Ordinal Value

An Ordinal Value is a set member of an ordered set.

• Context:
  • It can range from being a Coarse-Grained Ordinal Value to being a Fine-Grained Ordinal Value.
  • It can range from being an Abstract Ordinal Value to being an Ordinal Data Value.
  • It can be associated with: an Ordinal Variable; an Ordinal Data Attribute; an Ordinal-Output Function; Ranked Learning Record; ...
• Example(s):
  • Large ∈ {Large > Medium > Small};
  • Hard ∈ {Soft < Medium < Hard};
  • Second ∈ {First < Second < Third < … < Infinity} (members of cardinal numbers);
  • a Score Value;
  • a Dispersion Value.
  • …
• Counter-Example(s):
  • a Categorical Value, such as Red ∈ (Red, Yellow, Red, Black).
  • a Numerical Value.
  • a Continuous Number.
• See: Ordinal Space, Partial Order Relation, Ordinal Regression.

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References

2009a

• (WordNet, 2009) ⇒ http://wordnetweb.princeton.edu/perl/webwn?s=ordinal%20number
  • S: (n) ordinal number, ordinal, no. (the number designating place in an ordered sequence)

2009b

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Ordinal_number
  • In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals. Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated. The finite ordinals (and the finite cardinals) are the natural numbers: 0, 1, 2, …, since any two total orderings of a finite set are order isomorphic. The least infinite ordinal is ω which is identified with the cardinal number \aleph_0. However in the transfinite case, beyond ω, ordinals draw a finer distinction than cardinals on account of their order information. Whereas there is only one countably infinite cardinal, namely \aleph_0 itself, there are uncountably many countably infinite ordinals, namely ω, ω + 1, ω + 2, …, ω·2, ω·2 + 1, …, ω2, …, ω3, …, ωω, …, ωωω, …, ε0, … Here addition and multiplication are not commutative: in particular 1 + ω is ω rather than ω + 1, while 2·ω is ω rather than ω·2. The set of all countable ordinals constitutes the first uncountable ordinal ω1 which is identified with the cardinal \aleph_1 (next cardinal after \aleph_0). Well-ordered cardinals are identified with their initial ordinals, i.e. the smallest ordinal of that cardinality. The cardinality of an ordinal defines a many to one association from ordinals to cardinals.
  • Ordinals were introduced by Georg Cantor in 1897[1] to accommodate infinite sequences and to classify sets with certain kinds of order structures on them. [2]

2009c

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Ordinal_number_(linguistics)
  • In linguistics, ordinal numbers are the words representing the rank of a number with respect to some order, in particular order or position (i.e. first, second, third, etc.). Its use may refer to size, importance, chronology, etc. They are adjectives. They are different from the cardinal numbers (one, two, three, etc.) referring to the quantity. Ordinal numbers are alternatively written in English with numerals and letter suffixes: 1st, 2nd or 2d, 3rd or 3d, 4th, 11th, 21st, 477th, etc. In some countries, written dates omit the suffix, although it is nevertheless pronounced. For example: 4 July 1776 (pronounced "the fourth of July … "); July 4, 1776, ("July fourth ..."). When written out in full with "of", however, the suffix is retained: the 4th of July. In other languages, different ordinal indicators are used to write ordinal numbers.

2006

• (Li & Link 2006) ⇒ Ling Li, and Hsuan-Tien Lin. (2006). “Ordinal Regression by Extended Binary Classification.” In: Advances in Neural Information Processing Systems 19 (NIPS 2006).
  • QUOTE: In an ordinal regression problem, an example [math]\displaystyle{ (\mathbf{x}, y) }[/math] is composed of an input vector [math]\displaystyle{ {\bf x} \in \mathcal{X} }[/math] and an ordinal label (i.e., rank) [math]\displaystyle{ y \in \mathcal{Y} = {1, 2,...,K} }[/math]. Each example is assumed to be drawn i.i.d. from some unknown distribution [math]\displaystyle{ P({\bf x}, y) }[/math] on [math]\displaystyle{ \mathcal{X} \times \mathcal{Y} }[/math].

1998

• (Kohavi & Provost, 1998) ⇒ Ron Kohavi, and Foster Provost. (1998). “Glossary of Terms.” In: Machine Leanring 30(2-3).
  • Ordinal: A finite number of discrete values. The type nominal denotes that there is no ordering between the values, such as last names and colors. The type ordinal denotes that there is an ordering, such as in an attribute taking on the values low, medium, or high.

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