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 $i$.
For examples on how to load and use .mrdi files, please see our notions and notations page or refer to the official OSCAR documentation.
Tree 4-0-0-0-0-0-0
Evolutionary model
Summary
Dimension
5
Degree
8
Probability coordinates
8
Fourier coordinates
8
Dimension of Singular locus
24
Degree of Singular locus
2
ML degree
92
ED degree
92
Model parametrizations
Probability parametrization
$\begin{align}
p_{\texttt{ACCC}} &= \frac{1}{4}(2a_1b_2b_3b_4 + 2a_2a_3a_4b_1) \\[0.5em]
p_{\texttt{ACCA}} &= \frac{1}{4}(2a_1a_4b_2b_3 + 2a_2a_3b_1b_4) \\[0.5em]
p_{\texttt{ACAC}} &= \frac{1}{4}(2a_1a_3b_2b_4 + 2a_2a_4b_1b_3) \\[0.5em]
p_{\texttt{ACAA}} &= \frac{1}{4}(2a_1a_3a_4b_2 + 2a_2b_1b_3b_4) \\[0.5em]
p_{\texttt{AACC}} &= \frac{1}{4}(2a_1a_2b_3b_4 + 2a_3a_4b_1b_2) \\[0.5em]
p_{\texttt{AACA}} &= \frac{1}{4}(2a_1a_2a_4b_3 + 2a_3b_1b_2b_4) \\[0.5em]
p_{\texttt{AAAC}} &= \frac{1}{4}(2a_1a_2a_3b_4 + 2a_4b_1b_2b_3) \\[0.5em]
p_{\texttt{AAAA}} &= \frac{1}{4}(2a_1a_2a_3a_4 + 2b_1b_2b_3b_4)
\end{align}$
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 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
$\begin{align}
\text{Class } &0:\ q_{\texttt{AAAC}}, q_{\texttt{AACA}}, q_{\texttt{ACAA}}, q_{\texttt{ACCC}}, q_{\texttt{CAAA}}, q_{\texttt{CACC}}, q_{\texttt{CCAC}}, q_{\texttt{CCCA}} \\[0.5em]
\text{Class } &q_{\texttt{AAAA}}:\ q_{\texttt{AAAA}} \\[0.5em]
\text{Class } &q_{\texttt{AACC}}:\ q_{\texttt{AACC}} \\[0.5em]
\text{Class } &q_{\texttt{ACAC}}:\ q_{\texttt{ACAC}} \\[0.5em]
\text{Class } &q_{\texttt{ACCA}}:\ q_{\texttt{ACCA}} \\[0.5em]
\text{Class } &q_{\texttt{CAAC}}:\ q_{\texttt{CAAC}} \\[0.5em]
\text{Class } &q_{\texttt{CACA}}:\ q_{\texttt{CACA}} \\[0.5em]
\text{Class } &q_{\texttt{CCAA}}:\ q_{\texttt{CCAA}} \\[0.5em]
\text{Class } &q_{\texttt{CCCC}}:\ q_{\texttt{CCCC}}
\end{align}$
Phylogenetic invariants
Minimal generating set of the vanishing ideal
Ideal generated by $$ \begin{align}
-q_{\texttt{CCCC}}q_{\texttt{AAAA}} + q_{\texttt{CCAA}}q_{\texttt{AACC}} \\
-q_{\texttt{CCCC}}q_{\texttt{AAAA}} + q_{\texttt{CACA}}q_{\texttt{ACAC}} \\
-q_{\texttt{CCCC}}q_{\texttt{AAAA}} + q_{\texttt{CAAC}}q_{\texttt{ACCA}}
\end{align} $$
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 $$ \begin{align}
-q_{\texttt{CCCC}}q_{\texttt{AAAA}} + q_{\texttt{CCAA}}q_{\texttt{AACC}} \\
-q_{\texttt{CCCC}}q_{\texttt{AAAA}} + q_{\texttt{CACA}}q_{\texttt{ACAC}} \\
-q_{\texttt{CCCC}}q_{\texttt{AAAA}} + q_{\texttt{CAAC}}q_{\texttt{ACCA}}
\end{align} $$
Additional information
Cardinality of the smallest set of generators for the ideal
3
Cardinality of the degree reverse lexicographic reduced Gröbner basis
3
Largest degree in a minimal generating set for the ideal
2
Largest degree of a generator in the reduced Gröbner basis
2
Linear coordinate transformations
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}$