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Covariance matrix

A bivariate Gaussian probability density function centered at (0, 0), with covariance matrix given by
Sample points from a bivariate Gaussian distribution with a standard deviation of 3 in roughly the lower left-upper right direction and of 1 in the orthogonal direction. Because the x and y components co-vary, the variances of and do not fully describe the distribution. A covariance matrix is needed; the directions of the arrows correspond to the eigenvectors of this covariance matrix and their lengths to the square roots of the eigenvalues.

In probability theory and statistics, a covariance matrix, also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix, is a matrix whose element in the i, j position is the covariance between the i-th and j-th elements of a random vector. A random vector is a random variable with multiple dimensions. Each element of the vector is a scalar random variable. Each element has either a finite number of observed empirical values or a finite or infinite number of potential values. The potential values are specified by a theoretical joint probability distribution.

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.

Because the covariance of the i-th random variable with itself is simply that random variable's variance, each element on the principal diagonal of the covariance matrix is the variance of one of the random variables. Because the covariance of the i-th random variable with the j-th one is the same thing as the covariance of the j-th random variable with the i-th random variable, every covariance matrix is symmetric. Also, every covariance matrix is positive semi-definite.

The covariance matrix of a random vector is typically denoted by or .

DefinitionEdit

Throughout this article, boldfaced unsubscripted   and   are used to refer to random vectors, and unboldfaced 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[1]:p. 177

 

where the operator   denotes the expected value (mean) of its argument.

In other words,

 

The definition above is equivalent to the matrix equality

 

 

 

 

 

(Eq.1)

where  .

Generalization of the varianceEdit

This form (Eq.1) can be seen as a generalization of the scalar-valued variance to higher dimensions. Recall that for a scalar-valued random variable  

 

Indeed, the entries on the diagonal of the auto-covariance matrix   are the variances of each element of the vector  .

Conflicting nomenclatures and notationsEdit

Nomenclatures differ. Some statisticians, following the probabilist William Feller in his two-volume book An Introduction to Probability Theory and Its Applications,[2] 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

 

PropertiesEdit

Relation to the correlation matrixEdit

The auto-covariance matrix   is related to the autocorrelation matrix   by

 

where the autocorrelation matrix is defined as  .

Relation to the matrix of correlation coefficientsEdit

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   (i.e., a diagonal matrix of the variances of   for  ).

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 matrixEdit

The inverse of this matrix,  , if it exists, is the inverse covariance matrix, also known as the concentration matrix or precision matrix.[3]

Basic propertiesEdit

For   and  , where   is a  -dimensional random variable, the following basic properties apply:[4]

  1.  
  2.   is positive-semidefinite, i.e.  
  3.   is symmetric, i.e.  
  4. For any constant (i.e. non-random)   matrix   and constant   vector  , one has  
  5. If   is another random vector with the same dimension as  , then   where   is the cross-covariance matrix of   and  .

Block matricesEdit

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

 [5]

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 (OLS).

Partial covariance matrixEdit

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[6]

 

The partial covariance matrix   is effectively the simple covariance matrix   as if the uninteresting random variables   were held constant.

Covariance matrix as a parameter of a distributionEdit

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[6]

 

where   and   is the determinant of  .

Covariance matrix as a linear operatorEdit

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 (pseudo-)inverse covariance matrix provides an inner product  , which induces the Mahalanobis distance, a measure of the "unlikelihood" of c.[citation needed]

Which matrices are covariance matrices?Edit

From the identity just above, let   be a   real-valued vector, then

 

which must always be nonnegative, since it is the variance of a real-valued random variable. A covariance matrix is always a positive-semidefinite matrix, since

 

Conversely, every symmetric positive semi-definite matrix is a covariance matrix. To see this, suppose   is a   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 M1/2. Let   be any   column vector-valued random variable whose covariance matrix is the   identity matrix. Then

 

Complex random vectorsEdit

Covariance matrixEdit

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:[7]:p. 293

 ,

where   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,[8] with real numbers in the main diagonal and complex numbers off-diagonal.

Pseudo-covariance matrixEdit

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.

 

PropertiesEdit

  • The covariance matrix is a Hermitian matrix, i.e.  .[1]:p. 179
  • The diagonal elements of the covariance matrix are real.[1]:p. 179

EstimationEdit

If   and   are centred 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.

ApplicationsEdit

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[citation needed] or, from a different point of view, to find an optimal basis for representing the data in a compact way[citation needed] (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 mappingEdit

In covariance mapping the values of the   or   matrix are plotted as a 2-dimensional map. When vectors   and   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  , and   are acquired experimentally as rows of   samples, e.g.

 

where   is the i-th discrete value in sample j of the random function  . 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.[9]

 
Figure 1: Construction of a partial covariance map of N2 molecules undergoing Coulomb explosion induced by a free-electron laser.[10] Panels a and b map the two terms of the covariance matrix, which is shown in panel c. Panel d maps common-mode correlations via intensity fluctuations of the laser. Panel e maps the partial covariance matrix that is corrected for the intensity fluctuations. Panel f shows that 10% overcorrection improves the map and makes ion-ion correlations clearly visible. Owing to momentum conservation these correlations appear as lines approximately perpendicular to the autocorrelation line (and to the periodic modulations which are caused by detector ringing).

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.[10] The random function   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   such spectra,  , and averaging them over   produces a smooth spectrum  , which is shown in red at the bottom of Fig. 1. The average spectrum   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   and   are the same, except that the range of the time-of-flight   differs. Panel a shows  , panel b shows   and panel c shows their difference, which is   (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   is recorded at every shot, put into   and   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  . 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 spectroscopyEdit

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.[11]

See alsoEdit

ReferencesEdit

  1. ^ a b c Park,Kun Il (2018). Fundamentals of Probability and Stochastic Processes with Applications to Communications. Springer. ISBN 978-3-319-68074-3.
  2. ^ William Feller (1971). An introduction to probability theory and its applications. Wiley. ISBN 978-0-471-25709-7. Retrieved 10 August 2012.
  3. ^ Wasserman, Larry (2004). All of Statistics: A Concise Course in Statistical Inference. ISBN 0-387-40272-1.
  4. ^ Taboga, Marco (2010). "Lectures on probability theory and mathematical statistics".
  5. ^ Eaton, Morris L. (1983). Multivariate Statistics: a Vector Space Approach. John Wiley and Sons. pp. 116–117. ISBN 0-471-02776-6.
  6. ^ a b W J Krzanowski "Principles of Multivariate Analysis" (Oxford University Press, New York, 1988), Chap. 14.4; K V Mardia, J T Kent and J M Bibby "Multivariate Analysis (Academic Press, London, 1997), Chap. 6.5.3; T W Anderson "An Introduction to Multivariate Statistical Analysis" (Wiley, New York, 2003), 3rd ed., Chaps. 2.5.1 and 4.3.1.
  7. ^ Lapidoth, Amos (2009). A Foundation in Digital Communication. Cambridge University Press. ISBN 978-0-521-19395-5.
  8. ^ Brookes, Mike. "The Matrix Reference Manual". Cite journal requires |journal= (help)
  9. ^ L J Frasinski "Covariance mapping techniques" J. Phys. B: At. Mol. Opt. Phys. 49 152004 (2016), open access
  10. ^ a b O Kornilov, M Eckstein, M Rosenblatt, C P Schulz, K Motomura, A Rouzée, J Klei, L Foucar, M Siano, A Lübcke, F. Schapper, P Johnsson, D M P Holland, T Schlatholter, T Marchenko, S Düsterer, K Ueda, M J J Vrakking and L J Frasinski "Coulomb explosion of diatomic molecules in intense XUV fields mapped by partial covariance" J. Phys. B: At. Mol. Opt. Phys. 46 164028 (2013), open access
  11. ^ I Noda "Generalized two-dimensional correlation method applicable to infrared, Raman, and other types of spectroscopy" Appl. Spectrosc. 47 1329–36 (1993)

Further readingEdit