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. In statistics, copulas are multivariate cumulative distribution functions for which the marginal probability distribution of each variable is uniform on the interval [0, 1].

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."

Linguistics

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. 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

Density and Contour plot of a Bivariate Gaussian Distribution

Density and Contour plot of two Normal marginals joint with a Gumbel copula

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 $(X_{1},X_{2},\dots ,X_{d})$. Suppose its marginals are continuous, i.e. the marginal CDFs $F_{i}(x)=\Pr[X_{i}\leq x]$ are continuous functions. By applying the probability integral transform to each component, the random vector

$(U_{1},U_{2},\dots ,U_{d})=\left(F_{1}(X_{1}),F_{2}(X_{2}),\dots ,F_{d}(X_{d})\right)$

has marginals that are uniformly distributed on the interval [0, 1].

The copula of $(X_{1},X_{2},\dots ,X_{d})$ is defined as the joint cumulative distribution function of $(U_{1},U_{2},\dots ,U_{d})$:

$C(u_{1},u_{2},\dots ,u_{d})=\Pr[U_{1}\leq u_{1},U_{2}\leq u_{2},\dots ,U_{d}\leq u_{d}].$

The copula C contains all information on the dependence structure between the components of $(X_{1},X_{2},\dots ,X_{d})$ whereas the marginal cumulative distribution functions $F_{i}$ contain all information on the marginal distributions of $X_{i}$.

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 $(U_{1},U_{2},\dots ,U_{d})$ from the copula function, the required sample can be constructed as

$(X_{1},X_{2},\dots ,X_{d})=\left(F_{1}^{-1}(U_{1}),F_{2}^{-1}(U_{2}),\dots ,F_{d}^{-1}(U_{d})\right).$

The generalized inverses $F_{i}^{-1}$ are unproblematic almost surely, since the $F_{i}$ were assumed to be continuous. Furthermore, the above formula for the copula function can be rewritten as:

$C(u_{1},u_{2},\dots ,u_{d})=\Pr[X_{1}\leq F_{1}^{-1}(u_{1}),X_{2}\leq F_{2}^{-1}(u_{2}),\dots ,X_{d}\leq F_{d}^{-1}(u_{d})].$

In probabilistic terms, $C:[0,1]^{d}\rightarrow [0,1]$ is a d-dimensional copula if C is a joint cumulative distribution function of a d-dimensional random vector on the unit cube $[0,1]^{d}$ with uniform marginals.

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.