Discrete Choquet Integral as a Distance Metric

Insights and characterization of l<inf>1</inf>-norm based sparsity learning of a lexicographically encoded capacity vector for the Choquet integral

“…It has been used in different domains and problems such as image processing [12], multi-criteria decision making [13], skeletal age-at-death estimation in forensic anthropology [14], multi-source (e.g., feature, algorithm, sensor, confidence) fusion [15,16], used as a distance metric [17], classification [18], and pattern recognition [19,20]. The FI is most often used to combine the (often objective) support in some hypothesis, e.g., algorithm outputs or confidences, from multiple inputs with the (often subjective) worth of the different subsets of sources, encoded in a FM.…”

“…In [28], they showed that the discrete CI defines a metric if the corresponding measure satisfies certain monotonicity constraints, thereby completely characterizing the class of measures that induce a metric with the CI. In addition, the kernel-trick is a well-known way to map data from lower dimensions into higher dimensions in order to measure the similarity (inner product) of the data elements without ever explicitly performing the mapping.…”

“…shown that the -norm versus -norm leads to sparser models that can often be (more) easily interpreted [28]. In general, the -norm does not promote sparsity.…”

Decision Making in Reinsurance with Induced OWA Operators and Minkowski Distances

“…Note that in the literature there are a variety of new operators with distances among which are the OWA distance (Merigó & Gil-Lafuente, 2010, Xu & Chen, 2008, continuous distances (Zhou et al 2013), distances with Choquet integral (Bolton et al, 2008), the probabilistic weighted averaging distance (PWAD) (Merigó, 2013), the immediate probabilities (Merigó & Gil-Lafuente, 2012), the probabilistic OWA distance ) and moving distances (Merigó & Yager, 2013). Other models have focused on the use of imprecise information with distances including intervals numbers (Zeng, 2013a) and fuzzy numbers (Su et al 2013;Xu, 2013;Zeng, 2013b).…”

Indicators for the characterization of discrete Choquet integrals

“…Klement et al [13] presented a universal integral that covers the Choquet and the Sugeno integral for non-negative functions, while Torra and Narukawa [32] studied a generalization of the Choquet integral inspired by the Losonczi mean. Bolton et al [3] connected the Choquet integral with distance metrics and, more recently, Torra and Narukawa [33] introduced an operator that generalizes the Choquet integral and the Mahalanobis distance.…”