Phylogenetic tree 3-0-0

Evolutionary model

Kimura 2-parameter model with root distribution $\pi = \left(\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4} \right)$.
The transition matrix associated with edge $i$ is of the form $$M_j = \begin{pmatrix} a_i & b_i & c_i & b_i\\ b_i & a_i & b_i & c_i\\ c_i & b_i & a_i & b_i\\ b_i & c_i & b_i & a_i \end{pmatrix}$$ and the Fourier parameters are $\left(x^{(i)}_1, x^{(i)}_2, x^{(i)}_i, x^{(i)}_i\right)$ for all edges $i$.

Summary

Dimension: 7
Degree: 12
Probability coordinates: 10
Fourier coordinates: 10
dim of Singular locus: 22
deg of Singular locus: 4
MLdeg: 1964
EDdeg: 184

Probability parametrization

$\begin{align} p_{\texttt{ACA}} &= \frac{1}{2}(a_1a_3b_2 + a_2b_1b_3 + b_1b_3c_2 + b_2c_1c_3) \\[0.5em] p_{\texttt{AAA}} &= \frac{1}{2}(a_1a_2a_3 + 2b_1b_2b_3 + c_1c_2c_3) \\[0.5em] p_{\texttt{AGA}} &= \frac{1}{2}(a_1a_3c_2 + a_2c_1c_3 + 2b_1b_2b_3) \\[0.5em] p_{\texttt{ACC}} &= \frac{1}{2}(a_1b_2b_3 + a_2a_3b_1 + b_1c_2c_3 + b_2b_3c_1) \\[0.5em] p_{\texttt{ACG}} &= \frac{1}{2}(a_1b_2c_3 + a_2b_1b_3 + a_3b_2c_1 + b_1b_3c_2) \\[0.5em] p_{\texttt{AAC}} &= \frac{1}{2}(a_1a_2b_3 + a_3b_1b_2 + b_1b_2c_3 + b_3c_1c_2) \\[0.5em] p_{\texttt{AAG}} &= \frac{1}{2}(a_1a_2c_3 + a_3c_1c_2 + 2b_1b_2b_3) \\[0.5em] p_{\texttt{ACT}} &= \frac{1}{2}(a_1b_2b_3 + a_2b_1c_3 + a_3b_1c_2 + b_2b_3c_1) \\[0.5em] p_{\texttt{AGG}} &= \frac{1}{2}(a_1c_2c_3 + a_2a_3c_1 + 2b_1b_2b_3) \\[0.5em] p_{\texttt{AGC}} &= \frac{1}{2}(a_1b_3c_2 + a_2b_3c_1 + a_3b_1b_2 + b_1b_2c_3) \end{align}$

Fourier parametrization

$\begin{align} q_{\texttt{AAA}} &= x^{(1)}_{1}x^{(2)}_{1}x^{(3)}_{1} \\[0.5em] q_{\texttt{GGA}} &= x^{(1)}_{3}x^{(2)}_{3}x^{(3)}_{1} \\[0.5em] q_{\texttt{CCA}} &= x^{(1)}_{2}x^{(2)}_{2}x^{(3)}_{1} \\[0.5em] q_{\texttt{ACC}} &= x^{(1)}_{1}x^{(2)}_{2}x^{(3)}_{2} \\[0.5em] q_{\texttt{GAG}} &= x^{(1)}_{3}x^{(2)}_{1}x^{(3)}_{3} \\[0.5em] q_{\texttt{CGT}} &= x^{(1)}_{2}x^{(2)}_{3}x^{(3)}_{3} \\[0.5em] q_{\texttt{GCT}} &= x^{(1)}_{3}x^{(2)}_{2}x^{(3)}_{3} \\[0.5em] q_{\texttt{AGG}} &= x^{(1)}_{1}x^{(2)}_{3}x^{(3)}_{3} \\[0.5em] q_{\texttt{GTC}} &= x^{(1)}_{3}x^{(2)}_{3}x^{(3)}_{2} \\[0.5em] q_{\texttt{CAC}} &= x^{(1)}_{2}x^{(2)}_{1}x^{(3)}_{2} \end{align}$

Equivalent classes of probability parametrization

$\begin{align} \text{Class } &p_{\texttt{AAA}}:\ p_{\texttt{AAA}},\ p_{\texttt{CCC}},\ p_{\texttt{GGG}},\ p_{\texttt{TTT}} \\[0.5em] \text{Class } &p_{\texttt{AAC}}:\ p_{\texttt{AAC}},\ p_{\texttt{AAT}},\ p_{\texttt{CCA}},\ p_{\texttt{CCG}},\ p_{\texttt{GGC}},\ p_{\texttt{GGT}},\ p_{\texttt{TTA}},\ p_{\texttt{TTG}} \\[0.5em] \text{Class } &p_{\texttt{AAG}}:\ p_{\texttt{AAG}},\ p_{\texttt{CCT}},\ p_{\texttt{GGA}},\ p_{\texttt{TTC}} \\[0.5em] \text{Class } &p_{\texttt{ACA}}:\ p_{\texttt{ACA}},\ p_{\texttt{ATA}},\ p_{\texttt{CAC}},\ p_{\texttt{CGC}},\ p_{\texttt{GCG}},\ p_{\texttt{GTG}},\ p_{\texttt{TAT}},\ p_{\texttt{TGT}} \\[0.5em] \text{Class } &p_{\texttt{ACC}}:\ p_{\texttt{ACC}},\ p_{\texttt{ATT}},\ p_{\texttt{CAA}},\ p_{\texttt{CGG}},\ p_{\texttt{GCC}},\ p_{\texttt{GTT}},\ p_{\texttt{TAA}},\ p_{\texttt{TGG}} \\[0.5em] \text{Class } &p_{\texttt{ACG}}:\ p_{\texttt{ACG}},\ p_{\texttt{ATG}},\ p_{\texttt{CAT}},\ p_{\texttt{CGT}},\ p_{\texttt{GCA}},\ p_{\texttt{GTA}},\ p_{\texttt{TAC}},\ p_{\texttt{TGC}} \\[0.5em] \text{Class } &p_{\texttt{ACT}}:\ p_{\texttt{ACT}},\ p_{\texttt{ATC}},\ p_{\texttt{CAG}},\ p_{\texttt{CGA}},\ p_{\texttt{GCT}},\ p_{\texttt{GTC}},\ p_{\texttt{TAG}},\ p_{\texttt{TGA}} \\[0.5em] \text{Class } &p_{\texttt{AGA}}:\ p_{\texttt{AGA}},\ p_{\texttt{CTC}},\ p_{\texttt{GAG}},\ p_{\texttt{TCT}} \\[0.5em] \text{Class } &p_{\texttt{AGC}}:\ p_{\texttt{AGC}},\ p_{\texttt{AGT}},\ p_{\texttt{CTA}},\ p_{\texttt{CTG}},\ p_{\texttt{GAC}},\ p_{\texttt{GAT}},\ p_{\texttt{TCA}},\ p_{\texttt{TCG}} \\[0.5em] \text{Class } &p_{\texttt{AGG}}:\ p_{\texttt{AGG}},\ p_{\texttt{CTT}},\ p_{\texttt{GAA}},\ p_{\texttt{TCC}} \end{align}$

Equivalent classes of Fourier parametrization

$\begin{align} \text{Class } &0:\ q_{\texttt{CAA}}, q_{\texttt{GAA}}, q_{\texttt{TAA}}, q_{\texttt{ACA}}, q_{\texttt{GCA}}, q_{\texttt{TCA}}, q_{\texttt{AGA}}, q_{\texttt{CGA}}, q_{\texttt{TGA}}, q_{\texttt{ATA}}, q_{\texttt{CTA}}, q_{\texttt{GTA}}, q_{\texttt{AAC}}, q_{\texttt{GAC}}, q_{\texttt{TAC}}, q_{\texttt{CCC}}, q_{\texttt{GCC}}, q_{\texttt{TCC}}, q_{\texttt{AGC}}, q_{\texttt{CGC}}, q_{\texttt{GGC}}, q_{\texttt{ATC}}, q_{\texttt{CTC}}, q_{\texttt{TTC}}, q_{\texttt{AAG}}, q_{\texttt{CAG}}, q_{\texttt{TAG}}, q_{\texttt{ACG}}, q_{\texttt{CCG}}, q_{\texttt{GCG}}, q_{\texttt{CGG}}, q_{\texttt{GGG}}, q_{\texttt{TGG}}, q_{\texttt{ATG}}, q_{\texttt{GTG}}, q_{\texttt{TTG}}, q_{\texttt{AAT}}, q_{\texttt{CAT}}, q_{\texttt{GAT}}, q_{\texttt{ACT}}, q_{\texttt{CCT}}, q_{\texttt{TCT}}, q_{\texttt{AGT}}, q_{\texttt{GGT}}, q_{\texttt{TGT}}, q_{\texttt{CTT}}, q_{\texttt{GTT}}, q_{\texttt{TTT}} \\[0.5em] \text{Class } &q_{\texttt{AAA}}:\ q_{\texttt{AAA}} \\[0.5em] \text{Class } &q_{\texttt{ACC}}:\ q_{\texttt{ACC}} \\[0.5em] \text{Class } &q_{\texttt{AGG}}:\ q_{\texttt{AGG}},\ q_{\texttt{ATT}} \\[0.5em] \text{Class } &q_{\texttt{CAC}}:\ q_{\texttt{CAC}} \\[0.5em] \text{Class } &q_{\texttt{CCA}}:\ q_{\texttt{CCA}} \\[0.5em] \text{Class } &q_{\texttt{CGT}}:\ q_{\texttt{CGT}},\ q_{\texttt{CTG}} \\[0.5em] \text{Class } &q_{\texttt{GAG}}:\ q_{\texttt{GAG}},\ q_{\texttt{TAT}} \\[0.5em] \text{Class } &q_{\texttt{GCT}}:\ q_{\texttt{GCT}},\ q_{\texttt{TCG}} \\[0.5em] \text{Class } &q_{\texttt{GGA}}:\ q_{\texttt{GGA}},\ q_{\texttt{TTA}} \\[0.5em] \text{Class } &q_{\texttt{GTC}}:\ q_{\texttt{GTC}},\ q_{\texttt{TGC}} \end{align}$

Invariants in Fourier coordinates

Ideal generated by $$ \begin{align} q_{AAA}q_{CGT}q_{GTC} - q_{GGA}q_{AGG}q_{CAC} \\ -q_{GGA}q_{GCT}q_{CAC} + q_{CCA}q_{GAG}q_{GTC} \\ q_{AAA}q_{CGT}q_{GCT} - q_{CCA}q_{GAG}q_{AGG} \\ q_{ACC}q_{GAG}q_{CGT} - q_{GCT}q_{AGG}q_{CAC} \\ q_{GGA}q_{ACC}q_{CGT} - q_{CCA}q_{AGG}q_{GTC} \\ -q_{AAA}q_{GCT}q_{GTC} + q_{GGA}q_{ACC}q_{GAG} \\ q_{AAA}q_{CCA}q_{GTC}^2 - q_{GGA}^2q_{ACC}q_{CAC} \\ q_{AAA}q_{ACC}q_{CGT}^2 - q_{CCA}q_{AGG}^2q_{CAC} \\ -q_{AAA}q_{GCT}^2q_{CAC} + q_{CCA}q_{ACC}q_{GAG}^2 \end{align} $$

Gröbner basis of ideal of invariants

Gröbner basis with elements $$ \begin{align} q_{AAA}q_{CGT}q_{GTC} - q_{GGA}q_{AGG}q_{CAC} \\ -q_{GGA}q_{GCT}q_{CAC} + q_{CCA}q_{GAG}q_{GTC} \\ q_{AAA}q_{CGT}q_{GCT} - q_{CCA}q_{GAG}q_{AGG} \\ q_{ACC}q_{GAG}q_{CGT} - q_{GCT}q_{AGG}q_{CAC} \\ q_{GGA}q_{ACC}q_{CGT} - q_{CCA}q_{AGG}q_{GTC} \\ -q_{AAA}q_{GCT}q_{GTC} + q_{GGA}q_{ACC}q_{GAG} \\ q_{AAA}q_{CCA}q_{GTC}^2 - q_{GGA}^2q_{ACC}q_{CAC} \\ q_{AAA}q_{ACC}q_{CGT}^2 - q_{CCA}q_{AGG}^2q_{CAC} \\ -q_{AAA}q_{GCT}^2q_{CAC} + q_{CCA}q_{ACC}q_{GAG}^2 \end{align}$$ with respect to the ordering degrevlex $ q_{\texttt{AAA}}, q_{\texttt{GGA}}, q_{\texttt{CCA}}, q_{\texttt{ACC}}, q_{\texttt{GAG}}, q_{\texttt{CGT}}, q_{\texttt{GCT}}, q_{\texttt{AGG}}, q_{\texttt{GTC}}, q_{\texttt{CAC}}$

Additional information

The cardinality of the smallest set of generators that define the ideal of phylogenetic invariants: 9
The largest degree of a generator in the degree reverse lexicographic reduced Gröbner basis: 4
The largest degree of an element in a minimal generating set for the ideal of phylogenetic invariants: 4
The cardinality of the degree reverse lexicographic reduced Gröbner basis of the ideal of phylogenetic invariants: 9

Specialized Fourier transform

$\frac{1}{1} \begin{pmatrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & -1 & 1 & -1 & 1 & -1 & 1 & 1 & -1 & 1\\ 1 & 0 & -1 & 0 & 1 & 0 & -1 & -1 & 0 & 1\\ 1 & -1 & 1 & 1 & -1 & 1 & -1 & 1 & -1 & 1\\ 1 & 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 & 1\\ 1 & 0 & -1 & 0 & -1 & 0 & 1 & -1 & 0 & 1\\ 1 & 0 & -1 & 1 & 0 & -1 & 0 & 1 & 0 & -1\\ 1 & 0 & -1 & -1 & 0 & 1 & 0 & 1 & 0 & -1\\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & -1 & -1 & -1\\ 1 & -1 & 1 & 0 & 0 & 0 & 0 & -1 & 1 & -1 \end{pmatrix} ] $

Inverse specialized Fourier transform

$\frac{1}{16} \begin{pmatrix} 1 & 1 & 2 & 1 & 1 & 2 & 2 & 2 & 2 & 2\\ 2 & -2 & 0 & -2 & 2 & 0 & 0 & 0 & 4 & -4\\ 1 & 1 & -2 & 1 & 1 & -2 & -2 & -2 & 2 & 2\\ 2 & -2 & 0 & 2 & -2 & 0 & 4 & -4 & 0 & 0\\ 2 & 2 & 4 & -2 & -2 & -4 & 0 & 0 & 0 & 0\\ 2 & -2 & 0 & 2 & -2 & 0 & -4 & 4 & 0 & 0\\ 2 & 2 & -4 & -2 & -2 & 4 & 0 & 0 & 0 & 0\\ 1 & 1 & -2 & 1 & 1 & -2 & 2 & 2 & -2 & -2\\ 2 & -2 & 0 & -2 & 2 & 0 & 0 & 0 & -4 & 4\\ 1 & 1 & 2 & 1 & 1 & 2 & -2 & -2 & -2 & -2 \end{pmatrix} $