Copula
Copula is a word in the English language. It has applications in linguistics, music, and probability theory. In linguistics, it refers to a word or phrase that links the subject of a sentence to a subject complement, such as the word is in the sentence "The sky is blue" or the phrase was not being in the sentence "It was not being cooperative." Another word for copula is linking verb, referring to a copula's role of linking a subject with its predicate. The word copula derives from the Latin noun for a "link" or "tie" that connects two different things.[1][2] In statistics, copulas are multivariate cumulative distribution functions for which the marginal probability distribution of each variable is uniform on the interval [0, 1]. [3]
The word copula dates back to 1619 and comes from the Latin word with the same spelling, meaning "link" or "tie." It is formed with the prefix co- ("together") and the root word apere ("to fasten"). Another common English word, couple, also derives from the Latin word copula. In English, the general meaning of the word "copula" is "something that connects." [4]
Linguistics
[edit]In linguistics, a copula, also called a linking verb or copular verb, connects the subject of a sentence to the predicate. All forms of the verb to be (be, is, was, were) are copulas, as are other non-action verbs such as seem, become, and feel. [5] The verbs turn and become are copulas that denote change (i.e. "The leaves turn brown in autumn."), while stay and remain denote absence of change ("Wear a jacket so you can stay warm.")
Statistics
[edit]![](http://upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Gaussian_copula_gaussian_marginals.png/220px-Gaussian_copula_gaussian_marginals.png)
![](http://upload.wikimedia.org/wikipedia/commons/thumb/6/67/Biv_gumbel_dist.png/220px-Biv_gumbel_dist.png)
In statistics and probability theory, a copula is a function that represents the relationship between two or more variables, independent of the individual marginal distributions of the variables.
Consider a random vector . Suppose its marginals are continuous, i.e. the marginal CDFs are continuous functions. By applying the probability integral transform to each component, the random vector
has marginals that are uniformly distributed on the interval [0, 1].
The copula of is defined as the joint cumulative distribution function of :
The copula C contains all information on the dependence structure between the components of whereas the marginal cumulative distribution functions contain all information on the marginal distributions of .
The reverse of these steps can be used to generate pseudo-random samples from general classes of multivariate probability distributions. That is, given a procedure to generate a sample from the copula function, the required sample can be constructed as
The generalized inverses are unproblematic almost surely, since the were assumed to be continuous. Furthermore, the above formula for the copula function can be rewritten as:
In probabilistic terms, is a d-dimensional copula if C is a joint cumulative distribution function of a d-dimensional random vector on the unit cube with uniform marginals.[6]
In addition to the traditional approach of using correlation coefficients as a way to measure dependency between two variables, copula-based approaches are sometimes more effective at understanding the variables' joint distribution. [7]
Facts about the word
[edit]Copula is a beautiful, versatile word with many applications, including the aforementioned linguistics and statistics ones. It is a fairly common word; at the height of its usage, in 2016 and 2017, it had a frequency of 0.0000758% according to Google Ngram Viewer. [8] For every trillion words used in published books in 2017, "copula" (all lowercase) appeared 758,147 times, while "Copula" (capital C) appeared 90,389 times, and "COPULA" (all capitals) appeared 3,534 times. Of the 848,536 instances of the former two forms ("copula" and "Copula"), the lowercase version accounted for 89.3% of instances, and it was capitalized 10.7% of the time.
A September 2020 video about copulas, titled "English Grammar: Linking Verbs (Copula)", has 272,000 views on YouTube as of August 2024. [9]
References
[edit]- ^ See copula in the Online Etymology Dictionary for attestation of the use of the term, "copula", since the 1640s.
- ^ See the appendix to Moro 1997 and the references cited there for a short history of the copula.
- ^ Low, R.K.Y.; Faff, R.; Aas, K. (June 1, 2016). "Enhancing mean–variance portfolio selection by modeling distributional asymmetries" (PDF). Journal of Economics and Business. 85: 49–72. doi:10.1016/j.jeconbus.2016.01.003.
- ^ "Copula Definition and Meaning". April 13, 2016.
- ^ "What are copular verbs". English Grammar. November 15, 2010.
- ^ Nelsen, Roger B. (1999), An Introduction to Copulas, New York: Springer, ISBN 978-0-387-98623-4
- ^ "A Copula Approach to Volatility Index in India". September 3, 2014.
- ^ "Study shows increased awareness and usage of the word 'copula' in the past two years". January 2, 2018.
- ^ "English Grammar: Linking Verbs (Copula)". September 12, 2020.