Where does covariance collide?

In probability and statistics, the covariance of two random variables is a single number measuring how they vary together: it is the average product of their departures from their own means, positive when high values of one tend to accompany high values of the other, negative when one rises as the other falls, and zero when there is no linear co-movement. Karl Pearson's 1896 Mathematical contributions to the theory of evolution put this product-moment measure of joint variation on a firm mathematical footing, the quantity later named covariance and, across many variables at once, gathered into a covariance matrix. The referent here is a computed number summarising the joint spread of two measured quantities, not a property of physical laws and not a rule about types in a program.[1]

Unit

One number in the squared units of the two variables, or an entry of a covariance matrix

Recovery means

Not applicable

Measured by

Estimated from paired data as the average product of each variable's deviations from its mean

In physics, covariance is the requirement that the equations expressing a physical law keep the same form when the coordinates used to describe them are changed: a law written covariantly reads the same for observers using different frames, so the physics does not hang on an arbitrary choice of coordinates. Einstein's 1916 Die Grundlage der allgemeinen Relativitätstheorie (The foundation of the general theory of relativity) built the theory on general covariance, expressing the laws through tensors that transform in a fixed way, and physicists mark quantities as covariant or contravariant by how their components change under such a transformation. The referent here is a symmetry property of the form of laws, and it has nothing to do with a number measuring how two quantities co-vary or with a direction of subtyping in a type system.[2]

Unit

A property of an equation: its form survives a class of coordinate transformations

Recovery means

Not applicable

Measured by

Checked by writing the law with tensors and confirming its form is preserved under the transformation

In programming language theory, covariance describes one way the subtyping of a compound type tracks the subtyping of the parts it is built from: a type constructor is covariant in a position when a subtype there yields a subtype of the whole, as a list of cats can stand in for a list of animals, and contravariant when the direction reverses, as it does for a function's argument. Cardelli's 1988 A semantics of multiple inheritance gave the rules under which record and function types become subtypes of one another, the analysis behind the variance annotations later found in typed languages. The referent here is a rule about how type relationships compose, and it is unrelated to a number summarising joint variability or to the form-invariance of physical laws.[3]

Unit

A direction, covariant or contravariant, for one position of a type constructor

Recovery means

Not applicable

Measured by

Determined by the subtyping rules of the type system for that constructor and position