Covariance matrix

In probability theory and statistics, a covariance matrix is a square matrix giving the covariance between each pair of elements of a given random vector.
Intuitively, the covariance matrix generalizes the notion of variance to multiple dimensions. As an example, the variation in a collection of random points in two-dimensional space cannot be characterized fully by a single number, nor would the variances in the and directions contain all of the necessary information; a matrix would be necessary to fully characterize the two-dimensional variation.
Any covariance matrix is symmetric and positive semi-definite and its main diagonal contains variances.
The covariance matrix of a random vector is typically denoted by, or.

Definition

Throughout this article, boldfaced unsubscripted and are used to refer to random vectors, and Roman subscripted and are used to refer to scalar random variables.
If the entries in the column vector
are random variables, each with finite variance and expected value, then the covariance matrix is the matrix whose entry is the covariance
where the operator denotes the expected value of its argument.

Conflicting nomenclatures and notations

Nomenclatures differ. Some statisticians, following the probabilist William Feller in his two-volume book An Introduction to Probability Theory and Its Applications, call the matrix the variance of the random vector, because it is the natural generalization to higher dimensions of the 1-dimensional variance. Others call it the covariance matrix, because it is the matrix of covariances between the scalar components of the vector.
Both forms are quite standard, and there is no ambiguity between them. The matrix is also often called the variance-covariance matrix, since the diagonal terms are in fact variances.
By comparison, the notation for the cross-covariance matrix between two vectors is

Properties

Relation to the autocorrelation matrix

The auto-covariance matrix is related to the autocorrelation matrix by
where the autocorrelation matrix is defined as.

Relation to the correlation matrix

An entity closely related to the covariance matrix is the matrix of Pearson product-moment correlation coefficients between each of the random variables in the random vector, which can be written as
where is the matrix of the diagonal elements of .
Equivalently, the correlation matrix can be seen as the covariance matrix of the standardized random variables for.
Each element on the principal diagonal of a correlation matrix is the correlation of a random variable with itself, which always equals 1. Each off-diagonal element is between −1 and +1 inclusive.

Inverse of the covariance matrix

The inverse of this matrix,, if it exists, is the inverse covariance matrix, also known as the precision matrix.
Just as the covariance matrix can be written as the rescaling of a correlation matrix by the marginal variances:
So, using the idea of partial correlation, and partial variance, the inverse covariance matrix can be expressed analogously:
This duality motivates a number of other dualities between marginalizing and conditioning for Gaussian random variables.

Basic properties

For and, where is an -dimensional random variable, the following basic properties apply:

is positive-semidefinite, i.e.
is symmetric, i.e.
For any constant matrix and constant vector, one has
If is another random vector with the same dimension as, then where is the cross-covariance matrix of and.
Block matrices

The joint mean and joint covariance matrix of and can be written in block form
where, and.
and can be identified as the variance matrices of the marginal distributions for and respectively.
If and are jointly normally distributed,
then the conditional distribution for given is given by
defined by conditional mean
and conditional variance
The matrix is known as the matrix of regression coefficients, while in linear algebra is the Schur complement of in.
The matrix of regression coefficients may often be given in transpose form,, suitable for post-multiplying a row vector of explanatory variables rather than pre-multiplying a column vector. In this form they correspond to the coefficients obtained by inverting the matrix of the normal equations of ordinary least squares.

Partial covariance matrix

A covariance matrix with all non-zero elements tells us that all the individual random variables are interrelated. This means that the variables are not only directly correlated, but also correlated via other variables indirectly. Often such indirect, common-mode correlations are trivial and uninteresting. They can be suppressed by calculating the partial covariance matrix, that is the part of covariance matrix that shows only the interesting part of correlations.
If two vectors of random variables and are correlated via another vector, the latter correlations are suppressed in a matrix
The partial covariance matrix is effectively the simple covariance matrix as if the uninteresting random variables were held constant.

Standard deviation matrix

The standard deviation matrix is the extension of the standard deviation to multiple dimensions. It is the symmetric square root of the covariance matrix.

Covariance matrix as a parameter of a distribution

If a column vector of possibly correlated random variables is jointly normally distributed, or more generally elliptically distributed, then its probability density function can be expressed in terms of the covariance matrix as follows
where and is the determinant of, the so-called generalized variance.

Covariance matrix as a linear operator

Applied to one vector, the covariance matrix maps a linear combination c of the random variables X onto a vector of covariances with those variables:. Treated as a bilinear form, it yields the covariance between the two linear combinations:. The variance of a linear combination is then, its covariance with itself.
Similarly, the inverse covariance matrix provides an inner product, which induces the Mahalanobis distance, a measure of the "unlikelihood" of c.

Admissibility

From basic property 4. above, let be a real-valued vector, then
which must always be nonnegative, since it is the variance of a real-valued random variable, so a covariance matrix is always a positive-semidefinite matrix.
The above argument can be expanded as follows:where the last inequality follows from the observation that is a scalar.
Conversely, every symmetric positive semi-definite matrix is a covariance matrix. To see this, suppose is a symmetric positive-semidefinite matrix. From the finite-dimensional case of the spectral theorem, it follows that has a nonnegative symmetric square root, which can be denoted by M^1/2. Let be any column vector-valued random variable whose covariance matrix is the identity matrix. Then

Complex random vectors

The variance of a complex scalar-valued random variable with expected value is conventionally defined using complex conjugation:
where the complex conjugate of a complex number is denoted ; thus the variance of a complex random variable is a real number.
If is a column vector of complex-valued random variables, then the conjugate transpose is formed by both transposing and conjugating. In the following expression, the product of a vector with its conjugate transpose results in a square matrix called the covariance matrix, as its expectation:
The matrix so obtained will be Hermitian positive-semidefinite, with real numbers in the main diagonal and complex numbers off-diagonal.
;Properties

The covariance matrix is a Hermitian matrix, i.e..
The diagonal elements of the covariance matrix are real.
Pseudo-covariance matrix

For complex random vectors, another kind of second central moment, the pseudo-covariance matrix is defined as follows:
In contrast to the covariance matrix defined above, Hermitian transposition gets replaced by transposition in the definition.
Its diagonal elements may be complex valued; it is a complex symmetric matrix.

Estimation

If and are centered data matrices of dimension and respectively, i.e. with n columns of observations of p and q rows of variables, from which the row means have been subtracted, then, if the row means were estimated from the data, sample covariance matrices and can be defined to be
or, if the row means were known a priori,
These empirical sample covariance matrices are the most straightforward and most often used estimators for the covariance matrices, but other estimators also exist, including regularised or shrinkage estimators, which may have better properties.

Applications

The covariance matrix is a useful tool in many different areas. From it a transformation matrix can be derived, called a whitening transformation, that allows one to completely decorrelate the data or, from a different point of view, to find an optimal basis for representing the data in a compact way.
This is called principal component analysis and the Karhunen–Loève transform.
The covariance matrix plays a key role in financial economics, especially in portfolio theory and its mutual fund separation theorem and in the capital asset pricing model. The matrix of covariances among various assets' returns is used to determine, under certain assumptions, the relative amounts of different assets that investors should or are predicted to choose to hold in a context of diversification.

Use in optimization

The evolution strategy, a particular family of Randomized Search Heuristics, fundamentally relies on a covariance matrix in its mechanism. The characteristic mutation operator draws the update step from a multivariate normal distribution using an evolving covariance matrix. There is a formal proof that the evolution strategy's covariance matrix adapts to the inverse of the Hessian matrix of the search landscape, up to a scalar factor and small random fluctuations.
Intuitively, this result is supported by the rationale that the optimal covariance distribution can offer mutation steps whose equidensity probability contours match the level sets of the landscape, and so they maximize the progress rate.