Reflection coefficients

Reflection coefficients (RCs) are an alternate way of representing the information in an AR model. The RCs can be more convenient and easier to statistically model. Reflection coefficients (RCs) may be calculated by invertible transformation from AR coefficients or ACF estimates [31], and therefore we may use the results of Sections 5.2.5, 5.2.8, or 5.2.8. Computationally, the best method is to convert from ACF to RC because the RCs are obtained as a by-product of the Levinson algorithm.

ACF to RCs (module_acf2rc.m). The conversion from ACF to RCs is an invertible transformation that is characterized by a Jacobian matrix. The determinant of this matrix is the J-function of the transformation.

where ${\bf r}$ is the ACF vector, ${\bf r}\stackrel{\mbox{\tiny $\Delta$}}{=}[r_0, r_1, \ldots r_P]$, and ${\bf z}$ is the vector of reflection coefficients augmented by the variance (zero-th lag ACF sample),

$${\bf k}\stackrel{\mbox{\tiny$\Delta$}}{=}[r_0, k_1 \ldots k_P].$$

Note that we use $r_0$ and not the AR prediction error variance $\sigma^2_0$. This transformation is invertible and is characterized by the Jacobian

The above is implemented by software/module_acf2rc.m.

AR to RCs(module_ar2rc.m). You would probably never need to convert from AR coefficients to RC coefficients because the Levinson algorithm outputs both coefficients as a by-product. However, for completeness, the algorithm, which makes the conversion in-place is provided in software/module_ar2rc.m.

The Jacobian for converting from AR to RC coefficients is obtained by the following algorithm:

$$\log J = - \sum_{p\in {\rm\bf O}} \frac{(p-1)}{2} \; \log(1-k_p^2) - \sum_{p\in {\rm\bf E}} \frac{p}{2} \; \log(1-k_p^2),$$

where O is the set of odd values of $p$ in the range $2\leq p \leq P$, and where E is the set of even values of $p$ in the same range.