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Furthermore, requiring that A should also be self-dual leads to the following alternative characterization of invariant aggregation functions. Corollary 3. (10) An n-ary aggregation function A is invariant if and only if it is invariant under all involutive negators. Various characterizations of N -invariant aggregation functions, with N some involutive negator, have been proposed in (12). All of them invoke a [0, 1]2 → [0, 1] function C that allows to characterize the class of N -invariant aggregation functions in the following sense: A [0, 1]n → [0, 1] function A is an N -invariant aggregation function if and only if there exists an aggregation function B such that A(x1 , .

The former are essentially recursive implementations of off-line algorithms while in the latter case the off-line algorithms can be employed directly, with minor modifications. The approach followed here is the latter, the use of the off-line training method described in section 3, to a sliding window of data. Fig. 12. Evolution of the error. Fixed model. The following modifications were introduced to the off-line algorithm: • • • the data used for training, in instant k, is stored in a sliding window which is also updated in each instant, using a FIFO policy – the current pair input-target data replaces the oldest pair stored in the window; the initial values of the network parameters, for instant k, are the final values of the optimization conducted in instant k-1.

The second bisector. t. −id. t. a given monotone [0, 1] → [0, 1] bijection Φ. For instance, suppose that Φ contains part of a circle with center (x0 , y0 ) belonging to F . There does not exist a unique straight line perpendicular to Φ that contains (x0 , y0 ). To overcome this problem we have introduced in (11) the Φ-inverse of a set F : F Φ := {(x, y) ∈ [0, 1]2 | (Φ−1 (y), Φ(x)) ∈ F }. This definition has the following geometrical interpretation. Through every point (x, y) ∈ F we draw a line parallel to the X-axis and a line parallel to the Y-axis.

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