From a radial law to a distribution
An elliptical model is often specified by a center, a matrix, and a simpler spherical random vector. This is useful in calculations: the radius describes how far observations tend to lie from the center, while the matrix determines the directions in which the model is stretched. Once the ingredients are chosen, support, projections, moments, and characteristic functions can be worked out from the construction.
The constructions below use a uniform direction on the unit sphere. They include a fixed-radius spherical model, a uniform distribution on an ellipsoid, and a multivariate Student distribution. These examples have different radial laws, so their supports and tails differ even when their affine geometry is similar.
A radial law is the probability distribution assigned to \(R\). For example, a fixed radius gives a spherical distribution concentrated on a sphere; a radius that can take all values from zero to one can give a distribution supported on a ball. A linear transformation maps these sets to their corresponding ellipsoidal sets. The transformation rule for the characteristic function follows directly from the inner product, and will let us calculate affine examples from their spherical versions.
Worked construction: fixed radius and affine transformation
Worked Example: A Spherical Shell and Its Projection
Let \(U\) be uniform on the unit circle in \(\mathbb{R}^2\), and set \(S=2U\). Write \(U=(\cos\Theta,\sin\Theta)^\top\), where \(\Theta\) is uniform on \([0,2\pi)\). The radius is exactly 2, so the support of \(S\) is the circle of radius 2, not the disk it encloses.
The first coordinate is \(S_1=2\cos\Theta\). For \(-2<x<2\), the change of variable \(x=2\cos\theta\), accounting for the two angle values with that cosine, gives \[ f_{S_1}(x)=\frac{1}{\pi\sqrt{4-x^2}}. \] This density integrates to one: substituting \(x=2\cos\theta\) over \(0<\theta<\pi\) gives an integral of \(1/\pi\) over an interval of length \(\pi\). Its mean is zero by symmetry, and its variance is \(4E[\cos^2\Theta]=4(1/2)=2\).
For any \(t\in\mathbb{R}^2\), rotational invariance lets us rotate \(t\) onto the first coordinate axis. Thus \[ \varphi_S(t)=E[e^{it^\top S}] =\frac{1}{2\pi}\int_0^{2\pi}e^{2i\|t\|\cos\theta}\,d\theta =J_0(2\|t\|), \] where \(J_0(x)=(2\pi)^{-1}\int_0^{2\pi}e^{ix\cos\theta}\,d\theta\) is the Bessel function of the first kind of order zero. The integral is real because the sine terms cancel over a full turn.
Now take \(A=\begin{pmatrix}2&0\\0&1\end{pmatrix}\) and \(\mu=(1,-1)^\top\), and let \(X=\mu+AS\). Its support is the ellipse \[ \frac{(x_1-1)^2}{16}+\frac{(x_2+1)^2}{4}=1. \] For the projection \(c^\top X\), with \(c=(1,2)^\top\), the center is \(c^\top\mu=-1\). Since \(\operatorname{Cov}(S)=2I_2\), its variance is \[ \operatorname{Var}(c^\top X)=c^\top A(2I_2)A^\top c =2\|A^\top c\|^2=2(2^2+2^2)=16. \] Its characteristic function at scalar argument \(u\) is \(e^{-iu}J_0(2|u|\|A^\top c\|)=e^{-iu}J_0(4\sqrt{2}|u|)\).
Uniform ellipsoids from a uniform direction
A uniform distribution on a ball does not have a fixed radius. In \(d\) dimensions, if \(S\) is uniform in the unit ball, its direction is uniform on the sphere and independent of its radius \(R=\|S\|\). The radius has distribution \(P(R\leq r)=r^d\) for \(0\leq r\leq1\), hence density \(d r^{d-1}\) on that interval. This follows because the ball of radius \(r\) has \(r^d\) times the volume of the unit ball.
Worked Example: A Uniform Ellipse in Two Dimensions
Let \(S\) be uniform on the unit disk in \(\mathbb{R}^2\), and set \(X=\mu+AS\), where \(\mu=(1,-1)^\top\) and \(A=\operatorname{diag}(2,1)\). Because the transformation is nonsingular, \(X\) is uniform on the ellipse \[ \frac{(x_1-1)^2}{4}+(x_2+1)^2\leq1. \] The inequality includes the boundary; the boundary itself has probability zero.
Here \(R\) has density \(2r\) on \(0\leq r\leq1\), so \(E[R^2]=\int_0^1 2r^3\,dr=1/2\). For a uniform direction in two dimensions, \(E[UU^\top]=I_2/2\). Independence therefore gives \(E[S]=0\) and \[ \operatorname{Cov}(S)=E[R^2]E[UU^\top]=\frac14 I_2, \qquad \operatorname{Cov}(X)=A\operatorname{Cov}(S)A^\top =\begin{pmatrix}1&0\\0&1/4\end{pmatrix}. \] For \(c=(1,2)^\top\), the projection has mean \(c^\top\mu=-1\) and variance \(c^\top\operatorname{Cov}(X)c=1+4(1/4)=2\).
For \(t\in\mathbb{R}^2\), polar coordinates give the characteristic function of \(S\): \[ \varphi_S(t)=\int_0^1 2rJ_0(r\|t\|)\,dr =\frac{2J_1(\|t\|)}{\|t\|}, \] with value 1 at \(t=0\). The last equality follows from \(\frac{d}{dx}[xJ_1(x)]=xJ_0(x)\); the limit at zero uses \(J_1(x)/x\to1/2\). Consequently, \[ \varphi_X(t)=e^{it^\top\mu}\frac{2J_1(\|A^\top t\|)}{\|A^\top t\|}, \] again interpreted by continuity when \(t=0\).
General transformation calculations
The examples illustrate rules that apply to any spherical vector with finite second moments. If \(S\) has mean zero and covariance matrix \(\tau^2 I_d\), then \(X=\mu+AS\) has mean \(\mu\) and covariance \(\tau^2 AA^\top\). For a projection \(c^\top X\), its mean is \(c^\top\mu\), its variance is \(\tau^2\|A^\top c\|^2\), and its characteristic function at \(u\) is \(e^{iu c^\top\mu}\varphi_S(uA^\top c)\). Each statement follows by expanding the mean, covariance, or scalar inner product; no density is needed.
The matrix \(AA^\top\) describes the directions and relative scales of covariance when that covariance exists. It need not equal the covariance itself: the scalar \(\tau^2\) depends on the radial law. In particular, heavy-tailed models may have a well-defined scatter matrix while lacking a finite covariance. A support calculation likewise depends on the radial law, not just the matrix.
Worked construction: a multivariate Student model
A multivariate Student vector can be constructed from a standard normal vector and a positive radial scaling variable. Let \(Z\sim N(0,I_d)\), let \(W\) have the chi-square distribution with \(\nu>0\) degrees of freedom, and assume \(Z\) and \(W\) are independent. For positive-definite \(\Sigma\), define \[ T=\mu+\sqrt{\frac{\nu}{W}}\,\Sigma^{1/2}Z. \] The normal vector is spherical, and its random scale gives the Student radial behavior. Since \(W>0\) almost surely, the support is all of \(\mathbb{R}^d\).
To obtain the density, condition on \(W=w\). The conditional distribution is normal with mean \(\mu\) and covariance \((\nu/w)\Sigma\). Write \(\delta=(x-\mu)^\top\Sigma^{-1}(x-\mu)\). Multiplying its conditional density by the chi-square density and integrating over \(w>0\) gives \[ \frac{|\Sigma|^{-1/2}}{(2\pi\nu)^{d/2}2^{\nu/2}\Gamma(\nu/2)} \int_0^\infty w^{(\nu+d)/2-1} \exp\left\{-\frac{w}{2}\left(1+\frac{\delta}{\nu}\right)\right\}\,dw. \] The gamma integral \(\int_0^\infty w^{a-1}e^{-bw}\,dw=\Gamma(a)b^{-a}\), for \(a,b>0\), evaluates this expression to the displayed density.
For the covariance, symmetry gives \(E[T]=\mu\) whenever the mean exists. Conditional covariance is \((\nu/W)\Sigma\), so for \(\nu>2\), \[ E\left[\frac{\nu}{W}\right] =\frac{\nu}{2^{\nu/2}\Gamma(\nu/2)} \int_0^\infty w^{\nu/2-2}e^{-w/2}\,dw =\frac{\nu}{\nu-2}. \] The integral is finite precisely when \(\nu>2\), yielding the stated covariance. For \(0<\nu\leq2\), this covariance does not exist as a finite matrix.
Worked Example: Projecting a Bivariate Student Vector
Take \(\nu=5\), \(\mu=(1,0)^\top\), and \[ \Sigma=\begin{pmatrix}4&1\\1&2\end{pmatrix}. \] This matrix is positive definite because its leading diagonal entry is 4 and its determinant is \(8-1=7>0\). For \(c=(1,-1)^\top\), the projection \(c^\top T\) has location \(c^\top\mu=1\) and Student scale squared \(c^\top\Sigma c=4-2+2=4\). Thus it is a univariate Student distribution with 5 degrees of freedom, location 1, and scale 2. Its variance is \(5/3\) times its scale squared, or \(20/3\).
The full vector covariance is \((5/3)\Sigma\), and its support is all of \(\mathbb{R}^2\). To compute its characteristic function, let \(q=t^\top\Sigma t\). Conditional on \(W\), the centered normal part has characteristic function \(\exp\{-\nu q/(2W)\}\). Integrating against the chi-square density gives \[ \varphi_T(t)=e^{it^\top\mu} \frac{2^{1-\nu/2}}{\Gamma(\nu/2)} (\sqrt{\nu q})^{\nu/2}K_{\nu/2}(\sqrt{\nu q}), \] where \(K_a\) is the modified Bessel function of the second kind. This follows from the integral identity \[ \int_0^\infty w^{a-1}e^{-\beta w-\gamma/w}\,dw =2(\gamma/\beta)^{a/2}K_a(2\sqrt{\beta\gamma}), \quad a,\beta,\gamma>0. \] For this example, \(q=4t_1^2+2t_1t_2+2t_2^2\), and substituting \(\nu=5\) gives the characteristic function at every \(t\). At \(t=0\), its value is 1 by continuity.
Why the construction matters
A construction separates three questions that are easy to conflate: what radii are possible, how directions are distributed, and how the model is transformed. The fixed-radius example has support on a shell; the uniform-ball example fills an ellipse; the Student example has unbounded support and heavier tails. Their affine matrices can look similar without making these distributions interchangeable.
A common mistake is to infer the entire distribution from a covariance or scatter matrix. For instance, the fixed-radius circle and a normal distribution can both be spherical, but one is supported on a circle and the other has support throughout the plane. The correction is to specify the radial law as well as the matrix. When calculating a projection, another useful check is dimensional: its location is \(c^\top\mu\), and its variance, when finite, is a scalar \(c^\top\operatorname{Cov}(X)c\).
Check Your Understanding
Use the constructions and calculations above to answer the following.
- If \(S=3U\) with \(U\) uniform on the unit circle, what is the support of \(S\), and what is the variance of its first coordinate?
- For \(X=\mu+AS\), explain why the characteristic function involves \(A^\top t\), not \(At\).
- A vector is uniform on the unit ball in \(\mathbb{R}^4\). What is its radial distribution function for \(0\leq r\leq1\)?
- For a multivariate Student vector with \(\nu=4\) and scatter matrix \(\Sigma\), what is its covariance matrix?
- For a Student projection \(c^\top T\), identify its location and scale squared in terms of \(\mu,\Sigma,\) and \(c\).