id: "4-0-0-0-0-0-0-CFN"
treename: "4-leaf star tree"
modelname: "Cavender Felsenstein Neyman"
rooted: "False"
leaves: "4"
n_cycles: "0"
level: "0"
internal_edges: "0"
dim: "5"
deg: "8"
np: "8"
nq: "8"
mu: "3"
e: "2"
M: "3"
E: "2"
sD: "2"
sd: "24"
MLdeg: "92"
EDdeg: "92"
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: |-
    $\begin{align}
    p_{\texttt{ACCC}} &= \frac{1}{4}(2a_1b_2b_3b_4 + 2a_2a_3a_4b_1) \\[0.5em]
    p_{\texttt{ACCA}} &= \frac{1}{4}(2a_1a_4b_2b_3 + 2a_2a_3b_1b_4) \\[0.5em]
    p_{\texttt{ACAC}} &= \frac{1}{4}(2a_1a_3b_2b_4 + 2a_2a_4b_1b_3) \\[0.5em]
    p_{\texttt{ACAA}} &= \frac{1}{4}(2a_1a_3a_4b_2 + 2a_2b_1b_3b_4) \\[0.5em]
    p_{\texttt{AACC}} &= \frac{1}{4}(2a_1a_2b_3b_4 + 2a_3a_4b_1b_2) \\[0.5em]
    p_{\texttt{AACA}} &= \frac{1}{4}(2a_1a_2a_4b_3 + 2a_3b_1b_2b_4) \\[0.5em]
    p_{\texttt{AAAC}} &= \frac{1}{4}(2a_1a_2a_3b_4 + 2a_4b_1b_2b_3) \\[0.5em]
    p_{\texttt{AAAA}} &= \frac{1}{4}(2a_1a_2a_3a_4 + 2b_1b_2b_3b_4)
    \end{align}$
  equivalent_classes_prob: |-
    $\begin{align}
    \text{Class } &p_{\texttt{AAAA}}:\ p_{\texttt{AAAA}},\ p_{\texttt{CCCC}} \\[0.5em]
    \text{Class } &p_{\texttt{AAAC}}:\ p_{\texttt{AAAC}},\ p_{\texttt{CCCA}} \\[0.5em]
    \text{Class } &p_{\texttt{AACA}}:\ p_{\texttt{AACA}},\ p_{\texttt{CCAC}} \\[0.5em]
    \text{Class } &p_{\texttt{AACC}}:\ p_{\texttt{AACC}},\ p_{\texttt{CCAA}} \\[0.5em]
    \text{Class } &p_{\texttt{ACAA}}:\ p_{\texttt{ACAA}},\ p_{\texttt{CACC}} \\[0.5em]
    \text{Class } &p_{\texttt{ACAC}}:\ p_{\texttt{ACAC}},\ p_{\texttt{CACA}} \\[0.5em]
    \text{Class } &p_{\texttt{ACCA}}:\ p_{\texttt{ACCA}},\ p_{\texttt{CAAC}} \\[0.5em]
    \text{Class } &p_{\texttt{ACCC}}:\ p_{\texttt{ACCC}},\ p_{\texttt{CAAA}}
    \end{align}$
  fourier_parametrization: |-
    $\begin{align}
    q_{\texttt{CCCC}} &= y_1y_2y_3y_4 \\[0.5em]
    q_{\texttt{CCAA}} &= x_3x_4y_1y_2 \\[0.5em]
    q_{\texttt{CACA}} &= x_2x_4y_1y_3 \\[0.5em]
    q_{\texttt{CAAC}} &= x_2x_3y_1y_4 \\[0.5em]
    q_{\texttt{ACCA}} &= x_1x_4y_2y_3 \\[0.5em]
    q_{\texttt{ACAC}} &= x_1x_3y_2y_4 \\[0.5em]
    q_{\texttt{AACC}} &= x_1x_2y_3y_4 \\[0.5em]
    q_{\texttt{AAAA}} &= x_1x_2x_3x_4
    \end{align}$
  equivalent_classes_fourier: |-
    $\begin{align}
    \text{Class } &0:\ q_{\texttt{AAAC}}, q_{\texttt{AACA}}, q_{\texttt{ACAA}}, q_{\texttt{ACCC}}, q_{\texttt{CAAA}}, q_{\texttt{CACC}}, q_{\texttt{CCAC}}, q_{\texttt{CCCA}} \\[0.5em]
    \text{Class } &q_{\texttt{AAAA}}:\ q_{\texttt{AAAA}} \\[0.5em]
    \text{Class } &q_{\texttt{AACC}}:\ q_{\texttt{AACC}} \\[0.5em]
    \text{Class } &q_{\texttt{ACAC}}:\ q_{\texttt{ACAC}} \\[0.5em]
    \text{Class } &q_{\texttt{ACCA}}:\ q_{\texttt{ACCA}} \\[0.5em]
    \text{Class } &q_{\texttt{CAAC}}:\ q_{\texttt{CAAC}} \\[0.5em]
    \text{Class } &q_{\texttt{CACA}}:\ q_{\texttt{CACA}} \\[0.5em]
    \text{Class } &q_{\texttt{CCAA}}:\ q_{\texttt{CCAA}} \\[0.5em]
    \text{Class } &q_{\texttt{CCCC}}:\ q_{\texttt{CCCC}}
    \end{align}$
  invariants: |-
    Ideal generated by  $$ \begin{align}
    -q_{\texttt{CCCC}}q_{\texttt{AAAA}} + q_{\texttt{CCAA}}q_{\texttt{AACC}} \\
    -q_{\texttt{CCCC}}q_{\texttt{AAAA}} + q_{\texttt{CACA}}q_{\texttt{ACAC}} \\
    -q_{\texttt{CCCC}}q_{\texttt{AAAA}} + q_{\texttt{CAAC}}q_{\texttt{ACCA}}
    \end{align} $$
  grobner_basis: |-
    Gröbner basis with respect to the ordering
    $ \texttt{degrevlex}\ [ q_{\texttt{CCCC}}, q_{\texttt{CCAA}}, q_{\texttt{CACA}}, q_{\texttt{CAAC}}, q_{\texttt{ACCA}}, q_{\texttt{ACAC}}, q_{\texttt{AACC}}, q_{\texttt{AAAA}} ] $
    and with elements  $$ \begin{align}
    -q_{\texttt{CCCC}}q_{\texttt{AAAA}} + q_{\texttt{CCAA}}q_{\texttt{AACC}} \\
    -q_{\texttt{CCCC}}q_{\texttt{AAAA}} + q_{\texttt{CACA}}q_{\texttt{ACAC}} \\
    -q_{\texttt{CCCC}}q_{\texttt{AAAA}} + q_{\texttt{CAAC}}q_{\texttt{ACCA}}
    \end{align} $$
  coordinate_change: |-
    $\begin{align}
    q_{\texttt{CCCC}} &= -p_{\texttt{ACCC}} + p_{\texttt{ACCA}} + p_{\texttt{ACAC}} - p_{\texttt{ACAA}} + p_{\texttt{AACC}} - p_{\texttt{AACA}} - p_{\texttt{AAAC}} + p_{\texttt{AAAA}} \\[0.5em]
    q_{\texttt{CCAA}} &= -p_{\texttt{ACCC}} - p_{\texttt{ACCA}} - p_{\texttt{ACAC}} - p_{\texttt{ACAA}} + p_{\texttt{AACC}} + p_{\texttt{AACA}} + p_{\texttt{AAAC}} + p_{\texttt{AAAA}} \\[0.5em]
    q_{\texttt{CACA}} &= -p_{\texttt{ACCC}} - p_{\texttt{ACCA}} + p_{\texttt{ACAC}} + p_{\texttt{ACAA}} - p_{\texttt{AACC}} - p_{\texttt{AACA}} + p_{\texttt{AAAC}} + p_{\texttt{AAAA}} \\[0.5em]
    q_{\texttt{CAAC}} &= -p_{\texttt{ACCC}} + p_{\texttt{ACCA}} - p_{\texttt{ACAC}} + p_{\texttt{ACAA}} - p_{\texttt{AACC}} + p_{\texttt{AACA}} - p_{\texttt{AAAC}} + p_{\texttt{AAAA}} \\[0.5em]
    q_{\texttt{ACCA}} &= p_{\texttt{ACCC}} + p_{\texttt{ACCA}} - p_{\texttt{ACAC}} - p_{\texttt{ACAA}} - p_{\texttt{AACC}} - p_{\texttt{AACA}} + p_{\texttt{AAAC}} + p_{\texttt{AAAA}} \\[0.5em]
    q_{\texttt{ACAC}} &= p_{\texttt{ACCC}} - p_{\texttt{ACCA}} + p_{\texttt{ACAC}} - p_{\texttt{ACAA}} - p_{\texttt{AACC}} + p_{\texttt{AACA}} - p_{\texttt{AAAC}} + p_{\texttt{AAAA}} \\[0.5em]
    q_{\texttt{AACC}} &= p_{\texttt{ACCC}} - p_{\texttt{ACCA}} - p_{\texttt{ACAC}} + p_{\texttt{ACAA}} + p_{\texttt{AACC}} - p_{\texttt{AACA}} - p_{\texttt{AAAC}} + p_{\texttt{AAAA}} \\[0.5em]
    q_{\texttt{AAAA}} &= p_{\texttt{ACCC}} + p_{\texttt{ACCA}} + p_{\texttt{ACAC}} + p_{\texttt{ACAA}} + p_{\texttt{AACC}} + p_{\texttt{AACA}} + p_{\texttt{AAAC}} + p_{\texttt{AAAA}}
    \end{align}$
  inverse_coordinate_change: |-
    $\begin{align}
    p_{\texttt{ACCC}} &= -\frac{1}{8}q_{\texttt{CCCC}} - \frac{1}{8}q_{\texttt{CCAA}} - \frac{1}{8}q_{\texttt{CACA}} - \frac{1}{8}q_{\texttt{CAAC}} + \frac{1}{8}q_{\texttt{ACCA}} + \frac{1}{8}q_{\texttt{ACAC}} + \frac{1}{8}q_{\texttt{AACC}} + \frac{1}{8}q_{\texttt{AAAA}} \\[0.5em]
    p_{\texttt{ACCA}} &= \frac{1}{8}q_{\texttt{CCCC}} - \frac{1}{8}q_{\texttt{CCAA}} - \frac{1}{8}q_{\texttt{CACA}} + \frac{1}{8}q_{\texttt{CAAC}} + \frac{1}{8}q_{\texttt{ACCA}} - \frac{1}{8}q_{\texttt{ACAC}} - \frac{1}{8}q_{\texttt{AACC}} + \frac{1}{8}q_{\texttt{AAAA}} \\[0.5em]
    p_{\texttt{ACAC}} &= \frac{1}{8}q_{\texttt{CCCC}} - \frac{1}{8}q_{\texttt{CCAA}} + \frac{1}{8}q_{\texttt{CACA}} - \frac{1}{8}q_{\texttt{CAAC}} - \frac{1}{8}q_{\texttt{ACCA}} + \frac{1}{8}q_{\texttt{ACAC}} - \frac{1}{8}q_{\texttt{AACC}} + \frac{1}{8}q_{\texttt{AAAA}} \\[0.5em]
    p_{\texttt{ACAA}} &= -\frac{1}{8}q_{\texttt{CCCC}} - \frac{1}{8}q_{\texttt{CCAA}} + \frac{1}{8}q_{\texttt{CACA}} + \frac{1}{8}q_{\texttt{CAAC}} - \frac{1}{8}q_{\texttt{ACCA}} - \frac{1}{8}q_{\texttt{ACAC}} + \frac{1}{8}q_{\texttt{AACC}} + \frac{1}{8}q_{\texttt{AAAA}} \\[0.5em]
    p_{\texttt{AACC}} &= \frac{1}{8}q_{\texttt{CCCC}} + \frac{1}{8}q_{\texttt{CCAA}} - \frac{1}{8}q_{\texttt{CACA}} - \frac{1}{8}q_{\texttt{CAAC}} - \frac{1}{8}q_{\texttt{ACCA}} - \frac{1}{8}q_{\texttt{ACAC}} + \frac{1}{8}q_{\texttt{AACC}} + \frac{1}{8}q_{\texttt{AAAA}} \\[0.5em]
    p_{\texttt{AACA}} &= -\frac{1}{8}q_{\texttt{CCCC}} + \frac{1}{8}q_{\texttt{CCAA}} - \frac{1}{8}q_{\texttt{CACA}} + \frac{1}{8}q_{\texttt{CAAC}} - \frac{1}{8}q_{\texttt{ACCA}} + \frac{1}{8}q_{\texttt{ACAC}} - \frac{1}{8}q_{\texttt{AACC}} + \frac{1}{8}q_{\texttt{AAAA}} \\[0.5em]
    p_{\texttt{AAAC}} &= -\frac{1}{8}q_{\texttt{CCCC}} + \frac{1}{8}q_{\texttt{CCAA}} + \frac{1}{8}q_{\texttt{CACA}} - \frac{1}{8}q_{\texttt{CAAC}} + \frac{1}{8}q_{\texttt{ACCA}} - \frac{1}{8}q_{\texttt{ACAC}} - \frac{1}{8}q_{\texttt{AACC}} + \frac{1}{8}q_{\texttt{AAAA}} \\[0.5em]
    p_{\texttt{AAAA}} &= \frac{1}{8}q_{\texttt{CCCC}} + \frac{1}{8}q_{\texttt{CCAA}} + \frac{1}{8}q_{\texttt{CACA}} + \frac{1}{8}q_{\texttt{CAAC}} + \frac{1}{8}q_{\texttt{ACCA}} + \frac{1}{8}q_{\texttt{ACAC}} + \frac{1}{8}q_{\texttt{AACC}} + \frac{1}{8}q_{\texttt{AAAA}}
    \end{align}$
