Covariance matrix

Definition

Throughout this article, boldfaced unsubscripted X and Y are used to refer to random vectors, and unboldfaced subscripted X_i and Y_i are used to refer to random scalars.

If the entries in the column vector

are random variables, each with finite variance, then the covariance matrix Σ is the matrix whose (i, j) entry is the covariance

where

is the expected value of the i th entry in the vector X. In other words,

The inverse of this matrix, Σ − 1 , if it exists, is the inverse covariance matrix, also known as the concentration matrix or precision matrix.

Generalization of the variance

The definition above is equivalent to the matrix equality

This form can be seen as a generalization of the scalar-valued variance to higher dimensions. Recall that for a scalar-valued random variable X

Indeed, the entries on the diagonal of the covariance matrix Σ are the variances of each element of the vector X .

Correlation matrix

A quantity closely related to the covariance matrix is the correlation matrix, the matrix of Pearson product-moment correlation coefficients between each of the random variables in the random vector X , which can be written

where diag ( Σ ) is the matrix of the diagonal elements of Σ (i.e., a diagonal matrix of the variances of X i for i = 1 , … , n ).

Equivalently, the correlation matrix can be seen as the covariance matrix of the standardized random variables X i / σ ( X i ) for i = 1 , … , n .

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.

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 Σ the variance of the random vector X , 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 X .

Both forms are quite standard and there is no ambiguity between them. The matrix Σ is also often called the variance-covariance matrix since the diagonal terms are in fact variances.

By comparison, the notation for the cross-covariance between two vectors is

Properties

For Σ = E [ ( X − E [ X ] ) ( X − E [ X ] ) T ] and μ = E ( X ) , where X is a random p-dimensional variable and Y a random q-dimensional variable, the following basic properties apply:

1. Σ = E ( X X T ) − μ μ T
2. Σ is positive-semidefinite and symmetric.
3. var ⁡ ( A X + a ) = A var ⁡ ( X ) A T
4. cov ⁡ ( X , Y ) = cov ⁡ ( Y , X ) T
5. cov ⁡ ( X 1 + X 2 , Y ) = cov ⁡ ( X 1 , Y ) + cov ⁡ ( X 2 , Y )
6. If p = q, then var ⁡ ( X + Y ) = var ⁡ ( X ) + cov ⁡ ( X , Y ) + cov ⁡ ( Y , X ) + var ⁡ ( Y )
7. cov ⁡ ( A X + a , B T Y + b ) = A cov ⁡ ( X , Y ) B
8. If X and Y are independent (or somewhat less restrictedly, if every random variable in X is uncorrelated with every random variable in Y ), then cov ⁡ ( X , Y ) = 0

where X , X 1 and X 2 are random p×1 vectors, Y is a random q×1 vector, a is a q×1 vector, b is a p×1 vector, and A and B are q×p matrices of constants.

This 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 (see Rayleigh quotient for a formal proof and additional properties of covariance matrices). This is called principal components analysis (PCA) and the Karhunen-Loève transform (KL-transform).

Block matrices

The joint mean μ X , Y and joint covariance matrix Σ X , Y of X and Y can be written in block form

where Σ X X = var ( X ) , Σ Y Y = var ( Y ) , and Σ X Y = Σ Y X T = cov ( X , Y ) .

Σ X X and Σ Y Y can be identified as the variance matrices of the marginal distributions for X and Y respectively.

If X and Y are jointly normally distributed,

then the conditional distribution for Y given X is given by

defined by conditional mean

and conditional variance

The matrix Σ_YXΣ_XX⁻¹ is known as the matrix of regression coefficients, while in linear algebra Σ_Y|X is the Schur complement of Σ_XX in Σ_X,Y

The matrix of regression coefficients may often be given in transpose form, Σ_XX⁻¹Σ_XY, suitable for post-multiplying a row vector of explanatory variables x^T rather than pre-multiplying a column vector x. In this form they correspond to the coefficients obtained by inverting the matrix of the normal equations of ordinary least squares (OLS).

As a parameter of a distribution

If a vector of n 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 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: c T Σ = cov ⁡ ( c T X , X ) . Treated as a bilinear form, it yields the covariance between the two linear combinations: d T Σ c = cov ⁡ ( d T X , c T X ) . The variance of a linear combination is then c T Σ c , its covariance with itself.

Similarly, the (pseudo-)inverse covariance matrix provides an inner product, ⟨ c − μ | Σ + | c − μ ⟩ which induces the Mahalanobis distance, a measure of the "unlikelihood" of c.

Complex random vectors

The variance of a complex scalar-valued random variable with expected value μ is conventionally defined using complex conjugation:

where the complex conjugate of a complex number z is denoted z ∗ ; thus the variance of a complex number is a real number.

If Z 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, as its expectation:

where Z † denotes the conjugate transpose, which is applicable to the scalar case since the transpose of a scalar is still a scalar. The matrix so obtained will be Hermitian positive-semidefinite, with real numbers in the main diagonal and complex numbers off-diagonal.

Estimation

If M X and M Y are centred data matrices of dimension n-by-p and n-by-q respectively, i.e. with n rows of observations of p and q columns of variables, from which the column means have been subtracted, then, if the column means were estimated from the data, sample correlation matrices Q X and Q X Y can be defined to be

or, if the column means were known a-priori,

These empirical sample correlation matrices are the most straightforward and most often used estimators for the correlation matrices, but other estimators also exist, including regularised or shrinkage estimators, which may have better properties.

Applications

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 (in a normative analysis) or are predicted to (in a positive analysis) choose to hold in a context of diversification.