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 dimension 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 $\sum_k \mathrm P(\mathrm{pick}=k) = 1$.