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Many classes of aggregation functions we have considered so far provide a large arsenal of tools to be used in specific applications. The parameters of various families and the vectors of weights can be adjusted to fit numerical data. Yet for certain applications these classes are not flexible enough to be fully consistent with the desired inputs-outputs. Sometimes the problem specification is not sufficient to make a call as to what type of aggregation function must be used.In this Chapter we consider an alternative construction, which is based almost entirely on the empirical data, but also incorporates important application specific properties if required. The resulting aggregation function does not belong to any specific family: it is a general aggregation function, which, according to Definition 1.5 is simply a monotone non-decreasing function satisfying f (0) = 0 and f (1) = 1. Furthermore, this function is not given by a simple algebraic formula, but typically as an algorithm. Yet it is an aggregation function, and for computational purposes, which is the main use of aggregation functions, it is as good as an algebraic formula in terms of efficiency.Of course, having such a "black-box" function is not as transparent to the user of a system, nor is it easy to replicate calculations with pen and paper. Still many such black-box functions are routinely used in practice, for example neural networks for pattern recognition.What is important, though, is that the outputs of such an aggregation function are always consistent with the data and the required properties. This is not easy to achieve using standard off-the-shelf tools, such as neural network libraries. The issue here is that no application specific properties are incorporated into the construction process, and as a consequence, the solution may fail to satisfy them (even if the data does). For example, the function determined by a neural network may fail to be monotone, hence it is not an aggregation function. Construction based on spline functions Problem formulationMonotone tensor product splines are defined asThe univariate basis functions are chosen to be linear combinations of standard B-splines, as in [13, 15, 32], which ensures that the conditions of 1 See p. 22 for definitions of various norms. 6.2 Construction based on spline functions 273 monotonicity of f are expressed as linear conditions on spline coefficients c j1j2...jn .The computation of spline coefficients (there are J 1 × J 2 × . . . × J n of them, where J i is the number of basis functions in respect to each variable) is performed by solving a quadratic programming problem minimize K k=1 ⎛ ⎝ J1 j1=1 . . . Jn jn=1 c j1j2...jn B j1 (x 1k ) . . . B jn (x nk

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