A major aim of the RCBP approach is to use a goal weight path representing a transition sequence of FO machines within the same network with a generated activity path as internal states to provide the desired FO associations for untrained as well as trained inputs. In principle, generalization in RCBP is very similar to that in FCBP. All the interpolation techniques applied in FCBP for approximating a goal weight state path can be considered and employed here for approximating a goal weight state path and activity state path.
Generalisation in RCBP can also be considered in both spatial and temporal aspects when an appropriate space-time scheme is introduced (refer to the discussions in §4.3.2).
However, discussion of these aspects would be very similar to FCBP. Consequently, discussion of generalisation technique is restricted here to inieipolation aspects.
The linear-interpolation technique (LIT) as a generalization regime described in FCBP (§4.5) has also been explored and implemented in RCBP. The implementation of the LIT regime has been carried out in the simulator cbptool (§7.5.5).
.Linear interpolation technique for RCBP
For approximating a goal weight state for an untrained FO pattern associated with an untrained position, a linear interpolation is employed. This uses two learnt weight states associated with two trained positions which are the neighbouring trained positions of the position associated with the untrained pattern along the training or generalization path. The approximated weight state for the untrained pattern is carried out by using the learnt weight states. If the untrained pattern is the untrained pattern for recognition out of k such patterns regularly spaced between two neighbouring trained patterns, the approximated weight state can be calculated where each component of the weight has the form: w ^ (w^ - w^ ), where w^ and are the values of corresponding component weights in the two trained weight states.
For approximating an activity state associated with an untrained position, a slightly more complex way of approximating can be done. The untiained activities will be computed in the same way as the trained activities. That is, in order to approximate the activity associated with the untrained position (i+k), 0<k<l (where k is the k^ untrained pattern for recognition out of n such patterns regularly spaced between two neighbouring trained activity states A/ and A/+/), the previous activity state at position (i-J+k) is used in conjunction with the weight state and environmental input at (i+k). A recursive approximation of the previous activity would involve all previous moments and hence be time consuming. Instead, an interpolation similar to that for the weights is used based on
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the trained activities at (/-/) and i. That is, the activity state at is calculated I according to :
where aj and a2 are the values of the corresponding component activity values in the two ti'ained activity states at (i-l) and i.
This rule of finding suitable training positions can be applied to all the untrained positions except the positions between the first and second training positions. For these positions, there is no previous activity information available. For approximating any untrained activity states between the first and second training positions, the generated learnt activity states at the first and second positions are used to directly interpolate the required activity state using the above expression.
5.5 Conclusions
The RCBP approach has been introduced in this chapter. Through a close look and discussion of the approach from tlie structure design to the inherent features of the model, it is clear that RCBP is similar to the FCBP approach in the sense that both approaches are based on paths, but RCBP is a very different approach in comparison with FCBP, by controlling the dynamics of activity states of units in networks, RCBP provides an approach of path-based dynamic system.
The major features of RCBP have been summarised in §5.2.4. A comparison with the features of PBP and FCBP is as follows:
(1) Like FCBP, RCBP is a sequential technique which allows arbitrary approximations of a set of continuous I/O associations within a given topology. RCBP provides a dynamic system with input in contrast to FCBP's FO function.
(2) A trained network in RCBP produces multiple FO associations in the form of desired ^ output paths when the two constraints similar to that in FCBP are obeyed. The first
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constraint is that each of the paths has the same number of the training positions. The second is that there are no one-many associations involved at the first training position. The desired output paths can be produced in reaction to independent switches in the input paths and associated internal state paths during performance though it is much more restricted in the independence than that in FCBP. The restriction on the independence switches is that in RCBP the switch cannot happen at each moment but after complete traversal of a training path.
(3) In providing a version of a path-based approach with dynamic system features, RCBP offers an extension of the FCBP approach. The temporal stmcture and internal states allow variable approximation within a conti olled and fixed size network with controllable internal states.