Affine Maps and Marginals of Elliptical Vectors

Why Affine Maps and Marginals Matter

Elliptical models describe multivariate distributions whose contours are ellipsoids, while allowing shapes beyond the multivariate normal. In applications, one often changes the coordinates—for example, by forming sums, contrasts, or other linear combinations—or studies only a subset of the coordinates. The location and scatter have simple transformation rules, but the density generator has a subtler one: reducing dimension generally changes it.

We use the following density convention. For a $p$-dimensional vector $X$, location $\mu\in\mathbb R^p$, positive definite scatter matrix $\Sigma$, and generator $g_p$,

f_X(x)=|\Sigma|^{-1/2}g_p\!\left((x-\mu)'\Sigma^{-1}(x-\mu)\right).

The generator is normalized so that $\int_{\mathbb R^p}g_p(u'u)\,du=1$. It is nonnegative and measurable; this normalization ensures that the displayed expression is a probability density. Here scatter specifies the shape and scale of the contours. It need not equal the covariance matrix.

Definition: A vector with the density above is an elliptical vector with location $\mu$, scatter $\Sigma$, and generator $g_p$. Its density contours are sets on which $(x-\mu)'\Sigma^{-1}(x-\mu)$ is constant. The subscript on $g_p$ records the dimension in which the generator is used.

Affine Transformations

Let $X$ be $p$-dimensional and elliptical, and let $Y=c+AX$, where $c\in\mathbb R^q$ and $A$ is a $q\times p$ matrix. The transformed location is $c+A\mu$, and the transformed scatter is $A\Sigma A'$. If $A$ has full row rank $q$, then $A\Sigma A'$ is positive definite, and $Y$ has a $q$-dimensional density. If $A$ has lower rank, the transformed law is supported on a lower-dimensional affine subspace, so it does not have a density with respect to $q$-dimensional volume.

Theorem: If $X$ has the elliptical density above and $A$ has full row rank $q$, then $Y=c+AX$ has location $c+A\mu$, scatter $\Lambda=A\Sigma A'$, and density $$ f_Y(y)=|\Lambda|^{-1/2}g_q\!\left((y-c-A\mu)'\Lambda^{-1}(y-c-A\mu)\right), $$ where the marginal generator is $$ g_q(s)=\int_{\mathbb R^{p-q}}g_p(s+v'v)\,dv,\qquad s\geq 0. $$ The integral defines the generator up to changes on sets irrelevant to the density.

To see the location and scatter rule at the level of characteristic functions, write the characteristic function of $X$ in elliptical form as $\exp(it'\mu)\psi(t'\Sigma t)$, with $\psi$ a scalar function. Then

\varphi_Y(t)=\exp\!\left(it'(c+A\mu)\right)\psi\!\left(t'A\Sigma A't\right).

The density argument also shows why the transformed generator is $g_q$, rather than necessarily $g_p$. Set $U=\Sigma^{-1/2}(X-\mu)$, so $U$ has spherical density $g_p(u'u)$. Let $\Lambda=A\Sigma A'$, and put $T=\Lambda^{-1/2}A\Sigma^{1/2}$. Then $TT'=I_q$: the rows of $T$ are orthonormal. Complete them to the rows of a $p\times p$ orthogonal matrix $R$. The density of $RU$ is still $g_p(u'u)$, because orthogonal transformations preserve both length and volume. Its first $q$ coordinates are $TU$; integrating out the remaining $p-q$ coordinates gives density $\int_{\mathbb R^{p-q}}g_p(z'z+v'v)\,dv=g_q(z'z)$. Finally, $Y=c+A\mu+\Lambda^{1/2}TU$, whose change-of-variables determinant contributes $|\Lambda|^{-1/2}$. This proves the stated density and scatter. $\square$

Worked Example: A Two-Dimensional Affine Transformation

Suppose $X$ is elliptical with $\mu=(1,-1)'$ and $\Sigma=\begin{pmatrix}4&1\\1&2\end{pmatrix}$. Define $Y=c+AX$, where $c=(2,0)'$ and $A=\begin{pmatrix}1&1\\1&-1\end{pmatrix}$. Find the location and scatter of $Y$.

First, the transformed location is

c+A\mu= \begin{pmatrix}2\\0\end{pmatrix} + \begin{pmatrix}1&1\\1&-1\end{pmatrix} \begin{pmatrix}1\\-1\end{pmatrix} = \begin{pmatrix}2\\2\end{pmatrix}.

For the scatter, calculate its entries from the rows $(1,1)$ and $(1,-1)$ of $A$:

A\Sigma A' = \begin{pmatrix} (1,1)\Sigma(1,1)' & (1,1)\Sigma(1,-1)'\\ (1,-1)\Sigma(1,1)' & (1,-1)\Sigma(1,-1)' \end{pmatrix} = \begin{pmatrix}8&2\\2&4\end{pmatrix}.

For example, the upper-left entry is $4+1+1+2=8$, and the upper-right entry is $4-1+1-2=2$. The determinant of the resulting scatter is $8\cdot4-2\cdot2=28>0$, so it is positive definite. Thus $Y$ has a full-dimensional elliptical density with the displayed location and scatter. Its generator is the two-dimensional generator obtained from $g_2$ by the theorem, which here equals the original generator because the transformation is invertible and $p=q=2$.

Coordinate Marginals and Their Generators

A coordinate marginal is a special case of an affine transformation: selecting $q$ coordinates corresponds to a matrix $A$ whose rows select those coordinates. Its scatter is the corresponding principal submatrix of $\Sigma$. The generator, however, results from integrating over the discarded coordinates.

Marginal-generator formula: If $X$ is $p$-dimensional with generator $g_p$, then any $q$-coordinate marginal, for $1\leq q\leq p$, has generator $$ g_q(s)=\int_{\mathbb R^{p-q}}g_p(s+v'v)\,dv. $$ For $q=p$, the integral is over zero coordinates and means $g_q(s)=g_p(s)$. For $q<p$, the integral sums the joint density over the omitted directions. Nonnegativity permits this integration, and Tonelli’s theorem ensures the resulting marginal density integrates to one.

For a general coordinate split $X=(X_1',X_2')'$, with $X_1$ of dimension $q$, the marginal location is $\mu_1$ and the marginal scatter is the upper-left block $\Sigma_{11}$. This is a principal block, not the Schur complement that appears in conditional calculations. Integrating over $X_2$ produces the same generator formula above; the theorem’s affine argument establishes the result without requiring the joint density to factor into separate coordinate densities.

Worked Example: A Heavy-Tailed Kernel Changes After Marginalization

Consider a three-dimensional spherical density with generator $g_3(s)=C(1+s)^{-3}$. Find $C$, then derive the generator for a one-coordinate marginal.

Normalization requires $\int_{\mathbb R^3}C(1+u'u)^{-3}\,du=1$. Using the standard radial integral $\int_{\mathbb R^p}(1+u'u)^{-\alpha}du =\pi^{p/2}\Gamma(\alpha-p/2)/\Gamma(\alpha)$, valid for $\alpha>p/2$, the integral without $C$ is $\pi^{3/2}\Gamma(3/2)/\Gamma(3)=\pi^2/4$. Hence $C=4/\pi^2$.

For a one-coordinate marginal, $v\in\mathbb R^2$. Using polar coordinates in that plane gives

\begin{aligned} g_1(s) &=\frac{4}{\pi^2}\int_{\mathbb R^2}(1+s+v'v)^{-3}\,dv\\ &=\frac{4}{\pi^2}\,2\pi\int_0^\infty r(1+s+r^2)^{-3}\,dr\\ &=\frac{4}{\pi^2}\,2\pi\left(\frac{1}{4}(1+s)^{-2}\right)\\ &=\frac{2}{\pi}(1+s)^{-2}. \end{aligned}

The generator’s power changes from $3$ in dimension three to $2$ in dimension one. This is not a change in location or scatter; it is the effect of integrating out two coordinates. As a check, the resulting one-dimensional generator is normalized: $\int_{\mathbb R}(2/\pi)(1+t^2)^{-2}\,dt=1$.

Worked Example: The Gaussian Generator Under Marginalization

Let $X$ be bivariate normal with mean $\mu=(2,-1)'$ and scatter $\Sigma=\begin{pmatrix}1&0.6\\0.6&4\end{pmatrix}$. Find the distribution of $Y=2X_1-3$, and show how marginalization acts on the generator.

The bivariate normal generator under our convention is $g_2(s)=(2\pi)^{-1}\exp(-s/2)$. Integrating out the second standardized coordinate gives

g_1(s)=\int_{\mathbb R}\frac{1}{2\pi}\exp\!\left(-\frac{s+v^2}{2}\right)\,dv =\frac{1}{\sqrt{2\pi}}\exp(-s/2).

Thus $X_1\sim N(2,1)$. The affine map $Y=2X_1-3$ has location $2(2)-3=1$ and scatter $2^2\cdot1=4$, so $Y\sim N(1,4)$, where the second argument is the variance. In this case the functional form of the generator remains exponential, with its dimension-dependent normalizing constant changed.

Interpreting Scatter and Avoiding a Common Pitfall

Affine transformation rules make elliptical models useful for analyzing linear summaries and changing coordinate systems. They let an analyst determine the shape of a transformed distribution directly from the original scatter, while the marginal formula supplies the correct density for a subset of components. These rules apply to elliptical distributions whether or not their second moments exist.

Pitfall: assuming the generator is unchanged after dropping coordinates. The wrong reasoning is that a marginal merely removes unused entries from the vector, so it must keep $g_p$. But a marginal density integrates over every omitted coordinate. The correction is to use $g_q(s)=\int_{\mathbb R^{p-q}}g_p(s+v'v)\,dv$; the heavy-tailed example shows that this can change the functional form.
Pitfall: calling scatter covariance without checking moments. Scatter is the matrix in the elliptical density and transforms as $A\Sigma A'$. It is not automatically $\operatorname{Cov}(X)$; that covariance may not exist, and when it does exist it need not equal $\Sigma$ under the chosen generator convention.
Pitfall: using a full-dimensional density for a rank-deficient map. If $A$ does not have full row rank, $A\Sigma A'$ is singular. The transformed vector lies in a lower-dimensional affine subspace, so the full-dimensional density formula does not apply.

Key takeaway: Under $Y=c+AX$, location becomes $c+A\mu$ and scatter becomes $A\Sigma A'$. For a $q$-dimensional marginal, the generator is obtained by integrating the original generator over the $p-q$ omitted directions; do not assume that it stays the same.

Check Your Understanding

Use the definitions and derivations above to answer these questions.

If $A$ has full row rank, what are the location and scatter of $c+AX$?
Why does a rank-deficient affine map fail to have the stated full-dimensional density?
Write the marginal-generator formula for a $q$-coordinate marginal of a $p$-dimensional elliptical vector.
For the three-dimensional kernel in the worked example, what power appears in the one-dimensional marginal generator?
Why is scatter not automatically the covariance matrix?

Affine Maps and Marginals of Elliptical Vectors

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