copula

(kŏp'yə-lə)
n.

1. A verb, such as a form of be or seem, that identifies the predicate of a sentence with the subject. Also called linking verb.
2. Logic. The word or set of words that serves as a link between the subject and predicate of a proposition.

[Latin cōpula, link.]

copular cop'u·lar (-lər) adj.

A function that relates a joint cumulative distribution function to the distribution functions of the individual variables. If the individual distribution functions are known, but the joint distribution is unknown, then a copula can be used to suggest a suitable form for the joint distribution.

Let F be the multivariate distribution function for the random variables X₁, X₂,..., X_n and let the cumulative distribution function of X_j be F_j (for all j). Define random variables U₁, U₂,..., U_n by U_j=F_j(X_j) for each j, so that the marginal distribution of each U_j has a continuous uniform distribution in the interval (0, 1). Assume that for each value u_j there is a unique value x_j=F⁻¹(u_j) and let the joint cumulative distribution function of U₁, U₂,..., U_n be C. Then

, for all u₁, u₂,..., u_n in (0, 1), since U_j<u_j if and only if X_j < F_j^-1(u_j). The function C is called the copula. An equivalent equation to the above isC{F₁(x₁), F₂(x₂),..., F_n(x_n)}=F(x₁, x₂,..., x_n),for all x₁, x₂,..., x_n , where u_j=F_j(x_j) for each j. Sklar's theorem, formulated by Abe Sklar of the Illinois Institute of Technology and published in 1959, states that, for a given F, there is a unique C such that this equation holds.

Note that it may well be that it is not possible to express the inverse functions F_j^-1 in a simple form (an example is the multivariate normal distribution).

Assuming that the copula and the marginal distribution functions are differentiable, the corresponding result for probability density functions is thatf(x₁, x₂,..., x_n)=c{F₁(x₁), F₂(x₂),..., F_n(x_n)} f₁(x₁)f₂(x₂)...f_n(x_n).The trivial case where c {F₁(x₁), F(x₂),..., F(x_n)}=1 corresponds to the case where the n X-variables are independent. Thus the copula encapsulates the interdependencies between the X-variables and is therefore also known as the dependence function. If c(u₁, u₂,..., u_n) is the joint probability density function of U₁, U₂,..., U_n, thenc(u₁, u₂,..., u_n)=f(x₁, x₂,..., x_n)/{f₁(x₁)f₂(x₂)...f_n(x_n)},where


, for each j.

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In traditional logic a proposition not only contains a subject and a predicate, but also a coupling device or copula (is, are, is not, are not) binding them together. It is generally held that very different things are covered by the notion, such as the ‘is’ of predication (rain is wet), that of identity (Marilyn Monroe is Norma Jean), and that of composition (this statue is marble). The need for a copula arises from a picture in which all the other elements of a proposition are complete ‘terms’, so they need some glue to bind them together into a sentence. Since Frege it has been more common to think of the predicate in a subject-predicate sentence on the model of an expression for a mathematical function, which needs no coupling device to bind it to its argument. No such connection has to be postulated, any more than there is a connection between the square function and 2 in the expression 2².

Any connecting part or structure.

Dansk (Danish)
n. - kopula

Nederlands (Dutch)
koppelwerkwoord

Français (French)
n. - copule

Deutsch (German)
n. - Kopula, (Satzband)

Ελληνική (Greek)
n. - (ανατ.) σύνδεσμος, (γραμμ.) συνδετικό ρήμα

Italiano (Italian)
copula

Português (Portuguese)
n. - cópula (f), cartilagem (f) (Anat.)

Русский (Russian)
связка

Español (Spanish)
n. - cópula

Svenska (Swedish)
n. - kopula

中文（简体）(Chinese (Simplified))
连系辞, 接合部, 介体

中文（繁體）(Chinese (Traditional))
n. - 連系辭, 接合部, 介體

한국어 (Korean)
n. - 계합사, 접합부, (남녀의) 교접

日本語 (Japanese)
n. - 連辞, 繋辞

العربيه (Arabic)
‏(الاسم) وصله, الفعل الرابط بين مبتدأ و خبر‏

עברית (Hebrew)
n. - ‮אוגד (בדקדוק), מילת-קישור‬

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