Phylogenetic tree 3-0-0

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 $\left(x^{(i)}_1, x^{(i)}_2\right)$ for all edges $i$.

Summary

Dimension: 4
Degree: 1
Probability coordinates: 4
Fourier coordinates: 4
dim of Singular locus: -
deg of Singular locus: -
MLdeg: 1
EDdeg: 1

Probability parametrization

$\begin{align} p_{\texttt{010}} &= \frac{1}{2}(a_1a_3b_2 + a_2b_1b_3) \\[0.5em] p_{\texttt{000}} &= \frac{1}{2}(a_1a_2a_3 + b_1b_2b_3) \\[0.5em] p_{\texttt{011}} &= \frac{1}{2}(a_1b_2b_3 + a_2a_3b_1) \\[0.5em] p_{\texttt{001}} &= \frac{1}{2}(a_1a_2b_3 + a_3b_1b_2) \end{align}$

Fourier parametrization

$\begin{align} q_{\texttt{000}} &= x^{(1)}_{1}x^{(2)}_{1}x^{(3)}_{1} \\[0.5em] q_{\texttt{110}} &= x^{(1)}_{2}x^{(2)}_{2}x^{(3)}_{1} \\[0.5em] q_{\texttt{011}} &= x^{(1)}_{1}x^{(2)}_{2}x^{(3)}_{2} \\[0.5em] q_{\texttt{101}} &= x^{(1)}_{2}x^{(2)}_{1}x^{(3)}_{2} \end{align}$

Equivalent classes of probability parametrization

$\begin{align} \text{Class } &p_{\texttt{000}}:\ p_{\texttt{000}},\ p_{\texttt{111}} \\[0.5em] \text{Class } &p_{\texttt{001}}:\ p_{\texttt{001}},\ p_{\texttt{110}} \\[0.5em] \text{Class } &p_{\texttt{010}}:\ p_{\texttt{010}},\ p_{\texttt{101}} \\[0.5em] \text{Class } &p_{\texttt{011}}:\ p_{\texttt{011}},\ p_{\texttt{100}} \end{align}$

Equivalent classes of Fourier parametrization

$\begin{align} \text{Class } &0:\ q_{\texttt{100}}, q_{\texttt{010}}, q_{\texttt{001}}, q_{\texttt{111}} \\[0.5em] \text{Class } &q_{\texttt{000}}:\ q_{\texttt{000}} \\[0.5em] \text{Class } &q_{\texttt{011}}:\ q_{\texttt{011}} \\[0.5em] \text{Class } &q_{\texttt{101}}:\ q_{\texttt{101}} \\[0.5em] \text{Class } &q_{\texttt{110}}:\ q_{\texttt{110}} \end{align}$

Invariants in Fourier coordinates

Ideal generated by $\langle 0 \rangle$

Gröbner basis of ideal of invariants

Gröbner basis with $0$ elements

Additional information

The cardinality of the smallest set of generators that define the ideal of phylogenetic invariants: 0
The largest degree of a generator in the degree reverse lexicographic reduced Gröbner basis: 0
The largest degree of an element in a minimal generating set for the ideal of phylogenetic invariants: 0
The cardinality of the degree reverse lexicographic reduced Gröbner basis of the ideal of phylogenetic invariants: 0

Specialized Fourier transform

$\frac{1}{1} \begin{pmatrix} 1 & 1 & 1 & 1\\ 1 & -1 & -1 & 1\\ 1 & -1 & 1 & -1\\ 1 & 1 & -1 & -1 \end{pmatrix} ] $

Inverse specialized Fourier transform

$\frac{1}{4} \begin{pmatrix} 1 & 1 & 1 & 1\\ 1 & -1 & -1 & 1\\ 1 & -1 & 1 & -1\\ 1 & 1 & -1 & -1 \end{pmatrix} $