545

1 NAUTICALEVIDENCE

Navigational evidence embrace results of

observations as well as knowledge within discipline

known as nautical science [3]. Observations mainly

mean taking distances and or bearings. Horizontal

angles are also taken from time to time. Results of

observationsareimprecise.Itiswidelyassumedthat

any measurement contains systematic deflection

along with random error. It is also assumed tha

t

randomerrorsaregovernedbyGaussiandistribution

ortakeformofhistogramthatisempiricaldiagramof

varioustestsoutputs.Figure1showstwoschemesof

taking distance. Both presented cases marked as a)

andb)differwithsystematicdefections.Atpresented

example random deflections feat

ure the same

theoreticalorempiricalcharacteristics.

Figure1.Resultoftakingdistanceisanimprecisevaluethat

israndomlyandsystematicallydistorted

On Nautical Observation Errors E

aluation

W.Filipowicz

GdyniaMaritimeUniversity,Poland

ABSTRACT:MathematicalTheoryofEvidence(MTE)enablesupgradingmodelsandsolvingcrucialproblems

in many disciplines. MTE delivers new unique opportunity once one engages possibilistic concept. Since

fuzzinessiswidelyperceivedassomethingthatenablesencodingknowledgethusmodelsbuilduponfuzzy

platformsacceptsonesskillwithingivenfield.Atthesameti

meevidencecombiningschemeisamechanism

enabling enrichment initial data informative context. Therefore it can be exploited in many cases where

uncertaintyandlackofprecisionprevail.Innauticalapplications,forexample,itcanbeusedinordertohandle

data feature systematic and random deflections. Theoretical background was discussed and computer

applicat

ionwassuccessfullyimplementedinordertocopewitherroneousanduncertaindata.Outputofthe

applicationresultedinmakinga  fixandaposteriorievaluatingitsquality.Itwasalsoproventhatitcanbe

usefulforcalibratingmeasurementappliances.Uniquefeatureofthecombinat

ionschemeprovenbytheauthor

inhispreviouspaper,enablesidentifyingmeasurementsystematicdeflection.Basedonthetheoremthepaper

aimsatfurtherexplorationofpracticalaspectsoftheproblem.Itconcentratesonreductionofhypothesisframe

reductionandrandomalongwithsystematicerrorsidentifications.

http://www.transnav.eu

the International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 9

Number 4

December 2015

DOI:10.12716/1001.09.04.11

546

Figure2presentssituationinwhichtwodistances

for two landmarks are taken. Distances were

establishedforobjectslocatedatoppositesidesfrom

the observer position [9]. The scheme is usually

followedinordertoidentifysystematicdeflectionof

the measuring appliance, which usually is a radar.

Two circles being distance isolines projected on the

chart should be tangent at collinear gradient

directions unless they are distorted. Isolines are

separated by certain distance once measurement

deflections occurred. Breadth of resulted gap enable

reasoningonkindofinvolvederrors.Incasewhenit

is smaller tha

n sum of both random distributions

standarddeviations:



2occurrenceofasystematic

errorisveryunlikely.Figure2presentssituationfor

which the gap is bigger than sum of tripled

deviations: 3



1+3



2. It means that fixed deflection

occurred and is to be identified. Systematic error is

assumed as being the same for both observations

provided taken with the same appliance. Therefore

distance correction is estimated as half of the

observed gap what is valid with assumption that

randomerrorisnegligible.

Figure2. Graphical interpretation of two imprecise

measurements, distorted with random and systematic

errors,takentoobjectsatoppositedirections 

Seafarers know that mean error, related to

standarddeviationofthebellfunction,ofadistance

measured with radar variable range marker is a

functionoftheobtainedvalueandissaidtobewithin

the interval of [1%; 1.5%] of the measurement.

Taken distance of 10Nm is a random va

riable with

mean error inside the range of [1; 1.5] cables. In

figure 2 distance d

1 is assumed greater than d2 (see

alsoshapesofinserteddistributionfunctions).

Standard deviation is one of the most important

factor in observations accuracy evaluations. In

practiceitsexactevaluationisratherimpossible.Thus

crisp valued mean errors of measurements are

considered inadequate. Instead in recent navigation

books(forexamplesee[12])measurementmeanerror

is described as imprecise int

erval value usually as:

[



–

d; 



d] (herein letter d denotes taken distance).

Beinginterestedinanisolinepossible deflectionone

considers interval [m

–

d; m

d] established along

gradientdirection.Sincegradientmoduleisequal to

one for distance isoline both before mentioned

parameters are of the same meaning. With fuzzy

arithmeticnotation[12]thelatestcanberewrittenasa

quad(‐m

d;‐m

–

d;+m

–

d;+m

d).Itmeansfuzzyvaluewith

coreof[‐m

–

d;+m

–

d]cablesandsupportof[‐m

d;+m

d].

In order to include such data into upgraded model

onehastoengageadequateformalapparatus.

2 IMPRECISENAUTICALEVIDENCEANDITS

ENCODING

In possibilistic approach uncertain evidence is

represented with sets and masses of confidence

attributed to these sets. Sets embrace relations

between hypothesis and evidence spaces. Relations

can be bina

ry or fuzzy ones [1, 13, 14]. Fuzzy sets

embrace gradesexpressingpossibilities of belonging

ofconsecutivehypothesisitemstothesetsrelatedto

each piece of evidence. Therefore appropriate

relationsbetweenconsideredframesareencodedinto

evidence representation, which takes the form

specifiedbyformula(1).

m( ) {( ( ), ( ( ))), ,

( ( ), ( ( )))}

iikiik

in k i in k

exfex

xfe x









 (1)

Whatdoeshypothesis framemean fornavigator?

Whatareashoulditcoverincaseofdiscreteversion

oftheproblemisconsidered?Hypothesisframeisto

beconsideredassearchspacewhereforexampletrue

isoline,dueitserroneousnature,is beinglocated[5,

6].Itshouldalsobeperceivedasacollect

ionofchart

points that represent fixed position of the ship.

Results of combination of evidence related to two

random variables that projected on the plane are

separated by certain distance, were examined in

previous paper by the author [10]. Like at figure 2

there were considered varia bles  referred to isolines

relatedto distances ta

ken fortwo objects situatedat

opposite directions. It was assumed that va riables

could be distorted with systematic error apart from

random one. Identifying permanent measurement

shiftisanimportantnauticalissue.

Figure3showshistogramwithprobabilitiesofthe

true isoline being located in the vicinity of the

observedone.Thehistogramwaspreparedba

sedon

bellfunction,widthofeachbinisequaltoquarterof

normalized standard deviation. Figures within each

binindicateprobabilitythatthetruelinerelatedtothe

measurement is located within particular strip

provided random error is taken int

o account. It

should be noticed that due to discrepancies in

statistical investigations regarding measurements 

distributions parameters presented bin limits are

rather range valued [8, 9]. For normal distribution

widthofthesearchframeshouldbeconfinedto6m,

tosixfoldedstandarddeviationofgivenisoline(see

also figures related to extreme bins). It should be

pointedtha

titisnotalwaysthecase.Figure2shows

search space limited to area located in between

obtained distance isolines. In view of particular

evidence related to distances taken for landmarks

situated at opposite directions only half of the

distributionisvalid.Thetruelocat

ionoftheobserver

istobelocatedwithinshowngap.Onlymeaningless

supportcanbeattributedtocontrarystatement.

In presented application evidence representation

consists of pairs [5]: fuzzy vectors

)(

1 ki





representing locations of a set of each points { x

k}

547

withinsetsrelatedtoeachpieceofevidence–degrees

of confidence assigned to these vectors

))((

kini

xef



 . Degrees of confidence reflect

probabilityoftrueisolinebeing locatedwithingiven

strip area can be obtained from presented drawing

(seefigure3).



Figure3.Histogramshowingprobabilitiesofthetrueisoline

beinglocatedinthevicinityofthemeasuredone

Strip areas are related to confidence intervals

establishedforprobabilitydistributionfunctions and

areassumed tobeadjustedtothe evidenceat hand.

Forbellfunctiontheyarequiteoftenassumedas:half,

single,doubleandtripleofstandarddeviation.Figure

4showsprobabilitiesofthetrueisolinebeinglocated

 the vicinity of the measured one. Bins width are

assumedequaltoasingleunifiedstandarddeflection.

First of each pair of figures placed within bins are

thoseobtainedforunconfinedsearchframe.Thenthe

space was reduced to range [‐0.4; 4.8] of standard

deviation. Second figure in each

pair of numbers

placed within bins are modified due to available

evidencelimitation.



AlgorithmI

1 Sum up allprobabilities attributed to rangesthat

areoutsideofthesearchframe.Includereduction

of probabilities for partial inclusion (see extreme

binsofincludedrectangleatfigure4)

2 All modified probabilities that

 are greater than

zerodividebycomplementofthetotalcalculated

instep1



Algorithm I guarantees that only focal elements

are included into created belief structures [2]. Focal

items are those with non zeroed masses assigned.

Additionally total of all masses assigned to focal

elements,withoutuncertainty,isone.



Figure4.Histogramshowingprobabilitiesofthetrueisoline

being located in the vicinity of the measured one and its

modifiedversion.

3 CASESTUDY

The concept of exploiting  evidence that is meant as

encoded facts and knowledge, in supporting

decisions in navigation is based on measurement

distributions and fuzziness. Introduced confidence

intervals (see figure 3) define probabilities of true

isolines being located within appropriate strips

established along gradient directions. Modified

probabilitiesare

incorporatedintobeliefassignments

that enable the modelling of uncertain, imprecise

data. Imprecision is due to random errors but

systematicdeflectionsoccurquiteoften.Thiskindof

error is to be identified and eliminated. The

identificationofapermanentmeasurementshiftisan

importantpracticalnauticalissue.

In this chapter

considered are observations

engagingtwodistancesmadefortwoobjectssituated

at opposite directions as seen from the observer’s

position. Both observations resulted in isolines that

areassumedtobedistortedwithrandomerrorsand

include systematic deflection. Random errors

distribution means are supposed to be within the

rangeof

1%ofthemeasureddistance.Possiblelimits

oftheestimatedmeanarewithin15%oftheirvalue.

Data used in numerical experiment are gathered in

table1.

Table1.Summaryofdatausedinnumericalexperiment

_______________________________________________

observation1  observation2

_______________________________________________

distances30cables50cables

meanerrors  0.3cables 0.5cables

meanerrorlimits [0.255;0.345]cables[0.425;0.575]cables

subjective90%80%

 confidence

 evaluation

gapwidth0.58cables

 (seefigure2)

 forcasea)

gapwidth3cables

 forcaseb)

_______________________________________________



Baseduponpresentednauticalevidencenavigator

should reason on quality of measurements and

possibly identify systematic deflection. He is

supposed to answer two questions: what is the

systematicerroroftheappliedmeasuringdeviceand

howrandomerrormightaffectedhisevaluation.

548

Figure 5 shows two examples in which pairs of

observationsmadefortwoobjectssituatedatopposite

directions from ship position. Each of the

observations is marked with small circular shape

placedatabscissaaxisthatisassumedcollinearwith

gradient directions. Observation’s random error

distributionaredepictedwithtwobellfunct

ionsthat

represent extreme value of assumed standard

deviation (see also interval valued data in table 1).

Shapes emphasising interval valued limits of mean

errorarealsoincluded.Searchspacewasconfinedby

both isolines, its discrete points represent true

location of the vessel. Question which of them best

represents the true locat

ion is resolved through

reasoning base on results of evidence combination

scheme.

Left hand side of illustrations placed in figure 5

presents situation in which gap between isolines is

due to random errors. Case a) presents two

observations for which systematic deflection should

be rather excluded since gap between isolines is

smallertha

nsumofmeanerrors.Statementisrather

unlikely for right hand side case. The gap can be

estimated as sum of three folded mean errors. Thus

probability that systematic error was involved is

rather high. In order to cover the isolines gap,

consequently to create artificial free of systematic

error case, mea

n errors were increased during

iterativecombinationprocess.Finalstagesituationin

which enlarged observations mean errors cover the

gap as well as association result was presented at

figure6.

Itshouldbe stressed that figures 5and6 remain

closely related. Based on results of combinat

ion

illustrated at figure 6 (notice direct reference to

case5b) one can reason on solution to problem

presented at figure 5a). Note that for the latest case

location of true measurement in between extreme

observations can be easily evaluated. Therefore one

can reason on influence of random errors  on final

observations’evaluationas,forexample,presentedin

right case 5b

). Combination results are transferable

forthetwocases.Systematicerrorcanbeestimatedas

interval valued equal to observations gap mean

distortedwithrandomdeflection.Hereinthescheme

of approach was exploited in order to demonstrate

practicalaspectsofthemethodology.

It was proven [10] tha

t belief and plausibility

measures that are calculated based on results of the

combinationoftwopiecesofevidencerelatedtotwo

randomvariablesgovernedbyGaussiandistributions

with given approximate standard deviations for

whichappropriateisolinesareseparatedwithcertain

Euclideandistance(case5a)andthoseobtainedfrom

association of evidence related to random va

riables

governedbythesamedistributionswithapproximate

standard deviations magnified by certain constant

with isolines being separated with distance

incremented with the same value (case5b) are

mutuallydependentonthisconstant.Theproposition

was further exploited in order to calculate data 

includedinta

ble2.

Figure5. Two cases related to pairs of observations made

fortwoobjectssituatedatoppositedirections

Figure6. Case presented at figure 5b with proportionally

enlargedobservationsmeanerrors

Topractically proveabove proposition,results of

the combination of evidence related to two pairs of

random variables represented by distances taken to

different landmarks were examined. Example

variables referredtoisolines related to the distances

taken for two objects located at counter  bearings.

Unlikesecondpairthefirstonewaslikelytoremain

free from systematic error. Further permanent error

esti

mation was achieved with iterative imprecise

evidence combination scheme. In each step

proportional increment of isolines mean errors took

place. Iterations stopped once maximum belief and

plausibility measures are recorded for the same

hypothesis point while the mass of inconsistency

remained low (see data in ta

ble 2). Further looping

results in decreasing of belief and plausibility

measures.

Table2.Summaryofnumericalexperimentresults 

__________________________________________________________________________________________________

belief hypothesis  plausibility hypothesis   solution inconsistency meanerrorI

) meanerrorII

)

pointnumberpointnumber

__________________________________________________________________________________________________

‐‐ 0.157 1 0.050.714 0.30 0.50

0.131 23 0.710 23 1.150.003 1.18 1.95

0.098 23 0.699 23 1.150.001 1.30 2.15

0.064  230.690231.150.0011.372.26

__________________________________________________________________________________________________

) iterativelyincreasedvaluesarepresented

549

Figure 7 presents diagrams of plausibility values

variations during  iterative combination process.

Bottomcurverepresentsresultsoftheinitialstageof

calculations. They refer to situation illustrated at

figure5b.Randomlydistortedisolinesapproximately

separated with doubled permanent deflection. More

dataregardingthissituationaregatheredinthefirst

row of ta

ble 2. Row number one reads that

uncertainty,inthiscasemeantasinconsistencydueto

notoverlappingevidencecases,isveryhigh.Itsvalue

of0.714suggestsrathercontradictorydata,onepiece

of evidence supports hypothesis points separated

from those endorsed by the second one. Highest

plausibility value indicates solution point tha

t is

located at the first isoline. Mean error of this

measurement is smaller and assigned confidence is

higherthanforthesecondcase(seedataintable1).

Threeuppermostdiagramsatfigure7refertolast

stagesofiterativecombinationprocess.Combination

schemeshowsthesamepointofthehypothesisframe

tha

t is clearly distinguished as solution to the

problem. Iterative association engaged incremented

valuesofmeanerrors.Fromdatagatheredintable2

one can notice that at the final stage of processing

sum of increased mean distortions covers isolines

intersection gap. For all last three cases uncertainty

remainslow,beliefandplausibilit

yarehighandboth

these measures clearly indicate the same hypothesis

point. The latest also mean that solution remains

stable.

Figure7. Diagrams showing plausibility values variations

duringiterativecombinationprocess

Benefits that can come out of the presented

proposition were depicted within examples devoted

to distance error analyses. Two pieces of evidence,

onefreefromsystematicerrorandanotherdistorted

withthiskindofdeflection,wereassociated.Results

ofcombinationswereconfrontedinordertominefor

generalpracticalaspects.Examinationoftheoutcome

empirically proves the correctness of the presented

theorem and enables calibrat

ion of the nautical

appliance.Utilizationofthelemmaforpositionfixing

basedupon multipleobservations[7] takenwith the

sametoolandpossiblydistortedwithsystematicerror 

is straightforward. At first pair or pairs of

observations enabling permanent shift indicat

ion

shouldbeselected. Constantshouldbeextractedand

further used for standard deviations of all

observationsadjustment.Modifiedevidencearetobe

encodedandcombinedafterwards.

AlgorithmII

1 Assign initial data, evaluate  approximate

observations mean errors and their uncertainty

(knowndiscrepanciesintheirestimation)

2 Identifylimit

softhehypothesisframeandadjust

probabilitiesforselectedconfidence intervals(see

algorithmI)

3 Preparebeliefstructures,normalize

 andcombine

them

4 Calculatetotalofinconsistencymasses

5 Calculate belief, plausibility measures based on

results of combination. Locate belief and

plausibilitymaxima

6 Quitifbeliefandplausibilitymaximarefertothe

same point, consistency is below required

thresholdandsumofmodifiedmeanerrorscovers

isolinesgap

7 Modifymea

nerrorswithgivenconstantandgoto

step3

Output generated by software implementing

algorithm II for previously defined numerical

examplearepresentedintable3,inwhichapartfrom

constant C all data refer to cables  as distance unit.

Two distances for opposite locations objects were

taken with medium class radar. Mean errors were

esti

matedas: ±0.3and±0.5cables respectively.Their

possible random distortions were assumed to be

±15%. Presented data refer to four lastiterations for

whichmaximumofplausibilityvalueremainedhigh

and referred to the same solution indicated value is

1.15.Collecteddataincludemea

nerrorsmultiplierC

withcalculatedtworandomdeflections 



ialongwith

intervalvaluedsystematicerror.Basedonintroduced

lemma for each multiplier random errors were

estimated.Theevaluationisbasedonpropositionthat

enablemigrationto“freefromsystematicerrorcase”

(see both illustrations at figure 5). Please also note

thatdirectionof randomshifts cannot beindicated.

Availableevidencedonotallowtostatewhatsignsof

random deflection might be thus int

erval valued

permanenterrorswerecalculatedtakingintoaccount

both possible randomness directions (both negative

andpositiveextremevalues).

Table3.Fourlastiterationsresults

_______________________________________________

C  1  2  gapwidth S



 

+

_______________________________________________

3.933 0.292 0.470 0.7631.119 1.881

4.133 0.278 0.448 0.7261.137 1.863

4.333 0.265 0.427 0.6921.154 1.846

4.567 0.252 0.405 0.6571.172 1.828

_______________________________________________

Finalresultsext

rapolationsforvariousgap’swidth

areincludedintable4.Fromdatagatheredinthefirst

rowoftable4onecanreadthatforisolinesgapof0.8

cables probability of an measurement being within

thegapis0.721.Atthesametimethankstoevidence

combination(seedatainta

ble2)solutionforthecase

arerandomerrorsof0.295and0.505respectively.As

can be seen from table 4 appropriate random errors

tendtodecreasebutprobabilityofparticularcaseare

getting smaller and smaller. Probability that

 Generalideaofnormalisationwaspresentedin[15],specificityof

nauticalapplicationsinthisrespectwasdiscussedbytheauthorin

[8]

550

systematic error is within range of [1.40;1.60] is reducedto0.399.

Table4.Finalextrapolationresults

__________________________________________________________________________________________________

stddeviationratio distance probabilityI probabilityII  sumofprobabilities 1i  2i  S



  S



__________________________________________________________________________________________________

10.80.4970.4450.7210.295 0.505 1.10 1.90

0.750.60.4770.3850.6790.221 0.379 1.20 1.80

0.50.40.4200.2890.5870.148 0.253 1.30 1.70

0.250.20.2890.1560.3990.074 0.126 1.40 1.60

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

4 SUMMARY

Thanks to Mathematical Theory of Evidence

approaches towards theoretical evaluation of tasks

includingimprecisedataaretobereconsidered.One

of such problem is indication and evaluation of a

measurementsystematicerror.Innauticalpracticein

ordertocalculatecompasscorrectiononehastoknow

direction to

a landmark or celestial body.

Alternativelyonehastomakeobservationsforobjects

that are located at opposite bearings. Application of

MathematicalTheory ofEvidence inorder toreason

onnauticalappliancecalibrationwaspresentedinthe

paper.Atfirstrangeofhypothesisframewasreduced

in order to guarantee correctness

 of a posteriori

reasoning inselected nautical applications. Seafarers

know where true measurement is supposed to be

located. Observations are assumed to be made for

landmarks situated at opposite sides are examples

wheresuchlocationscanbeeasilyidentified.Dueto

proposed reduction combination inconsistency mass

remain small while belief

and plausibility are

relatively high. Usually high inconsistency mass

indicatespoorqualitynauticalevidence.Yetanother

reasonforlarge conflictingmassis wrongly defined

hypothesisframe,consequentlyitisnotsupportedby

evidenceathand.

In the second part of the paper proposition

regarding unique feature of nautical evidence

combination scheme

 was exploited. Statement

regarding behaviour of the association process was

presented and proven in recent paper delivered by

the author. Theorem enables reasoning on random

andsystematicerrorsofobservationsmadeforobjects

situated at opposite sites as seen from observer’s

position.

In presented numerical example two distance 

observations distorted

with random and systematic

errors wereconsidered. Obtained measurementdata

along with nautical knowledge were encoded into

belief structures which were further iteratively

combined.Iterationswerequittedoncestablesolution

was achieved. Given this solution reasoning

regarding combination of systematic deflection free

data was carried out. Thanks to MTE particular

distance

betweenisolinessolelyduetorandomerrors

couldbeachieved.Itisidentifiedbyhypothesispoint

with the highest support measures in view of

evidenceathand.Itsubsequentlygivesbasementfor

random errors estimations and systematic deflection

evaluation.Resultfixederrorappearsintervalvalued,

range of obtained values  depends

on required

threshold probability. To estimate limits results of

informal interpolation were introduced. For selected

decreasingisolinesgapsprobabilityofthetrueisoline

being located within were calculated. For each

presentedcasethetruelocationoftheshipcouldbe

alsoestimatedbasedonobtainedresults.

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