Elliptical Distributions as Affine Images

From Spherical Symmetry to Elliptical Distributions

A spherical distribution has no preferred direction: rotating a random vector does not change its law. An elliptical distribution allows directional stretching and shifting while retaining this spherical structure underneath. This gives a useful construction for multivariate models without requiring every example to have a density or finite moments.

The construction separates two ingredients. A spherical random vector supplies the underlying shape, while a matrix transforms that shape and a vector translates it. The matrix is summarized by a scatter parameter, but scatter is not automatically covariance; its meaning comes first from the affine representation.

Definition: A random vector \(Z\) in \(\mathbb{R}^d\) is spherical if \(QZ\) has the same distribution as \(Z\) for every \(d\times d\) orthogonal matrix \(Q\), where \(Q^TQ=I_d\). A random vector \(X\) in \(\mathbb{R}^d\) has an elliptical distribution if it can be represented in distribution as

X \overset{d}{=} \mu + AZ,

where \(\mu\in\mathbb{R}^d\), \(A\) is a \(d\times d\) matrix, and \(Z\) is spherical. The vector \(\mu\) is called a location parameter. The matrix \(S=AA^T\) is called a scatter matrix for this representation.

Unless stated otherwise, take \(A\) to be nonsingular, so \(S\) is positive definite. Indeed, for any nonzero vector \(v\), \(v^TSv=v^TAA^Tv=\lVert A^Tv\rVert^2>0\). One may also allow singular \(A\), in which case \(S\) is positive semidefinite; the resulting distribution can be concentrated on a lower-dimensional set.

The word “location” describes the translation in the construction; it does not, by itself, assert that \(E[X]\) exists or equals \(\mu\). Similarly, “scatter” names \(AA^T\), not a covariance matrix. These definitions require neither a density nor moments.

Why the Matrix Enters Through Scatter

The factor \(A\) is not usually unique. If \(Z\) is spherical and \(Q\) is orthogonal, then \(QZ\) has the same law as \(Z\), so replacing \(A\) by \(AQ\) does not change the distribution of \(AZ\). At the same time, this replacement leaves \(AA^T\) unchanged because \((AQ)(AQ)^T=AA^T\).

Theorem: If \(A\) and \(B\) are nonsingular \(d\times d\) matrices and \(AA^T=BB^T\), then \(B=AQ\) for an orthogonal matrix \(Q\). Consequently, for any spherical \(Z\), \(AZ\) and \(BZ\) have the same distribution.

Proof. Set \(Q=A^{-1}B\). Then

QQ^T =A^{-1}BB^T(A^{-1})^T =A^{-1}AA^T(A^{-1})^T =I_d.

Thus \(Q\) is orthogonal, and \(B=AQ\). Spherical symmetry gives \(QZ\overset{d}{=}Z\); multiplying by \(A\) yields \(BZ=AQZ\overset{d}{=}AZ\). \(\square\)

This result explains why \(AA^T\) is a natural summary of the linear deformation when the spherical base distribution is fixed. It does not make scatter an absolute scale independent of that base distribution. For example, \(AZ=(A/2)(2Z)\), and \(2Z\) is also spherical. These two representations have scatter matrices \(AA^T\) and \(AA^T/4\), respectively, even though they describe the same \(X\). A scatter parameter is therefore interpreted relative to a chosen convention for the spherical base law.

Multivariate Normal Distributions

The standard multivariate normal vector provides the principal example. Its spherical symmetry follows directly from its density, and applying a nonsingular affine transformation produces a multivariate normal distribution with the corresponding location and scatter.

Theorem: Let \(Z\sim N_d(0,I_d)\), and let \(A\) be nonsingular. Then \(Z\) is spherical and \(X=\mu+AZ\) has the multivariate normal distribution \(N_d(\mu,AA^T)\), whose density is

f_X(x) = \frac{1}{(2\pi)^{d/2}|AA^T|^{1/2}} \exp\!\left( -\frac{1}{2}(x-\mu)^T(AA^T)^{-1}(x-\mu) \right).

Proof. The density of \(Z\) is \((2\pi)^{-d/2}\exp(-z^Tz/2)\). For an orthogonal \(Q\), \((Qz)^T(Qz)=z^Tz\) and \(|\det Q|=1\). The change-of-variables formula therefore leaves the density unchanged under \(z\mapsto Qz\), proving that \(Z\) is spherical.

For \(x=\mu+Az\), the inverse transformation is \(z=A^{-1}(x-\mu)\), with Jacobian factor \(|\det A|^{-1}\). Substitution into the density of \(Z\) gives

f_X(x) = \frac{1}{(2\pi)^{d/2}|\det A|} \exp\!\left( -\frac{1}{2}(x-\mu)^T(A^{-1})^TA^{-1}(x-\mu) \right).

Since \((AA^T)^{-1}=(A^{-1})^TA^{-1}\) and \(|AA^T|^{1/2}=|\det A|\), this is the stated density. \(\square\)

Worked Example: An Affine Image of a Standard Normal Vector

Let \(Z\sim N_2(0,I_2)\), and define

X=\begin{pmatrix}1\\-2\end{pmatrix} + \begin{pmatrix}2&1\\0&1\end{pmatrix}Z.

The matrix \(A\) is nonsingular because its determinant is \(2\). Its scatter matrix is

AA^T = \begin{pmatrix}2&1\\0&1\end{pmatrix} \begin{pmatrix}2&0\\1&1\end{pmatrix} = \begin{pmatrix}5&1\\1&1\end{pmatrix}.

Thus \(X\sim N_2\!\left(\begin{pmatrix}1\\-2\end{pmatrix},\begin{pmatrix}5&1\\1&1\end{pmatrix}\right)\). The location in this representation is \((1,-2)^T\), and the scatter is the displayed positive-definite matrix. The construction establishes that this normal vector is elliptical.

A Nonnormal Radial Construction

Spherical vectors need not be normal. One convenient way to verify spherical symmetry for a continuous example is to use a density that depends only on the Euclidean norm. The following calculation also shows explicitly how an affine transformation changes such a density.

Worked Example: Transforming a Nonnormal Spherical Vector

In \(\mathbb{R}^2\), let \(Z\) have density

f_Z(z)=\frac{2}{\pi^{3/2}}\exp\!\left(-\lVert z\rVert^4\right).

First check that this is a probability density. Using polar coordinates with radius \(r\), its integral is

\frac{2}{\pi^{3/2}}\int_0^{2\pi}\int_0^\infty e^{-r^4}r\,dr\,d\phi = \frac{4}{\sqrt{\pi}}\int_0^\infty e^{-r^4}r\,dr = \frac{2}{\sqrt{\pi}}\int_0^\infty e^{-u^2}\,du =1,

where \(u=r^2\), so \(r\,dr=du/2\), and \(\int_0^\infty e^{-u^2}\,du=\sqrt{\pi}/2\). For any orthogonal \(Q\), \(\lVert Qz\rVert=\lVert z\rVert\) and the absolute Jacobian determinant is \(1\). Hence \(QZ\) has the same density as \(Z\), so \(Z\) is spherical. Its density is not the standard normal density.

Now set \(X=\mu+AZ\), with \(\mu=(1,0)^T\) and \(A=\begin{pmatrix}2&0\\0&1\end{pmatrix}\). For \(x=(x_1,x_2)^T\),

A^{-1}(x-\mu) = \begin{pmatrix}(x_1-1)/2\\x_2\end{pmatrix}, \qquad |\det A|=2.

The change-of-variables formula gives

f_X(x) = \frac{1}{\pi^{3/2}} \exp\!\left[ -\left\{ \left(\frac{x_1-1}{2}\right)^2+x_2^2 \right\}^{\!2} \right].

This is an elliptical distribution by the affine-image definition. Its location parameter is \((1,0)^T\), and its scatter matrix is

AA^T=\begin{pmatrix}4&0\\0&1\end{pmatrix}.

The construction verifies ellipticity without treating the resulting law as multivariate normal.

Equivalent Factors and a Common Pitfall

Worked Example: Two Factors, One Scatter Matrix

Use the spherical vector \(Z\) from the preceding example and let \(A=\begin{pmatrix}2&0\\0&1\end{pmatrix}\). Choose the orthogonal rotation

Q=\begin{pmatrix}0&-1\\1&0\end{pmatrix}, \qquad Q^TQ=I_2.

Then

AQ=\begin{pmatrix}0&-2\\1&0\end{pmatrix}, \qquad (AQ)(AQ)^T = \begin{pmatrix}4&0\\0&1\end{pmatrix} = AA^T.

Because \(QZ\overset{d}{=}Z\), the vectors \(AQZ\) and \(AZ\) have the same distribution. Thus the two different matrix factors describe the same affine-image law and give the same scatter matrix.

A common mistake is to call \(AA^T\) the covariance matrix simply because that is true for a standard normal base \(Z\). That conclusion does not follow from the definition of an elliptical distribution: a general spherical \(Z\) may not have finite second moments, and even when moments exist, their scale depends on the base distribution. The correction is to call \(AA^T\) the scatter matrix for the stated representation. Any claim about a covariance matrix requires separate moment assumptions and a calculation.

Key takeaway: An elliptical vector is a translated linear image of a spherical vector. The location records the translation, while \(AA^T\) records scatter relative to the chosen spherical base law. Normality is one important case, not part of the definition.

Check Your Understanding

Use the affine-image definition and the distinction between a matrix factor and its scatter matrix.

State the condition that makes a random vector spherical, using orthogonal matrices.
If \(X=\mu+AZ\) and \(Z\) is spherical, what is the scatter matrix associated with this representation?
Why does replacing \(A\) by \(AQ\), for orthogonal \(Q\), preserve both the distribution of \(AZ\) and the scatter matrix?
For the two-dimensional normal construction, calculate \(AA^T\) when \(A=\begin{pmatrix}1&1\\0&2\end{pmatrix}\).
Why does the affine-image definition not imply that an elliptical vector has a covariance matrix equal to its scatter matrix?