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1
Conflict-based Force Aggregation
John Cantwell
1
Johan Schubert
2
and Johan Walter
Department of Infrastructure and Planning Department of Data and Information Fusion
Royal Institute of Technology Division of Command and Control Warfare Technology
SE-100 44 Stockholm, Sweden Swedish Defence Research Agency
SE–172 90 Stockholm, Sweden
cantwell@infra.kth.se schubert@foi.se johanw@foi.se
Abstract
In this paper we present an application where
we put together two methods for clustering and
classification into a force aggregation method.
Both methods are based on conflicts between
elements. These methods work with different
type of elements (intelligence reports, vehicles,
military units) on different hierarchical levels
using specific conflict assessment methods on
each level. We use Dempster-Shafer theory for
conflict calculation between elements,
Dempster-Shafer clustering for clustering these
elements, and templates for classification. The
result of these processes is a complete force
aggregation on all levels handled.
1 Introduction
In this application oriented paper we report the
first result from an ongoing project at the
Swedish Defence Research Agency
investigating which automatic conclusions can
be drawn about force deployment based on low
level intelligence. The scenario studied is an
army scenario but in a future situation with a
high flow of automatically generated
intelligence reports concerning vehicles.
Our idea is to investigate the intelligence
reports through a series of aggregation
processes and draw automatic conclusions about
force strength, deployment and ongoing actions
to establish a solid basis for enemy prediction
and decision superiority. In the longer run our
aim is to automate the Intelligence Preparation
of the Battlefield (IPB) process [18].
In section 2 of this paper we give a detailed
problem description, followed by an overview in
section 3 of an actual scenario. In section 4 we
describe an aggregation procedure in three
major steps, aggregating the information
upwards level by level: first, we investigate
methods for assessing conflict (using
Dempster’s rule [16]) between elements, i.e.,
intelligence reports, vehicles or units,
depending on level. Secondly, we use these
conflicts to partition the elements into groups
(through Dempster-Shafer clustering [1
−2, 5−
14]) forming elements one level up in the
hierarchy. Finally, we use templates and
multiple hypothesis evaluation to classify
grouped elements into an element on the higher
level. This process is carried on level by level as
long as data permits, Figure 1. Results are
discussed in section 5 and conclusions are
drawn in section 6.
Figure 1: The aggregation process hierarchy.
Clustering
of intelligence
Clustering
of vehicles
Clustering
of platoons
Clustering
of companies
Classification
of vehicles
Classification
of platoons
Classification
of companies
Classification
of battalions
reports
1. This work was done while the author was with the 2. http://www.foi.se/fusion/
Swedish Defence Research Agency.
in Cd Proc. Sixth Int. Command and Control Research and Technology Symp., Track 7, Paper 031,
pp. 1−15, Annapolis, June 2001, US Dept. of Defence CCRP, Washington, DC, 2001.
2
2 Problem description
2.1 Unit model hierarchy
Our unit model consists of a mechanized
company. In this model, the company consists of
a company commander, three mechanized
platoons and a main battle tank (MBT) platoon
or an anti-tank missile platoon.
A mechanized platoon consists of four
armored personnel carriers (tracked), a main
battle tank (MBT) platoon consists of five
MBTs and an anti-tank missile platoon consists
of five anti-tank missile launchers (see Figure
2).
Figure 2: The hierarchy of units.
2.2 Environment
To generate sensor reports, we use the FbSim
simulator. FbSim has been developed to analyze
military units in combat situations
1
(see Figure
3).
Figure 3: The FbSim simulator.
The simulated units, called actuators, are
designed as agents, with decision and action
capabilities. They have information (through
sensors, communication with other agents, and a
battle plan) and use a set of rules to make
decisions based on this information, and then
respond with weapons, vehicle actions or by
giving orders. When a simulation is run, these
agents act independently.
When a vehicle gets into a visual contact
with another vehicle, a report is generated and
saved in a log-file. Whether a visual contact is
established depends on the distance between the
two vehicles and the terrain between the two
vehicles. The better the visual contact, the more
specific classification of the vehicle is obtained.
2.3 Intelligence reports
In FbSim, when a vehicle gets a visual contact
with another vehicle, a report is generated. This
report is composed of the following slots: [from,
name, position, time, classification, orientation]
• from is the name of the observer
• name is the name of the observed vehicle
(only to be used for validation and not in
the aggregation process)
• position is the position of the observed
vehicle
• time is the number of seconds since
simulation start
• classification is the type of the observed
vehicle, at a long range the classification
could be unknown, then perhaps tracked
vehicle and finally a tank
• orientation is the direction of the observed
vehicle.
2.4 What do we want to know?
An understanding of the current situation is
essential for evaluating threats and courses of
action. A reasonably accurate picture,
abstracted from individual intelligence reports
spread out in time, of the current number and
positions of the opposing forces is itself
valuable as it gives a measure of the scale of the
threat and a guide to possible objectives.
When it comes to incapacitating opposing
units with methods that require precision, single
units or tightly spaced groups of units is perhaps
the most useful level of analysis of a situation.
After all weapon deployment seeks to
1. FbSim was jointly developed by the Defence Re-
search Establishment, the Swedish Defence Materi-
el Administration, Bofors AB, Ericsson Microwave
Systems AB, Celsius Aerotech AB, Saab Dynamics
AB and Sjöland & Thyselius Datakonsulter AB.
Company
Mechanized
Platoon
Anti-tank robot
Platoon
MBT
Platoon
or
3
incapacitate physical objects and while the
ambition may be to incapacitate an
organizationally more complex unit, this can be
done only by incapacitating a sufficient number
of the concrete physical objects that constitute
the complex unit.
However if one seeks understanding of an
ongoing development, the relevant level of
analysis is typically not on the scale of single
physical objects but on organizationally relevant
groups of objects. Often enough the objectives
with a military assignment are such that no
particular single unit is necessary for the
assignment to be carried through. Furthermore a
group of single units can pose a qualitatively
different threat
−a far greater threat−than the sum
of the individual threats posed by the single
units. Thus quite apart from the need to present
a situation to a decision maker in a way that
avoids clutter, grouping vehicles together into
units is an important step towards exposing the
objectives of the opposing side: an important
step in finding effective counter-measures.
Tracking individual physical objects on the
ground can be extremely hard, in part due to
irregularities in the terrain which reduces sensor
coverage but often also in part due to the sheer
number of very similar objects within a
relatively small area.
When the opposing forces display
organizational structure and when this structure
is at least partially known one can go even
further. To be able to classify a group of objects
as being at least part of a unit of type X means
that one can access previous information about
units of type X: what function do units of that
type fill in the opponents organization? what
kind of capacity as regards movement and fire-
power do they have? how is the unit itself
organized and what kind of behavior can one
expect from the unit? In addition, if one has
only observed parts of the unit, a correct
classification gives one reason to suspect that
the remaining parts of the unit should be
somewhere in the vicinity, which shows that the
threat may be greater than it seems and gives
one reason to direct sensor- and other resources
towards the area.
3 Scenario
Our scenario consists of two forces, the red
force is moving on a road towards a small town,
the blue force is defending the town (see Figure
3). The area is about 10 000 x 10 000 meters,
the terrain is partially hills and forests. Both
forces consist of one company (see section 2.1).
Both companies consist of three mechanized
platoons, each with four armored personnel
carriers (tracked). The blue company also
contains a anti-tank robot platoon, with five
anti-tank missile launchers. The red company
also contains an MBT platoon, with five MBTs.
As a consequence we have 17 vehicles on both
sides.
Not all vehicles are detected and are hence
not reported. The scenario takes about 13 min.
and all together, 356 reports are generated,
which gives an average of 27 reports/min. There
are 14 blue vehicles detected that cause 204
reports and eight red vehicles detected that
cause 152 reports. So at best our aggregation
algorithm, would say that there are 14 blue
vehicles and eight red vehicles. The first thing
we can do is to plot all reports in IS Mark (see
Figure 4).
Figure 4: All reports in IS Mark.
4 Aggregation
Aggregation can take place on all levels: from
intelligence reports to vehicles, from vehicles to
platoons, and from any size of units to higher-up
units in the military hierarchy. In this paper we
focus only on the first three levels: intelligence
reports, vehicles and platoons.
4
In aggregation it is just as crucial to combine
the correct lower elements of all that are
possible, as it is to combine those elements into
a correct higher element. When there are many
different lower elements available in
comparison to the average number of elements
in a template we face an initial selection
problem of choosing which elements should be
combined together for many different parallel
aggregations, Figure 5.
Figure 5: Parallel aggregation.
Secondly, when there are several different types
of templates we must select which template to
choose for these elements. Obviously, these two
phases are not independent. Choosing which
elements to fuse together depends very strongly
on the template. We must avoid combining
elements that do not match any of the templates.
If this is the case, why do we not use the
templates from start to control which elements
to fuse?
The answer is simple. While there is nothing
to prevent us from doing that from a theoretical
point of view it is unwise from a computational
point of view. In order to partition a large set of
elements into groups which should be fused we
prefer to handle only pairwise relations between
the elements. As the templates are usually
relations between more that two elements their
use would have a very high computational load.
Instead we use a pairwise distance measure
between each pair of elements [5] that have
some of the most important characteristics of
the template. We define different conflict
measures between intelligence reports and
between vehicles, respectively. When any of
these are violated a conflict arises between two
elements, section 4.1. The different conflicts
from all violated conditions are combined with
Dempster’s rule into an overall conflict for the
two elements and serve as the actual distance
measure for both clustering and classification.
If the conflict is high among all the elements
of the template the hypothesis (corresponding to
the template) must be disregarded. In both
processes the criterion to find the best
aggregation is to minimize the conflict. With
this we first find which elements fit together by
using a very fast clustering method described in
4.2. As the clustering is based on some of the
most important characteristics of the templates
on the next level the cluster result is usually
quite good on average, but not necessarily
perfect. Secondly, we classify those that do fit
together by using templates on the clustering
result, as described in section 4.3.
4.1 Conflict
The types of objects that can be aggregated are
reports, vehicles and units, e.g., reports are
aggregated into vehicles. Thus it is always
assumed that a vehicle has caused the reports.
Furthermore, vehicles can be clustered into
units; i.e., the vehicles all belong to some unit.
And finally, any size of units can be aggregated
into larger units.
There are different aspects of conflicts,
depending on the characteristics of the two
objects. Aspects of a report conflict could be
conflict with regard to type, direction or
position. Instead of a position conflict we use a
speed conflict by calculating the speed at which
a vehicle must travel, in order to cause the two
reports.
4.1.1 Reports
A physical conflict between two reports is the
speed conflict. If a vehicle must travel at a
greater speed than is physically (type of vehicle,
terrain) possible, in order to cause the reports,
there is a conflict between the reports. Another
physical conflict is the type conflict. There is a
conflict between two reports if they describe
two different types of vehicles, but there is no
conflict if one report indicates an unknown type.
The third type of conflict is the direction
conflict. If two reports describe two vehicles,
traveling in opposite directions, there is a
conflict between the reports, and no conflict if
they travel in the same direction.
100%
67%
50%
5
The conflict is a value between 0 and 1,
where 0 means no conflict and 1 means an
absolute conflict. The overall conflict is
obtained by combining the conflicts of the
different aspects according to
. (1)
Speed Conflict
The speed conflict is given by calculating the
speed a vehicle must have in order to cause the
two reports. The speed conflict is then given by
eq. (2) and shown in Figure 6. We have
. (2)
Figure 6: The conflict function.
The parameter x is given the speed value. If we
knew the maximum speed at which a particular
vehicle could travel, given a certain terrain, we
could set the conflict to one for values greater
than the maximum speed. However, this is not
the case. What we can say is that x > x
2
is
impossible and x < x
1
is certainly possible. We
would also like to express that it is more
probable that two reports belong to the same
vehicle, the lower the speed requirement is.
Therefore we introduce the parameter p.We
used the following values: p = 0.01, x
1
= 22 m/s,
x
2
= 25 m/s.
Type Conflict
All vehicle classifications are hierarchically
ordered in a tree, with unknown as root and the
specific vehicle types as the leaves. The conflict
function then returns zero if one classification is
a descendant of the other and otherwise returns
one. We used the hierarchy shown in Figure 7.
Figure 7: Classification of vehicles.
Direction Conflict
The direction conflict is not an absolute conflict,
since it is possible for a vehicle to change
direction between two reports. Assume we have
ten reports describing ten identical vehicles that
are close to each other. Five reports describing a
vehicle moving south and five reports describe a
vehicle moving north. It is then more likely that
two vehicles have caused the reports instead of
one.
Minor variations in the directions should be
allowed, since the vehicles, for instance, might
be following a non-straight road. Also, if too
long time has elapsed between the two reports,
the direction conflict becomes obsolete, eq. (3),
(3)
where is the difference in directions and
is the difference in time. If the
difference in direction is considered to be to
small, to influence the conflict. If to
long time has elapsed, to influence the conflict.
C
11C
a
–()
a∀
∏
–=
conf x p x
1
x
2
,, ,
()=
xp
x
1
------
, xx
1
<
xx
1
–()px
2
x–()+
x
2
x
1
–
------------------------------------------------
, x
1
xx
2
<≤
1, xx
2
>
=
1
p
x
1
x
2
x
Unknown
Vehicle
Combat
vehicle
Anti-tank robot
vehicle
MBT
Tracked
conf
δ
δ
d
δ
t
δ
d
0
δ
t
0
k
,, , ,
()=
k δd
π k δt+()
----------------------
, δt δt
0
≤()δd δd
0
≥()∧
0, otherwise
=
δ
d
δ
t
δ
d
δ
d
0
<
δ
t
δ
t
0
>
6
We use the following values: and
s. The parameter k is used to get a
slower increase of the conflict. We use the value
.
4.1.2 Vehicles
When vehicles are to be clustered into units,
there are no physical conflicts since the
structure of a unit depends on templates. For
instance a tank platoon template contain five
tanks, driving 50
−200 meters apart. All vehicles
in a unit also tend to drive in the same direction.
So, when clustering vehicles into units, we
consider their relative distance and their
direction.
A difficulty is that we do not know where the
vehicles are, for every specific time. We only
know where they have been, according to
reports. The same holds for their direction.
Instead we plot the vehicle position according to
its reports and combine them with lines, as
shown in Figure 8.
Figure 8: Plotting the reports.
In Figure 8 we have two vehicles with a total of
16 reports, with time stamps t
1
to t
15
, where t
i
<
t
j
for i < j.
Distance Conflict
There are a number of different distances that
could be calculated. The most trivial would be
to calculate the distance between the respective
vehicles last known positions (last reports
positions). However, a better approach is to
consider all reports. We know, for instance,
where vehicle B is at t
4
. We do not know where
vehicle A is at t
4
, but we do know where vehicle
A is at t
2
and t
6
. If we assume that vehicle A
travels in a straight line and at constant speed,
we can calculate where vehicle A is at t
4
. We use
this method to calculate all distances at time t
3
to t
14
, the maximum starting time and minimum
finishing time, respectively, of the two vehicles
and get twelve distances. We call the time
interval t
3
to t
14
the common time interval. A
distance candidate to use in the conflict function
could be the minimum distance, the maximum
distance, the average distance or the median
distance. We used the median distance to avoid
that a few wrongly clustered reports would
influence the result.
The distance is then fed into the conflict
function given by eq. (2). Here, we used:
p = 0.01, x
1
= 300 m, x
2
= 1000 m, with x the
median distance.
Direction Conflict
To calculate the difference in direction for
vehicle A and vehicle B, based on their
respective reports, we calculate the direction
vehicle A must take in order to travel at a
straight line from its position at t
3
to its position
at t
14
. We do the same for vehicle B and
calculate the difference in direction and feed the
value into eq. (2). We use the following values:
p = 0, , , with x the direction.
4.2 Clustering
The clustering method developed may cluster
intelligence reports, vehicles and units on all
hierarchical levels. Initially, we consider
intelligence reports regarding observations of
vehicles that come from multiple sources. Here,
it is not known apriori if two different
intelligence reports refer to the same vehicle.
All reports concerning one vehicle should be
fused separately from all other intelligence
reports.
We use the clustering process to separate the
intelligence into subsets for each vehicle. We
combine Dempster-Shafer theory with the Potts
Spin Neural Network model [17] into a
powerful solver for very large Dempster-Shafer
clustering problems [1]. The Potts model has
proven useful in many complex optimization
problems [4]. We believe this method can serve
as a general solution for preprocessing of
intelligence data in information fusion.
Let i be an index for the element and a the
index for the clusters. Then S
ia
=0,1isa
δ
d
0
π
4
⁄
=
δ
t
0
8=
k 10=
Vehicle B
Vehicle A
t
1
t
2
t
6
t
9
t
11
t
3
t
4
t
5
t
7
t
8
t
10
t
12
t
14
t
13
t
15
x
1
0=
x
2
π
=
7
discrete vector with the constraint S
ia
=
1
∀i where S
ia
= 1 means that element i is in
cluster a. Then the energy function that defines
the Potts model is
. (4)
This model can serve as a data clustering
algorithm if J
ij
is used as a penalty factor of
element i and j being in the same cluster;
elements in different clusters get no penalty.
The problem consists of minimizing this
energy function by changing the states of the
S
ia
. This process is carried out with simulated
annealing. Simulated annealing uses
temperature as an important factor. We start at a
high temperature where the S
ia
change state
more or less at random, and are only marginally
biased by their interactions J
ij
. As the
temperature is lowered parts of the system
become constrained in one way or the other,
they freeze. Finally, when the complete system
is frozen, the spins are completely biased by the
interactions (J
ij
) so that, hopefully, the
minimum of the energy function is reached. For
computational reasons we will use a mean field
model, where spins are deterministic [4].
In each cluster we use the conflict of
Dempster’s rule when all elements within a
subset are combined as an indication of whether
these elements belong together. The higher this
conflict is, the less credible that they belong
together.
In [5] a criterion function of overall conflict
called the metaconflict function was established.
The metaconflict was derived as the plausibility
that the partitioning is correct.
D
EFINITION. Let the metaconflict function,
(5)
be the conflict against a partitioning of n pieces
of evidence of a set
χ
into q disjoint subsets
χ
i
.
Here, c
i
is the conflict in subset i.
In Dempster-Shafer clustering we use the
minimizing of the metaconflict function as the
method of partitioning all elements into separate
subsets. After this, each subset refers to a
different event and the reasoning can take place
with each event treated separately.
We will logarithmize the metaconflict in
order to be able to use it as interaction in the
Potts model.
We have
(6)
where is a weight [16] of
evidence, i.e., in this context a weight of
conflict.
Since the minimum of Mcf (= 0) is obtained
when the final sum is minimal (= 0), the
minimization of the final sum yields the same
result as a minimization of Mcf would have
done.
In Dempster-Shafer theory one defines a
simple support function, where the evidence
points precisely and unambiguously to a single
nonempty subset A of
Θ.IfS is a simple support
function focused on A, then the basic probability
numbers are denoted m(A)=s, and m(
Θ)=1− s.
If two simple support functions, S
1
and S
2
,
focused on A
1
and A
2
respectively, are
combined, the weight of conflict between them
is
(7)
which may be rewritten as
(8)
with being defined so that it is equal to
one for and zero otherwise.
The complete energy function including
constraints that we are considering is
a
1
=
q
∑
E
1
2
---
J
ij
S
ia
S
j
a
a 1=
q
∑
ij, 1=
N
∑
=
∆
M
cf q S
1
S
2
… S
n
,,,,()11c
i
–()
i 1=
q
∏
,
–=
min Mcf
⇔
max 1 Mcf–()log max 1 c
i
–()
i
∏
log=
max 1 c
i
–()log
i
∑
min 1 c
i
–()log–
i
∑
==
1 c
i
–
()
log–0
∞
[, ]
∈
Con S
1
S
2
,()
1 s
1
s
2
–()log– A
1
A
2
∩∅=,
0 otherwise,
=
Con S
1
S
2
,
() 1s
1
s
2
–
()
log–
δ
A
1
A
2
∩
=
δ
A
1
A
2
∩
A
1
A
2
∩∅
=
8
(9)
where the first term is the standard clustering
cost. Using mean field theory we can find the
minimum of this energy function. The Potts
mean field equations are derived as [4]:
(10)
where
(11)
with V
ia
= 〈S
ia
〉.
In order to minimize eq. (9) we use eqs. (10)
and (11) recursively until a stationary
equilibrium state has been reached for each
temperature. To apply it to Dempster-Shafer
clustering we use interactions J
ij
= −log (1 − s
i
s
j
)
δ
|Ai∩Aj|
.
The algorithm for simulating these spins
works roughly as follows: Use a precomputed
highest critical temperature, T
c
, as the starting
temperature. Choose the mean field spins to be
in their symmetric high temperature state; V
ia
=
1/K
∀i, a. At each temperature, iterate eqs. (10),
(11) until a fix point has been reached. The
temperature is lowered by a constant factor until
every spin has frozen, i.e., V
ia
= 0, 1, Figure 9.
In order to find the correct number of clusters
the parameter K must be varied and the
remaining conflict after clustering evaluated
against some threshold. When K is increased the
remaining conflict after clustering decreases.
When it falls below the threshold the best K is
considered found. In Figure 10 an example of
Figure 9: The clustering algorithm.
clustering 204 intelligence reports is
investigated. Here, the logarithm of the total
weight of conflict for 1
−20 clusters is
calculated. The total weight of conflict falls
below the threshold for K = 10.
E
S[]
1
2
---
J
ij
S
ia
S
ja
ij, 1=
N
∑
a 1=
K
∑
γ
2
---
S
i
a
2
i 1=
N
∑
a 1=
K
∑
–=
+
α
2
---
S
ia
i 1=
N
∑
2
a 1=
K
∑
V
ia
e
H
ia
– V
[]
T
⁄
e
H
ib
– V[]
T
⁄
b 1=
K
∑
-------------------------------
----
=
H
ia
V[]
EV
[]∂
V
ia
∂
----------------==
J
ij
V
ja
j 1=
N
∑
γV
ia
α V
j
a
j 1=
N
∑
+–
G
a
--------------------------------------------------------------------
----
=
INITIALIZE
K (# clusters); N (# elements);
J
ij
= −log (1 − s
i
s
j
);
s = 0; t = 0; ε = 0.001; τ = 0.9; α (for K
≤
7: α =0,K =8:α =10
−6
, K =9:α =0,K =
10: α =3
.
10
−7
, K = 11: α =3
.
10
−8
); γ =
0.5;
T
0
= T
c
(a critical temperature) =
, where and
are the extreme eigenvalues of M,
where ;
;
REPEAT
• REPEAT
−2
• ∀a ;
• ∀i Do:
•;
•;
•
;
•;
UNTIL−2
;
•;
•;
UNTIL
;
RETURN
;
δ
A
i
A
j
∩
ij
,∀
1
K
----
max λ
min
– λ
max
,(
)=
λ
mi
n
λ
ma
x
M
ij
J
ij
αγδ
ij
–+=
V
ia
0
1
K
---- ε rand 01[,] i
a
,∀⋅+=
G
a
s
K
N
----
V
i
a
s
i 1=
N
∑
⋅=
H
ia
s
J
ij
α+()V
ja
s
j 1=
N
∑
γV
ia
s
–
G
a
s
----------------------------------------------------------
a
∀=
F
i
s
e
H
ia
s
– T
t
⁄
a 1=
K
∑
=
V
ia
s 1+
e
H
ia
s
–
T
t
⁄
F
i
s
------------------ ε rand 01[,]
a
∀⋅+=
ss1+=
1
N
-
---
V
ia
s
V
ia
s 1–
–
ia,
∑
0.0
1
≤
T
t 1+
τ T
t
⋅=
tt1+=
1
N
-
---
V
ia
s
()
2
ia,
∑
0.9
9
≥
χ
a
S
i
χ
a
.∈ ba≠ V
ia
s
V
ib
s
>∀∀
{}
correction inserted: figure 9
9
In Figure 11 and 12 the clustering process of
clustering 204 intelligence reports into ten
clusters is illustrated. In Figure 11 the neuron
output from 2040 neurons over 19 iterations is
shown. Each intelligence report is represented
by ten neurons showing the degree to which the
intelligence report belongs to the clusters. As
shown, most of the clustering takes place after
the twelfth iteration. In Figure 12 the seven last
iterations are shown. Each rectangle
corresponds to one iteration, the ten columns
correspond to ten different clusters and each
row corresponds to one report. The output signal
of a neuron is indicated by the size of the
square. In the final iteration (rightmost
rectangle) all values are close to one and the
clustering has terminated. In the top row of the
rightmost rectangle we find that intelligence
report number one is in cluster number two, etc.
Figure 10: The logarithm of total weight of
conflict when clustering into 1
−20 clusters.
Figure 11: Neuron output from 2040 neurons
over 19 iterations.
Figure 12: From left to right: The seven last
iterations of the clustering process.
10
On a test problem when clustering N (= 2
K
− 1)
pieces of evidence into K subsets where the
evidence supports all subsets of the frame
Θ =
{1, 2, 3,
…, K} the Potts spin Dempster-Shafer
clustering was shown to have a computational
complexity of O(N
2
log
2
N).
A further description of the cluster methods
used in intelligence analysis was given in [15].
4.3 Classification
The next goal is to combine vehicles into units
and classify the units. While we employ some
basic methods and terminology from the
Dempster-Shafer framework the methods
involved are quite different from the methods
described in the previous section.
Units are described in templates. A unit
template might state that a unit of type X
consists of three vehicles of type A and two
vehicles of type B.
Hypotheses are of the form “vehicles v
1
, ...,v
n
together form a unit of type T
1
or ... or T
m
”. In
such an hypothesis it is implicitly understood
that no other observed vehicle than v
1
, ...,v
n
belongs to the unit (but of course there may be
other units of the same type present).
Hypotheses may conflict, this happens if and
only if a vehicle occurring in one hypothesis
also occurs in the other.
The overall objective is to find the best
consistent (no hypotheses conflict) and
complete (every vehicle is accounted for) set of
hypotheses. This involves generating and
assessing the worth of not only single
hypotheses but sets of hypotheses.
Note that an hypothesis is disjunctive in
character so while a number of hypotheses
regarding which objects belong to the same unit
will be discarded along the way no commitment
is made to the exact unit-type classification of a
group of objects until it is presented to the user.
Even then judgement may be withheld if the
competing alternative classifications are close
enough: a disjunction can be presented to the
operator.
4.3.1 Generating hypotheses
Hypotheses are generated as follows, Figure 13.
Figure 13: Generation of hypotheses.
In the worst case the above algorithm will
generate 2
n
hypotheses. This is worrisome, in
particular as all subsequent processing depends
on the number of hypotheses generated.
However, in the sample applications we have
studied (20
−100 vehicles) this has not been a
problem (total run-time has been 2
−10 seconds).
There are mainly two reasons:
1) A careful pruning of the hypotheses
generated
−hypotheses involving vehicles
that are too far apart are discarded, as are
hypotheses where a vehicle of a particular
type does not belong to a unit of a
particular type (see Step 2 above).
2) The problem space can often be
partitioned into a set of independent
problem spaces that can be solved one by
one (see below).
In the sample applications these factors have
been sufficient to reduce the problem space to
manageable proportions. In the general case,
however, one can set an upper limit to the
number of hypotheses of a particular size
involving any particular vehicle. Taking the m
best hypotheses of any size for any vehicle
means that at most (m
− 1) n
2
hypotheses of any
size will have to be generated for comparison.
In this way the generating algorithm becomes
polynomial in time.
4.3.2 Evaluating hypotheses
Hypotheses are evaluated with regard to two
parameters:
Step 1:Generate all hypotheses of size
1.
Step 2:For each surviving hypothesis of
size n and for each vehicle not in the
hypothesis, generate a hypothesis of
size n+1 by adding the vehicle to the
hypothesis. Evaluate the hypothesis,
if it is good enough, keep it.
Step 3:Repeat step 2 until no new
hypotheses are generated.
11
1) How the type of vehicles and the number
of vehicles of each type fits the
description of a unit of that type.
2) How the positions of the vehicles fit the
hypothesis that they belong to the same
unit.
These are combined using the method of
orthogonal combination from the Dempster-
Shafer framework. An hypothesis is “good
enough” if its conflict value is below some
threshold.
In a disjunctive hypothesis each disjunct is
evaluated independently of the others and the
entire hypothesis is given the conflict value of
the disjunct with the minimal conflict. This can
be justified by appealing to the non-probabilistic
nature of the conflict values: the values should
be interpreted as degrees of fit. So for instance
two disjuncts can both be given a value of say
0.75.
Evaluation occurs at two places: when
hypotheses are generated one needs to estimate
their worth and when sets of hypotheses are
compared one needs to evaluate the worth of a
set of hypotheses. These problems, of course,
are not independent.
The general method described in the previous
section leaves it open how evaluation is done,
but it does impose some constraints. The
computational need to prune off “bad”
hypotheses means that it is not sufficient that we
can rank hypotheses according to their worth.
We need some criterion of good and bad, for
instance, if numerical methods are employed, a
threshold. For this reason we choose to rely on
the Dempster-Shafer method of orthogonal
combination rather than Bayesian
conditionalisation. Below this will be given
further motivation.
We should emphasize that the problem of
hypothesis evaluation is still in need of
development. One might say that at the present
stage we are more interested in localizing and
isolating the issues and problems involved than
in finding completely satisfactory solutions to
the problems.
A simple hypothesis (an hypothesis without
disjunctions) makes two assertions: (1) the
vehicles in the hypothesis satisfy the spatial
constraints of being that part of a unit that has
been observed, (2) the unit is of type X.
When we speak of the conflict of an
hypothesis we refer to the degree to which the
vehicles mentioned in the hypothesis deviate
from an ideal unit of type X. Thus, we assume,
in the ideal unit no vehicles are missing and
they group themselves in a particular way. The
more the vehicles deviate from this the higher
the conflict assigned to the hypothesis.
In our present modelling we treat the
formation/spacing conflict as independent of the
classification conflict. Thus we do not take into
consideration the possibility that different units
may place different spatial constraints on the
vehicles in the unit. This however should not be
too difficult to correct.
Classification
We have used a simple, indeed simplistic,
method for classifying units. The fit is based on
the ratio of observed vehicles/expected vehicles.
Thus if the hypothesis is of the form “tanks a, b
and c form a unit of type X” where a unit of type
X in its standard setup consists of four tanks, the
classification conflict is 0.25 (1
− ratio) and the
support is 0.75. If the hypothesis is of the more
indeterminate form “vehicle a and tanks b and c
form a unit of type X” the classification conflict
is still 0.25, but the direct support is only 0.5.
The problems with such a method are fairly
obvious. No consideration is taken to base
frequencies of unit types and no consideration is
taken to the probability of detection or clutter.
While it would be possible to let these factors
influence the result this would have to be done
in an ad hoc fashion as there is no underlying
theory.
In several respects the present classification
method is inferior to the Bayesian method
proposed in [3] where base frequencies and
probability of detection is dealt with in a
principled manner. However, there are two
problems with the Bayesian approach.
The first is how to establish the probability of
detection. Quite apart from the fact that the
probability of detection varies with the kind of
sensors employed, the kind of objects that are to
be detected, whether it’s day or night, the
12
weather and so on, we have the problem that
sensors move around and terrain, hence
visibility, varies. To deal with the varying
probability of detection across time one needs
recourse to geographical information, detailed
information about sensors, information about
which routes are possible for which vehicles and
more still. Even though these problems need not
be insurmountable, they are problems that we
have not addressed here.
A more fundamental problem can be
described as follows. While a Bayesian method
of force aggregation such as [3] can give us the
probability of the data given the hypothesis
P(D|H) it will not as easily give the probability
of the hypothesis given the data P(H|D) which is
what we are interested in (the data here is of the
form “vehicles a, b and c have been observed”,
while the hypothesis is that these form a unit of
some particular type). From Bayes’ rule
(12)
we need (1) the prior probability of the data
P(D), and (2) the prior probability of the
hypothesis P(H). Estimating P(D) seems
hopelessly difficult, but if we know the relative
frequency of different unit types and if we were
only interested in comparing P(H|D) for
different hypotheses H, then P(D) merely
becomes a normalizing constant.
Now the vehicles mentioned in D is a subset
of the total number of vehicles observed. It is an
hypothesis in its own right that the vehicles in D
together form a unit. There might be a different
method of carving out units which gives us a
different set of data D´ (for instance “vehicles a,
b and d have been observed”) which means that
we must be able to compare the different ways
of carving out objects which in turn means that
we cannot treat P(D) as a normalizing constant.
Remember: we are using the results from
classification to determine an optimal way of
clustering the data. The latter problem is not
addressed in [3].
There is a further possibility for the Bayesian
approach. If one can estimate the probability
that the data is a result of a hitherto unknown
unit type one can determine the value of P(H|D)
(as then one has an exhaustive list of
hypotheses). However without recourse to any
deeper analysis of the probability of an
unknown unit type one ends up applying a
method with a firm theoretical basis to
assumptions that are more or less arbitrary. It is
not obvious that anything has been gained by
such a move.
Combining classification and formation
conflict
The formation conflict of a group of vehicles is
treated as the average pairwise conflict of the
vehicles (as described in section 4.2). Again
there are other conceivable methods but this
particular method does not penalize larger units
which a number of other methods of
combination do.
The classification conflict c
0
and the
formation conflict c
1
using the probabilistic sum
giving the hypothesis conflict
. (13)
4.3.3 Partitioning the problem space
Two hypotheses A and B belong to the same
sub-problem if there are hypotheses H
0
, ..., H
n
such that A and H
0
are jointly inconsistent
(conflict), H
i
and H
i+1
are jointly inconsistent
for each i<n, and H
n
is inconsistent with B.
Thus A and B need not be jointly inconsistent to
belong to the same sub-problem, it is sufficient
that there is a chain of inconsistency linking
them.
The algorithm used for partitioning the
problem space is fairly obvious: take any
hypothesis and place it and all the hypotheses
that are inconsistent with it in a partition, repeat
the process for each new hypothesis added to
the partition until no new hypotheses are added.
The result is a partition. Take one of the
remaining hypotheses and construct a new
partition using the same method and repeat until
all hypotheses have been added to some
partition, Figure 14. The complexity of the
algorithm is O(n
2
) where n is the number of
hypotheses.
P
HD()
PH
()
PDH
()
PD()
------------------------------
----
=
CH
()
11c
0
–
()
1 c
1
–
()
–=
13
Figure 14: A problem space that has been
divided into three sub-problems. Each x
represents a vehicle.
4.3.4 Generating maximal consistent sets
For each sub-problem one generates the set of
maximal and consistent sets of hypotheses and
selects the best such set. The complexity of the
problem is reduced by the fact that to check
whether a set of hypotheses is consistent it is
enough to check all pairs of hypotheses are
jointly consistent (this is due to the particular
nature of the hypotheses).
The algorithm employed can be described as
a recursive procedure F, Figure 15, that takes
three lists of hypotheses and returns an optimal,
maximal and consistent list of hypotheses. The
initial call to F is F(nil, S, nil), where S is a
partition of the problem space.
Figure 15: Procedure F.
The conflict value of a set or list of hypotheses
S ={H
0
, ..., H
m
} is given by eq. (14)
. (14)
The above algorithm has an upper bound of
complexity O(kn
4
) where n is the number of
hypothesis and k some constant.
5 Results
5.1 Quantitative analysis
First, let us investigate aggregation of
intelligence reports into vehicles. After running
our aggregation algorithm we end up with the
following. The blue side: five armored
personnel carriers (tracked), three unspecified
tracked vehicles and two anti-tank robot
vehicles. The red side: four armored personnel
carriers (tracked), one unspecified tracked
vehicle and four MBT (see Figure 16).
Figure 16: All vehicles in IS Mark.
Hence the aggregation algorithm concludes that
there are ten blue vehicles, when there are
actually 14 vehicles within sight (of the 17 in
the scenario). This is due to the fact that the
algorithm is based on minimizing conflicts
between reports and that ten vehicles are
sufficient to cause all blue reports. The
algorithm also concludes nine red vehicles,
when there are actually eight within sight. Here
eight vehicles could have explained the reports,
but that was deemed unlikely (too close to
conflict).
The next step is to aggregate the vehicles into
platoons. On the blue side, three armored
personnel carriers (tracked) and one unspecified
tracked vehicle are aggregated into a
mechanized platoon, the two anti-tank robot
vehicles and the two remaining unspecified
tracked vehicles are aggregated into a anti-tank
robot platoon, while the final two armored
x
x
x
x
x
x
x
x
xx
x
x
x
x
x
x
x
x
x
x
x
x
F(CURRENT, REST, BEST):
IF for every object there is an hypothesis in CURRENT
where the object occurs
THEN RETURN CURRENT or BEST depending on whic
h
has lowest conflict
ELSE
FOR each H
i
in REST
CURRENT(H
i
) : = APPEND(CURRENT, H
i
);
REST(H
i
) : = {H
j
| H
j
is consistent with
CURRENT(H
i
) and i < j};
BEST : = F(CURRENT(H
i
), REST(H
i
), BEST);
RETURN BEST
C
S() 11CH
0
()–[]
…
1 CH
m
()–[]–=
14
personnel carriers (tracked) remain
unaggregated. On the red side, the four MBTs
and one unspecified tracked vehicle are
aggregated into an MBT Platoon and the four
armored personnel carriers (tracked) are
aggregated into a mechanized platoon (see
Figure 17).
Figure 17: All platoons in IS Mark.
5.2 Limitations and possibilities
There is an inherent problem in the methods
described above: what if there are several
different ways of clustering objects that are
roughly equal in value? Our experience shows
that this is more or less the normal state of
affairs and not the exception. Choosing one of
these and presenting it as the truth, or as the
description of the situation that is most likely to
be true given the evidence, will be more or less
arbitrary.
It is not a new problem: how should one
present uncertain conclusions when there are
rival conclusions to be drawn? The problem is
compounded by the time critical nature of the
endeavour. A user will often not have time for a
deep analysis of the material available: if he did
he would probably have limited use of a method
for automatic force aggregation to begin with.
Thus one cannot get around the problem by
saying that no automatic decision about what
hypotheses to present should be made at all.
Even if there is a risk that the wrong conclusion
is drawn, this risk must be taken.
6 Conclusions
In this papers we have shown that it is possible
to perform automatic force aggregation for the
lowest levels (intelligence reports to vehicles,
and vehicles to platoons) by clustering and
classification using only conflicts between
elements.
For the future we aim to study higher levels
in the hierarchy. We believe it will then be
necessary to use both positive relations (reasons
elements might belong together) as well as the
negative conflicting relations used in this paper
for clustering together with more advanced
templates for classification.
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