International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 5

Number 2

June 2011

225

1 INTRODUCTION

In his recent papers the author presented application

of Mathematical Theory of Evidence (MTE) in nav-

igation. The Theory appeared to be flexible enough

to be used for reasoning on the fix. Contrary to the

traditional approach, it enables embracing

knowledge into calculations. Knowledge regarding

position fixing includes: characteristics of random

distributions of measurements as well as ambiguity

and imprecision in obtained parameters of the distri-

butions. Relation between observations errors and

lines of position deflection is also important. Uncer-

tainty can be additionally expressed by subjectively

evaluated masses of confidence attributed to each of

observations.

New scheme enabling inclusion of knowledge in-

to the fixing process was presented by Filipowicz

2009c. Way of computation of belief and plausibility

as well as location vectors grades can be found in

other papers by Filipowicz 2009a, 2009b. Location

vectors were constructed assuming normal distribu-

tion of measurement errors. The latest was rather a

result of limitation imposed on the publications. In

order to fill up the hiatus empirical distributions are

discussed herein.

Those interested in computational complexity of

the fixing algorithms and ways of detecting local

maxima should refer to another paper Filipowicz

2010a.

This paper is devoted to a new idea in position

fixing in terrestrial navigation. Therefore character-

istics of measurements errors are discussed, relation

between imprecision of the measured values and

lines of position or isolines is also presented.

During computation process abnormally high in-

accuracy should be detected. In proposed approach

the condition results in large mass of inconsistency,

which occurs when no zero mass is assigned to emp-

ty sets. High inconsistency mass leads to rejection of

the fix or undertaking steps towards fix adjustment.

Selected position can be evaluated based on the final

inconsistency but also on plausibility and belief val-

ues. It should be noted that constant errors are of

primary importance when quality of the fix is con-

sidered. Using methods that remove systematic de-

flection of a measurement is recommended. Exploit-

ing horizontal angles instead of bearings makes the

fixed position independent from constant errors. The

latest is a reason that part of the paper is devoted to

the horizontal angle isoline.

MTE exploits belief and plausibility measures, it

operates on belief structures. Belief structures are

subject to combination in order to increase their ini-

tial informative context. The structures can be crisp,

Fuzzy Evidence in Terrestrial Navigation

W. Filipowicz

Gdynia Maritime University, Poland

ABSTRACT: Measurements taken in terrestrial navigation are random values. Mean errors are within certain

ranges what means imprecision in their estimation. Measurements taken to different landmarks can be subjec-

tively diversified. Measurements errors affect isolines deflections. The type of the relation: observation error

– line of position deflection, depends on isolines gradients. All the mentioned factors contribute to an overall

evidence to be considered once vessel’s position is being fixed. Traditional approach is limited in its ability of

considering mentioned factors while making a fix. In order to include evidence into a calculation scheme one

has to engage new ideas and methods. Mathematical Theory of Evidence extended for fuzzy environment

proved to be universal platform for wide variety of new solutions in navigation.

226

interval and fuzzy valued. Mainly crisp valued struc-

tures were presented and discussed in the author’s

previous papers. The structures consist of sets of

normal location vectors along with crisp masses of

confidence attributed to them. Vectors normality can

be achieved through transformation procedure called

normalization. Approaches known as Dempster and

Yager methods are widely used. Advantages and

disadvantages of the two proposals are discussed

from nautical usage point of view. Being stuck to the

original proposals proved to be not adequate while

position fixing. For this reason a modified normali-

zation procedure is proposed in this paper.

2 FUZZY EVIDENCE

Crisp valued standard deviation of a measurement is

inadequate. In recent navigation books mean error is

described as imprecise interval value usually as:

[±

, ±

]. Mean error of a distance measured

with radar variable range marker is within the inter-

val of [±1% ÷ ±1.5%]. In the same condition mean

error of a bearing taken with medium class radar is

within [±1° ÷ ±2°] as presented by Jurdziński 2008

& Gucma 1995. Using fuzzy arithmetic notation it

can be written as a quad (-2, -1, 1, 2). The latest

means fuzzy value with core of [-1°, 1°] and support

of [-2°, 2°], and reflects the statement that the error

is within [±1° ÷ ±2°]. Graphic interpretation of the

proposition is shown in Figure 1. The scheme en-

gages probability and possibility theory. Observa-

tional errors are assumed to follow a normal distri-

bution. Mean error estimates standard deviation

(square root of a variance) of the distribution. The

picture shows two confidence intervals related to

two different distribution functions. A confidence in-

terval is an interval in which a measurement falls

within a range with selected probability. It is as-

sumed that the confidence intervals are symmetrical-

ly placed around the mean. A confidence interval

with probability equal to 0.683, for the Gauss proba-

bility density function is the interval [α - σ, α + σ]

where α is a mean and σ is a standard deviation.

Two confidence intervals introduce imprecision

that is usually expressed by an interval or fuzzy val-

ue that is a synonym of fuzzy set.

Figure 1 shows trapezoid-like membership func-

tion that locates adjacent bearings within the defined

set. The function returns possibility regarding giv-

en x, it attributes x degree of inclusion within the set.

For example abscissa: x = α+0.5 fully belongs to the

given set, contrary to x = α+1.5, its inclusion within

the set is partial with degree of membership equal

to 0.5. Different membership functions intended for

nautical application were discussed by the author in

his previous paper Filipowicz 2009a.

gradient

probability/possibility

1°

2°

-2°

-1°

-σ

Figure 1. Graphic interpretation of the proposition “bearings

mean error is between ±1°÷±2°”

Empirical parameters are estimated based on ob-

servations. Empirical probability is widely used in

practice. In terrestrial navigation it is exploited quite

often. Theoretical probabilities are estimated by

those calculated from experiments and observations.

Empirical probability is the ratio of the number of

those results that fall into a selected category to the

total number of observations.

The empirical probability estimates statistical

probability. Avoiding any assumptions regarding ob-

tained data is the main advantage of estimating

probabilities using empirical data. Histograms are

widely used as graphical representation of empirical

probabilities. Histogram is a diagram of the distribu-

tion of experimental data. Usually histogram con-

sists of rectangles, placed over non-overlapping in-

tervals also known as bins. The histogram is

normalized and displays relative frequencies. It then

shows the proportion of cases that fall into each of

several bins. In normalized histogram total area of

rectangles equals to one. The bins or intervals are

usually chosen to be of the same size. There is no

universal rule to calculate number of bins. In the

presented application their quantity equals to the

number of ranges established around measured value

assumed as governed by normal distribution. Empir-

ical distribution of observational errors with impre-

cise bin width and relative frequencies is shown in

Figure 2.

Family of sets {{l

}

} of measured values are giv-

en as a result of experiments. Therefore sets of mean

values {l

} and the bin width s can be obtained. Ex-

treme deflection of means ∆l

–

and ∆l

can be also

known. Modal value

is calculated based on ex-

treme means. Consequently empirical mean and bin

widths are interval valued with above mentioned

limits. Relative frequencies {p

} for each of consid-

Modal value is defined for a fuzzy set. Usually it is calculated as a

mean of the set’s core, Piegat 2003. It should be noted that modal val-

ue is of secondary meaning in distribution characteristics.

ered bins are obtained as crisp or imprecise valued.

Formulas from 1 to 4 define complete set of parame-

ters for empirical distributions.

})({min})({max

:where

})({max,})({min,

llllll













−−=













∆∆

+−

(1)

})}({{min})}({{max

−

(2)

























+−

})}({{max}),}({{min,

ijk

pppp

(3)













∆∆=













∆∆

+−+−

llss ,,

(4)

+−

'1'1

+−

'3'3

+−

'2'2

−

Figure 2. Empirical distribution with imprecise bin width and

relative frequencies

3 ISOLINES AND THEIR GRADIENTS

Results of measurements plotted at a chart appear as

lines of position. From the mathematic point of view

the lines of position are isolines or in many cases

lines tangent to them. An isoline for a function of

two variables is a curve connecting points where the

measurement has the same value. In terrestrial navi-

gation, an isoline joins points of equal bearing, dis-

tance or horizontal angel. A bearing is the direction

one object is from a vessel. Isoline of a bearing is a

line, the same distance from an object produces cir-

cle. Isoline of the horizontal angle is also a circle

since all inscribed angles that subtend the same arc

are equal. The arc joins observed objects. Figure 3

presents isoline of a horizontal angle. A horizontal

angle obtained as difference of two bearings is a

valuable thing for navigator since it does not contain

constant error.

The gradient of a function is a vector which com-

ponents are the partial derivatives of the function.

For function of two variables gradient is defined by

Formula 5.













∂

=∇=

yxfyxg ,),(),(

(5)

0 1 2 3 4 5 6 7 8 9 10

– g

Figure 3. Isoline of a horizontal angle and its gradient in select-

ed point (g

and g

refers to gradients of the first and second

bearing, first bearing is taken to the left object)

Product of the gradient at given point with a vec-

tor gives the directional derivative of the function in

the direction of the vector. The direction of gradient

of the function is always perpendicular to the isoline.

Gradients, measurements error and lines of position

deflections are dependent values. Formula 6 shows

the relation.

),(

yxg

yxM

(6)

Table 1. Parameters of the horizontal angle isoline

__________________________________________________

isoline parameter formula

__________________________________________________

isoline radius

)sin(2

r =

center coordinates (x

+r cos(90-

, y

+r sin(90-

))

gradient module













rad

__________________________________________________

– distance between observed objects

– inclination, related to x axis, of the line passed through

the objects (

= 0 in Figure 3)

, y

– coordinates of the left object (see Figure 3)

– horizontal angle calculated as difference of bearings

(

> 0)

– distance to i-th object (d

≠ 0)

Error of the measurement divided by the module

or length of the gradient in selected point gives de-

flection of the isoline at the point. In the proposed

solution limits of introduced strips and possible iso-

lines coverage are to be calculated accordingly.

228

Radius length, coordinates of the center and gra-

dient module for horizontal angle isoline can be cal-

culated with formulas presented in Table 1.

Table 2 contains data regarding isoline shown in

Figure 3. The data embrace distances, gradients

modules and isoline errors calculated for measure-

ments standard deviation of ±1°. Appropriate values

were obtained for selected points placed in the iso-

line.

Table 2. Selected points at the horizontal angle isoline, gradi-

ents and isoline errors

__________________________________________________

0 1 3 4 5 6 8 9

3.2 4.9 6.2 6.4 6.4 6.2 4.9 3.2

[Nm]

3.2 5.0 6.8 7.5 8.1 8.6 9.4 9.6

[Nm] 9.6 9.4 8.6 8.1 7.5 6.8 5.0 3.2















16.7 11.1 8.8 8.5 8.5 8.8 11.1 16.7

±M [cables] 0.60 0.90 1.14 1.18 1.18 1.14 0.90 0.60

__________________________________________________

isoline error M was calculated for measurement mean error

= ±1°

Isoline of a horizontal angle and its limits calcu-

lated for interval [

- 3σ,

+ 3σ] is shown in Figure

4. Limits of an isoline shows its extreme shifts due

to measurements errors. These limits can be of the

same size along the line as for example for distanc-

es. For bearings and horizontal angles limits vary

depending on the position of the observer. Within

the limits strips related to confidence intervals are

established. Levels of confidence and way of select-

ing stripes are discussed in the paper by Filipowicz

2010b.

0 1 2 3 4 5 6 7 8 9 10

isoline of a

horizontal

angle

limits of the

isoline

Figure 4. Isoline of the horizontal angle and its limits

4 SCHEME OF A POSITION FIXING

Let us consider three rectangular ranges related to

three isolines as shown in Figure 5. Within ranges

six strips were distinguished. Widths of the strips are

calculated based on measurement errors and the iso-

line gradients. Each strip has fuzzy borders depend-

ing on imprecision in estimations of the isoline er-

rors distribution. Theoretical or empirical

probabilities of containing the true isoline within

strips are given. Having particular point and all be-

fore mentioned evidence support on representing

fixed position for given point should be found. This

is quite different from traditional approach where

single point should be found and available evidence

hardly exploited.

The scheme of approach is as follows:

Given: available evidence obtained thanks to nauti-

cal knowledge

Question: what is a support that particular point can

be considered as fixed position of the ship?

Figure 5. Three isolines with strips established around them

Figure 5 shows common area of intersection of

three areas associated with three isolines. Six strips

were selected around each isoline, the strips were

numbered as shown in the figure. Number 3’ refers

to the far most section, number 3 indicates closest

range according to gradient direction and regarding

observed object(s). Assuming normal or empirical

distribution probabilities attributed to each of the

strips might be as shown in Table 3.

Table 3. Example probability values

__________________________________________________

strip 3’ 2’ 1’ 1 2 3

__________________________________________________

normal

distribution

0.021 0.136 0.342 0.342 0.136 0.021

empirical

distribution 0.05 0.15 0.30 0.35 0.10 0.05

__________________________________________________

Figure 5 also shows magnified fragment of the

area with two points situated within it. Points are

marked with a and b. For both points hypothesis that

they represent fixed position will be calculated.

Support that point a can be considered as a fix is jus-

tified by the following probabilities related to (note

that point a is entirely situated within crossing

strips):

− membership within strip 1 regarding isoline I

− membership within strip 2’ regarding isoline II

− location within strip 1’ regarding isoline III

Position of point b is partial within strips related

to isolines II and III. Its memberships are estimated

as follows: II/2’→0.3, II/1’→0.7, III/1’→0.9,

III/2’→0.1. Thus support that point b can be consid-

ered as a fix is justified by the following:

− full membership within strip 1 with reference to

isoline I

− partial location within strip 2’ regarding isoline II

− partial location within strip 1’ regarding isoline II

− partial membership within strip 1’ with reference

to isoline III

− partial location within strip 2’ with reference to

isoline III

Evaluation of each of the measurements should

also be included in calculation. Navigator knows

which observation is good or bad, which are prefer-

able to the others. Usually the opinion is subjective

and can be expressed as linguistic term or a crisp

value.

Table 4. Example probability values

__________________________________________________

mass of sets memberships

strip evidence ref. I ref. II. ref. III

__________________________________________________

3’ m

3’

= 0.05

i3’

0 0 0 0 0 0

2’ m

2’

= 0.15

i2’

0 0 1 0.3 0 0.9

1’ m

1’

= 0.35

i1’

0 0 0 0.7 1 0.1

1 m

= 0.30

1 0.5 0 0 0 0

2 m

= 0.10

0 0.5 0 0 0 0

3 m

= 0.05

0 0 0 0 0 0

__________________________________________________

uncertainty 0.3 0.2 0.1

__________________________________________________

ref. stands for reference to:

index i indicates isolines (I, II or III)

Table 4 contains preliminary results of the exam-

ple analysis. The table contains fuzzy points loca-

tions within selected strips, locations are given with

reference to each of the isolines. Example empirical

probabilities are included in column 2. Last row pre-

sents uncertainty, weights of doubtfulness, which is

a complement of credibility, attributed to each

measurement.

Belief structure is a mapping or an assignment of

masses to normal location sets. Location vectors are

to be normal it means that their highest grade must

be one. Subnormal sets should be converted to their

normal state using normalization procedure. Vectors

are supplemented with all one set, which expresses

uncertainty. It says that each location is equally pos-

sible. Mass attributed to this vector shows lack of

confidence to a particular measurement. Thanks to

this value all observations can be subjectively differ-

entiated. All location vectors have assigned mass of

confidence. Appropriate values are calculated as a

product of empirical probability assigned to particu-

lar strip and complement of uncertainty related to

given measurement. It should be noted that the sum

of all masses within a single belief structure is to be

equal to one. Table 5 presents three normalized be-

lief structures constructed based on data from Ta-

ble 4.

Belief structures are subject of combination in or-

der to obtain knowledge base enabling reasoning on

the position of the ship. It is known that combination

of belief structures increase their initial informative

context. By taking several distances and/or bearings

a navigator is supposed to be confident on true loca-

tion of the ship.

Plausibility and belief of the proposition repre-

sented by a fuzzy vector included in collection of re-

sult sets are calculated. In position fixing plausibility

is of primary importance, for discussion on this topic

see Filipowicz 2009a, 2010c. To calculate final

plausibility and belief one has to use formulas pre-

sented by Denoeux 2000, the expressions were fur-

ther simplified by the author Filipowicz 2010c. In

presented example plausibility values that given

points can be selected as a fixed position are: pl

0.62, pl

= 0.60. Obviously a dense mesh of points is

to be considered in practical implementations.

Table 5. Final normalized belief structures

__________________________________________________

b.s. I b.s.II b.s. III

__________________________________________________

{1 0.5} 0.21 {1 0.3} 0.12 {0 1} 0.08

{0 1} 0.07 {0 1} 0.28 {1 0.1} 0.31

{1 1} 0.72 {1 1} 0.60 {1 1} 0.61

__________________________________________________

b.s. stands for belief structure

5 NOTES ON NORMALIZATION OF PSEUDO

BELIEF STRUCTURES

Two strips that do not embrace the common points

are disjunctive and their intersection is empty. Re-

sult of combination of the disjunctive vectors is a

null set. Therefore product of masses attributed to

both combined disjunctive vectors is assigned to

empty set what means occurrence of inconsistency.

Inconsistency results in a pseudo belief structure that

must be converted to its normal state. Two normali-

zation procedures are used: one was proposed by

Dempster another one by Yager. At first both of

them considered crisp vectors. Further extensions for

230

fuzzy environment were suggested by Yager 1995.

Although it is quite often that many authors refer to

them using original methods inventor names. Nor-

malization procedures are quite different in two as-

pects, namely in allocation of inconsistency masses

and modification of fuzzy sets contents called

grades. Masses of inconsistency in Dempster ap-

proach increase weights attributed to not null sets. In

Yager proposal the masses increase uncertainty. In

case of subnormal sets Dempster suggested division

by highest grade. It preserves allocation of points

within selected strips. Yager proposed adding com-

plement of the largest grade to all elements of the

set. It corrupts allocation of points within selected

strips. Therefore results of subnormal belief struc-

tures conversion to their normal state using the two

methods are different, see Table 6 for case study.

Fuzzy sets are location vectors containing fuzzy

memberships of a search space points within select-

ed strips. Thus Dempster transformation causes that

points with not null locations increase their member-

ships, empty grades are not changed. In Yager nor-

malization all considered points gain some degrees

of membership. Unfortunately it may adversely af-

fect computational process and ability of evaluation

of the obtained fix. Therefore modified normaliza-

tion method is proposed. In the approach incon-

sistency masses increase uncertainty very much like

in Yager method. Conversion of subnormal sets re-

mains in line with Dempster proposal. In order to

obtain proper grades all of them are divided by the

highest one. Modified method preserves location of

search space points. The method also enables identi-

fication of all inconsistency cases as depicted by Fil-

ipowicz 2010b.

Table 6. Two example fuzzy sets, their normalizations and

combinations

__________________________________________________

Location vectors m(..)

__________________________________________________

{0 0.8 0 0 0 0 0 0.6 0} 0.41

{0.2 1 0.2 0.2 0.2 0.2 0.2 0.8 0.2} 0.41

{0 1 0 0 0 0 0 0.75 0} 0.48

)

{0 1 0 0 0 0 0 0.75 0} 0.33

{0 0 0 0.67 0 1 0 0 0} 0.20

∧µ

{0 0 0 0.2 0 0.2 0 0 0} 0.08

∧µ

{0 0 0 0 0 0 0 0 0} 0.10

∧µ

{0 0 0 0 0 0 0 0 0} 0.07

__________________________________________________

) - according to Dempster proposal masses of non empty sets

are modified during normalization

- fuzzy set

normalized with Yager method

- fuzzy set

normalized with Dempster method

- fuzzy set

normalized with modified method

∧µ

- result of combination of fuzzy sets

and

∧µ

- result of combination of fuzzy sets

and

∧µ

- result of combination of fuzzy sets

and

Table 7. Dempster versus Yager versus modified approaches

________________________________________________________________________________________________________

Dempster normalization

(Yager smooth normalization)

) Yager normalization modified normalization

________________________________________________________________________________________________________

way of modification of increased by a factor calculated reduced by complement of

masses assigned to not null sets using inconsistency values remain unchanged the highest grade

________________________________________________________________________________________________________

solely depend on initial uncertainty is increased by increased by reduction of

result uncertainty uncertainties total mass of inconsistency not null sets masses

________________________________________________________________________________________________________

modification of membership general image of location vectors null grades of location vectors general image of location

grades is preserved, null grades remain gain some membership vectors is preserved

unchanged

________________________________________________________________________________________________________

ability to detect all

inconsistency cases possible impossible possible

________________________________________________________________________________________________________

recommendation belief structures with fuzzy belief structures with binary belief structures with fuzzy

location vectors location vectors location vectors

________________________________________________________________________________________________________

not recommended for belief structures with binary belief structures with fuzzy belief structures with binary

vectors and high inconsistency vectors and high vectors and high

inconsistency inconsistency

________________________________________________________________________________________________________

computational complexity rather high rather low rather low

________________________________________________________________________________________________________

final solution affected by high might adversely affect final

inconsistency not observed solution not observed

________________________________________________________________________________________________________

) original method name suggested by Yager 1995

231

Table 6 embraces example of two fuzzy sets that

are excerpted from belief structures. First of the sets

is subnormal and needs to be converted. Their nor-

mal states obtained by three different methods are

also presented. Results of combinations of the con-

verted sets with the second one are included in last

three rows of the table.

Combination is carried out using minimum opera-

tor and product of masses involved. Formula 7 de-

livers proper expressions.

))(())(())((

))(),(min()(

iii

xmxmxm

xxx

µµµ

µµ

⋅=

∧

(7)

Masses of credibility assigned to all vectors and

to results of their combinations are shown in the last

column of Table 6.

Table 7 contains comparison of Dempster, Yager

and modified normalizations taking into account

practical aspects presented in first column. It should

be noted that position fixing engages fuzzy location

vectors therefore modified normalization should be

recommended. Most important feature of the Demp-

ster and modified methods is ability to preserve gen-

eral shape of location vectors, null grades remain

unchanged. Consequently all inconsistency cases

can be detected.

6 SUMMARY AND 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 enrichment of their informative con-

text. The Theory extension to a fuzzy platform pro-

posed by Yen 1990 enables wider and more complex

applications.

Based on the Theory concept new method of po-

sition fixing in terrestrial navigation is proposed.

The method enables reasoning on position fixing

based on measured distances and/or bearings. It was

assumed that measured values are random ones with

theoretical or empirical distribution. Knowledge on

used aids and observed objects is included into com-

bination scheme. Relation between measurement er-

ror and deflection of the isoline was also depicted. It

was suggested that instead of bearings concept of

horizontal angles should be used, obtained isoline is

constant error free.

The true isoline of distance, bearing or horizontal

angle is somewhere in the vicinity of the isoline

linked to a measurement. To define true observation

location probabilities six ranges were introduced.

Probability levels assigned to each strip can be cal-

culated based on features of normal distribution or

they can be delivered from experiments. Standard

deviation of the distribution is assumed to be within

known range. Empirical data also varies within some

range. In both cases imprecise interval valued limits

of ranges are to be adopted. Sigmoid membership

functions are used for establishing points of interest

levels of locations within established ranges. Calcu-

lated locations are elements of fuzzy sets called lo-

cation vectors. Vectors supplemented with the one

expressing uncertainty compose one part of belief

structure. Another part embraces masses of initial

believes assigned to location vectors and uncertain-

ty. Complete belief structure is related to each of

measurements. Mass assigned to uncertainty ex-

presses subjective assessment of measuring condi-

tions. One has to take into account: radar echo signa-

ture, height of objects, visibility and so on to include

measurement evaluation. Fuzzy values such as poor,

medium or good can be used instead of crisp figures.

Imprecise masses values engage different way of

calculation and will be discussed in a future paper.

Belief structures are combined. During associa-

tion process search space points within common in-

tersection region are selected. Result of association

is to be explored for reasoning on the fix. All associ-

ated items are to be taken into account in order to se-

lect final solution.

Mathematical Theory of Evidence requires that

mass of evidence assigned to null set is to be zero

and fuzzy sets are to be normal. Assignment for

which above requirements are not observed is pseu-

do belief structure and is to be normalized. Pseudo

belief structures can occur at the structures prepara-

tion stage as well as during association process.

Usually null sets are results of combination of two

ranges or areas without common search space

points. The occurrences indicate abnormality in

computation that might result from extraordinary er-

roneous measurements and/or wrongly adjusted

search space. Therefore all null assignment cases are

to be recorded and analyzed. Two normalization

procedures proposed by Dempster and Yager are

widely used. Converting procedures are quite differ-

ent in two aspects. Masses of inconsistency in

Dempster approach increase weights attributed to

not null sets. In Yager proposal the masses increase

uncertainty. In case of subnormal sets Dempster

suggested division by highest grade, Yager proposed

adding complement of the largest grade to all ele-

ments of the set. The latter causes that none of these

approaches should be perceived as superior in case

of position fixing. Therefore modified scheme was

proposed. It takes best things from both proposals.

Way of conversion of subnormal sets is taken from

232

Dempster method and managing of inconsistency

comes from Yager approach.

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Filipowicz, W. 2009b. Mathematical Theory of Evidence and

its Application in Navigation. In Adam Grzech (eds.)

Knowledge Engineering and Expert Systems: 599-614.

Warszawa: Exit.

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