Tree 4-0-0-0-0-1-0

Evolutionary model

Cavender Felsenstein Neyman model with root distribution $\pi = \left(\frac{1}{2}, \frac{1}{2}\right)$.
The transition matrix associated with edge $i$ is of the form $$M_j = \begin{pmatrix} a_i & b_i\\ b_i & a_i \end{pmatrix}$$ and the Fourier parameters are $\left(x_i, y_i\right)$ for all edges $i$.

Summary

Probability parametrization

$\begin{align} p_{\texttt{ACCC}} &= \frac{1}{4}(2a_1a_3a_4b_2b_5 + 2a_1a_5b_2b_3b_4 + 2a_2a_3a_4a_5b_1 + 2a_2b_1b_3b_4b_5) \\[0.5em] p_{\texttt{ACCA}} &= \frac{1}{4}(2a_1a_3b_2b_4b_5 + 2a_1a_4a_5b_2b_3 + 2a_2a_3a_5b_1b_4 + 2a_2a_4b_1b_3b_5) \\[0.5em] p_{\texttt{ACAC}} &= \frac{1}{4}(2a_1a_3a_5b_2b_4 + 2a_1a_4b_2b_3b_5 + 2a_2a_3b_1b_4b_5 + 2a_2a_4a_5b_1b_3) \\[0.5em] p_{\texttt{ACAA}} &= \frac{1}{4}(2a_1a_3a_4a_5b_2 + 2a_1b_2b_3b_4b_5 + 2a_2a_3a_4b_1b_5 + 2a_2a_5b_1b_3b_4) \\[0.5em] p_{\texttt{AACC}} &= \frac{1}{4}(2a_1a_2a_3a_4b_5 + 2a_1a_2a_5b_3b_4 + 2a_3a_4a_5b_1b_2 + 2b_1b_2b_3b_4b_5) \\[0.5em] p_{\texttt{AACA}} &= \frac{1}{4}(2a_1a_2a_3b_4b_5 + 2a_1a_2a_4a_5b_3 + 2a_3a_5b_1b_2b_4 + 2a_4b_1b_2b_3b_5) \\[0.5em] p_{\texttt{AAAC}} &= \frac{1}{4}(2a_1a_2a_3a_5b_4 + 2a_1a_2a_4b_3b_5 + 2a_3b_1b_2b_4b_5 + 2a_4a_5b_1b_2b_3) \\[0.5em] p_{\texttt{AAAA}} &= \frac{1}{4}(2a_1a_2a_3a_4a_5 + 2a_1a_2b_3b_4b_5 + 2a_3a_4b_1b_2b_5 + 2a_5b_1b_2b_3b_4) \end{align}$

Fourier parametrization

$\begin{align} q_{\texttt{CCCC}} &= x_5y_1y_2y_3y_4 \\[0.5em] q_{\texttt{CCAA}} &= x_3x_4x_5y_1y_2 \\[0.5em] q_{\texttt{CACA}} &= x_2x_4y_1y_3y_5 \\[0.5em] q_{\texttt{CAAC}} &= x_2x_3y_1y_4y_5 \\[0.5em] q_{\texttt{ACCA}} &= x_1x_4y_2y_3y_5 \\[0.5em] q_{\texttt{ACAC}} &= x_1x_3y_2y_4y_5 \\[0.5em] q_{\texttt{AACC}} &= x_1x_2x_5y_3y_4 \\[0.5em] q_{\texttt{AAAA}} &= x_1x_2x_3x_4x_5 \end{align}$

Equivalent classes of probability parametrization

$\begin{align} \text{Class } &p_{\texttt{AAAA}}:\ p_{\texttt{AAAA}},\ p_{\texttt{CCCC}} \\[0.5em] \text{Class } &p_{\texttt{AAAC}}:\ p_{\texttt{AAAC}},\ p_{\texttt{CCCA}} \\[0.5em] \text{Class } &p_{\texttt{AACA}}:\ p_{\texttt{AACA}},\ p_{\texttt{CCAC}} \\[0.5em] \text{Class } &p_{\texttt{AACC}}:\ p_{\texttt{AACC}},\ p_{\texttt{CCAA}} \\[0.5em] \text{Class } &p_{\texttt{ACAA}}:\ p_{\texttt{ACAA}},\ p_{\texttt{CACC}} \\[0.5em] \text{Class } &p_{\texttt{ACAC}}:\ p_{\texttt{ACAC}},\ p_{\texttt{CACA}} \\[0.5em] \text{Class } &p_{\texttt{ACCA}}:\ p_{\texttt{ACCA}},\ p_{\texttt{CAAC}} \\[0.5em] \text{Class } &p_{\texttt{ACCC}}:\ p_{\texttt{ACCC}},\ p_{\texttt{CAAA}} \end{align}$

Equivalent classes of Fourier parametrization

$\begin{align} \text{Class } &0:\ q_{\texttt{AAAC}}, q_{\texttt{AACA}}, q_{\texttt{ACAA}}, q_{\texttt{ACCC}}, q_{\texttt{CAAA}}, q_{\texttt{CACC}}, q_{\texttt{CCAC}}, q_{\texttt{CCCA}} \\[0.5em] \text{Class } &q_{\texttt{AAAA}}:\ q_{\texttt{AAAA}} \\[0.5em] \text{Class } &q_{\texttt{AACC}}:\ q_{\texttt{AACC}} \\[0.5em] \text{Class } &q_{\texttt{ACAC}}:\ q_{\texttt{ACAC}} \\[0.5em] \text{Class } &q_{\texttt{ACCA}}:\ q_{\texttt{ACCA}} \\[0.5em] \text{Class } &q_{\texttt{CAAC}}:\ q_{\texttt{CAAC}} \\[0.5em] \text{Class } &q_{\texttt{CACA}}:\ q_{\texttt{CACA}} \\[0.5em] \text{Class } &q_{\texttt{CCAA}}:\ q_{\texttt{CCAA}} \\[0.5em] \text{Class } &q_{\texttt{CCCC}}:\ q_{\texttt{CCCC}} \end{align}$

Minimal generating set of the vanishing ideal

Ideal generated by $$ \begin{align} -q_{\texttt{CCCC}}q_{\texttt{AAAA}} + q_{\texttt{CCAA}}q_{\texttt{AACC}} \\ -q_{\texttt{CACA}}q_{\texttt{ACAC}} + q_{\texttt{CAAC}}q_{\texttt{ACCA}} \end{align} $$

Gröbner basis of the vanishing ideal

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 $$ \begin{align} -q_{\texttt{CCCC}}q_{\texttt{AAAA}} + q_{\texttt{CCAA}}q_{\texttt{AACC}} \\ -q_{\texttt{CACA}}q_{\texttt{ACAC}} + q_{\texttt{CAAC}}q_{\texttt{ACCA}} \end{align} $$

Additional information

Probability $\to$ Fourier

$\begin{align} q_{\texttt{CCCC}} &= -p_{\texttt{ACCC}} + p_{\texttt{ACCA}} + p_{\texttt{ACAC}} - p_{\texttt{ACAA}} + p_{\texttt{AACC}} - p_{\texttt{AACA}} - p_{\texttt{AAAC}} + p_{\texttt{AAAA}} \\[0.5em] q_{\texttt{CCAA}} &= -p_{\texttt{ACCC}} - p_{\texttt{ACCA}} - p_{\texttt{ACAC}} - p_{\texttt{ACAA}} + p_{\texttt{AACC}} + p_{\texttt{AACA}} + p_{\texttt{AAAC}} + p_{\texttt{AAAA}} \\[0.5em] q_{\texttt{CACA}} &= -p_{\texttt{ACCC}} - p_{\texttt{ACCA}} + p_{\texttt{ACAC}} + p_{\texttt{ACAA}} - p_{\texttt{AACC}} - p_{\texttt{AACA}} + p_{\texttt{AAAC}} + p_{\texttt{AAAA}} \\[0.5em] q_{\texttt{CAAC}} &= -p_{\texttt{ACCC}} + p_{\texttt{ACCA}} - p_{\texttt{ACAC}} + p_{\texttt{ACAA}} - p_{\texttt{AACC}} + p_{\texttt{AACA}} - p_{\texttt{AAAC}} + p_{\texttt{AAAA}} \\[0.5em] q_{\texttt{ACCA}} &= p_{\texttt{ACCC}} + p_{\texttt{ACCA}} - p_{\texttt{ACAC}} - p_{\texttt{ACAA}} - p_{\texttt{AACC}} - p_{\texttt{AACA}} + p_{\texttt{AAAC}} + p_{\texttt{AAAA}} \\[0.5em] q_{\texttt{ACAC}} &= p_{\texttt{ACCC}} - p_{\texttt{ACCA}} + p_{\texttt{ACAC}} - p_{\texttt{ACAA}} - p_{\texttt{AACC}} + p_{\texttt{AACA}} - p_{\texttt{AAAC}} + p_{\texttt{AAAA}} \\[0.5em] q_{\texttt{AACC}} &= p_{\texttt{ACCC}} - p_{\texttt{ACCA}} - p_{\texttt{ACAC}} + p_{\texttt{ACAA}} + p_{\texttt{AACC}} - p_{\texttt{AACA}} - p_{\texttt{AAAC}} + p_{\texttt{AAAA}} \\[0.5em] q_{\texttt{AAAA}} &= p_{\texttt{ACCC}} + p_{\texttt{ACCA}} + p_{\texttt{ACAC}} + p_{\texttt{ACAA}} + p_{\texttt{AACC}} + p_{\texttt{AACA}} + p_{\texttt{AAAC}} + p_{\texttt{AAAA}} \end{align}$

Fourier $\to$ Probability

$\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}$