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Fig. 2. Generalized recognition of trained and untrained sequences.
The PB values of the
two trained 2-D time series using Eq. 8 with
θ
=90
and
θ
= 180
, respectively, are marked
using white circles. In contrast, the PB values obtained by feeding the network with untrained
sequences generated with varying
θ
values are drawn as black dots. These values are arranged in
a structured way, emphasizing the self-organization of the PB space.
the network is able to capture a reciprocal relationship between a time series and its
associated PB value.
These generalized recognition and generation capabilities of the modified RNNPB
are demonstrated in a more complex example. For this purpose, consider the 2-D sinu-
soidal sequences described by the following equation:
x
1
x
2
=0
.
5
cos θ
sin
π·t
6
cos
π·t
12
sin θ
sin θ cos θ
−
(8)
Plotting
x
1
vs.
x
2
results in a figure-eight shape that is rotated according to the angular
value specified by
θ
. Two 2-D time series of length
t
=25
were generated using
θ
=90
and
θ
= 180
, respectively. In contrast to the robot experiments presented below, the
network only has a single PB unit.
After training, the network is able to recognize those sequences based on their trained
PB values (PB
θ
:90
=
0
.
65604
), which differ only by a
small amount (
θ
:90
=0
.
0005
and
θ
:180
=0
.
002
) from the ones obtained during
storage. The PB values of the two trained sequences are plotted in Fig. 2 using white
circular markers. Next to the trained sequences, the network is also fed with novel,
previously untrained, sequences. These are generated using Eq. 8 with varying
θ
values.
The network also generates stable PB values for those unknown sequences (black dots
in Fig. 2). It can be seen that the PB values are ordered according to the angular value of
the underlying time series. This reciprocal relationship can be used to infer the angular
value of a sequence generated with an unknown
θ
value. Thus, varying the PB value
results in a rotation of the figure-eight shape.
−
0
.
01107
and PB
θ
:180
=
−
