##

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 [25], 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 is the ACF vector,
, and
is the vector of reflection coefficients
augmented by the variance (zero-th lag ACF sample),

Note that we use and *not* the AR prediction
error variance
.
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:

where **O** is the set of odd values of
in the range
, and
where **E** is the set of even values of
in the same range.

Baggenstoss
2017-05-19