorded cases with
southwest bound vessels of similar tonnage were ex-
amined. For all n stored cases x out of the all have
chosen route r
1
. Appropriate masses related to each
of the events can be calculated according to formula
(1). Assuming that data stored in traffic related data-
base gives x=24 and n=39 masses of likelihood that
this time particular route is taken should be assigned
as shown in formula (1)
025.0
1
1
}),({m
375.0
1
})({m
600.0
1
})({m
21
2
1
=
+
=
=
+
−
=
=
+
=
n
rr
n
xn
r
n
x
r
(1)
It is easy to find out that all masses sum up to one so
probability requirement is satisfied. The theory also
requires that: m({r
1
, r
2
})= 1 - m({r
1
})- m({r
2
}). Note
that set {r
1
, r
2
} expresses some sort of uncertainty
since it reflects that both available routes can be tak-
en with the same credibility level.
Let us again consider example on guessing which
of routes r
1
or r
2
will be taken by the vessel. This
time we assume different evidence supporting rea-
soning on intended ship’s itinerary. We assume that
various samples of recorded cases are available.
Registered routes for similar ships referred to differ-
ent weather conditions. Number of records in the
samples varied within range of [20, 50]. Data anal-
yses discovered that number of southwest bound
ships that have chosen route r
1
was around 70% of
all stored cases. The percentage never fell below
60% and did not exceed 80% of the total number.
Under these assumptions one is not able to calculate
masses of evidence using before presented way of
reasoning. The task is seemingly unsolved due to
limitation imposed by crisp values. Interval values
are to be used instead. Counting all pros and cons
and numbers of records in the samples interval-
valued masses presents formula (2).
]048.0,020.0[
21
1
,
51
1
],[}),({m
]381.0,196.0[
21
8
,
51
10
],[})({m
]784.0,571.0[
51
40
,
21
12
],[})({m
3
-
321
2
-
22
1
-
11
=
==
=
==
=
==
+
+
+
mmrr
mmr
mmr
(2)
In this case all masses cannot sum up to one so basic
probability requirement cannot be satisfied. The ap-
proach stipulates that exists a set of sub ranges with-
r
2
Island
r
1
425
in defined intervals within which summation to one
is observed. More formally conditions 1 and 2 in
definition (1) are to be true. Definition (1) refers to
interval-valued probability assignment that is also
called as interval-valued belief structure.
Definition (1):
Interval-valued masses attributed to respective el-
ements of the frame of discernment, namely: [m
-
1
,
m
+
1
], [m
-
2
, m
+
2
],… , [m
-
n
, m
+
n
] define adequate
probability assignment if there is a set m such that
for m∈m following are satisfied:
− within each interval there is a value: m
-
i
≤
m
i
≤
m
+
i
, for each i∈{1, … , n}
− for all such values:
1
1
=
∑
=
n
i
i
m
For example three interval-valued masses the set
of legal probability assignment is shown as two di-
mensional shape in figure 2. Procedure of establish-
ing such shape can also lead to tightening interval
bounds since some values may appear as unreacha-
ble.
Figure 2. Graphical presentation of the set of valid probability
assignment in interval-valued belief structure
We again consider above example on guessing
which of routes r
1
or r
2
will be taken by the vessel
using fuzzy approach. We assume that it is experi-
enced radar observer who reasons on intended ship’s
itinerary. His subjective way of thinking is like this:
the vessel is a medium one, visibility is rather good,
wind moderate so to his best knowledge it is “likely”
the vessel will take route r
1
. He also observed quite
many similar vessels have taken route r
2
so it is
“fairly likely” that this route will be chosen this
time. Judging from his experience uncertainty of his
opinion is very low. Formal expression of the above
statement requires introduction of meaning terms:
“likely”, “fairly likely” and “very low”. All of them
are linguistic terms referring to fuzzy reasoning.
Such terms are characterized by membership func-
tions.
2.1 Theoretical membership functions
Set with elements like: “very unlikely”, “unlikely”,
“fairly likely”, “likely”, “very likely” and “certain”
consists of linguistic terms which human beings use
for estimated reasoning. To evaluate uncertainty one
can use “very low”, “low”, “medium”, “high”, “very
high” and “totally uncertain” as the highest term.
Both sets contain six elements and membership
functions can be used interchangeably depending on
the context.
Counting elements from 0 up to n
c
-1 one can use
formula (3) to calculate normalized and regular
fuzzy membership functions. Trapezoid shapes ob-
tained for w
T
=0.8 are presented in figure 3 and tri-
angular ones for w
T
=0 in figure 4.
−=−−
−<<
++
−−
=
=
1)1,1,*1,1(
10
)*)1(,**
,**,*)1((
0),*,0,0(
cT
c
T
T
T
k
nkifwww
nkif
wkwwwk
wwwkwk
kifwww
F
(3)
where:
−
1
1
−
=
c
n
w
− n
c
- is a number of selected terms
− w
T
∈ [0, 1]– is the shape parameter, w
T
= 0
means that membership function is a triangular
one and w
T
= 1 means rectangular shape.
Formula (4) defines trapezoid fuzzy-valued mass-
es assignment for the third discussed case of proba-
bility assignment on expected route taken by the
spotted craft. The formula contains membership
functions for terms respectively “likely” (k=3), “fair-
ly likely” (k=2) and “very low” (k=0). Functions are
quads calculated with formula (1) for listed above k
value. Membership functions are also presented as
intervals for three selected possibility levels α = 0,
0.5 and 1. Possibility equal to zero denotes support
of a fuzzy value. Possibility equal to one refers to
the core of imprecise value.
Figure 3. Trapezoid membership functions (w
T
=0.8)
426
Figure 4. Triangular membership functions expressing six lin-
guistic terms (w
T
=0)
=
=
=
≈=
=
=
=
≈=
=
=
=
≈=
]16.0,0[1
]18.0,0[5.0
]2.0,0[0
)2.0,16.0,0,0(}),({m
]56.0,24.0[1
]58.0,22.0[5.0
]6.0,2.0[0
)6.0,56.0,24.0,2.0(})({m
]76.0,44.0[1
]78.0,42.0[5.0
]8.0,4.0[0
)8.0,76.0,44.0,4.0(})({m
21
2
1
α
α
α
α
α
α
α
α
α
rr
r
r
(4)
3 COMBINATION OF TWO BELIEF
STRUCTURES
Probability assignments that examples are showed
above can be combined in order to increase result in-
formation context. Probability assignment to events
from frame of discernment at hand is called as belief
structure. Belief structures are supposed to verify
certain constraints (see for example definition (1)).
Depending on type of assigned masses basic, inter-
val-valued and fuzzy-valued structures are distin-
guished. It is said that combination of belief struc-
tures creates new assignment characterized by
enrichment of engaged data. To take benefit of this
enrichment other sources of data are to be available.
In the above interval-valued example on guessing
which of routes r
1
or r
2
will be taken by the vessel
single source of data was assumed. Let us consider
yet another archive that contains different sets of
recorded cases. Registered routes for similar ships
referred to similar weather conditions were ana-
lyzed. Number of records in the samples varied
within range of [40, 60]. Data analyses revealed that
number of southwest bound ships that have chosen
route r
1
never fell below 50% and did not exceed
65% of the total number. Masses attributed to each
event are shown in formula (5).
]024.0,016.0[],[}),({m
]492.0,341.0[],[})({m
]639.0,488.0[],[})({m
23
-
23212
22
-
2222
21
-
2112
==
==
==
+
+
+
mmrr
mmr
mmr
(5)
Combination procedure for ranges and fuzzy val-
ues extends original Dempster-Shafer method initial-
ly proposed for crisp masses in basic belief struc-
tures. Comprehensive way of two sources
combinations is summarized below. The scheme was
further used for example combination of the two
discussed sources results are shown in table 1.
Dempster-Shafer rules of combination:
1 Create table with rows that refer to events em-
braced in second source.
Columns refer to the events of first source.
Each event has mass of evidence (fuzzy or inter-
val-valued) that is assigned to it
2 For each intersection of a row and a column
product of masses involved is calculate and at-
tributed to a common, for the two sets, event.
In case of crisp events inconsistency occurs if the
two sets have empty intersection. Therefore, for
particular cell, the product of masses of evidence
is assigned to an empty set
In case of fuzzy events conjunctive operator is
applied and search for minimum values on mem-
bership functions involved carried out
3 Calculate masses for each resulting set of events
4 Calculate belief functions (and if required plausi-
bility) values
Definition (2):
There are two sets of interval-valued masses at-
tributed to elements of the same frame of discern-
ment, namely: m
1
, m
2
. Each of them embraces cer-
tain set of events referred to as: F(m
1
) and F(m
2
).
Their combination defines probability assignment as
a set m such that for m∈m appropriate limits (Deno-
eux 1999) are given by formula (6).
∑
∑
=∩
×∈
+
=∩
×∈
−
=
=
ACB
21
m(m
ACB
21
m(m
CmBmAm
CmBmAm
21
21
)()(max)(
)()(min)(
)(),
)(),
21
21
mm
mm
(6)
Table 1. Combination of two sources of crisp data
__________________________________________________
Source I
m
1
({s
1
}) m
1
({s
2
}) m
1
({s
1
, s
2
})
[0.571, 0.784][0.196, 0.381] [0.020, 0.048]
__________________________________________________
m
2
({s
1
}) m
1-2
({s
1
}) m
1-2
({∅}) m
1-2
({s
1
})
[0.488, 0.639] [0.488, 0.639] [0.096, 0.243] [0.010, 0.031]
m
2
({s
2
}) m
1-2
({∅}) m
1-2
({s
2
}) m
1-2
({s
2
})
[0.341, 0.492] [0.195, 0.386] [0.067, 0.187] [0.007, 0.024]
m
2
({s
1
, s
2
}) m
1-2
({s
1
}) m
1-2
({s
2
}) m
1-2
({s
1
, s
2
})
[0.016, 0.024] [0.009, 0.019] [0.003, 0.009] [0.000, 0.001]
__________________________________________________
427
Using formula (6) one can obtain limits of joint
masses that are as follows:
− m
-
1-2
({s
1
}) = 0.279 + 0.009 + 0.01 = 0.298
− m
+
1-2
({s
1
}) = 0.501 + 0.019 + 0.031 = 0.551
− m
-
1-2
({s
2
}) = 0.067 + 0.003 + 0.007 = 0.077
− m
+
1-2
({s
2
}) = 0.187 + 0.009 + 0.024 = 0.220
− m
-
1-2
({s
1
, s
2
}) = 0.0003
− m
+
1-2
({s
1
, s
2
}) = 0.0012
Since in two cases there were empty intersections
therefore inconsistency occurred. Limits of the emp-
ty set are as below:
− m
-
(∅) = 0.096 + 0.195 = 0.291
− m
+
(∅) = 0.386 + 0.243 = 0.529
Result belief structure with its interval-valued
probability assignment enables determination of evi-
dential functions. Lower and upper limits of belief
function can be calculated with formula (7).
∑∑
∑∑
≠⊄
−−
∅≠⊆
++
≠⊄
++
∅≠⊆
−−
∅−−=
∅−−=
ABABBAB
ABABBAB
mBmBmAbel
mBmBmAbel
;;
;;
))()(1),(min()(
))()(1),(max()(
(7)
Taking into account limits of empty sets obtained
during combination ranges of believes for each of
the events are as shown in table 2.
Table 2. Joint masses, belief function values and tighten
bounds
__________________________________________________
Event Joint masses Interval-valued Tighten
beliefs intervals
__________________________________________________
{s
1
} [0.298, 0.551] [0.298, 0.551] [0.313, 0.529]
{s
2
} [0.077, 0.220] [0.077, 0.220] [0.078, 0.219]
{s
1
, s
2
} [0.0003, 0.0012] [0.471, 0.709] [0.0003, 0.0012]
__________________________________________________
MTE defines belief function in terms of the mass
of evidence assigned to each event and its constitu-
ents, if available. Thus in order to obtain total belief
committed to the set, masses of evidence associated
with all the sets that are subsets of the given set must
be added. Consequently beliefs of atomic event re-
main unchanged and equal to combined values. Joint
events increase their belief values according to con-
stituents masses (see last row in table 2).
3.1 Evidence combination as optimization problem
Presented procedure is an extension of initial Demp-
ster proposal intended for structures with crisp
events as well as crisp masses assigned to the events.
Extension of the approach substitute crisp values
with interval-valued probabilities. Subsequently
principles of adequate mathematics are to be ap-
plied.
Unfortunately such direct modification can lead
to results that are too broad. The new approach to-
ward data association is to be considered since its re-
sults are to be tightened. Problem of combination of
interval-valued structures can be introduced as fol-
lowing optimization task (Denoeux 1999).
Search for lower and upper limits of combined
structure:
)(*)(max),(
)(*)(min),(
2121
2121
CmBmmmm
CmBmmmm
ACB
A
ACB
A
∑
∑
=∩
+
=∩
−
=
=
(8)
Under constraints:
)()()()(
)()()()(
1)
(
1)(
2111
1111
)(
1
)(
1
2
1
m
CCmCmCm
mBBmBmBm
Cm
Bm
mC
mB
F
F
F
F
∈∀≤≤
∈∀≤≤
=
=
+
−
+−
∈
∈
∑
∑
(9)
Adequate optimization problem was solved using
available software and results are shown in the
rightmost part of table 2. It is seen that optimization
leads to results falling within limits established with
previous method. All further results of combination
presented in the paper were obtained using software
available at website:
http://www.hds.utc.fr/~tdenoeux/.
The software implements procedures solving
above defined optimization problem.
4 BELIEF STRUCTURES IN MARITIME
NAVIGATION
Previously presented case of guessing which route
will be taken by unknown vessel, although interest-
ing, is not very much representative for maritime
navigation. Its typical problems are related to posi-
tion fixing. The aim of the position interpretation is
to find out what the distance from nearest obstacle
could be. The distance given as crisp value is not of
primary importance instead it subjective assessment
really matters. Subjectivity should embrace local
condition. Confined water distance of 4Nm must be
differently perceived than the same distance in the
open sea. Nevertheless safe or sufficient distance
value is to be maintained everywhere and all the
time. Example of the set of fuzzy-valued subjective
distances is shown in figure 5.
428
very close
sufficient
close
safe
very safe
Figure 5. Distances from an obstacle expressed as fuzzy values
Fixing can be directly transferred into appropriate
state referring to the obstacle. Being within the state
can be treated as an event in MTE terminology.
From figure 5 it is also clear that limits of a state are
imprecise values therefore event is not crisp any
longer. In figure 5 circle around position cross re-
flects error, standard deviation attributed to particu-
lar system. Marked spot is somewhere in between
“close” and “sufficient” distance if proposed limits
are assumed. Instead of establishing borders one can
ask experts what they think about, for example, 4Nm
off the buoy. They are to use scale that covers five
terms from “very close” to “very safe”. Table 3 con-
tains results of the inquiry with 16 unity intervals
scale. Each linguistic term covers four adjacent unity
intervals. Extreme interval is assumed to be shared
with neighbour term.
Table 3. Meaning of 4Nm off safe water buoy in the given area
__________________________________________________
very close sufficient very safe
close safe
Expert 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
__________________________________________________
1 + + +
2 + + + +
3 + + + +
4 + + +
5 + +
frequency 0.2 0.8 0.6
0.2 1.0 0.4
__________________________________________________
Last two rows of the table 3 embrace relative fre-
quencies of answers for non-zero unity intervals. Set
of these figures creates irregular membership func-
tion that will be written as:
µ
d1
(x
i
) = {0.2/6, 0.2/7, 0.8/8, 1/9, 0.6/10, 0.4/11}
Unlike theoretical membership functions these
similar to presented in table 3 are called empirical
membership functions.
Accuracy of distance measured by a navigational
aid depends on method and appliance involved. Dif-
ferent credibility is attributed to various aids. To
conclude reasoning regarding measured distance one
has to attribute mass of credibility to engaged sys-
tem. Let us assume that example system’s credibility
is high. In this case using suggested 6-grade scale
and formula 3 factor k will be assumed as equal to 4
(trapezoid regular membership function with w
T
=0.8
are further used). Doubtfulness regarding proper
functionality of the aid and outcome of expert opin-
ions is rather low (k=1).
Above statements define following belief struc-
ture.
Measured distance to the obstacle expressed sub-
jectively:
µ
d1
(x
i
) = (0.2/6, 0.2/7, 0.8/8, 1/9, 0.6/10, 0.4/11)
Mass of credibility attributed to navigational aid
and quality of expert opinions:
m
1
(d
1
) = (0.8, 0.84, 0.96, 1)
The last can be approximately expressed as:
α = 1 [0.84, 0.96]
m
1
(d
1
)≈ α =0.5 [0.82, 0.98]
α =0.0 [0.80, 1]
Mass of uncertainty attributed to navigational aid
and to quality of expert opinions:
m
1
(any) = (0, 0.04, 0.36, 0.4)
This can be equivalent to:
α = 1 [0.04, 0.36]
m
1
(any)≈ α =0.5 [0.02, 0.38]
α =0.0 [0, 0.40]
The latest reflects statement that contradicts
membership function shown in table 5. It expresses
conclusion that engaged system might not work
properly and indicates wrong data. Consequently
every distance is equally possible. Membership
function attached to such uncertainty consists of all
one:
µ
any
(x
i
) = (1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9,
1/10, 1/11, 1/12, 1/13, 1/14, 1/15, 1/16).
Assuming approximation of fuzzy values by in-
terval values at selected possibility levels conditions
of definition 1 are observed for all levels thus the
above assignment is appropriate belief structure.
In order to enrich knowledge and reduce uncer-
tainty regarding distance from the obstacle we as-
sume that there is another navigational aid that indi-
cates different distance and the aid is also reputed in
different way. Another belief structure is as follows.
Measured distance to the obstacle expressed in
subjective way:
µ
d2
(x
i
) = (0.2/5, 0.4/6, 0.6/7, 1/8, 0.6/9, 0.2/10)
429
Mass of credibility attributed to another naviga-
tional aid and quality of new expert opinions as-
sumed as trapezoid fuzzy value (k = 3 and w
T
= 0.8):
m
2
(d
2
) = (0.4, 0.44, 0.76, 0.8)
α = 1 [0.44, 0.76]
m
2
(d
2
)≈ α =0.5 [0.42, 0.78]
α =0.0 [0.40, 0.80]
Mass of uncertainty attributed to this positioning
system and quality of other expert opinions ex-
pressed as trapezoid fuzzy value with k = 3 and w
T
=
0.8):
m
2
(any) = (0.2, 0.24, 0.56, 0.6)
α = 1 [0.24, 0.56]
m
2
(any)≈ α =0.5 [0.22, 0.58]
α =0.0 [0.20, 0.60]
Same as before fuzzy values were approximated
by interval values at three selected possibility levels.
Conditions of definition 1 are observed for each of
the levels thus the second assignment is also correct
belief structure.
Indications coming from two sources were asso-
ciated using extended Dempster-Shafer scheme and
optimization approach. Obtained results are shown
in table 4.
Table 4. Combination of two navigational aids
__________________________________________________
m
1
(µ
d1
) m
1
(any)
α = 1 [0.84, 0.96] [0.04, 0.36]
α =0.5 [0.82, 0.98] [0.02, 0.38]
α =0.0 [0.80, 1] [0, 0.40]
__________________________________________________
m
1-2
(µ
d1
∧µ
d2
) m
1-2
(µ
d2
)
α = 1 [0.44, 0.76] [0.37, 0.73] [0.018, 0.27]
m
2
(µ
d2
) α =0.5 [0.42, 0.78] [0.34, 0.76] [0.008 0.30]
α =0.0 [0.40, 0.80] [0.32, 0.80] [0.0, 0.32]
m
1-2
(µ
d1
) m
1-2
(any)
α = 1
[0.24, 0.56] [0.20, 0.54] [0.01, 0.20]
m
2
(any) α =0.5 [0.22, 0.58] [0.18, 0.57] [0.004 0.22]
α =0.0 [0.20, 0.60] [0.16, 0.60] [0.0 0.24]
__________________________________________________
In table 4 there is expression m
1-2
(µ
d1
∧µ
d2
) that
remains to be explained. It is at the intersection of
m
2
(µ
d2
) row and m
1
(µ
d1
) column and mean joint con-
fidence regarding distances to the same obstacle
measured by different navigational aid. In case of
crisp events the mass would be assigned to empty set
(∅). In case when events are fuzzy the expression
should be written as m
1-2
(µ
d1
(x
i
)∧µ
d2
(x
i
)) and inter-
preted as a mass of confidence attributed to conjunc-
tion of two fuzzy values respectively µ
d1
(x
i
) and
µ
d2
(x
i
). In this case µ
d1
(x
i
)∧µ
d2
(x
i
) =(0/5, 0.2/6,
0.2/7, 0.8/8, 1/9, 0.6/10, 0.4/11)∧(0.2/5, 0.4/6, 0.6/7,
1/8, 0.6/9, 0.2/10, 0/11) = (0/5, 0.2/6, 0.2/7, 0.8/8,
0.6/9, 0.2/10, 0/11). Note that conjunction ∧ means
minimum operation in the two sets. As a result of
combination of fuzzy events apart from initial sets
appear yet another membership functions. The more
sources are combined the more numerous count of
such extra events. Note that such events bring some
support for certain classes of fuzzy events.
Seemingly this phenomenon makes the approach
vague. To some extent the statement is true. At the
other hand result of combination could be treated as
an encoded knowledge base. Having such database
one is supposed to ask questions and get answers. As
a matter of fact this is main advantage of the ap-
proach.
Kind of questions that can be submitted to the
knowledge base depend on the problem at hand. In
discussed case it could be interesting to know sup-
port for a statement that the distance from the obsta-
cle is safe or sufficient one. Table 5 contains interval
values of belief functions for different regular fuzzy
functions related to considered scale of distances.
Figure 6. Bundle of benchmark membership functions
Benchmark membership functions used in table 5
are regular trapezoid ones presented in figure 6.
They are based on sixteen unity interval scale as pre-
sented in table 3. First of the functions reflects term
“safe”, second one is shifted left (closer to the obsta-
cle) by 1 unit and so on. In this way fourth function
is related to sufficient distance and seventh to close
condition.
Fuzzy belief functions values are given as α-cuts
for α=1, 0.5 and 0 in top to bottom order.
Figure 7 shows diagrams of three belief values
marked with asterisk in table 5. They represent in-
terval-valued beliefs that the distance is close, suffi-
cient and safe, for the highest possibility level. The
highest credibility with upper limit approaching 0.74
receives sufficient distance.
Table 5. Fuzzy beliefs for obtained combination results and se-
lected fuzzy distances
__________________________________________________
Pattern fuzzy value Belief function
__________________________________________________
α = 1 [0.074, 0.146]*
1 (0.5/10, 1/11, 1/12, 0.5/13) safe α =0.5 [0.069, 0.153]
430
α =0.0 [0.064, 0.160]
α = 1 [0.114, 0.195]
2 (0.5/9, 1/10, 1/11, 0.5/12) α =0.5 [0.105, 0.197]
α =0.0 [0.096, 0.200]
α = 1 [0.376, 0.493]
3 (0.5/8, 1/9, 1/10, 0.5/11) α =0.5 [0.364, 0.497]
α =0.0 [0.352, 0.500]
α = 1 [0.510, 0.714]*
4 (0.5/7, 1/8, 1/9, 0.5/10) sufficient α =0.5 [0.493, 0.727]
α =0.0 [0.476, 0.740]
α = 1 [0.358, 0.475]
5 (0.5/6, 1/7, 1/8, 0.5/9) α =0.5 [0.347, 0.480]
α =0.0 [0.336, 0.484]
α = 1 [0.155, 0.315]
6 (0.5/5, 1/6, 1/7, 0.5/8) α =0.5 [0.141, 0.326]
α =0.0 [0.128, 0.336]
α = 1 [0.074, 0.146]*
7 (0.5/4, 1/5, 1/6, 0.5/7) close α =0.5 [0.069, 0.153]
α =0.0 [0.064, 0.160]
__________________________________________________
Figure 7. Belief intervals for close, sufficient and safe distances
5 CONCLUSIONS
Bridge officer has to use different navigational aids
in order to refine position of the vessel. To combine
various sources he uses his common sense or relies
on traditional way of data association. So far Kal-
man filter proved to be most famous method of data
integration. Mathematical Theory of Evidence deliv-
ers new ability. It can be used for data combination
that results in their enrichment. Dempster-Shafer
scheme initially designed for crisp data association
now is widely used to cope with imprecision, which
is expressed by intervals or fuzzy values. Assign-
ment of masses of evidence to each of events at hand
creates belief structure. Crisp, interval-valued and
fuzzy-valued belief structures are distinguished.
In the paper interval-valued belief structure is de-
fined. It is also shown that transition from interval to
fuzzy-valued structure is straightforward. Example
of such structures for position fixing was presented.
The structures were then combined and results dis-
cussed. The most important conclusion that can be
drawn from included example is that with help of
MTE quantification of imprecise statement is possi-
ble. With at least two navigational aids engaged
credibility that the distance from an obstacle is safe
receives its unique, although interval or fuzzy-
valued belief.
REFERENCES
Denoeux, T. Fuzzy Sets and Systems, Modelling vague beliefs
using fuzzy valued belief structures, 1999
Filipowicz, W. Intelligent VTS. Weintrit, A. (ed.) 2007 Ad-
vances in Marine Navigation and Safety of Sea Transporta-
tion.: 151-159, Gdynia
Filipowicz Wł., Mikulski J. (ed.), 2008 Advances in Transport
Systems Telematics, Wydawnictwo Komunikacji i
Łączności, Mathematical Theory Of Evidence And Naviga-
tional Situation Evaluation: 81-92, Warszawa
0,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
close distance
sufficient distance
safe distance
