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depends on the initial positions of the particles. Consider a swarm which is initialized at
a local optimum. Once a better fitness value is found, global best fitness will change. But
this may not happen in the few iterations that are observed. Therefore the swarm is ini-
tialized by a pivot element chosen from a set of evaluated points.
Incremental Swarming
considers a set of
k
= 100
evaluated solutions and the position which evaluates to the
worst fitness value is chosen as pivot element. This is important because if we choose
a pivot element randomly, it is possible to find a local optimum and the behavior of the
swarm results in no movement. The other particle is initialized in a defined distance to
the pivot element. Similarly to
Incremental Probing
we use increments to define the dis-
tance between the particles. The increment values
ε
1
=1
,
ε
2
=2
,
ε
5
=5
and
ε
10
=10
are used to create 20 features. For each increment the feature
Swarming Slope
de-
scribes the development of the global best fitness as a linear model that fits the relation-
ship between the iteration and the global best fitness value (see figure 4). For the feature
μ
IS.Slope
the slope of the straight line is divided by the spread of the global best fitness.
Swarming Max Slope
describes the greatest change of the global best fitness value
between two successive iterations. For normalization the value of
μ
IS.Max
is divided
by the spread of the global best fitness. The other three features, which are computed
for each increment, are
Swarming Delta Lin
μ
IS.Lin
,
Swarming Delta Phi
μ
IS.Phi
,and
Swarming Delta Sgm
μ
IS.Sgm
. They describe to what degree the ob-
served development of the global best fitness value differs from sequences that represent
idealistic developments.
Swarming Delta Lin
implies a measure of linearity, thus
quantifies how much the observed development differs from a linear decrease of the
global best fitness. Let
=
x
0
,...,x
t
denote the observed sequence of the global
best fitness value. We compute this feature with equation 4.
x
t
x
i
−
2
k
t
−
i
+1
μ
IS.Lin
=
(4)
t
−
1
i
=0
Similarly we create two additional ideal sequences and compute the features
μ
IS.Phi
and
μ
IS.Sgm
by the equations 5-6:
k
x
i
−
φ
i−
1
2
μ
IS.Phi
=
(5)
i
=0
x
i
−
2
k
1
1+exp
(
i−
1)
φ
μ
IS.Sgm
=
(6)
i
=0
2
where
φ
=
1+
√
5
. The factor
φ
was selected in order to mediate between a linear and an
exponential developing. The development of the global best fitness is used to calculate
the features of
Incremental Swarming
. The pivot element for the initialization of the
swarm is chosen from a set of
k
solutions and since the swarm of
m
=2
particles is
applied for
t
=20
iterations, overall there are
k
+
m
+
mt
evaluations of the objective
function. We choose the pivot element from the set
M
X
which was created for the
features of
Random Probing
. By this we reduce the number of additional evaluation to
m
+
mt
=42
.
