Extension of bivariate means to weighted means of several arguments by using binary trees
Introduction
Aggregation functions play an important role in many applications including decision making, fuzzy systems and image processing [1], [2], [3]. Averaging functions, whose prototypical examples are the arithmetic mean and the median, allow compensation between low values of some inputs and high values of the others. Such functions are also important for building decision models in weighted compensative logic [4], where the concepts of Generalized Conjunction/Disjunction (GCD) play a role [5], [6].
The major class of averaging functions is the class of weighted quasi-arithmetic means (QAM) [1], [2]. These functions are well studied and are convenient to work with as they have a natural definition for any number of arguments. Yet there are many other means, that often generalize quasi-arithmetic means, which are defined with respect to two arguments only, and do not offer a straightforward multivariate extension. A basic example here is the logarithmic mean [7] which belongs to a rather broad class of Cauchy means. Cauchy means are defined with respect to two differentiable generating functions g and h such that h′ ≠ 0, by the use the Cauchy mean value theorem.
In turn, Cauchy means have several prominent subclasses, such as the Lagrangian mean, the generalized logarithmic mean, Stolarsky means, and also quasi-arithmetic means. Frequently used members include the already mentioned logarithmic means, the identric mean and the Stolarsky means.
Another class of bivariate means are the neo-Pythagorean means [7], which are defined in terms of ratios between the inputs and the outputs. Here again, no obvious multivariate extension is present. One particular case is the Heronian mean recently studied in several works [8], [9], [10], which can be “naturally” extended to the multivariate case in several different ways.
There have been numerous attempts to extend various classes of bivariate means to n variables, and to incorporate the importance weights into this process. In particular, the logarithmic mean and the Stolarsky means were extended in different ways [11], [12], [13], [14], [15], notably through their representations as definite integrals of certain functions. Another way of extending the logarithmic mean and some related functions is through a series representation [12]. Finally, the divided differences approach is also useful in this respect [12].
The mentioned extensions, some of which we briefly present in the sequel, are undoubtedly useful for various applications, but they are not sufficiently general as to apply them to an arbitrary bivariate mean, without using its specific properties or alternative representations. Such a generic approach for extending bivariate means would be valuable for applications in decision making, so that alternative aggregators could be tried and added to the arsenal of aggregation tools.
In this contribution we propose one generic approach for extending the bivariate means and incorporating weighting vectors based on repetitive application of the given bivariate function, originally reported in [16]. It does not require knowledge of the properties of the bivariate function or its alternative representations, and is not based on an analytic formula but on an efficient computational procedure.
Attempts using repetitive application of the bivariate functions to generate multivariate ones have been made in the past. Of course, associativity of the triangular norms and conorms, as well as uninorms and nullnorms allows their straightforward extension to the multivariate case [1], [2].
Recursive means were presented by Cutello and Montero in [17], [18] in the framework of OWA functions, and later used in [19]. These authors used a sequence of bivariate functions and the relation where fn denotes an n-variate function. In particular, taking we obtain the n-variate arithmetic mean fn. Weighted quasi-arithmetic means can be generated in a similar way.
The approach we present here is also a recursive application of the bivariate function but in a different order, by constructing a binary tree with a suitable number of levels, where at each node the bivariate function is applied to the arguments of the child nodes. By using the idempotency of the means, we prune this tree to design a computationally efficient procedure. On the other hand, we are able to incorporate the weighting vectors by repeating the arguments as needed, following the approach of Calvo et al. [20].
Our binary tree approach is generic in terms of the starting bivariate idempotent function being used, but it is not exact, in the sense that it is aimed at approximating a weighted multivariate mean (with any desired accuracy). Indeed, the binary tree will not reproduce exactly the weighting vectors with irrational coefficients, or coefficients that do not have finite binary representation (e.g., ), in a finite number of iterations. We argue, however, that for computational purposes this is not inferior than even the explicit formulas: after all, all weighting vectors have finite binary representation in machine arithmetic, which we can match exactly.
We would like to emphasize that unlike the existing multivariate extensions which are custom-built for a specific mean, our approach is applicable to an arbitrary bivariate symmetric mean, transparent to the user, and automatically preserves some very useful properties of that bivariate mean. The availability of a computationally efficient procedure for evaluating the multivariate weighted mean together with the generality of the proposed construction makes our approach broadly applicable in practice.
The remainder of this article is structured as follows. In Section 2, we provide the necessary mathematical foundations which we rely on in the subsequent sections. We discuss multivariate extension of several means in Section 3. In Section 4 we provide our main definitions and study the fundamental properties of our construction. In Section 5 we present the computational algorithms and discuss their complexity. We illustrate the efficiency of the algorithms numerically in Section 6. Our conclusions are presented in Section 7.
Section snippets
Preliminaries
Consider now the following definitions adopted from [1], [2], [3], [21]. Let although other intervals can be accommodated easily.
Definition 1 A function is monotone (increasing) if then f(x) ≤ f(y), with the vector inequality understood componentwise. Definition 2 A function is idempotent if for every input the output is . Definition 3 A function is averaging if for every x it is bounded by min (x) ≤ f(x) ≤ max (x).
Averaging functions are necessarily idempotent, and monotone
Multivariate extensions
Some of the mentioned bivariate means have been extended to several variables. In particular we have one extension when all xi are mutually different positive numbers [13], see also [11]. Mustonen [12] provides a recursive formula for
An alternative formula for extending the logarithmic mean is [14]:
The generalized logarithmic mean can be extended by
Binary tree construction
We want to approximate (or construct for this matter) a weighted n-variate idempotent function fn with the weighting vector w, by using only an unweighted bivariate idempotent function f [16]. To introduce the weights we use the approach from [20] where each argument xi is replicated a suitable number of times. That is, we consider an auxiliary vector of arguments so that x1 is taken k1 times, x2 is taken k2 times and so on, and and where L
Efficient binary tree algorithm for idempotent functions
One practical disadvantage of the ABL algorithm is that its computational complexity is O(2L) in terms of the number of invocations of f. This makes it much slower than an explicit formula like (6). However it is possible to appropriately prune the binary tree by relying on idempotency of f, see Figs. 1 and 3. Indeed no invocation of f is necessary if both of its arguments are equal. Such a pruning was presented in [16] in the special case of where the aim was to construct a weighted
Numerical experiments
It is interesting to appreciate numerical performance of the PTA algorithm in comparison to the explicit extensions such as (6). Of course, it would be rather counterproductive to evaluate functions such as quasi-arithmetic means using the binary tree construction, as the explicit formula would work much faster. Yet it makes sense to appreciate the speed of evaluation when the binary tree construction is applied to bivariate means without known multivariate extension. Such an estimate of
Conclusion
Idempotent functions are valuable for building decision support and fuzzy logic based systems. There is a number of bivariate means, which have no obvious multivariate extension, like the Cauchy means, or to the contrary, have several competing “natural” multivariate extensions, as we have seen on the example of Heronian mean. The existing analytical extensions are tailored to specific means and hence are not universal. Often there is no obvious way to incorporate weights into the mentioned
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