International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 5

Number 3

September 2011

329

1 INTRODUCTION

Radar target tracking is one of the key issue influ-

encing navigational safety of vessels at sea. For sev-

eral dozen of years radar has been present on board

of the ships establishing its position as a very im-

portant device on the bridge. It has been commonly

used for observation of navigational and collision

situation in the vicinity of own vessel. In the re-

stricted visibility it is even the basic source of in-

formation, while remaining additional and comple-

mentary (to visual observation) source in good

visibility.

1.1 Tracking of maneuvering targets in radars

Radar’s functionality increased rapidly after imple-

menting of target tracking facilities in ARPA sys-

tems. Since then it has become possible to support

navigator’s work by replacing manual plotting with

automatic target tracking. The quality of tracking

depends however on the implemented tracking algo-

rithm. At the beginning relatively simple numerical

algorithms, like α-β were used. In time those were

replaced by more complex numerical algorithms

based on statistical estimation, like Kalman Filter

(Bole et al. 2005). It’s main deficiency is the as-

sumption of linear movement of the target, which

leads to large errors and delays of tracking during

target’s and own ship’s manoeuvres. These limita-

tions are commonly known to the navigators. Vari-

ous modifications of Kalman Filter (e.g. Extended

Kalman Filter, Unscented Kalman Filter) improved

the quality of tracking significantly. The main goal

was to include non-linear movement of maneuvering

vessels into the algorithm. Thus better quality of

tracking was achieved.

One of the possible solution of non-linearity

problem is to create a few different filters for differ-

ent motion stages (linear/ non-linear). This approach

is called multiple-model filtering and is thoroughly

examined for example in (Bar-Shalom & Li 1998).

Another possibility is to use typically non-linear

methods for tracking, for example artificial intelli-

gence. An interesting example might be Intelligent

Kalman Filter presented in (Lee et al. 2006).

For several years the research focused on use of

artificial neural networks in radar target tracking has

been carried out in Maritime University of Szczecin.

Particularly interesting results were obtained while

using General Regression Neural Network (GRNN),

which was presented for example during TransNav

2007 (Stateczny & Kazimierski 2007).

Statistical Analysis of Simulated Radar Target's

Movement for the Needs of Multiple Model

Tracking Filter

W. Kazimierski

Maritime University of Szczecin, Poland

ABSTRACT: The quality of radar target tracking has a great impact on navigational safety at sea. There are

many tracking filters used in maritime radars. Large group of them are multiple model filters in which differ-

ent filter parameters are used for different states (models) of vessel movement. One of possible filter is multi-

ple model neural filter based on General Regression Neural Network. Tuning of such filter means to adjust its

parameters for a suitable target movement model. This paper shows the results of an experiment aiming at de-

termining such models based on statistical analysis of target's movement parameters. The research has been

carried out with PC-based simulator in which typical radar measuring errors were implemented. Different

manoeuvres of targets have been examined. Based on this, the possibility of movement models description

has been stated as conclusion.

330

1.2 Research project and paper scope

The experience on target tracking with neural net-

works gained so far, resulted in preparing new re-

search project in Maritime University of Szczecin,

called Elaborating of methods for radar tracking of

maritime targets with the use of multiple model neu-

ral filtration. The main goal of the project is to com-

bine neural tracking filters with multiple model phi-

losophy, traditionally used for numerical filters.

Different neural filters will be adjusted to track tar-

gets with different dynamics of movement. This

means that, as the first stage of the project, several

models of target’s movement has to be declared. The

aim of research presented in this paper was to per-

form statistical analysis of different target’s move-

ment to conclude on how to find these unique

movement models.

2 GRNN FILTER FOR TRACKING IN MARINE

RADARS

The filter proposed for radar target tracking and ex-

amined in presented research was based on General

Regression Neural Network invented by D. F.

Specht (Specht 1991), which is basically neural im-

plementation of kernel regression algorithms pre-

sented in (Nadaraya 1964) or (Watson 1964). The

structure of the network is strictly defined, but it

needs some kind of adjusting to solve particular

problem. This means mainly determining of input

and output vectors, teaching sequence, radial neu-

rons activation function and smoothing factor of it.

2.1 Tracking with GRNN

The concept of using GRNN to track radar targets in

maritime navigational radars was shown in

(Juszkiewicz & Stateczny 2000), (Stateczny &

Kazimierski 2005) and (Kazimierski 2008). The fil-

ter proposed in these papers consists of two parallel

GRNNs. One of them is to etimate Vx and the other

Vy. For additional smoothing of signal, which

means more stable vector of target on the radar

screen, the second filtration stage, with another pair

of the same networks is used. To ensure proper func-

tioning of the filter, since the beginning of observa-

tion, the dynamic increase of number of radial neu-

ron in hidden layer and elements of teaching

sequence is introduced. Observed (measured) values

of movement vectors are used as input and teaching

values while estimated movement vector is the out-

put. Movement vector is defined as (1).

[ ]

VVV =

(1)

where V

= speed vector over x axis, V

= speed vec-

tor over y axis.

Both of the networks can be joined into one -

more complex structure presented in the figure 1.

Such a network has two basic parameters – the

smoothing factor and the length of teaching se-

quence, usually both adjusted empirically.

Figure 1. Two-stage GRNN for target tracking.

The smoothing factor determines the range of

gaussian function in radial neurons and the teaching

sequence determines how many observed vectors are

included in estimating the state vector.

GRNN performs kernel regression, resulting in

computing weighted average of teaching vectors.

The weights are the values of Gaussian kernel func-

tion for the distances of input vector to teaching vec-

tor. Thus the estimation of movement vector is cal-

culated according to following equation

(Kazimierski 2008):













⋅













∑













−













−













−













−

eVyo

eVxo

Vye

Vxe

(5)

where Vxe and Vye = estimated speed vector on axis

x and y, Vxo and Vyo – observed speed vector on

axis x and y, σ = smoothing factor of Gaussian ker-

nel function, t = actual time step, t

= former time

steps.

2.2 Multiple model filtering

Multiple model approach is the development of so

called decision based filters. The main idea is simi-

lar. The filter consists of a few elementary filters,

each of them tuned to track target in unique move-

ment stage, called model. They are running simulta-

neously. The final estimation can be a chosen output

of one of elementary filters (in the decision based

331

methods) or a combination of elementary estimates

(in multiple model approach).

There are several particular algorithms of multi-

ple model tracking, in which different interaction

methods between elementary filters is used. Usually

the probability of target being in each particular

mode state is the criterion. Thus the estimated state

vector is weighted average of elementary estimates.

Fine description of most popular multiple model

methods is given in (Bar-Shalom & Li 2001) and (Li

& Jilkov 2005).

2.3 GRNN multiple model filter

The empirical research (Stateczny & Kazimierski

2006) or (Kazimierski 2007) has shown that differ-

ent values of smoothing factor and of teaching

length are needed in GRNN filter for different

movement characteristics. For uniform motion –

longer teaching sequences and bigger smoothing

factors and for maneuvers shorter teaching sequenc-

es and smaller values of smoothing factor are ex-

pected. This gave the idea of creating multiple mod-

el neural filter, which can be implemented as

decision based filter as well. A suitable patent appli-

cation was issued. An example of such a filter is

given in figure 2.

Figure 2. GRNN filter for target tracking.

Main problem in such an approach is to tune ele-

mentary filters for suitable movement model. This of

course shows the need of defining such models.

3 NUMERICAL EXPERIMENT

The research based on simulation presented in this

paper is just an initial phase and preparation for fur-

ther parts in which real data will be involved. This

time PC- based tracking radar simulator was used.

3.1 Experiment overview

The main goal of the experiment was to find any sta-

tistical dependency, that can be useful for defining

tracked targets movement models. To ensure useful-

ness of experiment results for any tracking method,

the unfiltered data were analyzed. These were ob-

tained in the simulator by implementing suitable

noise of measurements prior to any filtration.

To obtain statistical information, 100 Monte Car-

lo runs were performed for each research scenario.

For each run, an average value and a standard devia-

tion of ship’s course, speed, Vx, Vy and increments

of Vx and Vy as well as covariance between Vx and

Vy were calculated. After the simulations, obtained

values were examined and analyzed with the use MS

Excel

The research scenarios were planned in such a

way to examine both uniform motion state and ma-

neuver state.

3.1.1 Simulator description

The simulator used in research showed in this pa-

per is a PC-based application, prepared by the author

in MS Visual Studio.

The idea of radar target simulation used in the

simulator derives from (Kantak et al. 1988) and is

based on adding to non-cluttered measurement, pro-

cess noise. Thus the position of simulated target is

obtained. The noise is calculated as a product of

maximum sensor noise and pseudo-random value.

Start point of random numbers is changing, which

allows carrying out Monte Carlo simulation.

Own ship movement is also simulated and typical

errors of gyrocompass (0,5°) and log (0,05 kn) are

included. The auto-correlation function factors were

established based on (Stateczny et al. 1987).

The simulator has also other possibilities and

functionalities, which were not used for the research

for this paper, however they can be used for many

other purposes.

3.1.2 Research scenarios

The idea of the research is to find different

movement models based on statistical analysis of

non-filtered target data. The research scenarios

therefore include both - uniform target movement

and maneuvers.

The first part of research focused on finding sta-

tistics for linear movement as the basis for compari-

son with maneuvering stages. Five different scenari-

os were examined for uniform movement. Initial

situation was the same for each of them, except of

course and speed values which differ for particular

scenarios. The scenarios are described in Table 1.

332

Table 1. Scenarios for uniform movement

______________________________________________

Scenario no 1 2 3 4 5

______________________________________________

Initial situation

______________________________________________

Bearing 030°

Range 8 Nm

Own ship course 000°

Own ship speed 10 kn

______________________________________________

Target movement parameters

______________________________________________

Target course [°] 135 180 270 135 135

Target speed [kn] 10 10 10 20 30

_____________________________________________

These research scenarios allowed to check the in-

fluence of various speed and courses on statistical

dependences of target movement during steady

movement.

The second part of research aimed at finding re-

sults during maneuver of target. The maneuver of

course changing was examined as the most and ad-

vised in COLREG popular way of collision avoid-

ance. The maneuver was applied with different rate

of turn in different scenarios. As the same change of

course was assumed, the maneuvers were lasting for

different time in each scenario. The statistics were

calculated only for the time during maneuver. Detail

description of scenarios can be found in Table 2.

Table 2. Scenarios for maneuvering target

________________________________________________

Scenario no 1 2 3 4 5

________________________________________________

Initial situation

________________________________________________

Bearing 030°

Range 8 Nm

Own ship course 000°

Own ship speed 10 kn

Target course 135°

Target speed 10 kn

________________________________________________

Course maneuver

________________________________________________

Course change 90° to starboard

New course 225°

Rate of turn [°/min] 10 20 30 40 50

________________________________________________

Examining of the maneuvers with different dy-

namics allowed to answer the question if there is any

statistic dependent of turn rate, which can become a

basis for establishing movement models in multi

model filter.

Each scenario covers 200 measurement steps,

which means about 10 minutes of simulation time.

3.2 Results of experiment

The simulator used for experiment prepares the out-

put statistics for each of 100 Monte Carlo runs in

ASCII file. In the next step it was imported to MS

Excel to prepare graphs and to perform further anal-

ysis. The results are divided into two parts – uniform

motion and maneuver. The conclusions are stated for

each part separately and then jointly for all simula-

tions.

3.2.1 Uniform motion

The scenarios in which the course was different

were analyzed together and the scenarios in which

the speed was different were also analyzed jointly.

Figure 3 shows the standard deviation of Vx dur-

ing simulation for each of 100 runs. Scenarios 1, 2

and 3 were included. It can be noticed, that the value

of standard deviation does not vary significantly for

the scenarios, although in case of scenario 3 the val-

ues of standard deviation is slightly bigger than in

other scenarios.

Similar results were obtained for other measured

values – standard deviation of Vy, course and speed.

this can lead to the conclusion that standard devia-

tion of movement vector parameters does not change

significantly in case of uniform movement with dif-

ferent courses.

In figure 4 the same standard deviation of Vx is

presented but for the scenarios in which the target

was moving uniformly but with different initial

speed. Once again the value of standard deviation

seems not to vary much in different scenarios.

Figure 3. Standard deviation of Vx during 100 Monte Carlo

runs for uniform motion of target with different course.

The values for scenario 5 in which the speed was

the biggest are usually a bit smaller. As similar re-

sults were obtained for other parameters it can be

concluded, that standard value of them does not

change significantly for the uniform motion, even if

the speed is different.

0 10 20 30 40 50 60 70 80 90 100run no

scenario 1 scenario 2 scenario 3

333

Figure 4. Standard deviation of Vx during 100 Monte Carlo

runs for uniform motion of target with different speed

An interesting issue of statistical analysis of

movement vector can be covariance of Vx and Vy

vectors. It was measured as the covariance of ran-

dom samples Vx and Vy. Figure 5 shows the average

value of average covariance in each of Monte Carlo

runs. It can be noticed that, although average value

vary for different scenarios, the standard deviation

remains on the same level. This means that covari-

ance value vary in different scenarios, but it remains

in the same “statistical frame” for all of them and it

could not be easily stated which scenario is it, based

only on the results.

The most important conclusion of the first part of

experiment is that one movement model for non-

maneuvering target is sufficient. Changes of course

and speed do not influence significantly on statistical

factors for movement vector parameters. The next

question is if this is also true for maneuvering target.

Figure 5. Average value and standard deviation for average co-

variance in 100 Monte Carlo runs for non-maneuvering target.

3.2.2 Maneuver

In the second part of research, maneuvering target

was observed. Only course maneuver was imple-

mented. Based on earlier works it was assumed, that

conclusions for speed maneuver would be similar.

One maneuver was examined but with five differ-

ent dynamics, represented by rate of turn.

Figure 6 contains four graphs in fact. Two of

them present average values of Vx and Vy during

maneuver and two other presents standard deviations

of these. The values are presented for 100 Monte

Carlo runs. Only two selected scenarios are present-

ed on this figure, namely scenario 1 (rate of turn =

10°/min) and scenario 5 (rate of turn = 50°/min).

Graph for these extreme values presents the nature

of statistics sufficiently and adding other (middle

valued) scenarios to the graph would only decreased

its readability.

Figure 6. Average values and standard deviations of Vx and Vy

in the scenarios with target course maneuver

It can be derived from figure 6, that average val-

ues of both Vx and Vy for different scenarios are

more or less on the same level, however the variance

of them is definitely bigger for more dynamic ma-

neuvers. Similar observation can be made for stand-

ard deviation of Vx and Vy. Although in this case it

is not so obvious, but larger spread of standard devi-

ation values for maneuvers with bigger rate of turns

can be noticed.

The conclusions derived from figure 6 should be

confirmed with the analysis presented on figure 7.

Figure 7. Statistics for covariance of Vx and Vy in 100 Monte

Carlo runs.

It can be easily noticed, that average value of co-

variance remains on the same level for each scenar-

0 10 20 30 40 50 60 70 80 90 100run no

scenario 1 scenario 4 scenario 5

100

150

200

250

Average SD

scenario 1 scenario 2 scenario 3 scenario 4 scenario 5

-2

0 10 20 30 40 50 60 70 80 90 100

run no

ROT = 10

ROT = 50

Standard deviation of Vy

Standard deviation of Vx

Average Vy [m/s]

Average Vx [m/s]

100

150

200

250

300

Average SD min95% max95%

ROT = 10 ROT = 20 ROT = 30 ROT = 40 ROT = 50

334

io. What seems to be very interesting, the value of

standard deviation is changing at the same time and

the tendency is obvious. The faster the turn is (rate

of turn is bigger), the bigger standard deviation – the

covariance is more spread.

In the figure 7 the boundaries of 95% confidence

levels are additionally shown. It was calculated

based on average and standard deviation values.

These two graphs visualize directly how the region

of 95% confidence is enlarging with the increase of

rate of turn value.

An important conclusion derives from figures 6

and 7, namely that dynamic of the maneuver can be

noticed during statistical analysis. Especially the

value of covariance of Vx and Vy can be very useful

in determining maneuver rate.

3.3 Conclusions

To conclude jointly the research figures 5 and 7 shall

be compared. The average value of calculated covar-

iance is basically the same for non-maneuvering and

for maneuvering targets. The standard deviation on

the other hand is clearly increasing as the rate of turn

is getting bigger. So for the steady motion standard

deviation is small and for fast maneuvers is bigger.

Statistical analysis of movement vector parame-

ters (course, speed, Vx, Vy) can also be used for dif-

fering steady motion from maneuvers, the analysis

however is not so obvious.

This leads to a conclusion, that covariance be-

tween Vx and Vy is the best value to define move-

ment models and the definition should be based on

standard deviation analysis.

4 SUMMARY

The idea of building multiple model neural filter

seems to be promising alternative for numerical fil-

ters in the light of earlier research.

Definition of movement models is of the key is-

sue for this project. The paper presented the analysis

of possibility of determining such models, based on

statistic dependences. The results of the research

showed that statistical analysis of covariance of

movement vector elements (Vx, Vy) can be particu-

larly useful for this purpose. It has been proven that

standard deviation of such a covariance is increasing

when the target is maneuvering faster.

The determination of particular models, based on

standard deviation threshold, shall be the subject of

empirical research. This will probably be one of the

future steps for continuation of presented research.

However prior to these more simulation research

shall be conducted. These shall include especially

the influence of own ship – target geometry for the

statistics observations.

An important conclusion derived from research is

also the fact that one movement model is sufficient

for describing uniform motion, while for the maneu-

vers a few models shall be established. The number

of them should be the result of empirical research.

Another problem will be to “translate” statistical

model to GRNN, thus to adjust network parameters

accordingly.

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