The transformation matrices T.a and T.c are chosen by the analyst
and not estimated. The T matrices must be invertible square
matrices of size outcomes-1. As a shortcut, either T matrix
can be specified as "trend" for a Fourier basis or as "id" for an
identity basis. The response probability function is$$a_k = T_a \alpha_k$$
$$c_k = T_c \gamma_k$$
$$\mathrm P(\mathrm{pick}=k|a,a_k,c_k) = C\ \frac{1}{1+\exp(-(a \theta a_k + c_k))}$$
where $a_k$ and $c_k$ are the result of multiplying two vectors
of free parameters $\alpha$ and $\gamma$ by fixed matrices $T_a$ and $T_c$, respectively;
$a_0$ and $c_0$ are fixed to 0 for identification;
and $C$ is a normalizing constant to ensure that $\sum_k \mathrm P(\mathrm{pick}=k) = 1$.