Covariance matrix

A bivariate Gaussian probability density function centered at (0, 0), with covariance matrix [1.00, 0.50; 0.50, 1.00]

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 x and y do not fully describe the distribution. A 2×2 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 dispersion 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 x and y directions contain all of the necessary information; a 2×2 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 one, every covariance matrix is symmetric. In addition, every covariance matrix is positive semi-definite.

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 scalar random variables.

If the entries in the column vector

\mathbf {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}=\operatorname {cov} (X_{i},X_{j})=\operatorname {E} [(X_{i}-\mu _{i})(X_{j}-\mu _{j})]=\operatorname {E} [X_{i}X_{j}]-\mu _{i}\mu _{j},

where the operator E denotes the expected (mean) value of its argument, and

\mu _{i}=\operatorname {E} (X_{i})

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

\Sigma ={\begin{bmatrix}\operatorname {E} [(X_{1}-\mu _{1})(X_{1}-\mu _{1})]&\operatorname {E} [(X_{1}-\mu _{1})(X_{2}-\mu _{2})]&\cdots &\operatorname {E} [(X_{1}-\mu _{1})(X_{n}-\mu _{n})]\\\\\operatorname {E} [(X_{2}-\mu _{2})(X_{1}-\mu _{1})]&\operatorname {E} [(X_{2}-\mu _{2})(X_{2}-\mu _{2})]&\cdots &\operatorname {E} [(X_{2}-\mu _{2})(X_{n}-\mu _{n})]\\\\\vdots &\vdots &\ddots &\vdots \\\\\operatorname {E} [(X_{n}-\mu _{n})(X_{1}-\mu _{1})]&\operatorname {E} [(X_{n}-\mu _{n})(X_{2}-\mu _{2})]&\cdots &\operatorname {E} [(X_{n}-\mu _{n})(X_{n}-\mu _{n})]\end{bmatrix}}.

The inverse of this matrix, $\Sigma ^{-1},$ if it exists, is the inverse covariance matrix, also known as the concentration matrix or precision matrix.^[1]

Generalization of the variance

The definition above is equivalent to the matrix equality

\Sigma =\operatorname {E} \left[(\mathbf {X} -\operatorname {E} [\mathbf {X} ])(\mathbf {X} -\operatorname {E} [\mathbf {X} ])^{\rm {T}}\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}=\operatorname {var} (X)=\operatorname {E} [(X-\operatorname {E} (X))^{2}]=\operatorname {E} [(X-\operatorname {E} (X))\cdot (X-\operatorname {E} (X))].

Indeed, the entries on the diagonal of the covariance matrix $\Sigma$ are the variances of each element of the vector $\mathbf {X}$ .

Correlation matrix

An entity 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 $\mathbf {X}$ , which can be written as

\operatorname {corr} (\mathbf {X} )={\big (}\operatorname {diag} (\Sigma ){\big )}^{-{\frac {1}{2}}}\,\Sigma \,{\big (}\operatorname {diag} (\Sigma ){\big )}^{-{\frac {1}{2}}},

where $\operatorname {diag} (\Sigma )$ is the matrix of the diagonal elements of $\Sigma$ (i.e., a diagonal matrix of the variances of $X_{i}$ for $i=1,\dots ,n$ ).

Equivalently, the correlation matrix can be seen as the covariance matrix of the standardized random variables $X_{i}/\sigma (X_{i})$ for $i=1,\dots ,n$ .

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,^[2] call the matrix $\Sigma$ 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 $X$ .

\operatorname {var} (\mathbf {X} )=\operatorname {cov} (\mathbf {X} )=\operatorname {E} \left[(\mathbf {X} -\operatorname {E} [\mathbf {X} ])(\mathbf {X} -\operatorname {E} [\mathbf {X} ])^{\rm {T}}\right].

Both forms are quite standard, and there is no ambiguity between them. The matrix $\Sigma$ 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

\operatorname {cov} (\mathbf {X} ,\mathbf {Y} )=C_{\mathbf {X} ,\mathbf {Y} }=\operatorname {E} \left[(\mathbf {X} -\operatorname {E} [\mathbf {X} ])(\mathbf {Y} -\operatorname {E} [\mathbf {Y} ])^{\rm {T}}\right].

Properties

Basic properties

For $\Sigma =\operatorname {var} (\mathbf {X} )=\operatorname {E} \left[\left(\mathbf {X} -\operatorname {E} [\mathbf {X} ]\right)\left(\mathbf {X} -\operatorname {E} [\mathbf {X} ]\right)^{\rm {T}}\right]$ and ${\boldsymbol {\mu }}=\operatorname {E} ({\textbf {X}})$ , where X is a p-dimensional random variable, the following basic properties apply:^[3]

$\Sigma =\operatorname {E} (\mathbf {XX^{\rm {T}}} )-{\boldsymbol {\mu }}{\boldsymbol {\mu }}^{\rm {T}}$
$\Sigma \,$ is positive-semidefinite and symmetric
$\operatorname {var} (\mathbf {AX} +\mathbf {a} )=\mathbf {A} \,\operatorname {var} (\mathbf {X} )\,\mathbf {A^{\rm {T}}}$
If Y is another random vector with the same dimension as X, then $\operatorname {var} (\mathbf {X} +\mathbf {Y} )=\operatorname {var} (\mathbf {X} )+\operatorname {cov} (\mathbf {X} ,\mathbf {Y} )+\operatorname {cov} (\mathbf {Y} ,\mathbf {X} )+\operatorname {var} (\mathbf {Y} )$

where $\mathbf {a}$ is a p×1 constant vector, $\mathbf {A}$ is a p×p matrix of constants, and $\operatorname {cov} (\mathbf {X} ,\mathbf {Y} )$ is a cross-covariance matrix.

Block matrices

The joint mean ${\boldsymbol {\mu }}_{X,Y}$ and joint covariance matrix ${\boldsymbol {\Sigma }}_{X,Y}$ of ${\boldsymbol {X}}$ and ${\boldsymbol {Y}}$ can be written in block form

{\boldsymbol {\mu }}_{X,Y}={\begin{bmatrix}{\boldsymbol {\mu }}_{X}\\{\boldsymbol {\mu }}_{Y}\end{bmatrix}},\qquad {\boldsymbol {\Sigma }}_{X,Y}={\begin{bmatrix}{\boldsymbol {\Sigma }}_{\mathit {XX}}&{\boldsymbol {\Sigma }}_{\mathit {XY}}\\{\boldsymbol {\Sigma }}_{\mathit {YX}}&{\boldsymbol {\Sigma }}_{\mathit {YY}}\end{bmatrix}}

where ${\boldsymbol {\Sigma }}_{XX}=\operatorname {var} ({\boldsymbol {X}}),{\boldsymbol {\Sigma }}_{YY}=\operatorname {var} ({\boldsymbol {Y}}),$ and ${\boldsymbol {\Sigma }}_{XY}={\boldsymbol {\Sigma }}_{\mathit {YX}}^{T}=\operatorname {cov} ({\boldsymbol {X}},{\boldsymbol {Y}})$ .

${\boldsymbol {\Sigma }}_{XX}$ and ${\boldsymbol {\Sigma }}_{YY}$ can be identified as the variance matrices of the marginal distributions for ${\boldsymbol {X}}$ and ${\boldsymbol {Y}}$ respectively.

If ${\boldsymbol {X}}$ and ${\boldsymbol {Y}}$ are jointly normally distributed,

{\boldsymbol {X}},{\boldsymbol {Y}}\sim \ {\mathcal {N}}({\boldsymbol {\mu }}_{X,Y},{\boldsymbol {\Sigma }}_{X,Y})

,

then the conditional distribution for ${\boldsymbol {Y}}$ given ${\boldsymbol {X}}$ is given by

{\boldsymbol {Y}}|{\boldsymbol {X}}\sim \ {\mathcal {N}}({\boldsymbol {\mu }}_{Y|X},{\boldsymbol {\Sigma }}_{Y|X})

,^[4]

defined by conditional mean

{\boldsymbol {\mu }}_{Y|X}={\boldsymbol {\mu }}_{Y}+{\boldsymbol {\Sigma }}_{YX}{\boldsymbol {\Sigma }}_{XX}^{-1}\left(\mathbf {x} -{\boldsymbol {\mu }}_{X}\right)

and conditional variance

{\boldsymbol {\Sigma }}_{Y|X}={\boldsymbol {\Sigma }}_{YY}-{\boldsymbol {\Sigma }}_{\mathit {YX}}{\boldsymbol {\Sigma }}_{\mathit {XX}}^{-1}{\boldsymbol {\Sigma }}_{\mathit {XY}}.

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).

Covariance matrix 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.^[5]

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: $\mathbf {c} ^{\rm {T}}\Sigma =\operatorname {cov} (\mathbf {c} ^{\rm {T}}\mathbf {X} ,\mathbf {X} )$ . Treated as a bilinear form, it yields the covariance between the two linear combinations: $\mathbf {d} ^{\rm {T}}\Sigma \mathbf {c} =\operatorname {cov} (\mathbf {d} ^{\rm {T}}\mathbf {X} ,\mathbf {c} ^{\rm {T}}\mathbf {X} )$ . The variance of a linear combination is then $\mathbf {c} ^{\rm {T}}\Sigma \mathbf {c}$ , its covariance with itself.

Similarly, the (pseudo-)inverse covariance matrix provides an inner product $\langle c-\mu |\Sigma ^{+}|c-\mu \rangle$ , which induces the Mahalanobis distance, a measure of the "unlikelihood" of c.

Which matrices are covariance matrices?

From the identity just above, let $\mathbf {b}$ be a $(p\times 1)$ real-valued vector, then

\operatorname {var} (\mathbf {b} ^{\rm {T}}\mathbf {X} )=\mathbf {b} ^{\rm {T}}\operatorname {var} (\mathbf {X} )\mathbf {b} ,\,

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

w^{\rm {T}}\operatorname {E} \left[(\mathbf {X} -\operatorname {E} [\mathbf {X} ])(\mathbf {X} -\operatorname {E} [\mathbf {X} ])^{\rm {T}}\right]w=\operatorname {E} \left[w^{\rm {T}}(\mathbf {X} -\operatorname {E} [\mathbf {X} ])(\mathbf {X} -\operatorname {E} [\mathbf {X} ])^{\rm {T}}w\right]=\operatorname {E} \left[(w^{\rm {T}}(\mathbf {X} -\operatorname {E} [\mathbf {X} ]))^{2}\right]\geq 0

.

Conversely, every symmetric positive semi-definite matrix is a covariance matrix. To see this, suppose M is a p×p positive-semidefinite matrix. From the finite-dimensional case of the spectral theorem, it follows that M has a nonnegative symmetric square root, which can be denoted by M^1/2. Let $\mathbf {X}$ be any p×1 column vector-valued random variable whose covariance matrix is the p×p identity matrix. Then

\operatorname {var} (\mathbf {M} ^{1/2}\mathbf {X} )=\mathbf {M} ^{1/2}\,\operatorname {var} (\mathbf {X} )\,\mathbf {M} ^{1/2}=\mathbf {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} (\mathbf {Z} )=\operatorname {E} [(\mathbf {Z} -\mathbf {\mu } )(\mathbf {Z} -\mathbf {\mu } )^{*}],

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

If $\mathbf {Z}$ 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:

\operatorname {E} \left[(\mathbf {Z} -\mathbf {\mu } )(\mathbf {Z} -\mathbf {\mu } )^{\dagger }\right],

where $\mathbf {Z} ^{\dagger }$ 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,^[6] with real numbers in the main diagonal and complex numbers off-diagonal.

Estimation

If $\mathbf {M} _{\mathbf {X} }$ and $\mathbf {M} _{\mathbf {Y} }$ are centred data matrices of dimension n×p and n×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 $\mathbf {Q} _{\mathbf {X} }$ and $\mathbf {Q} _{\mathbf {XY} }$ can be defined to be

\mathbf {Q} _{\mathbf {X} }={\frac {1}{n-1}}\mathbf {M} _{\mathbf {X} }^{\rm {T}}\mathbf {M} _{\mathbf {X} },\qquad \mathbf {Q} _{\mathbf {XY} }={\frac {1}{n-1}}\mathbf {M} _{\mathbf {X} }^{\rm {T}}\mathbf {M} _{\mathbf {Y} }

or, if the column means were known a priori,

\mathbf {Q} _{\mathbf {X} }={\frac {1}{n}}\mathbf {M} _{\mathbf {X} }^{\rm {T}}\mathbf {M} _{\mathbf {X} },\qquad \mathbf {Q} _{\mathbf {XY} }={\frac {1}{n}}\mathbf {M} _{\mathbf {X} }^{\rm {T}}\mathbf {M} _{\mathbf {Y} }.

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 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.

References

↑ Wasserman, Larry (2004). All of Statistics: A Concise Course in Statistical Inference. ISBN 0-387-40272-1.
↑ William Feller (1971). An introduction to probability theory and its applications. Wiley. ISBN 978-0-471-25709-7. Retrieved 10 August 2012.
↑ Taboga, Marco (2010). "Lectures on probability theory and mathematical statistics".
↑ Eaton, Morris L. (1983). Multivariate Statistics: a Vector Space Approach. John Wiley and Sons. pp. 116–117. ISBN 0-471-02776-6.
↑ Frahm, G.; Junker, M.; Szimayer, A. (2003). "Elliptical copulas: Applicability and limitations". Statistics & Probability Letters. 63 (3): 275–286. doi:10.1016/S0167-7152(03)00092-0.
↑ Brookes, Mike. "The Matrix Reference Manual".

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