887

1 INTRODUCTION

In recent papers delivered by the author, fuzzy

systems and Mathematical Theory of Evidence were

used as a platform for processing uncertainty [4, 5, 6].

Models require methods to obtain objective

evaluation of uncertainty. In nautical science, sets of

random variables instances are exploited as main

source of knowledge on observations. Usually they

were perceived as governed by Gaussian dispersion

patterns. Knowing the magnitude of standard

deviations enables introduction of an observation

rough assessment. This attitude is popular among

navigators. Modern computer procedures accept

uncertainty as an element of a processing scheme.

Identification of doubtfulness was discussed in the

author’s publications [7, 8]. Proposed approach

exploited evidence proximity exploration and

engaged principles of fuzzy systems. Suggestion was

that an application should include analyses of

available raw recorded instances in order to extract

required range of useful parameters. Introducing

uncertainty model enriches the approach. This paper

contains presentation of dealing with doubtfulness

using its popular model. Association of uncertainty

items are included and result with its increased

informative context discussed. Further, histograms

conversion is recalled and their embedded belief and

uncertainty extracted. Concluded part concentrates on

numerical example being an output of the application

implementing presented approach. For those who

want to get deeper insight into the terminology and

the engaging scheme of reasoning the author

recommends recent excellent book [1].

Position Fixing and Uncertainty

W. Filipowicz

Gdynia Maritime University, Gdynia, P

oland

ABSTRACT: Taken random observations are usually accompanied by rectified knowledge regarding their

behaviour. In modern computer applications, raw data sets are usually exploited at learning phase. At this

stage, available data are explored in order to extract necessary parameters required within the inference scheme

computations. Crude data processing enables conditional dependencies extraction. It starts with upgrading

histograms and their uncertainty estimation. Exploiting principles of fuzzy systems one can obtain modified

step-wise structure in the form of locally injective density functions. They can be perceived as conditional

dependency diagrams with identified uncertainty that enables constructing basic probability assignments.

Belief, uncertainty and plausibility measures are extracted from initial raw data sets. The paper undertakes

problem of belief structures upgraded from uncertainty model in order to solve the position fixing problem. The

author intention is presenting position fixing as an inference scheme. The scheme engages evidence, hypothesis

and revokes concept of conditional relationships.

http://www.transnav.eu

the

International Journal

on M

arine Navigation

and Safety of S

ea Transportation

Volume 17

Number 4

December 2023

DOI: 10.12716/1001.17.04.

888

2 MODELLING UNCERTAINTY

Let us consider a problem of evaluation of a truth of a

statement. It is popular that to some extent the

proposition is considered true. There is also interval

where it is treated as uncertain. Finally, range of false

truth of the statement might exist. Figure 1 presents

probability vs possibility diagram as uncertainty

representation. The polyline is a membership function

that specifies fuzzy probability set of the proposition

truth. One can use interval [a, b] to define the function

[10]. The abbreviation indicates three subsets: [0, a);

[a, b); [b, 1] that feature the diagram. It should be

noticed that a and b are equivalent to belief and

plausibility measures adopted in Mathematical

Theory of Evidence (MTE) [11]. One can upgrade

models for popular statements such as “I am

convinced that something is true, at the same time

some doubtfulness exists and lack of acceptance

might also be present”. In formal way, the proposition

can be written using Equation (1). Formula (1a) could

be followed for mentioned uncertainty model [2].

( ) ( )

( )

{ } { }

( )

{ }

m , , , , , , , e T mT FmF TF m TF=

(1)

( ) ( ) ( ) { }

( )

{ }

m , , , 1 , , , e T a F b TF b a=−−

(1a)

Figure 1. Diagram of uncertainty representation

Fragments of evidence and hypotheses exist in

inference schemes. To some extent each piece of

available data supports given hypothesis, uncertainty

of the backing is unavoidable. Quite often lack of

support also occurs. In nautical science, there are

observations and sets of points considered as potential

true locations of a vessel. Each measurement supports

the true location to certain degree. Mentioned

relations can be meant as conditional dependencies.

Conditional relationships can be considered as a

function that identifies belief and plausibility of

support measures for hypothesis items embedded

within each of the evidence fragments. Observations

neighbourhood explorations were subject of the recent

publication by the author [8].

3 COMBINING UNCERTAINTIES

Many persons or methods might evaluate the same

statement. It is usual that extent the proposition is

considered true varies. The same refers to uncertainty

and false truth of the statement. Problem of combined

assessment of a truth of the statement assessed by

different experts or delivered from various sources

appears practical. Figure 2 presents diagrams of

uncertainty representations and result of their

combination. Intervals [ai, bi] were used to define the

respective membership functions. Note that

assignment feature lower uncertainty contributes

more decisively to the result of combination. It is in

line with popular meaning of weighted contribution

from various quality inputs. The idea is native for

MTE’s scheme of combination.

Figure 2. Two diagrams of uncertainty representations and

result of their combination

In formal way, the two propositions can be written

using Equation (2). Formula (2a) could be followed for

mentioned uncertainty models.

( ) ( )

( )

{ } { }

( )

{ }

( ) ( )

( )

{ } { }

( )

{ }

1 1 1 1 1 11 11

2 2 2 2 2 2 2 22

m , , , , , , ,

e T mT F mF T F m T F

(2)

( ) (

) ( )

{ }

(

)

{ }

( )

(

) {

}

(

)

{ }

1 11 1 1 1 1 1 1

2 2 2 2 2 22 2 2

m , , , 1 , , ,

e Ta F b T F b a

e T a F b TF b a

=−−

(2a)

Combination scheme [3, 10] adopted for

association of two structures being belief distributions

and illustrated at Figure 2 is presented in Table 1. First

structure is shown in shaded part of the first raw,

second one presents first column. Each cell contains

two elements: involved set and a mass attributed to

the set. Note that for an assignment total mass is equal

to 1. Single element sets identify range of true (T) and

false statement (F). Sets consisting of two items

represent uncertainty. Association partial results are

included in other cells of the Table. Each result

contains two elements: combined involved sets and

product of their masses. Intersection of engaged sets

are required while conjunctive structures association

is carried out. In considered case two sets common

par

t is empty when {T} and {F} are being combined.

Instead of empty set ∅ uncertainty equivalent

molecule {T, F} was used in respective cells. In this

way non-null generating associations are obtained. It

should be noted that the idea follows the Hau-

Kashyap and Yager concepts [9, 12] of normalization.

The one is recommended for discussed scope of

applications.

889

Table 1. Combination scheme

________________________________________________

set {T1} {F1} {T1, F1} Result

mass 0.100 0.150 0.750

________________________________________________

{T2} {T} {T, F} {T} {Tr}

0.45 0.045 0.068 0.338 0.403

2} {T, F} {F} {F} {Fr}

0.35 0.035 0.053 0.263 0.345

2, F2} {T} {F} {T, F} {Tr, Fr}

0.20 0.020 0.030 0.150 0.253

________________________________________________

Ti – range of i-th true statement equals to ai

i – range of i-th false statement equals to 1-bi

Ti, Fi – range of i-th uncertainty equals to bi - ai

One expert claims that he beliefs that given

statement is true with probability at most 0.10. The

upper limit of accepting the proposition as true is 0.85.

Thus, the range of uncertainty is 0.75 and simple

model takes the form [0.10, 0.85]. Other expert beliefs

that the statement is true with probability up to 0.45.

The upper limit of accepting the proposition as true is

0.65. This time the range of uncertainty is 0.20 and

simple model takes the form [0.45, 0.65]. Obtaining

overall opinion on the truth of the statement is the

challenge. Adequate solution delivers combination of

available expertise, obtained model is [0.403, 0.656]

(see result diagram at Figure 2).

In nautical practice, there are randomly distorted

indications. Referring to one of the observations given

position represents the true observer location with

probability at most a

1. Note that probability is meant

as product of density and width of adjacent area.

Hereto unitary range is assumed. The upper limit of

accepting the representation as true is b

1. Thus, the

range of uncertainty is b

1- a1 and simple model takes

the form [a

1, b1]. Other observation indicates that the

statement is true with probability up to a

2. The upper

limit of accepting the proposition as true is b

2. This

time the range of uncertainty is b

2 - a2 and simple

model takes the form [a

2, b2]. Association of available

indications evaluates the truth regarding given

location. Combination result diagram is very much

like the one shown at Figure 2. Note that more

assertive individual, this with lower uncertainty,

dominates the final solution. Challenging are

proposals of methods estimating uncertainty models

elements [a

i, bi].

4 DISCOVERING DEPENDENCIES AND THEIR

UNCERTAINTIES

Figure 3 illustrates the idea of proposed processing

scheme aiming at dependencies extracting. Part a)

displays a piece of evidence (labelled o2) with a set of

instances showing its dispersion. The four hypothesis

points labelled with Hi is also presented. The

statement that a given location Hi is the true position

gain some endorsement from this piece of available

evidence. From the Figure, one can perceive that H1 is

the point with the highest support in this matter

provided lack of a systematic deflection. For the case,

conditional dependence P(H│O) is to be considered

as a function that identify measure of support for

hypothesis item Hi embedded within an evidence

fragment o

j. Thanks to the relationships appropriate

supports can be obtained and evaluated. Respective

metric is calculated and analysed. To obtain the

support, x- and y-axis histograms are upgraded based

on crude data, instances related to the indication.

Then they are converted to gain stipulated continuous

shape.

Part b) of Figure 1 shows the result of processing

the initial instance set. At first raw data are converted

to step-wise histograms. For these structures,

evaluation of trustfulness of an observation at hand is

to take place. Uncertainty of 0.37 is estimated for

presented case. The value is obtained based on

vertical and horizontal expansion of each histogram.

Respective partial metricises are also presented.

Figure 3. (a) Single observation with an example of its two-

dimension dispersion set and four hypothesis locations; (b)

horizontal and vertical axis histograms; (c) histograms and

their converted to continuous function versions.

Based on obtained results, further on rectangular

cells structures are transformed to injective diagrams

of density functions shown at part c) of the Figure.

Taking advantage of fuzzy sets and Bayesian

conditional dependencies, an adequate converting

method was implemented. Locally injective function

is required in order to obtain solutions for many

problems engaging random variables [7]. Figure 3

presents a basic scheme followed at the first stage of

imprecise data handling. At the stage uncertainty

embedded in initial data set are to be discovered and

injective density functions identified. The function

diagram shows support plausibility of density a

position being located approximately particular point.

890

5 HISTOGRAMS EVALUATION AND

CONVERSION

Randomly distorted evidence can be accompanied by

sets of recorded instances, which are traditionally

converted to a histogram. It displays a diagram of the

distribution of observed data. A histogram consists of

adjacent rectangles, primarily erected over non-

overlapping intervals. The histogram is usually

normalized and displays relative frequencies

considered as empirical probability densities. It shows

the proportion of cases that fall into each of bins. The

intervals are usually chosen to be of the same width,

percentage of the value limits the range very often. It

is assumed that family of instances sets of relative

frequencies are given as a result of a long term

observations. Histograms quality should be evaluated

in order to get knowledge and variables processed in

many applications. Although histograms are popular

and widely used, attempt of their quality assessment

was proposed by the author [8].

Histograms should be subject to evaluation,

intuitively and objectively. MTE’s scheme of

combination stipulates knowledge of uncertainty

embedded within engaged structures. Histograms

quality differs. Differentiations refer to their bin

heights and ranges as well as to the whole structure

expansion. Uncertainty refers to a certain feature

within the discussed scope. The number of items with

the same or almost the same value of the feature

defines uncertainty. In this view, uncertainty might be

related to distinguishability. One should note that

points within single cell are not distinguishable.

Following this way of reasoning one can conclude

that the wider is histogram the higher is its

uncertainty. For this reason, uncertainty of

rectangular cells histogram is higher compare to

continuous version of density distribution (see Figure

3).

Si i-th histogram cell, crisp valued limited area showing

number of observations falling within the range.

i is membership function for i-th cell.

g1 is a diagram of converted histogram.

Figure 4. Three bins histogram with membership functions

for each cell for uncertainty level 0.05

The idea of bin-to-bin additive method is crucial

for the histograms transformations concept [7].

Modern approach enables treating rectangular cells as

fuzzy density sets. Limitations of such sets are

established by membership functions which diagrams

are uncertainty dependent. Figure 4 presents

diagrams of membership functions for three cells

histogram with rather low uncertainty level (assumed

0.05). Table 2 gathers data referring to items presented

in the Figure. The Table contains membership grades

for each of the marked points within consecutive cell.

The assigned cell densities are included in the last

raw. Result measures, highest densities approximately

i-th point are included in the second last column.

Included values are plausibility measures since way

of their calculations refers to fuzzy systems well-

known formula [3]. Note that density contributed by a

cell is a product of the cell density and grade of

belonging to the bin. One should perceive the value as

bi, the title used in Figure 2. Thus to get belief (a

i) one

should subtract uncertainty from plausibility value.

The values are included in the last column.

Table 2. Grades of belonging to the histogram cells and

result of bin to bin product summation for example set of

points

________________________________________________

point

i(S1)

i(S2)

i(S3) pl(xi) bel(xi)

________________________________________________

x1 0.830 0.000 0.000 0.291 0.241

2 0.910 0.100 0.000 0.371 0.321

3 0.000 0.002 0.990 0.130 0.080

________________________________________________

m(Si) 0.350 0.520 0.130

________________________________________________

i(Sk) grade of i-th point belonging to k-th bin

pl(x

i) plausible, highest probable density measure in the

vicinity of i-th point

bel(x

i) belief, highest certain density measure in the vicinity

of i-th point

Si i-th histogram cell, crisp valued limited area showing

number of observations falling within the range.

i is membership function for i-th cell.

g2 is converted histogram.

Figure 5. Three bins histogram with membership functions

for consecutive cells for uncertainty level 0.44

Similar to Figure 4 is the next Figure 5, which

presents diagrams of membership functions for three

cells histogram with much higher uncertainty level

assumed equal to 0.44. Due to higher uncertainty,

ranges of membership functions expand compare to

those presented at Figure 4. For discussion on relation

between scope of range and doubtfulness, refer to [8].

Table 3 contains data referring to items presented in

the Figure. The Table contains membership grades for

each of marked points within consecutive cell. As

before the assigned cell densities are included in the

last raw. Result plausibility measures also referred to

as b

i, highest possible densities approximately i-th

point are included in the second last column. Result

belief measures also referred to as a

i, lowest densities

approximately i-th point are included in the last

column. Grades of belonging to adjacent cells are

much higher compare to those included in table 2.

Figure 4 and 5 present three cells histogram and its

two converted versions obtained for various

uncertainty levels (see curves g

1 and g2 at respective

891

Figure). Both were obtained using presented scheme

of calculations regarding three example points.

Obviously, number of locations involved was much

greater.

Table 3. Example set of points, their belonging to the

histogram cells and result plausibility

________________________________________________

point

i(S1)

i(S2)

i(S3) pl(xi) bel(xi)

________________________________________________

x1 0.960 0.410 0.005 0.550 0.110

x2 0.980 0.910 0.120 0.832 0.392

3 0.040 0.590 0.980 0.448 0.008

m(S

i) 0.350 0.520 0.130

________________________________________________

i(Sk) grade of i-th point belonging to k-th bin

pl(x

i) plausible highest, probable density measure in the

vicinity of i-th point

bel(x

i) belief highest, certain density measure in the vicinity

of i-th point

6 EVALUATING UNCERTAINTY

Random data evaluation aims at the discovery of

certain patterns included within available sets of their

instances. MTE combination scheme stipulates

probability assignments, which include uncertainty.

Differentiations refer to the histogram bin heights and

ranges as well as to the structure expansion.

Uncertainty refers to a certain feature within the

discussed domain. In discussed field, uncertainty is

related to discernibility. The more discernible abscissa

points the less amount of uncertainty contains

histogram. Thus, the diversity of bin heights can be

proposed as a factor to measure distinguishability.

The average of cases falling below and above the

mean line of a histogram can be perceived as an

objective measure of a bin heights diversity. Inserted

numbers in Figure 2 are respective metrics for each

the presented cases. The greater the numbers and

smaller the horizontal extension the “better” is a

histogram. Total of points that fall above and below

presented horizontal line may be zero when uniform

distribution of the feature is involved. Note that this is

in line with the popular understanding of uncertainty.

One can perceive doubtfulness as thinking of

something that might be somewhere within a given

scope but there is no hint as to where in particular.

One can perceive histogram with the same cells

heights as extremely unreliable. It should be noted

that in such case total of instances below and above

the central line is zero. As mentioned above, the

number of items with the same value of the feature

defines uncertainty. Thus, considering two histograms

with the same numbers of bins and different bin

widths, one can assume that wider structure embeds

greater uncertainty.

Figure 6. Results of case study engaging four observations,

selected histograms converted versions and four hypotheses

points. (a) four observations with their recorded dispersion

sets, hypotheses points are also included (b) dispersion set

for observation number 4, its histogram and converted

diagram (c) converted histograms for observations 1 and 4

with exploded insertion containing belief and uncertainty

measures for example mesh of points.

Results of a case study engaging four observations

with their instances dispersions, selected sets

converted versions; four hypotheses points

histograms and result of exploration of included space

of discernment are presented at Figure 6. Exploded

insertion contains belief measures and uncertainties

for selected mesh of points.

Set of hypothesis points from figure 6, their

supporting by depicted observations and measures

that they are the true location of the ship are gathered

in the Table 4. Available indications feature included

relative uncertainty. Given the input data one should

consider point x

3 as the best approximation of the ship

location. The point feature the highest belief and

reasonable uncertainty.

6.1 Normalizing a pool of data

Position fixing problem exploits a pool of various

quality observations. They need to be evaluated prior

to the final usage. Preparing a pool of data requires

their normalization in order to achieve uniform

density distributions [8]. This enable construction of

adequate conditional dependency functions. Also

expected is relative uncertainties vector to enable

definitions of basic probability assignments. Fixing

aims at the selection of a point that represents the true

location of a ship in best way. An evidence fragment

supports the choice of each item from the considered

set of hypothesis. Degrees of support are expressed as

conditional dependability that rely on credibility

attributed to the evidence items. Those with low

uncertainty should contribute to the selection more

892

decisively. To implement the idea, one has to create a

hierarchy for available evidence.

It is quite often that relative rather than absolute

uncertainty is of primary importance. Position fixing

is an example problem where relativity of

uncertainties really matters. The idea is establishing

grades affecting the final solution by each piece of

evidence compared to other ones. The relative weight

of contributing to the final solution is an important

issue. Given a set of histograms, their ranges and bin

heights for each structure, adequate measures can be

calculated [8].

Table 4. Example set of points, their supporting by

observations at hand and measures that they are the true

location of the ship

________________________________________________

plausibility of support from data on

observation: representing a fixed

position

________________________________________________

point o1/0.15 o2/0.29 o3/0.28 o4/0.26 bel(xi) pl(xi) un(xi)

________________________________________________

x1 0.613 0.537 0.463 0.416 0,145 0.585 0,44

2 0.474 0.514 0.540 0.514 0,192 0.582 0,39

x3 0.363 0.473 0.594 0.593 0,214 0.604 0,39

4 0.000 0.364 0.659 0.627 0,157 0.557 0,40

________________________________________________

oi/X i-th observation with its relative uncertainty

bel(x

i) belief measure that xi is the true location of the ship

(refers to a

i in the uncertainty model)

pl(xi) plausibility measure that xi is the true location of the

ship (b

i in the uncertainty model)

un(x

i) uncertainty that xi represents true location of the ship

(bi - ai) in the uncertainty model)

Vertical and horizontal layout uncertainty of a

histogram are considered. Measures referred to

variety of bin heights and to their widths enabled

estimation of an overall uncertainty. Doubtfulness

amount and shape of the membership functions are

mutually dependent. Both are main factor that decide

on shape of converted histograms, which can be seen

as conditional dependencies adequate diagrams. Pool

of data required for solving the problem engaging

distorted data, needs additional normalization.

Processing introduces comparable probability density

distributions. For this purpose, expected is ranking

regarding decisiveness on affecting the solution. Items

with lower uncertainty are of greatest influence in this

respect. Final ranking list regarding amounts of

embedded uncertainty for the poll of four

observations is {o1/0.15, o4/0.26, o3/0.28, o2/0.29}.

Indication o1 mostly decide on the final selection.

Figure 6 part c) includes example application

screenshot with belief and uncertainty calculated for a

rectangular frame of discernment. The distinguished

fragment contains items with the highest beliefs along

with rather low uncertainties.

7 CONCLUSIONS

In nautical practice, there are randomly distorted

indications or observations. Referring to one of the

available items one can discover support that given

position represents the true observer location. Range

the proposition is considered true varies from

observation to observation. The same refers to

uncertainty and false truth of the statement. Items

such as belief, plausibility and uncertainty are

included in uncertainty model that is presented at the

beginning of the paper. Combining models,

assessments of a truth of the statement extracted from

various observations delivers fixed position. At the

beginning, the paper contains combination of two

structures being belief functions. It should be noted

that result diagram is very much like less uncertain

compound. More accurate observations dominate

others while position fixing or other problems

involving randomly distorted data. Individuals that

are more assertive dominate final opinion.

Exploiting presented model one requires methods

of extracting data from sets of recorded instances of

given random variable. Proposal of exploration of the

raw data deliver good estimates of uncertainty

models. Partial results of processing are items of the

popular uncertainty model architecture. For example,

locally injective transformed histogram shows

plausibility measures. It can upgraded with reference

to given uncertainty. Overall evaluation of reasoning

on the true location can be delivered by combination

of the assignments created based on available

indications. Calculated belief and uncertainty

measures are helpful when the fixed position is

selected. Solution is an item feature highest belief, in

case of ambiguity one has to choose smallest

uncertainty.

Uncertainty regarding vertical and horizontal

layout of a histogram are considered. Measures

referred to variety of bin heights and to their widths

enabled estimation of an overall and relative

uncertainty ranked among the available observations.

Shape of the cell’s membership functions depends on

doubtfulness feature by given piece of evidence.

Converted histograms can be seen as conditional

dependencies diagrams [7]. Pool of data used for

position fixing needs to be normalized. Additional

processing introduces uniform, relatively balanced

density distributions. Initial reference vector is

required in order to obtain the hierarchy and

subsequently adequate probability assignments.

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