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10.12 Cholesky factors of correlation matrices

A $K \times K$ correlation matrix $\Omega$ is positive definite and has a unit diagonal. Because it is positive definite, it can be Cholesky factored to a $K \times K$ lower-triangular matrix $L$ with positive diagonal elements such that $\Omega = L\,L^{\top}$. Because the correlation matrix has a unit diagonal,

$$\Omega_{k,k} = L_k\,L_k^{\top} = 1,$$

each row vector $L_k$ of the Cholesky factor is of unit length. The length and positivity constraint allow the diagonal elements of $L$ to be calculated from the off-diagonal elements, so that a Cholesky factor for a $K \times K$ correlation matrix requires only $\binom{K}{2}$ unconstrained parameters.

Cholesky factor of correlation matrix inverse transform

It is easiest to start with the inverse transform from the $\binom{K}{2}$ unconstrained parameters $y$ to the $K \times K$ lower-triangular Cholesky factor $x$. The inverse transform is based on the hyperbolic tangent function, $\tanh$, which satisfies $\tanh(x) \in (-1,1)$. Here it will function like an inverse logit with a sign to pick out the direction of an underlying canonical partial correlation; see the section on correlation matrix transforms for more information on the relation between canonical partial correlations and the Cholesky factors of correlation matrices.

Suppose $y$ is a vector of $\binom{K}{2}$ unconstrained values. Let $z$ be a lower-triangular matrix with zero diagonal and below diagonal entries filled by row. For example, in the $3 \times 3$ case,

$$z = \left[ \begin{array}{ccc} 0 & 0 & 0 \\ \tanh y_1 & 0 & 0 \\ \tanh y_2 & \tanh y_3 & 0 \end{array} \right]$$

The matrix $z$, with entries in the range $(-1, 1)$, is then transformed to the Cholesky factor $x$, by taking¹⁵

$$x_{i,j} = \left\{ \begin{array}{lll} 0 & \mbox{ if } i < j & \mbox{ [above diagonal]} \\ \sqrt{1 - \sum_{j' < j} x_{i,j'}^2} & \mbox{ if } i = j & \mbox{ [on diagonal]} \\ z_{i,j} \ \sqrt{1 - \sum_{j' < j} x_{i,j'}^2} & \mbox{ if } i > j & \mbox{ [below diagonal]} \end{array} \right.$$

In the $3 \times 3$ case, this yields

$$x = \left[ \begin{array}{ccc} 1 & 0 & 0 \\ z_{2,1} & \sqrt{1 - x_{2,1}^2} & 0 \\ z_{3,1} & z_{3,2} \sqrt{1 - x_{3,1}^2} & \sqrt{1 - (x_{3,1}^2 + x_{3,2}^2)} \end{array} \right],$$

where the $z_{i,j} \in (-1,1)$ are the $\tanh$-transformed $y$.

The approach is a signed stick-breaking process on the quadratic (Euclidean length) scale. Starting from length 1 at $j=1$, each below-diagonal entry $x_{i,j}$ is determined by the (signed) fraction $z_{i,j}$ of the remaining length for the row that it consumes. The diagonal entries $x_{i,i}$ get any leftover length from earlier entries in their row. The above-diagonal entries are zero.

Cholesky factor of correlation matrix transform

Suppose $x$ is a $K \times K$ Cholesky factor for some correlation matrix. The first step of the transform reconstructs the intermediate values $z$ from $x$,

$$z_{i,j} = \frac{x_{i,j}}{\sqrt{1 - \sum_{j' < j}x_{i,j'}^2}}.$$

The mapping from the resulting $z$ to $y$ inverts $\tanh$,

$$y \ = \ \tanh^{-1} z \ = \ \frac{1}{2} \left( \log (1 + z) - \log (1 - z) \right).$$

Absolute Jacobian determinant of inverse transform

The Jacobian of the full transform is the product of the Jacobians of its component transforms.

First, for the inverse transform $z = \tanh y$, the derivative is

$$\frac{d}{dy} \tanh y = \frac{1}{(\cosh y)^2}.$$

Second, for the inverse transform of $z$ to $x$, the resulting Jacobian matrix $J$ is of dimension $\binom{K}{2} \times \binom{K}{2}$, with indexes $(i,j)$ for $(i > j)$. The Jacobian matrix is lower triangular, so that its determinant is the product of its diagonal entries, of which there is one for each $(i,j)$ pair,

$$\left| \, \mbox{det} \, J \, \right| \ = \ \prod_{i > j} \left| \frac{d}{dz_{i,j}} x_{i,j} \right|,$$

where

$$\frac{d}{dz_{i,j}} x_{i,j} = \sqrt{1 - \sum_{j' < j} x^2_{i,j'}}.$$

So the combined density for unconstrained $y$ is

$$p_Y(y) = p_X(f^{-1}(y)) \ \ \prod_{n < \binom{K}{2}} \frac{1}{(\cosh y)^2} \ \ \prod_{i > j} \left( 1 - \sum_{j' < j} x_{i,j'}^2 \right)^{1/2},$$

where $x = f^{-1}(y)$ is used for notational convenience. The log Jacobian determinant of the complete inverse transform $x = f^{-1}(y)$ is given by

$$\log \left| \, \det J \, \right| = -2 \sum_{n \leq \binom{K}{2}} \log \cosh y \ + \ \frac{1}{2} \ \sum_{i > j} \log \left( 1 - \sum_{j' < j} x_{i,j'}^2 \right) .$$

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15. For convenience, a summation with no terms, such as $\sum_{j' < 1} x_{i,j'}$, is defined to be 0. This implies $x_{1,1} = 1$ and that $x_{i,1} = z_{i,1}$ for $i > 1$.↩︎