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 3-0-0-0-0-0-0
Evolutionary model
Summary
Dimension
4
Degree
1
Probability coordinates
4
Fourier coordinates
4
Dimension of Singular locus
-
Degree of Singular locus
-
ML degree
1
ED degree
1
Model parametrizations
Probability parametrization
$\begin{align}
p_{\texttt{ACC}} &= \frac{1}{4}(2a_1b_2b_3 + 2a_2a_3b_1) \\[0.5em]
p_{\texttt{ACA}} &= \frac{1}{4}(2a_1a_3b_2 + 2a_2b_1b_3) \\[0.5em]
p_{\texttt{AAC}} &= \frac{1}{4}(2a_1a_2b_3 + 2a_3b_1b_2) \\[0.5em]
p_{\texttt{AAA}} &= \frac{1}{4}(2a_1a_2a_3 + 2b_1b_2b_3)
\end{align}$
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 of probability parametrization
$\begin{align}
\text{Class } &p_{\texttt{AAA}}:\ p_{\texttt{AAA}},\ p_{\texttt{CCC}} \\[0.5em]
\text{Class } &p_{\texttt{AAC}}:\ p_{\texttt{AAC}},\ p_{\texttt{CCA}} \\[0.5em]
\text{Class } &p_{\texttt{ACA}}:\ p_{\texttt{ACA}},\ p_{\texttt{CAC}} \\[0.5em]
\text{Class } &p_{\texttt{ACC}}:\ p_{\texttt{ACC}},\ p_{\texttt{CAA}}
\end{align}$
Equivalent classes of Fourier parametrization
$\begin{align}
\text{Class } &0:\ q_{\texttt{AAC}}, q_{\texttt{ACA}}, q_{\texttt{CAA}}, q_{\texttt{CCC}} \\[0.5em]
\text{Class } &q_{\texttt{AAA}}:\ q_{\texttt{AAA}} \\[0.5em]
\text{Class } &q_{\texttt{ACC}}:\ q_{\texttt{ACC}} \\[0.5em]
\text{Class } &q_{\texttt{CAC}}:\ q_{\texttt{CAC}} \\[0.5em]
\text{Class } &q_{\texttt{CCA}}:\ q_{\texttt{CCA}}
\end{align}$
Phylogenetic invariants
Minimal generating set of the vanishing ideal
Ideal generated by $\langle 0 \rangle$
Gröbner basis of the vanishing ideal
Gröbner basis with respect to the ordering
$ \texttt{degrevlex}\ [ q_{\texttt{CCA}}, q_{\texttt{CAC}}, q_{\texttt{ACC}}, q_{\texttt{AAA}} ] $
and with elements $$ \begin{align}
0
\end{align} $$
Additional information
Cardinality of the smallest set of generators for the ideal
0
Cardinality of the degree reverse lexicographic reduced Gröbner basis
1
Largest degree in a minimal generating set for the ideal
0
Largest degree of a generator in the reduced Gröbner basis
-1
Linear coordinate transformations
Probability $\to$ Fourier
$\begin{align}
q_{\texttt{CCA}} &= -p_{\texttt{ACC}} - p_{\texttt{ACA}} + p_{\texttt{AAC}} + p_{\texttt{AAA}} \\[0.5em]
q_{\texttt{CAC}} &= -p_{\texttt{ACC}} + p_{\texttt{ACA}} - p_{\texttt{AAC}} + p_{\texttt{AAA}} \\[0.5em]
q_{\texttt{ACC}} &= p_{\texttt{ACC}} - p_{\texttt{ACA}} - p_{\texttt{AAC}} + p_{\texttt{AAA}} \\[0.5em]
q_{\texttt{AAA}} &= p_{\texttt{ACC}} + p_{\texttt{ACA}} + p_{\texttt{AAC}} + p_{\texttt{AAA}}
\end{align}$
Fourier $\to$ Probability
$\begin{align}
p_{\texttt{ACC}} &= -\frac{1}{4}q_{\texttt{CCA}} - \frac{1}{4}q_{\texttt{CAC}} + \frac{1}{4}q_{\texttt{ACC}} + \frac{1}{4}q_{\texttt{AAA}} \\[0.5em]
p_{\texttt{ACA}} &= -\frac{1}{4}q_{\texttt{CCA}} + \frac{1}{4}q_{\texttt{CAC}} - \frac{1}{4}q_{\texttt{ACC}} + \frac{1}{4}q_{\texttt{AAA}} \\[0.5em]
p_{\texttt{AAC}} &= \frac{1}{4}q_{\texttt{CCA}} - \frac{1}{4}q_{\texttt{CAC}} - \frac{1}{4}q_{\texttt{ACC}} + \frac{1}{4}q_{\texttt{AAA}} \\[0.5em]
p_{\texttt{AAA}} &= \frac{1}{4}q_{\texttt{CCA}} + \frac{1}{4}q_{\texttt{CAC}} + \frac{1}{4}q_{\texttt{ACC}} + \frac{1}{4}q_{\texttt{AAA}}
\end{align}$