id: "3-0-0-0-0-0-0-CFN"
treename: "3-leaf star tree"
modelname: "Cavender Felsenstein Neyman"
rooted: "False"
leaves: "3"
n_cycles: "0"
level: "0"
internal_edges: "0"
dim: "4"
deg: "1"
np: "4"
nq: "4"
mu: "0"
e: "0"
M: "1"
E: "-1"
sD: "-"
sd: "-"
MLdeg: "1"
EDdeg: "1"
details:
  transition_matrix: |-
    $$M_j = \begin{pmatrix}
    a_i & b_i\\
    b_i & a_i
    \end{pmatrix}$$
  root_distributon: |-
    $\pi = \left(\frac{1}{2}, \frac{1}{2}\right)$
  fourier_parameters: |-
    $\left(x_i, y_i\right)$
  probability_parametrization: |-
    $\begin{align}
    p_{\texttt{ACC}} &= \frac{1}{4}(2a_1b_2b_3 + 2a_2a_3b_1) \\[0.5em]
    p_{\texttt{ACA}} &= \frac{1}{4}(2a_1a_3b_2 + 2a_2b_1b_3) \\[0.5em]
    p_{\texttt{AAC}} &= \frac{1}{4}(2a_1a_2b_3 + 2a_3b_1b_2) \\[0.5em]
    p_{\texttt{AAA}} &= \frac{1}{4}(2a_1a_2a_3 + 2b_1b_2b_3)
    \end{align}$
  equivalent_classes_prob: |-
    $\begin{align}
    \text{Class } &p_{\texttt{AAA}}:\ p_{\texttt{AAA}},\ p_{\texttt{CCC}} \\[0.5em]
    \text{Class } &p_{\texttt{AAC}}:\ p_{\texttt{AAC}},\ p_{\texttt{CCA}} \\[0.5em]
    \text{Class } &p_{\texttt{ACA}}:\ p_{\texttt{ACA}},\ p_{\texttt{CAC}} \\[0.5em]
    \text{Class } &p_{\texttt{ACC}}:\ p_{\texttt{ACC}},\ p_{\texttt{CAA}}
    \end{align}$
  fourier_parametrization: |-
    $\begin{align}
    q_{\texttt{CCA}} &= x_3y_1y_2 \\[0.5em]
    q_{\texttt{CAC}} &= x_2y_1y_3 \\[0.5em]
    q_{\texttt{ACC}} &= x_1y_2y_3 \\[0.5em]
    q_{\texttt{AAA}} &= x_1x_2x_3
    \end{align}$
  equivalent_classes_fourier: |-
    $\begin{align}
    \text{Class } &0:\ q_{\texttt{AAC}}, q_{\texttt{ACA}}, q_{\texttt{CAA}}, q_{\texttt{CCC}} \\[0.5em]
    \text{Class } &q_{\texttt{AAA}}:\ q_{\texttt{AAA}} \\[0.5em]
    \text{Class } &q_{\texttt{ACC}}:\ q_{\texttt{ACC}} \\[0.5em]
    \text{Class } &q_{\texttt{CAC}}:\ q_{\texttt{CAC}} \\[0.5em]
    \text{Class } &q_{\texttt{CCA}}:\ q_{\texttt{CCA}}
    \end{align}$
  invariants: |-
    Ideal generated by $\langle 0 \rangle$
  grobner_basis: |-
    Gröbner basis with respect to the ordering
    $ \texttt{degrevlex}\ [ q_{\texttt{CCA}}, q_{\texttt{CAC}}, q_{\texttt{ACC}}, q_{\texttt{AAA}} ] $
    and with elements  $$ \begin{align}
    0
    \end{align} $$
  coordinate_change: |-
    $\begin{align}
    q_{\texttt{CCA}} &= -p_{\texttt{ACC}} - p_{\texttt{ACA}} + p_{\texttt{AAC}} + p_{\texttt{AAA}} \\[0.5em]
    q_{\texttt{CAC}} &= -p_{\texttt{ACC}} + p_{\texttt{ACA}} - p_{\texttt{AAC}} + p_{\texttt{AAA}} \\[0.5em]
    q_{\texttt{ACC}} &= p_{\texttt{ACC}} - p_{\texttt{ACA}} - p_{\texttt{AAC}} + p_{\texttt{AAA}} \\[0.5em]
    q_{\texttt{AAA}} &= p_{\texttt{ACC}} + p_{\texttt{ACA}} + p_{\texttt{AAC}} + p_{\texttt{AAA}}
    \end{align}$
  inverse_coordinate_change: |-
    $\begin{align}
    p_{\texttt{ACC}} &= -\frac{1}{4}q_{\texttt{CCA}} - \frac{1}{4}q_{\texttt{CAC}} + \frac{1}{4}q_{\texttt{ACC}} + \frac{1}{4}q_{\texttt{AAA}} \\[0.5em]
    p_{\texttt{ACA}} &= -\frac{1}{4}q_{\texttt{CCA}} + \frac{1}{4}q_{\texttt{CAC}} - \frac{1}{4}q_{\texttt{ACC}} + \frac{1}{4}q_{\texttt{AAA}} \\[0.5em]
    p_{\texttt{AAC}} &= \frac{1}{4}q_{\texttt{CCA}} - \frac{1}{4}q_{\texttt{CAC}} - \frac{1}{4}q_{\texttt{ACC}} + \frac{1}{4}q_{\texttt{AAA}} \\[0.5em]
    p_{\texttt{AAA}} &= \frac{1}{4}q_{\texttt{CCA}} + \frac{1}{4}q_{\texttt{CAC}} + \frac{1}{4}q_{\texttt{ACC}} + \frac{1}{4}q_{\texttt{AAA}}
    \end{align}$
