In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
and
statistics
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** '' The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
giving the
covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the le ...
between each pair of elements of a given
random vector. Any
covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the le ...
matrix is
symmetric and
positive semi-definite and its main diagonal contains
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
s (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
and
directions contain all of the necessary information; a
matrix would be necessary to fully characterize the two-dimensional variation.
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 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
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
and
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
, then the covariance matrix
is the matrix whose
entry is the
covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the le ...
:
where the operator
denotes the expected value (mean) of its argument.
Conflicting nomenclatures and notations
Nomenclatures differ. Some statisticians, following the probabilist
William Feller
William "Vilim" Feller (July 7, 1906 – January 14, 1970), born Vilibald Srećko Feller, was a Croatian-American mathematician specializing in probability theory.
Early life and education
Feller was born in Zagreb to Ida Oemichen-Perc, a C ...
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
(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 matrix
The inverse of this matrix,
, if it exists, is the inverse covariance matrix (or inverse concentration matrix), also known as the ''
precision matrix'' (or ''concentration 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
In probability theory and statistics, partial correlation measures the degree of association between two random variables, with the effect of a set of controlling random variables removed. When determining the numerical relationship between two ...
, 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 a
-dimensional random variable, the following basic properties apply:
#
#
is
positive-semidefinite, i.e.
#
is
symmetric, i.e.
# For any constant (i.e. non-random)
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 distribution
In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variab ...
s for
and
respectively.
If
and
are
jointly normally distributed,
:
then the
conditional distribution
In probability theory and statistics, given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a particular value; in some cases the ...
for
given
is given by
:
defined by
conditional mean
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – give ...
:
and
conditional variance
:
The matrix
is known as the matrix of
regression coefficients, while in linear algebra
is the
Schur complement In linear algebra and the theory of matrices, the Schur complement of a block matrix is defined as follows.
Suppose ''p'', ''q'' are nonnegative integers, and suppose ''A'', ''B'', ''C'', ''D'' are respectively ''p'' × ''p'', ''p'' × ''q'', ''q'' ...
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
In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown statistical parameter, parameters in a linear regression model (with fixed level-one effects of a linear function of a set of explanatory ...
of
ordinary least squares
In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model (with fixed level-one effects of a linear function of a set of explanatory variables) by the ...
(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
and
are correlated via another vector
, the latter correlations are suppressed in a matrix
[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.]
:
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 distribution
If a column vector
of
possibly correlated random variables is
jointly normally distributed, or more generally
elliptically distributed, then its
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
can be expressed in terms of the covariance matrix
as follows
:
where
and
is the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of
.
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 (pseudo-)inverse covariance matrix provides an inner product
, which induces the
Mahalanobis distance, a measure of the "unlikelihood" of ''c''.
Which matrices are covariance matrices?
From the identity just above, let
be a
real-valued vector, then
:
which must always be nonnegative, since it is the
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
of a real-valued random variable, so a covariance matrix is always a
positive-semidefinite matrix
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a ...
.
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
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful be ...
, it follows that
has a nonnegative symmetric
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
...
, 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
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
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 (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.
Its diagonal elements may be complex valued; it is a complex symmetric matrix.
Estimation
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.
Applications
The covariance matrix is a useful tool in many different areas. From it a transformation matrix can be derived, called a whitening transformation A whitening transformation or sphering transformation is a linear transformation that transforms a vector of random variables with a known covariance matrix into a set of new variables whose covariance is the identity matrix, meaning that they are u ...
, 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
In mathematics, the Rayleigh quotient () for a given complex Hermitian matrix ''M'' and nonzero vector ''x'' is defined as:
R(M,x) = .
For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the ...
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
Financial economics, also known as finance, is the branch of economics characterized by a "concentration on monetary activities", in which "money of one type or another is likely to appear on ''both sides'' of a trade". William F. Sharpe"Financia ...
, especially in portfolio theory
Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk. It is a formalization and extension of diversificati ...
and its mutual fund separation theorem In portfolio theory, a mutual fund separation theorem, mutual fund theorem, or separation theorem is a theorem stating that, under certain conditions, any investor's optimal portfolio can be constructed by holding each of certain mutual funds in a ...
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
Diversification may refer to:
Biology and agriculture
* Genetic divergence, emergence of subpopulations that have accumulated independent genetic changes
* Agricultural diversification involves the re-allocation of some of a farm's resources to n ...
.
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 (proven for a single-parent strategy and a static model, as the population size increases, relying on the quadratic approximation).
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.
Covariance mapping
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
Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individual, a group ...
. 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
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementat ...
.[L J Frasinski "Covariance mapping techniques" ''J. Phys. B: At. Mol. Opt. Phys.'' 49 152004 (2016)]
open access
/ref>
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.[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
/ref> 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 spectroscopy
Two-dimensional infrared spectroscopy employs correlation analysis to obtain 2D spectra of the condensed phase
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, th ...
. There are two versions of this analysis: synchronous and asynchronous
Asynchrony is the state of not being in synchronization.
Asynchrony or asynchronous may refer to:
Electronics and computing
* Asynchrony (computer programming), the occurrence of events independent of the main program flow, and ways to deal wit ...
. Mathematically, the former is expressed in terms of the sample covariance matrix and the technique is equivalent to covariance mapping.[I Noda "Generalized two-dimensional correlation method applicable to infrared, Raman, and other types of spectroscopy" ''Appl. Spectrosc.'' 47 1329–36 (1993)]
See also
* Covariance function In probability theory and statistics, the covariance function describes how much two random variables change together (their ''covariance'') with varying spatial or temporal separation. For a random field or stochastic process ''Z''(''x'') on a doma ...
* Multivariate statistics
Multivariate statistics is a subdivision of statistics encompassing the simultaneous observation and analysis of more than one outcome variable.
Multivariate statistics concerns understanding the different aims and background of each of the dif ...
* Lewandowski-Kurowicka-Joe distribution
* Gramian matrix
* Eigenvalue decomposition
* Quadratic form (statistics)
* Principal components
Principal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while preserving the maximum amount of information, and ...
References
Further reading
*
*
Covariance Matrix Explained With Pictures
, an easy way to visualize covariance matrices!
*
*
{{DEFAULTSORT:Covariance Matrix
Covariance and correlation
Matrices
Summary statistics