Covariance

Covariance is a measure of how much something varies with another. It is a generalization of variance: Var(X) = Cov(X,X).

Description

Covariance is calculated as:

Cov(X,Y) = E[(X - E[X])(Y - E[y])]

Covariance is related to correlation as:

Corr(X,Y) = Cov(X,Y)/σ_Xσ_Y

Letting X̅ be the mean of X, and letting Y̅ be the mean of Y, the calculation becomes:

Cov(X,Y) = E[(X - X̅)(Y - Y̅)]

E[XY - X̅Y - XY̅ + X̅Y̅]

E[XY] - X̅E[Y] - E[X]Y̅ + X̅Y̅

E[XY] - X̅Y̅ - X̅Y̅ + X̅Y̅

E[XY] - X̅Y̅

This gives a trivial proof that independent variables have zero correlation and zero covariance. Necessarily E[XY] = E[X]E[Y], so E[XY] - X̅Y̅ = 0

In the context of linear algebra, the calculation is notated as:

Cov(X,Y) = E[(X - E[X])(Y - E[y])^T]

Letting m_X be the mean vector of X and m_Y be the mean vector of Y, the calculation becomes:

Cov(X,Y) = E[(X - m_X)(Y - m_Y)^T]

Properties

Covariance is symmetric: Cov(X,Y) = Cov(Y,X)

Transformations

Covariance linearly transforms with scalars.

Cov(aX,Y) = E[aXY] - E[aX]E[Y]

a E[XY] - a E[X]E[Y]

a (E[XY] - E[X]E[Y])

a Cov(X,Y)

Covariance is linear with inputs.

Cov(X+Y,Z) = E[(X+Y)Z] - E[X+Y]E[Z]

E[XZ+YZ] - E[X+Y]E[Z]

(E[XZ] + E[YZ]) - (E[X] + E[Y]) E[Z]

(E[XZ] + E[YZ]) - (E[X]E[Z] + E[Y]E[Z])

(E[XZ] - E[X]E[Z] + E[YZ] - E[Y]E[Z]

Cov(X,Z) + Cov(Y,Z)

This gives a trivial proof that constant additions cancel out.

Cov(a+X,Y) = Cov(X,Y) + Cov(a,Y) = Cov(X,Y) + 0

Altogether: Cov(a+bX,c+dY) = b d Cov(X,Y)

Matrix

A covariance matrix describes multivariate covariances. Cell (i,j) is the covariance of the ith variable with the jth variable. On the diagonal are variances (i.e., covariance of a variable with itself). The matrix is usually notated as Σ.

The inverse covariance matrix, Σ^-1, is also called the precision matrix.

Properties

A covariance matrix is necessarily square, symmetric, and positive semi-definite.

• Σ = Σ^T

• the determinant is bound by |Σ| >= 0

• Σ^0.5 can always be evaluated

Linear Algebra

The covariance matrix linearly transforms with the inputs.

Cov(AX,AY) = E[(AX - Am_X)(AY - Am_Y)^T]

E[A(X - m_X)(Y - m_Y)^TA^T]

AE[(X - m_X)(Y - m_Y)^T]A^T

AΣA^T

Trivially, if the transformation is a scalar like aI:

aIΣaI^T

aΣa

a²Σ

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