137
1 INTRODUCTION
To control the position of the vessel are held the
navigation parameters’ measurements, based on
which are calculated vessel’s coordinates. For safety
ship handling need to refine the methods of
determining the observed coordinates and estimates
of measurement and calculation errors.
If the excessive measurements of determination of
a vehicle trajectory in various environments (sea, land,
air, etc.) take place, coordinates calculation is usually
executed by the method of least squares (LSQ) [3, 4,
17]. But LSQ is effective only when errors have
distributed according to the Laplace–Gauss
distribution (further – normal distribution). Studies
have shown that errors of navigation measurements
frequently aren't obeyed to the normal distribution,
especially during critical situations, e. g. in the threat
of vessels’ collision, etc. [9, 10]. In such cases, the use
of LSQ leads to a loss of coordinates’ accuracy.
Therefore, it is necessary to analyze the possibility
of vessels’ coordinates calculating in the presence of
excessive measurements by alternative methods. Such
method is first of all a method of maximum likelihood
estimation (MLE) [6, 8].
2 STATE OF THE ART
Over the past decade, the issue of increasing the
accuracy of determining the vessel position has been
discussed in detail in many research works [15, 16].
Results of verification of statistical hypotheses of error
distribution laws navigation measurements set out in
publications [5, 12]. The researchers have found that
the errors of measurements of radar azimuth
orientation and distances are obeyed mainly to mixed
distribution laws errors of the first and second types
[1]. Simulation modeling is applied for the evaluation
of efficiency using the above methods, the
corresponding communicative and computing costs
[8].
The analysis of errors of navigation measurements'
statistical data received during practical observation,
set out in the work [7]. This study shows that the
Simulation Modeling for Evaluation of Efficiency of
Observed Ship Coordinates
I. Vorokhobin, I. Burmaka, I. Fusar & O. Burmaka
National University “Odessa Maritime Academy”, Odessa, Ukraine
ABSTRACT: Simulation computer modeling was used to evaluate the efficiency of the vessel’s observed
coordinates using the mixed laws of distribution errors of the first and second type for lines of position (LOP).
Simulation modeling showed good convergence of evaluation of efficiency calculated by analytical expressions
and obtained by simulation. A graphical depiction of the observed points’ deviation relative to the
mathematical expectation in the case of distribution of LOP errors of both types according to mixed laws is
obtained by the method of least squares and the method of maximum likelihood estimation.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 16
Number 1
March 2022
DOI: 10.12716/1001.16.01.15
138
errors of navigation measurements are distributed at
non-normal laws.
If the law of errors’ distribution differs from
Gaussian distribution, then using the LSQ to calculate
the vessel coordinates does not provide effective
evaluations. At excessive lines of position (LOP),
evaluation of the efficiency of vessel's coordinates has
held in the [2]. In this work is also shown that when
using mixed distribution laws, the efficiency is lower
«1», and with the growth of a significant parameter, it
approaches the value of «1».
Results of the statistical materials' analysis of
accuracy of a vessel position’s definition through
satellite radio navigation system has held in the [13].
It follows that the hypothesis of the random errors'
distribution in determining latitude and longitude by
Gaussian distribution is not correct and requires the
use of other distribution laws.
Mixed error distribution laws of both types, which
are an alternative to the normal law, are proposed to
describe the random errors of navigation
measurements proposed in the work [11]. Poisson's
distribution law is considered for the same purpose in
[14].
The analysis of the considered works shows that to
prevent loss of accuracy of the calculated coordinates
it is necessary to use methods which alternative to
LSQ.
The purpose of the article is to verify the
determination of the effectiveness of the ship's
coordinates by simulation computer modeling in the
case of distribution of measurement errors according
to mixed laws of errors distribution of the first and
second types.
3 METHODOLOGY
For evaluation of the efficiency of the observed
coordinates, obtained at excessive LOP and calculated
by LSQ, simulation computer modeling was
performed. In doing so, cases were considered when
LOP errors were obeyed to the normal law, as well as
to mixed laws of errors distribution of the first and
second type.
Simulation modeling was performed according to
the following algorithm. Initially, a sample of LOP
errors consisting of 1000 members was generated
according to the chosen distribution law. Calculation
of the each observed point's coordinates were done on
8 LOPs. Moreover, the LOP elements (transfers ri and
gradients’ directions αi) were set relatively the true
vessel position. That's why, LOPs' transfers ri equal to
their errors ξi. During simulation modeling gradients’
directions αi were chosen equal 30°, 75°, 120°, 165°,
210°, 255°, 300° and 345°. Using the generated sample,
it is possible to obtain 125 observed points. The
increment of their coordinates X і Y are projections of
vectorial error. This allows calculating the covariance
matrix of vectorial error of observation.
The formation of 125 observed points was repeated
four times, and the obtained coordinates were saved.
As a result, the sample S500 was accumulated in
vectorial error’s coordinates number of 500 terms.
Using the obtained sample were calculated the
expected values MX, MY and variances DX, DY of
projections X and Y of vectorial error. Simulation
computer application provides the graphic depiction
of the observed coordinates relative to the expected
value. This allows making a visual evaluation of their
deviation. It had should be take into account that a
vectorial error is determined relative to the true vessel
position. Then during simulation modeling for
evaluation of the efficiency of the observed
coordinates, it is necessary to use not variances of
projections X and Y, but their second initial moments.
The computer application for simulation modeling
provides calculation of observed points both LSQ and
MLE. In simulation modeling standard deviation
2
of LOP error was taken as equal to “5”.
First, a sample S500 was generated for LOPs errors
distributed under normal law. The coordinates of
observed points (xt, yt) have calculated by LSQ. In this
case, the second initial moments of vectorial error’s
components have turned out to be equal
2
24.48
xG
a =
and
2
0.8617
yG
a =
. The second initial moment of the
vectorial error’s module is calculated as in (1).
2 2 2
25.86
RG xG yG
a a a= + =
(1)
Fig. 1 shows the positions of observed points
relative to the expected value. At that, in Fig, 1 and
others maximal deviation coordinates’ values equal to
the standard deviation. At modeling, its value is
accepted as “5”.
Figure 1. Distribution of observed points under normal law
The observed coordinates of vessel were calculated
by LSQ. In doing so the errors of the LOP were
obeyed to mixed distribution. For evaluation of
efficiency of observed coordinates, the samples of
LOPs’ errors were generated accordance to mixed
distribution.
139
Then the observed coordinates have been
calculated by LSQ and the sample
500
LSQ
S
of vectorial
error’s coordinates was formed. At once, using the
values of the same errors, the calculation of observed
coordinates by MLE was performed. Then the sample
500
ML
S
of vectorial errors’ components was formed.
Based on the sample’s data the expected values,
variances, and the second initial moments
2
LSQ
x
,
2
LSQ
y
and
2
MLE
x
a
,
2
MLE
y
a
, components X і Y of vectorial
error were calculated. Then the values of the second
initial moments are calculated accordingly using LSQ
as in (2), and MLE as in (3).
2 2 2
LSQ LSQ LSQ
R x y
a a a=+
(2)
2 2 2
MLE MLE MLE
R x y
a a a=+
(3)
Obviously, during simulation modeling (SM)
efficiency of the observed coordinates eSM obtained by
LSQ, determined by the ratio of the second initial
moments
2
LSQ
R
a
and
2
MLE
R
a
as in (4).
2
2
LSQ
R
SM
MLE
R
a
e
a
=
. (4)
The eSM are compared with the corresponding
efficiency’s value eT calculated theoretically by
analytical expressions held in the work [2].
4 RESULTS OF SIMULATION MODELING
As a result of simulation modeling, the efficiency
value
(1)
SM
e
is obtained for the distribution of LOP’s
errors according to the mixed law of error distribution
of the first type. They are compared with the
calculated values
(1)
T
e
as indicated in Table 1.
Table 1 and Fig. 2 shown dependences
(1)
SM
e
obtained by simulation modeling, and
(1)
T
e
calculated
analytically, from the value of the essential parameter
m.
Table 1. Efficiencies
(1)
T
e
and
(1)
SM
e
of the mixed distribution
of errors of the first type
_______________________________________________
m 1 2 3 4 5 6
_______________________________________________
(1)
T
e
0.500 0.800 0.893 0.934 0.955 0.968
(1)
SM
e
0.507 0.636 0.710 0.792 0.800 0.888
( )
1
e
(%) 14.0 20.5 20.5 15.2 16.2 8.0
_______________________________________________
In Fig. 2 the values of efficiencies
(1)
T
e
are depicted
by blue points, and efficiencies
(1)
SM
e
– red points.
From Table 1 and Fig. 2 can be seen that the
percentage discrepancy
(1)
e
between efficiencies
(1)
T
e
and
(1)
SM
e
does not exceed 20.5%.
During the simulation in addition to the calculated
data are given graphic depiction deviation observed
points relative to mathematical expectation. The
variance of the vectorial error's modulus was
calculated by the methods LSQ and MLE.
Figure 2. Dependence of efficiencies
(1)
T
e
and
(1)
SM
e
on
parameter m
Fig. 3 shows a comparative characteristic of
deviation of the observed points obtained by LSQ and
MLE at m = 2 for the same LOPs.
Figure 3. Comparative characteristic of points’ deviation at
m = 2
Also, results of simulation modeling were received
for the case of distribution of LOPs errors according to
the mixed distribution law of errors of the second
type. In doing so, efficiency’s value
(2)
SM
e
obtained by
simulation computer modeling were compared with
the calculated values
(2)
T
e
shown in Table 2 and Fig.
4, depending on the value of essential parameter m.
Table 2. Efficiencies
(2)
T
e
and
(2)
SM
e
of the mixed
distribution of errors of the second type
_______________________________________________
m 1 2 3 4 5
_______________________________________________
(2)
T
e
0.700 0.857 0.917 0.945 0.962
(2)
SM
e
0.671 0.668 0.751 0.829 0.870
( )
2
e
(%) 4.1 22.0 18.1 12.3 9.6
_______________________________________________
Analyzing Table 2 and Fig. 4, can be seen that the
divergence
(2)
e
between efficiencies
(2)
T
e
and
(2)
SM
e
in percentage less than 22.0%. The study showed good
agreement between the evaluations of efficiencies
calculated by analytical expressions and obtained
during simulation modeling. This confirms the
correctness of the analytical method of the evaluation
of the efficiency of observed coordinates, calculated
by LSQ.
140
Figure 4. Dependence of efficiencies
(2)
T
e
and
(2)
SM
e
on
parameter m
The distribution of LOPs’ errors according to the
mixed distribution law of errors of the second type,
received at calculation by LSQ and MLE is considered.
Also was obtained deviation’s graphic depiction of
the observed points relative to the expected value.
The comparative characteristic of deviation the
observed points, coordinates of which are calculated
by LSQ and MLE, shown in Fig. 5.
Figure 5. Deviation coordinates’ values of the observed
points when m = 2.
5 CONCLUSIONS
In the study, the methodology of simulation modeling
is offered to evaluate the efficiency of the vessel’s
observed coordinates by mixed distribution laws of
errors of the first and second type for the lines of
position (LOP).
The conducted simulation modeling showed a
good convergence of evaluations of efficiency,
calculated by analytical expressions and obtained by
simulation modeling.
According to the results of the work, a deviation
graphic depiction of the observed points relative to
the expected value was obtained in the case of LOP
error distribution according to mixed error
distribution laws of both types when using the least
squares (LSQ) method and maximum likelihood
estimation (MLE).
ACKNOWLEDGEMENT
This work has received funding from the Ministry of
Education and Science of Ukraine (State Reg. No.
0117U005133, Supervisor Burmaka I. O.).
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