Covariance matrix

In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of a given random vector. Any covariance matrix is symmetric and positive semi-definite and its main diagonal contains variances (i.e., the covariance of each element with itself).
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 formula_1 and formula_2 directions contain all of the necessary information; a formula_3 matrix would be necessary to fully characterize the two-dimensional variation.
The covariance matrix of a random vector formula_4 is typically denoted by formula_5 or formula_6.
Definition.
Throughout this article, boldfaced unsubscripted formula_4 and formula_8 are used to refer to random vectors, and unboldfaced subscripted formula_9 and formula_10 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 formula_5 is the matrix whose formula_13 entry is the covariance
where the operator formula_15 denotes the expected value (mean) of its argument.
In other words,
The definition above is equivalent to the matrix equality
where formula_17.
Generalization of the variance.
This form () can be seen as a generalization of the scalar-valued variance to higher dimensions. Remember that for a scalar-valued random variable formula_18
Indeed, the entries on the diagonal of the auto-covariance matrix formula_5 are the variances of each element of the vector formula_4.
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 formula_5 the variance of the random vector formula_4, 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 formula_4.
Both forms are quite standard, and there is no ambiguity between them. The matrix formula_5 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 formula_5 is related to the autocorrelation matrix formula_29 by
where the autocorrelation matrix is defined as formula_31.
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 formula_4, which can be written as
where formula_34 is the matrix of the diagonal elements of formula_5 (i.e., a diagonal matrix of the variances of formula_9 for formula_37).
Equivalently, the correlation matrix can be seen as the covariance matrix of the standardized random variables formula_38 for formula_37.
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, formula_41, if it exists, is the inverse covariance matrix, also known as the concentration matrix or precision matrix.
Basic properties.
For formula_42 and formula_43, where formula_44 is a formula_45-dimensional random variable, the following basic properties apply:
Block matrices.
The joint mean formula_62 and joint covariance matrix formula_63 of formula_4 and formula_8 can be written in block form
where formula_67, formula_68 and formula_69.
formula_70 and formula_71 can be identified as the variance matrices of the marginal distributions for formula_72 and formula_73 respectively.
If formula_4 and formula_8 are jointly normally distributed,
then the conditional distribution for formula_8 given formula_4 is given by
defined by conditional mean
and conditional variance
The matrix formula_82 is known as the matrix of regression coefficients, while in linear algebra formula_83 is the Schur complement of formula_70 in formula_85.
The matrix of regression coefficients may often be given in transpose form, formula_86, suitable for post-multiplying a row vector of explanatory variables formula_87 rather than pre-multiplying a column vector formula_72. In this form they correspond to the coefficients obtained by inverting the matrix of the normal equations of ordinary least squares (OLS).
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 formula_4 and formula_8 are correlated via another vector formula_91, the latter correlations are suppressed in a matrix
The partial covariance matrix formula_93 is effectively the simple covariance matrix formula_94 as if the uninteresting random variables formula_91 were held constant.
Covariance matrix as a parameter of a distribution.
If a column vector formula_72 of formula_97 possibly correlated random variables is jointly normally distributed, or more generally elliptically distributed, then its probability density function formula_98 can be expressed in terms of the covariance matrix formula_99 as follows
where formula_101 and formula_102 is the determinant of formula_99.
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: formula_104. Treated as a bilinear form, it yields the covariance between the two linear combinations: formula_105. The variance of a linear combination is then formula_106, its covariance with itself.
Similarly, the (pseudo-)inverse covariance matrix provides an inner product formula_107, which induces the Mahalanobis distance, a measure of the "unlikelihood" of "c".
Which matrices are covariance matrices?
From the identity just above, let formula_108 be a formula_109 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:formula_111
Conversely, every symmetric positive semi-definite matrix is a covariance matrix. To see this, suppose formula_112 is a formula_113 symmetric positive-semidefinite matrix. From the finite-dimensional case of the spectral theorem, it follows that formula_112 has a nonnegative symmetric square root, which can be denoted by M1/2. Let formula_4 be any formula_116 column vector-valued random variable whose covariance matrix is the formula_113 identity matrix. Then
Complex random vectors.
Covariance matrix.
The variance of a complex "scalar-valued" random variable with expected value formula_119 is conventionally defined using complex conjugation:
where the complex conjugate of a complex number formula_121 is denoted formula_122; thus the variance of a complex random variable is a real number.
If formula_123 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:
where formula_125 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.
Pseudo-covariance matrix.
For complex random vectors, another kind of second central moment, the pseudo-covariance matrix (also called relation matrix) is defined as follows. In contrast to the covariance matrix defined above Hermitian transposition gets replaced by transposition in the definition.
Estimation.
If formula_128 and formula_129 are centred data matrices of dimension formula_130 and formula_131 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 formula_132 and formula_133 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 (see Rayleigh quotient for a formal proof and additional properties of covariance matrices).
This is called principal component analysis (PCA) and the Karhunen–Loève transform (KL-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 (in a normative analysis) or are predicted to (in a positive analysis) choose to hold in a context of diversification.
Covariance mapping.
In covariance mapping the values of the formula_136 or formula_137 matrix are plotted as a 2-dimensional map. When vectors formula_72 and formula_73 are discrete random functions, the map shows statistical relations between different regions of the random functions. Statistically independent regions of the functions show up on the map as zero-level flatland, while positive or negative correlations show up, respectively, as hills or valleys.
In practice the column vectors formula_140, and formula_141 are acquired experimentally as rows of formula_97 samples, e.g.
where formula_144 is the "i"-th discrete value in sample "j" of the random function formula_145. The expected values needed in the covariance formula are estimated using the sample mean, e.g.
and the covariance matrix is estimated by the sample covariance matrix
where the angular brackets denote sample averaging as before except that the Bessel's correction should be made to avoid bias. Using this estimation the partial covariance matrix can be calculated as
where the backslash denotes the left matrix division operator, which bypasses the requirement to invert a matrix and is available in some computational packages such as Matlab.
Fig. 1 illustrates how a partial covariance map is constructed on an example of an experiment performed at the FLASH free-electron laser in Hamburg. The random function formula_145 is the time-of-flight spectrum of ions from a Coulomb explosion of nitrogen molecules multiply ionised by a laser pulse. Since only a few hundreds of molecules are ionised at each laser pulse, the single-shot spectra are highly fluctuating. However, collecting typically formula_150 such spectra, formula_151, and averaging them over formula_152 produces a smooth spectrum formula_153, which is shown in red at the bottom of Fig. 1. The average spectrum formula_154 reveals several nitrogen ions in a form of peaks broadened by their kinetic energy, but to find the correlations between the ionisation stages and the ion momenta requires calculating a covariance map.
In the example of Fig. 1 spectra formula_151 and formula_156 are the same, except that the range of the time-of-flight formula_157 differs. Panel a shows formula_158, panel b shows formula_159 and panel c shows their difference, which is formula_160 (note a change in the colour scale). Unfortunately, this map is overwhelmed by uninteresting, common-mode correlations induced by laser intensity fluctuating from shot to shot. To suppress such correlations the laser intensity formula_161 is recorded at every shot, put into formula_141 and formula_163 is calculated as panels d and e show. The suppression of the uninteresting correlations is, however, imperfect because there are other sources of common-mode fluctuations than the laser intensity and in principle all these sources should be monitored in vector formula_141. Yet in practice it is often sufficient to overcompensate the partial covariance correction as panel f shows, where interesting correlations of ion momenta are now clearly visible as straight lines centred on ionisation stages of atomic nitrogen.
Two-dimensional infrared spectroscopy.
Two-dimensional infrared spectroscopy employs correlation analysis to obtain 2D spectra of the condensed phase. There are two versions of this analysis: synchronous and asynchronous. Mathematically, the former is expressed in terms of the sample covariance matrix and the technique is equivalent to covariance mapping.