Covariance matrix

In statistics and probability theory, the covariance matrix is a matrix of covariances between elements of a vector. It is the natural generalization to higher dimensions of the concept of the variance of a scalar-valued random variable.

Definition
If entries in the column vector


 * $$X = \begin{bmatrix}X_1 \\ \vdots \\ X_n \end{bmatrix}$$

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



\Sigma_{ij} =\mathrm{E}\begin{bmatrix} (X_i - \mu_i)(X_j - \mu_j) \end{bmatrix} $$

where


 * $$\mu_i = \mathrm{E}(X_i)\,$$

is the expected value of the ith entry in the vector X. In other words, we have



\Sigma = \begin{bmatrix} \mathrm{E}[(X_1 - \mu_1)(X_1 - \mu_1)] & \mathrm{E}[(X_1 - \mu_1)(X_2 - \mu_2)] & \cdots & \mathrm{E}[(X_1 - \mu_1)(X_n - \mu_n)] \\ \\ \mathrm{E}[(X_2 - \mu_2)(X_1 - \mu_1)] & \mathrm{E}[(X_2 - \mu_2)(X_2 - \mu_2)] & \cdots & \mathrm{E}[(X_2 - \mu_2)(X_n - \mu_n)] \\ \\ \vdots & \vdots & \ddots & \vdots \\ \\ \mathrm{E}[(X_n - \mu_n)(X_1 - \mu_1)] & \mathrm{E}[(X_n - \mu_n)(X_2 - \mu_2)] & \cdots & \mathrm{E}[(X_n - \mu_n)(X_n - \mu_n)] \end{bmatrix}. $$

As a generalization of the variance
The definition above is equivalent to the matrix equality



\Sigma=\mathrm{E} \left[ \left( \textbf{X} - \mathrm{E}[\textbf{X}] \right) \left( \textbf{X} - \mathrm{E}[\textbf{X}] \right)^\top \right] $$

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



\sigma^2 = \mathrm{var}(X) = \mathrm{E}[(X-\mu)^2], \, $$

where


 * $$\mu = \mathrm{E}(X).\,$$

The matrix $$\Sigma$$ is also often called the variance-covariance matrix since the diagonal terms are in fact variances.

Conflicting nomenclatures and notations
Nomenclatures differ. Some statisticians, following the probabilist William Feller, call this matrix 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 $$X$$. Thus

\operatorname{var}(\textbf{X}) = \operatorname{cov}(\textbf{X}) = \mathrm{E} \left[ (\textbf{X} - \mathrm{E} [\textbf{X}]) (\textbf{X} - \mathrm{E} [\textbf{X}])^\top \right] $$

However, the notation for the "cross-covariance" between two vectors is standard:

\operatorname{cov}(\textbf{X},\textbf{Y}) = \mathrm{E} \left[ (\textbf{X} - \mathrm{E}[\textbf{X}]) (\textbf{Y} - \mathrm{E}[\textbf{Y}])^\top \right] $$

The $$var$$ notation is found in William Feller's two-volume book An Introduction to Probability Theory and Its Applications, but both forms are quite standard and there is no ambiguity between them.

Properties
For $$\Sigma=\mathrm{E} \left[ \left( \textbf{X} - \mathrm{E}[\textbf{X}] \right) \left( \textbf{X} - \mathrm{E}[\textbf{X}] \right)^\top \right]$$ and $$ \mu = \mathrm{E}(\textbf{X})$$ the following basic properties apply:
 * 1) $$ \Sigma = \mathrm{E}(\mathbf{X X^\top}) - \mathbf{\mu}\mathbf{\mu^\top} $$
 * 2) $$ \mathbf{\Sigma}$$ is positive semi-definite
 * 3) $$ \operatorname{var}(\mathbf{A X} + \mathbf{a}) = \mathbf{A}\, \operatorname{var}(\mathbf{X})\, \mathbf{A^\top} $$
 * 4) $$ \operatorname{cov}(\mathbf{X},\mathbf{Y}) = \operatorname{cov}(\mathbf{Y},\mathbf{X})^\top$$
 * 5) $$ \operatorname{cov}(\mathbf{X_1} + \mathbf{X_2},\mathbf{Y}) = \operatorname{cov}(\mathbf{X_1},\mathbf{Y}) + \operatorname{cov}(\mathbf{X_2}, \mathbf{Y})$$
 * 6) If p = q, then $$\operatorname{var}(\mathbf{X} + \mathbf{Y}) = \operatorname{var}(\mathbf{X}) + \operatorname{cov}(\mathbf{X},\mathbf{Y}) + \operatorname{cov}(\mathbf{Y}, \mathbf{X}) + \operatorname{var}(\mathbf{Y})$$
 * 7) $$\operatorname{cov}(\mathbf{AX}, \mathbf{BY}) = \mathbf{A}\, \operatorname{cov}(\mathbf{X}, \mathbf{Y}) \,\mathbf{B}^\top$$
 * 8) If $$\mathbf{X}$$ and $$\mathbf{Y}$$ are independent, then $$\operatorname{cov}(\mathbf{X}, \mathbf{Y}) = 0$$

where $$\mathbf{X}, \mathbf{X_1}$$ and $$\mathbf{X_2}$$ are a random $$\mathbf{(p \times 1)}$$ vectors, $$\mathbf{Y}$$ is a random $$\mathbf{(q \times 1)}$$ vector, $$\mathbf{a}$$ is $$\mathbf{(p \times 1)}$$ vector, $$\mathbf{A}$$ and $$\mathbf{B}$$ are $$\mathbf{(p \times q)}$$ matrices.

This covariance matrix (though very simple) is a very useful tool in many very different areas. From it a transformation matrix can be derived 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) in statistics and Karhunen-Loève transform (KL-transform) in image processing.

Which matrices are covariance matrices
From the identity


 * $$\operatorname{var}(\mathbf{a^\top}\mathbf{X}) = \mathbf{a^\top} \operatorname{var}(\mathbf{X}) \mathbf{a}\,$$

and the fact that the variance of any real-valued random variable is nonnegative, it follows immediately that only a nonnegative-definite matrix can be a covariance matrix. The converse question is whether every nonnegative-definite symmetric matrix is a covariance matrix. The answer is "yes". To see this, suppose M is a p&times;p nonnegative-definite symmetric matrix. From the finite-dimensional case of the spectral theorem, it follows that M has a nonnegative symmetric square root, which let us call M1/2. Let $$\mathbf{X}$$ be any p&times;1 column vector-valued random variable whose covariance matrix is the p&times;p identity matrix. Then


 * $$\operatorname{var}(M^{1/2}\mathbf{X}) = M^{1/2} (\operatorname{var}(\mathbf{X})) M^{1/2} = M.\,$$

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



\operatorname{var}(z) = \operatorname{E} \left[ (z-\mu)(z-\mu)^{*} \right] $$

where the complex conjugate of a complex number $$z$$ is denoted $$z^{*}$$.

If $$Z$$ is a column-vector of complex-valued random variables, then we take the conjugate transpose by both transposing and conjugating, getting a square matrix:



\operatorname{E} \left[ (Z-\mu)(Z-\mu)^{*} \right] $$

where $$Z^{*}$$ denotes the conjugate transpose, which is applicable to the scalar case since the transpose of a scalar is still a scalar.

LaTeX provides useful features for dealing with covariance matrices. These are available through the extendedmath package.

Estimation
The derivation of the maximum-likelihood estimator of the covariance matrix of a multivariate normal distribution is perhaps surprisingly subtle. It involves the spectral theorem and the reason why it can be better to view a scalar as the trace of a 1 &times; 1 matrix than as a mere scalar. See estimation of covariance matrices.