Abstract
We shall now comment upon some other problems, considered in the broadly conceived domain of data analysis, as seen in the perspective of the bi-partial objective function . We shall show the applicability of the concept of bi-partial objective function to these (and, indeed, yet other) problems with, whenever appropriate, illustrations for the potential form of the respective objective function.
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Notes
- 1.
One would expect a realistic empirical distribution to approximate some known forms, while here it is, apparently, “haphazard”.
- 2.
Actually, the proper Lorenz curve, used to represent the distribution of wealth or income (xi) in some population, introduced in 1905 by M. O. Lorenz, concerns the normalised values, so that it always starts from 0 and ends with 1.
- 3.
We use the classical name of the k-means algorithm, although throughout this book the number of clusters, referred to in this name as “k”, is denoted p.
- 4.
- 5.
Although the rules extracted take usually the form of “if… then…” clauses, the reference to causal implication is often explicitly avoided. The expressions, appearing in the rules are referred to, alternatively, respectively as: “antecedent”, “premise” or “condition”, and “consequent”, “conclusion”, “decision” or “hypothesis”.
- 6.
It should be noted that simplicity of rules is closely associated with their “intuitive appeal” and with practicability (effectiveness) of use in many circumstances (e.g. when they are used or at least consulted by human operators).
- 7.
It is common to treat the minimum length problem and approach as (practically) indistinguishable from the minimum description length (MDL) problem formulation, first coined in by Rissanen (1978). The thin distinction—in the opinion of this author—is that MDL looks for the minimum length ‘coding’ of a definite set of data items, rather than for the simplest and still effective model (and is thus quite analogous to the rule extraction problem of Sect. 4.6). A very telling quotation from Grünwald (1998) on MDL says that it is “based on the following insight: any regularity in a given set of data can be used to compress the data, i.e. to describe it using fewer symbols than needed to describe the data literally”. Because of the quite close meaning and possibility of different interpretations, MDL is also used in the clustering context, see, e.g., Figueiredo, Leitão, and Jain (1999), or Böhm et al. (2006), in a way very much like that of MML. Actually, the two examples, provided in this section, can be interpreted in the perspective of both MML and MDL.
- 8.
Another case of interest, which can be effectively treated in the framework here proposed, is the one of a sports tournament, in which teams play in pairs, achieving various scores in their matches, but not all pairs of teams actually meet and play with each other.
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Owsiński, J.W. (2020). Formulations and Rationales for Other Problems in Data Analysis. In: Data Analysis in Bi-partial Perspective: Clustering and Beyond. Studies in Computational Intelligence, vol 818. Springer, Cham. https://doi.org/10.1007/978-3-030-13389-4_4
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