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In this short communication, we will show that a quasi-arithmetic mean induced by an additive generator of a strict t-norm (strict t-conorm, respectively) can be represented by the composition of a unary operator (called the tau function) and the strict t-norm (strict t-conorm, respectively), both induced by the same generator function as the quasi arithmetic mean. Here, we will also state a connection between the idempotency of a transformed strict t-norm (strict t-conorm, respectively) and the tau function. [ABSTRACT FROM AUTHOR]

Highlights: [•] Bayes error expressed using the total variation distance. [•] Chernoff bound interpreted as the minimization of a geometric weighted mean. [•] Chernoff bound construction generalized using quasi-arithmetic weighted means. [Copyright &y& Elsevier]

Abstract: In many practical applications, the need arises to aggregate data of varying dimension. Following from the self-identity property, some recent studies have looked at the stability of aggregation operators in terms of their behavior as the dimensionality is increased from to n. We use the penalty-based representation of aggregation functions in order to investigate the conditions for weighting vectors associated with some important weighted families, extending on the results already established for quasi-arithmetic means. In particular, we obtain results for quasi-medians and functions that involve a reordering of the inputs such as the OWA and order statistics. [Copyright &y& Elsevier]

Abstract: This article discusses Lipschitz properties of generated aggregation functions. Such generated functions include triangular norms and conorms, quasi-arithmetic means, uninorms, nullnorms and continuous generated functions with a neutral element. The Lipschitz property guarantees stability of aggregation operations with respect to input inaccuracies, and is important for applications. We provide verifiable sufficient conditions to determine when a generated aggregation function holds the k-Lipschitz property, and calculate the Lipschitz constants of power means. We also establish sufficient conditions which guarantee that a generated aggregation function is not Lipschitz. We found the only 1-Lipschitz generated function with a neutral element . [Copyright &y& Elsevier]

Abstract: This article studies a large class of averaging aggregation functions based on minimizing a distance from the vector of inputs, or equivalently, minimizing a penalty imposed for deviations of individual inputs from the aggregated value. We provide a systematization of various types of penalty based aggregation functions, and show how many special cases arise as the result. We show how new aggregation functions can be constructed either analytically or numerically and provide many examples. We establish connection with the maximum likelihood principle, and present tools for averaging experimental noisy data with distinct noise distributions. [Copyright &y& Elsevier]

Abstract: This article examines the construction of aggregation functions from data by minimizing the least absolute deviation criterion. We formulate various instances of such problems as linear programming problems. We consider the cases in which the data are provided as intervals, and the outputs ordering needs to be preserved, and show that linear programming formulation is valid for such cases. This feature is very valuable in practice, since the standard simplex method can be used. [Copyright &y& Elsevier]

Abstract: A variant of Jensen’s operator inequality for convex functions, which is a generalization of Mercer’s result, is proved. Obtained result is used to prove a monotonicity property for Mercer’s power means for operators, and a comparison theorem for quasi-arithmetic means for operators. [Copyright &y& Elsevier]

Standard ordering of aggregation operators in several distinguished classes is characterized. Lower and upper bounds of (finite) systems in those classes of aggregation operators are discussed. Several examples are given. [Copyright &y& Elsevier]