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

Evolutionary model

Jukes Cantor 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 $$M_j = \begin{pmatrix} a_i & b_i & b_i & b_i\\ b_i & a_i & b_i & b_i\\ b_i & b_i & a_i & b_i\\ b_i & b_i & b_i & a_i \end{pmatrix}$$ and the Fourier parameters are $\left(x_i, y_i, y_i, y_i\right)$ for all edges $i$.

Summary

Probability parametrization

$\begin{align} p_{\texttt{ACGT}} &= \frac{1}{4}(a_1a_3b_2b_4b_5 + a_1a_4b_2b_3b_5 + a_1a_5b_2b_3b_4 + a_1b_2b_3b_4b_5 + a_2a_3b_1b_4b_5 + a_2a_4b_1b_3b_5 + a_2a_5b_1b_3b_4 + a_2b_1b_3b_4b_5 + a_3a_5b_1b_2b_4 + a_3b_1b_2b_4b_5 + a_4a_5b_1b_2b_3 + a_4b_1b_2b_3b_5 + 4b_1b_2b_3b_4b_5) \\[0.5em] p_{\texttt{ACGG}} &= \frac{1}{4}(a_1a_3a_4b_2b_5 + a_1a_5b_2b_3b_4 + 2a_1b_2b_3b_4b_5 + a_2a_3a_4b_1b_5 + a_2a_5b_1b_3b_4 + 2a_2b_1b_3b_4b_5 + a_3a_4a_5b_1b_2 + a_3a_4b_1b_2b_5 + a_5b_1b_2b_3b_4 + 5b_1b_2b_3b_4b_5) \\[0.5em] p_{\texttt{ACGC}} &= \frac{1}{4}(a_1a_3b_2b_4b_5 + a_1a_4b_2b_3b_5 + a_1a_5b_2b_3b_4 + a_1b_2b_3b_4b_5 + a_2a_3b_1b_4b_5 + a_2a_4a_5b_1b_3 + 2a_2b_1b_3b_4b_5 + a_3a_5b_1b_2b_4 + a_3b_1b_2b_4b_5 + 2a_4b_1b_2b_3b_5 + a_5b_1b_2b_3b_4 + 3b_1b_2b_3b_4b_5) \\[0.5em] p_{\texttt{ACGA}} &= \frac{1}{4}(a_1a_3b_2b_4b_5 + a_1a_4a_5b_2b_3 + 2a_1b_2b_3b_4b_5 + a_2a_3b_1b_4b_5 + a_2a_4b_1b_3b_5 + a_2a_5b_1b_3b_4 + a_2b_1b_3b_4b_5 + a_3a_5b_1b_2b_4 + a_3b_1b_2b_4b_5 + 2a_4b_1b_2b_3b_5 + a_5b_1b_2b_3b_4 + 3b_1b_2b_3b_4b_5) \\[0.5em] p_{\texttt{ACCG}} &= \frac{1}{4}(a_1a_3b_2b_4b_5 + a_1a_4b_2b_3b_5 + a_1a_5b_2b_3b_4 + a_1b_2b_3b_4b_5 + a_2a_3a_5b_1b_4 + a_2a_4b_1b_3b_5 + 2a_2b_1b_3b_4b_5 + 2a_3b_1b_2b_4b_5 + a_4a_5b_1b_2b_3 + a_4b_1b_2b_3b_5 + a_5b_1b_2b_3b_4 + 3b_1b_2b_3b_4b_5) \\[0.5em] p_{\texttt{ACCC}} &= \frac{1}{4}(a_1a_3a_4b_2b_5 + a_1a_5b_2b_3b_4 + 2a_1b_2b_3b_4b_5 + a_2a_3a_4a_5b_1 + 3a_2b_1b_3b_4b_5 + 2a_3a_4b_1b_2b_5 + 2a_5b_1b_2b_3b_4 + 4b_1b_2b_3b_4b_5) \\[0.5em] p_{\texttt{ACCA}} &= \frac{1}{4}(a_1a_3b_2b_4b_5 + a_1a_4a_5b_2b_3 + 2a_1b_2b_3b_4b_5 + a_2a_3a_5b_1b_4 + a_2a_4b_1b_3b_5 + 2a_2b_1b_3b_4b_5 + 2a_3b_1b_2b_4b_5 + 2a_4b_1b_2b_3b_5 + 2a_5b_1b_2b_3b_4 + 2b_1b_2b_3b_4b_5) \\[0.5em] p_{\texttt{ACAG}} &= \frac{1}{4}(a_1a_3a_5b_2b_4 + a_1a_4b_2b_3b_5 + 2a_1b_2b_3b_4b_5 + a_2a_3b_1b_4b_5 + a_2a_4b_1b_3b_5 + a_2a_5b_1b_3b_4 + a_2b_1b_3b_4b_5 + 2a_3b_1b_2b_4b_5 + a_4a_5b_1b_2b_3 + a_4b_1b_2b_3b_5 + a_5b_1b_2b_3b_4 + 3b_1b_2b_3b_4b_5) \\[0.5em] p_{\texttt{ACAC}} &= \frac{1}{4}(a_1a_3a_5b_2b_4 + a_1a_4b_2b_3b_5 + 2a_1b_2b_3b_4b_5 + a_2a_3b_1b_4b_5 + a_2a_4a_5b_1b_3 + 2a_2b_1b_3b_4b_5 + 2a_3b_1b_2b_4b_5 + 2a_4b_1b_2b_3b_5 + 2a_5b_1b_2b_3b_4 + 2b_1b_2b_3b_4b_5) \\[0.5em] p_{\texttt{ACAA}} &= \frac{1}{4}(a_1a_3a_4a_5b_2 + 3a_1b_2b_3b_4b_5 + a_2a_3a_4b_1b_5 + a_2a_5b_1b_3b_4 + 2a_2b_1b_3b_4b_5 + 2a_3a_4b_1b_2b_5 + 2a_5b_1b_2b_3b_4 + 4b_1b_2b_3b_4b_5) \\[0.5em] p_{\texttt{AACG}} &= \frac{1}{4}(a_1a_2a_3b_4b_5 + a_1a_2a_4b_3b_5 + a_1a_2a_5b_3b_4 + a_1a_2b_3b_4b_5 + a_3a_5b_1b_2b_4 + 2a_3b_1b_2b_4b_5 + a_4a_5b_1b_2b_3 + 2a_4b_1b_2b_3b_5 + a_5b_1b_2b_3b_4 + 5b_1b_2b_3b_4b_5) \\[0.5em] p_{\texttt{AACC}} &= \frac{1}{4}(a_1a_2a_3a_4b_5 + a_1a_2a_5b_3b_4 + 2a_1a_2b_3b_4b_5 + a_3a_4a_5b_1b_2 + 2a_3a_4b_1b_2b_5 + 2a_5b_1b_2b_3b_4 + 7b_1b_2b_3b_4b_5) \\[0.5em] p_{\texttt{AACA}} &= \frac{1}{4}(a_1a_2a_3b_4b_5 + a_1a_2a_4a_5b_3 + 2a_1a_2b_3b_4b_5 + a_3a_5b_1b_2b_4 + 2a_3b_1b_2b_4b_5 + 3a_4b_1b_2b_3b_5 + 2a_5b_1b_2b_3b_4 + 4b_1b_2b_3b_4b_5) \\[0.5em] p_{\texttt{AAAC}} &= \frac{1}{4}(a_1a_2a_3a_5b_4 + a_1a_2a_4b_3b_5 + 2a_1a_2b_3b_4b_5 + 3a_3b_1b_2b_4b_5 + a_4a_5b_1b_2b_3 + 2a_4b_1b_2b_3b_5 + 2a_5b_1b_2b_3b_4 + 4b_1b_2b_3b_4b_5) \\[0.5em] p_{\texttt{AAAA}} &= \frac{1}{4}(a_1a_2a_3a_4a_5 + 3a_1a_2b_3b_4b_5 + 3a_3a_4b_1b_2b_5 + 3a_5b_1b_2b_3b_4 + 6b_1b_2b_3b_4b_5) \end{align}$

Fourier parametrization

$\begin{align} q_{\texttt{CGTA}} &= x_4y_1y_2y_3y_5 \\[0.5em] q_{\texttt{CGCG}} &= y_1y_2y_3y_4y_5 \\[0.5em] q_{\texttt{CGAT}} &= x_3y_1y_2y_4y_5 \\[0.5em] q_{\texttt{CCCC}} &= x_5y_1y_2y_3y_4 \\[0.5em] q_{\texttt{CCAA}} &= x_3x_4x_5y_1y_2 \\[0.5em] q_{\texttt{CAGT}} &= x_2y_1y_3y_4y_5 \\[0.5em] q_{\texttt{CACA}} &= x_2x_4y_1y_3y_5 \\[0.5em] q_{\texttt{CAAC}} &= x_2x_3y_1y_4y_5 \\[0.5em] q_{\texttt{ACGT}} &= x_1y_2y_3y_4y_5 \\[0.5em] q_{\texttt{ACCA}} &= x_1x_4y_2y_3y_5 \\[0.5em] q_{\texttt{ACAC}} &= x_1x_3y_2y_4y_5 \\[0.5em] q_{\texttt{AACC}} &= x_1x_2x_5y_3y_4 \\[0.5em] q_{\texttt{AAAA}} &= x_1x_2x_3x_4x_5 \end{align}$

Equivalent classes of probability parametrization

$\begin{align} \text{Class } &p_{\texttt{AAAA}}:\ p_{\texttt{AAAA}},\ p_{\texttt{CCCC}},\ p_{\texttt{GGGG}},\ p_{\texttt{TTTT}} \\[0.5em] \text{Class } &p_{\texttt{AAAC}}:\ p_{\texttt{AAAC}},\ p_{\texttt{AAAG}},\ p_{\texttt{AAAT}},\ p_{\texttt{CCCA}},\ p_{\texttt{CCCG}},\ p_{\texttt{CCCT}},\ p_{\texttt{GGGA}},\ p_{\texttt{GGGC}},\ p_{\texttt{GGGT}},\ p_{\texttt{TTTA}},\ p_{\texttt{TTTC}},\ p_{\texttt{TTTG}} \\[0.5em] \text{Class } &p_{\texttt{AACA}}:\ p_{\texttt{AACA}},\ p_{\texttt{AAGA}},\ p_{\texttt{AATA}},\ p_{\texttt{CCAC}},\ p_{\texttt{CCGC}},\ p_{\texttt{CCTC}},\ p_{\texttt{GGAG}},\ p_{\texttt{GGCG}},\ p_{\texttt{GGTG}},\ p_{\texttt{TTAT}},\ p_{\texttt{TTCT}},\ p_{\texttt{TTGT}} \\[0.5em] \text{Class } &p_{\texttt{AACC}}:\ p_{\texttt{AACC}},\ p_{\texttt{AAGG}},\ p_{\texttt{AATT}},\ p_{\texttt{CCAA}},\ p_{\texttt{CCGG}},\ p_{\texttt{CCTT}},\ p_{\texttt{GGAA}},\ p_{\texttt{GGCC}},\ p_{\texttt{GGTT}},\ p_{\texttt{TTAA}},\ p_{\texttt{TTCC}},\ p_{\texttt{TTGG}} \\[0.5em] \text{Class } &p_{\texttt{AACG}}:\ p_{\texttt{AACG}},\ p_{\texttt{AACT}},\ p_{\texttt{AAGC}},\ p_{\texttt{AAGT}},\ p_{\texttt{AATC}},\ p_{\texttt{AATG}},\ p_{\texttt{CCAG}},\ p_{\texttt{CCAT}},\ p_{\texttt{CCGA}},\ p_{\texttt{CCGT}},\ p_{\texttt{CCTA}},\ p_{\texttt{CCTG}},\ p_{\texttt{GGAC}},\ p_{\texttt{GGAT}},\ p_{\texttt{GGCA}},\ p_{\texttt{GGCT}},\ p_{\texttt{GGTA}},\ p_{\texttt{GGTC}},\ p_{\texttt{TTAC}},\ p_{\texttt{TTAG}},\ p_{\texttt{TTCA}},\ p_{\texttt{TTCG}},\ p_{\texttt{TTGA}},\ p_{\texttt{TTGC}} \\[0.5em] \text{Class } &p_{\texttt{ACAA}}:\ p_{\texttt{ACAA}},\ p_{\texttt{AGAA}},\ p_{\texttt{ATAA}},\ p_{\texttt{CACC}},\ p_{\texttt{CGCC}},\ p_{\texttt{CTCC}},\ p_{\texttt{GAGG}},\ p_{\texttt{GCGG}},\ p_{\texttt{GTGG}},\ p_{\texttt{TATT}},\ p_{\texttt{TCTT}},\ p_{\texttt{TGTT}} \\[0.5em] \text{Class } &p_{\texttt{ACAC}}:\ p_{\texttt{ACAC}},\ p_{\texttt{AGAG}},\ p_{\texttt{ATAT}},\ p_{\texttt{CACA}},\ p_{\texttt{CGCG}},\ p_{\texttt{CTCT}},\ p_{\texttt{GAGA}},\ p_{\texttt{GCGC}},\ p_{\texttt{GTGT}},\ p_{\texttt{TATA}},\ p_{\texttt{TCTC}},\ p_{\texttt{TGTG}} \\[0.5em] \text{Class } &p_{\texttt{ACAG}}:\ p_{\texttt{ACAG}},\ p_{\texttt{ACAT}},\ p_{\texttt{AGAC}},\ p_{\texttt{AGAT}},\ p_{\texttt{ATAC}},\ p_{\texttt{ATAG}},\ p_{\texttt{CACG}},\ p_{\texttt{CACT}},\ p_{\texttt{CGCA}},\ p_{\texttt{CGCT}},\ p_{\texttt{CTCA}},\ p_{\texttt{CTCG}},\ p_{\texttt{GAGC}},\ p_{\texttt{GAGT}},\ p_{\texttt{GCGA}},\ p_{\texttt{GCGT}},\ p_{\texttt{GTGA}},\ p_{\texttt{GTGC}},\ p_{\texttt{TATC}},\ p_{\texttt{TATG}},\ p_{\texttt{TCTA}},\ p_{\texttt{TCTG}},\ p_{\texttt{TGTA}},\ p_{\texttt{TGTC}} \\[0.5em] \text{Class } &p_{\texttt{ACCA}}:\ p_{\texttt{ACCA}},\ p_{\texttt{AGGA}},\ p_{\texttt{ATTA}},\ p_{\texttt{CAAC}},\ p_{\texttt{CGGC}},\ p_{\texttt{CTTC}},\ p_{\texttt{GAAG}},\ p_{\texttt{GCCG}},\ p_{\texttt{GTTG}},\ p_{\texttt{TAAT}},\ p_{\texttt{TCCT}},\ p_{\texttt{TGGT}} \\[0.5em] \text{Class } &p_{\texttt{ACCC}}:\ p_{\texttt{ACCC}},\ p_{\texttt{AGGG}},\ p_{\texttt{ATTT}},\ p_{\texttt{CAAA}},\ p_{\texttt{CGGG}},\ p_{\texttt{CTTT}},\ p_{\texttt{GAAA}},\ p_{\texttt{GCCC}},\ p_{\texttt{GTTT}},\ p_{\texttt{TAAA}},\ p_{\texttt{TCCC}},\ p_{\texttt{TGGG}} \\[0.5em] \text{Class } &p_{\texttt{ACCG}}:\ p_{\texttt{ACCG}},\ p_{\texttt{ACCT}},\ p_{\texttt{AGGC}},\ p_{\texttt{AGGT}},\ p_{\texttt{ATTC}},\ p_{\texttt{ATTG}},\ p_{\texttt{CAAG}},\ p_{\texttt{CAAT}},\ p_{\texttt{CGGA}},\ p_{\texttt{CGGT}},\ p_{\texttt{CTTA}},\ p_{\texttt{CTTG}},\ p_{\texttt{GAAC}},\ p_{\texttt{GAAT}},\ p_{\texttt{GCCA}},\ p_{\texttt{GCCT}},\ p_{\texttt{GTTA}},\ p_{\texttt{GTTC}},\ p_{\texttt{TAAC}},\ p_{\texttt{TAAG}},\ p_{\texttt{TCCA}},\ p_{\texttt{TCCG}},\ p_{\texttt{TGGA}},\ p_{\texttt{TGGC}} \\[0.5em] \text{Class } &p_{\texttt{ACGA}}:\ p_{\texttt{ACGA}},\ p_{\texttt{ACTA}},\ p_{\texttt{AGCA}},\ p_{\texttt{AGTA}},\ p_{\texttt{ATCA}},\ p_{\texttt{ATGA}},\ p_{\texttt{CAGC}},\ p_{\texttt{CATC}},\ p_{\texttt{CGAC}},\ p_{\texttt{CGTC}},\ p_{\texttt{CTAC}},\ p_{\texttt{CTGC}},\ p_{\texttt{GACG}},\ p_{\texttt{GATG}},\ p_{\texttt{GCAG}},\ p_{\texttt{GCTG}},\ p_{\texttt{GTAG}},\ p_{\texttt{GTCG}},\ p_{\texttt{TACT}},\ p_{\texttt{TAGT}},\ p_{\texttt{TCAT}},\ p_{\texttt{TCGT}},\ p_{\texttt{TGAT}},\ p_{\texttt{TGCT}} \\[0.5em] \text{Class } &p_{\texttt{ACGC}}:\ p_{\texttt{ACGC}},\ p_{\texttt{ACTC}},\ p_{\texttt{AGCG}},\ p_{\texttt{AGTG}},\ p_{\texttt{ATCT}},\ p_{\texttt{ATGT}},\ p_{\texttt{CAGA}},\ p_{\texttt{CATA}},\ p_{\texttt{CGAG}},\ p_{\texttt{CGTG}},\ p_{\texttt{CTAT}},\ p_{\texttt{CTGT}},\ p_{\texttt{GACA}},\ p_{\texttt{GATA}},\ p_{\texttt{GCAC}},\ p_{\texttt{GCTC}},\ p_{\texttt{GTAT}},\ p_{\texttt{GTCT}},\ p_{\texttt{TACA}},\ p_{\texttt{TAGA}},\ p_{\texttt{TCAC}},\ p_{\texttt{TCGC}},\ p_{\texttt{TGAG}},\ p_{\texttt{TGCG}} \\[0.5em] \text{Class } &p_{\texttt{ACGG}}:\ p_{\texttt{ACGG}},\ p_{\texttt{ACTT}},\ p_{\texttt{AGCC}},\ p_{\texttt{AGTT}},\ p_{\texttt{ATCC}},\ p_{\texttt{ATGG}},\ p_{\texttt{CAGG}},\ p_{\texttt{CATT}},\ p_{\texttt{CGAA}},\ p_{\texttt{CGTT}},\ p_{\texttt{CTAA}},\ p_{\texttt{CTGG}},\ p_{\texttt{GACC}},\ p_{\texttt{GATT}},\ p_{\texttt{GCAA}},\ p_{\texttt{GCTT}},\ p_{\texttt{GTAA}},\ p_{\texttt{GTCC}},\ p_{\texttt{TACC}},\ p_{\texttt{TAGG}},\ p_{\texttt{TCAA}},\ p_{\texttt{TCGG}},\ p_{\texttt{TGAA}},\ p_{\texttt{TGCC}} \\[0.5em] \text{Class } &p_{\texttt{ACGT}}:\ p_{\texttt{ACGT}},\ p_{\texttt{ACTG}},\ p_{\texttt{AGCT}},\ p_{\texttt{AGTC}},\ p_{\texttt{ATCG}},\ p_{\texttt{ATGC}},\ p_{\texttt{CAGT}},\ p_{\texttt{CATG}},\ p_{\texttt{CGAT}},\ p_{\texttt{CGTA}},\ p_{\texttt{CTAG}},\ p_{\texttt{CTGA}},\ p_{\texttt{GACT}},\ p_{\texttt{GATC}},\ p_{\texttt{GCAT}},\ p_{\texttt{GCTA}},\ p_{\texttt{GTAC}},\ p_{\texttt{GTCA}},\ p_{\texttt{TACG}},\ p_{\texttt{TAGC}},\ p_{\texttt{TCAG}},\ p_{\texttt{TCGA}},\ p_{\texttt{TGAC}},\ p_{\texttt{TGCA}} \end{align}$

Equivalent classes of Fourier parametrization

$\begin{align} \text{Class } &0:\ q_{\texttt{AAAC}}, q_{\texttt{AAAG}}, q_{\texttt{AAAT}}, q_{\texttt{AACA}}, q_{\texttt{AACG}}, q_{\texttt{AACT}}, q_{\texttt{AAGA}}, q_{\texttt{AAGC}}, q_{\texttt{AAGT}}, q_{\texttt{AATA}}, q_{\texttt{AATC}}, q_{\texttt{AATG}}, q_{\texttt{ACAA}}, q_{\texttt{ACAG}}, q_{\texttt{ACAT}}, q_{\texttt{ACCC}}, q_{\texttt{ACCG}}, q_{\texttt{ACCT}}, q_{\texttt{ACGA}}, q_{\texttt{ACGC}}, q_{\texttt{ACGG}}, q_{\texttt{ACTA}}, q_{\texttt{ACTC}}, q_{\texttt{ACTT}}, q_{\texttt{AGAA}}, q_{\texttt{AGAC}}, q_{\texttt{AGAT}}, q_{\texttt{AGCA}}, q_{\texttt{AGCC}}, q_{\texttt{AGCG}}, q_{\texttt{AGGC}}, q_{\texttt{AGGG}}, q_{\texttt{AGGT}}, q_{\texttt{AGTA}}, q_{\texttt{AGTG}}, q_{\texttt{AGTT}}, q_{\texttt{ATAA}}, q_{\texttt{ATAC}}, q_{\texttt{ATAG}}, q_{\texttt{ATCA}}, q_{\texttt{ATCC}}, q_{\texttt{ATCT}}, q_{\texttt{ATGA}}, q_{\texttt{ATGG}}, q_{\texttt{ATGT}}, q_{\texttt{ATTC}}, q_{\texttt{ATTG}}, q_{\texttt{ATTT}}, q_{\texttt{CAAA}}, q_{\texttt{CAAG}}, q_{\texttt{CAAT}}, q_{\texttt{CACC}}, q_{\texttt{CACG}}, q_{\texttt{CACT}}, q_{\texttt{CAGA}}, q_{\texttt{CAGC}}, q_{\texttt{CAGG}}, q_{\texttt{CATA}}, q_{\texttt{CATC}}, q_{\texttt{CATT}}, q_{\texttt{CCAC}}, q_{\texttt{CCAG}}, q_{\texttt{CCAT}}, q_{\texttt{CCCA}}, q_{\texttt{CCCG}}, q_{\texttt{CCCT}}, q_{\texttt{CCGA}}, q_{\texttt{CCGC}}, q_{\texttt{CCGT}}, q_{\texttt{CCTA}}, q_{\texttt{CCTC}}, q_{\texttt{CCTG}}, q_{\texttt{CGAA}}, q_{\texttt{CGAC}}, q_{\texttt{CGAG}}, q_{\texttt{CGCA}}, q_{\texttt{CGCC}}, q_{\texttt{CGCT}}, q_{\texttt{CGGA}}, q_{\texttt{CGGG}}, q_{\texttt{CGGT}}, q_{\texttt{CGTC}}, q_{\texttt{CGTG}}, q_{\texttt{CGTT}}, q_{\texttt{CTAA}}, q_{\texttt{CTAC}}, q_{\texttt{CTAT}}, q_{\texttt{CTCA}}, q_{\texttt{CTCC}}, q_{\texttt{CTCG}}, q_{\texttt{CTGC}}, q_{\texttt{CTGG}}, q_{\texttt{CTGT}}, q_{\texttt{CTTA}}, q_{\texttt{CTTG}}, q_{\texttt{CTTT}}, q_{\texttt{GAAA}}, q_{\texttt{GAAC}}, q_{\texttt{GAAT}}, q_{\texttt{GACA}}, q_{\texttt{GACC}}, q_{\texttt{GACG}}, q_{\texttt{GAGC}}, q_{\texttt{GAGG}}, q_{\texttt{GAGT}}, q_{\texttt{GATA}}, q_{\texttt{GATG}}, q_{\texttt{GATT}}, q_{\texttt{GCAA}}, q_{\texttt{GCAC}}, q_{\texttt{GCAG}}, q_{\texttt{GCCA}}, q_{\texttt{GCCC}}, q_{\texttt{GCCT}}, q_{\texttt{GCGA}}, q_{\texttt{GCGG}}, q_{\texttt{GCGT}}, q_{\texttt{GCTC}}, q_{\texttt{GCTG}}, q_{\texttt{GCTT}}, q_{\texttt{GGAC}}, q_{\texttt{GGAG}}, q_{\texttt{GGAT}}, q_{\texttt{GGCA}}, q_{\texttt{GGCG}}, q_{\texttt{GGCT}}, q_{\texttt{GGGA}}, q_{\texttt{GGGC}}, q_{\texttt{GGGT}}, q_{\texttt{GGTA}}, q_{\texttt{GGTC}}, q_{\texttt{GGTG}}, q_{\texttt{GTAA}}, q_{\texttt{GTAG}}, q_{\texttt{GTAT}}, q_{\texttt{GTCC}}, q_{\texttt{GTCG}}, q_{\texttt{GTCT}}, q_{\texttt{GTGA}}, q_{\texttt{GTGC}}, q_{\texttt{GTGG}}, q_{\texttt{GTTA}}, q_{\texttt{GTTC}}, q_{\texttt{GTTT}}, q_{\texttt{TAAA}}, q_{\texttt{TAAC}}, q_{\texttt{TAAG}}, q_{\texttt{TACA}}, q_{\texttt{TACC}}, q_{\texttt{TACT}}, q_{\texttt{TAGA}}, q_{\texttt{TAGG}}, q_{\texttt{TAGT}}, q_{\texttt{TATC}}, q_{\texttt{TATG}}, q_{\texttt{TATT}}, q_{\texttt{TCAA}}, q_{\texttt{TCAC}}, q_{\texttt{TCAT}}, q_{\texttt{TCCA}}, q_{\texttt{TCCC}}, q_{\texttt{TCCG}}, q_{\texttt{TCGC}}, q_{\texttt{TCGG}}, q_{\texttt{TCGT}}, q_{\texttt{TCTA}}, q_{\texttt{TCTG}}, q_{\texttt{TCTT}}, q_{\texttt{TGAA}}, q_{\texttt{TGAG}}, q_{\texttt{TGAT}}, q_{\texttt{TGCC}}, q_{\texttt{TGCG}}, q_{\texttt{TGCT}}, q_{\texttt{TGGA}}, q_{\texttt{TGGC}}, q_{\texttt{TGGG}}, q_{\texttt{TGTA}}, q_{\texttt{TGTC}}, q_{\texttt{TGTT}}, q_{\texttt{TTAC}}, q_{\texttt{TTAG}}, q_{\texttt{TTAT}}, q_{\texttt{TTCA}}, q_{\texttt{TTCG}}, q_{\texttt{TTCT}}, q_{\texttt{TTGA}}, q_{\texttt{TTGC}}, q_{\texttt{TTGT}}, q_{\texttt{TTTA}}, q_{\texttt{TTTC}}, q_{\texttt{TTTG}} \\[0.5em] \text{Class } &q_{\texttt{AAAA}}:\ q_{\texttt{AAAA}} \\[0.5em] \text{Class } &q_{\texttt{AACC}}:\ q_{\texttt{AACC}},\ q_{\texttt{AAGG}},\ q_{\texttt{AATT}} \\[0.5em] \text{Class } &q_{\texttt{ACAC}}:\ q_{\texttt{ACAC}},\ q_{\texttt{AGAG}},\ q_{\texttt{ATAT}} \\[0.5em] \text{Class } &q_{\texttt{ACCA}}:\ q_{\texttt{ACCA}},\ q_{\texttt{AGGA}},\ q_{\texttt{ATTA}} \\[0.5em] \text{Class } &q_{\texttt{ACGT}}:\ q_{\texttt{ACGT}},\ q_{\texttt{ACTG}},\ q_{\texttt{AGCT}},\ q_{\texttt{AGTC}},\ q_{\texttt{ATCG}},\ q_{\texttt{ATGC}} \\[0.5em] \text{Class } &q_{\texttt{CAAC}}:\ q_{\texttt{CAAC}},\ q_{\texttt{GAAG}},\ q_{\texttt{TAAT}} \\[0.5em] \text{Class } &q_{\texttt{CACA}}:\ q_{\texttt{CACA}},\ q_{\texttt{GAGA}},\ q_{\texttt{TATA}} \\[0.5em] \text{Class } &q_{\texttt{CAGT}}:\ q_{\texttt{CAGT}},\ q_{\texttt{CATG}},\ q_{\texttt{GACT}},\ q_{\texttt{GATC}},\ q_{\texttt{TACG}},\ q_{\texttt{TAGC}} \\[0.5em] \text{Class } &q_{\texttt{CCAA}}:\ q_{\texttt{CCAA}},\ q_{\texttt{GGAA}},\ q_{\texttt{TTAA}} \\[0.5em] \text{Class } &q_{\texttt{CCCC}}:\ q_{\texttt{CCCC}},\ q_{\texttt{CCGG}},\ q_{\texttt{CCTT}},\ q_{\texttt{GGCC}},\ q_{\texttt{GGGG}},\ q_{\texttt{GGTT}},\ q_{\texttt{TTCC}},\ q_{\texttt{TTGG}},\ q_{\texttt{TTTT}} \\[0.5em] \text{Class } &q_{\texttt{CGAT}}:\ q_{\texttt{CGAT}},\ q_{\texttt{CTAG}},\ q_{\texttt{GCAT}},\ q_{\texttt{GTAC}},\ q_{\texttt{TCAG}},\ q_{\texttt{TGAC}} \\[0.5em] \text{Class } &q_{\texttt{CGCG}}:\ q_{\texttt{CGCG}},\ q_{\texttt{CGGC}},\ q_{\texttt{CTCT}},\ q_{\texttt{CTTC}},\ q_{\texttt{GCCG}},\ q_{\texttt{GCGC}},\ q_{\texttt{GTGT}},\ q_{\texttt{GTTG}},\ q_{\texttt{TCCT}},\ q_{\texttt{TCTC}},\ q_{\texttt{TGGT}},\ q_{\texttt{TGTG}} \\[0.5em] \text{Class } &q_{\texttt{CGTA}}:\ q_{\texttt{CGTA}},\ q_{\texttt{CTGA}},\ q_{\texttt{GCTA}},\ q_{\texttt{GTCA}},\ q_{\texttt{TCGA}},\ q_{\texttt{TGCA}} \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}} \\ -q_{\texttt{CGTA}}q_{\texttt{ACAC}} + q_{\texttt{CGAT}}q_{\texttt{ACCA}} \\ -q_{\texttt{CAGT}}q_{\texttt{ACAC}} + q_{\texttt{CAAC}}q_{\texttt{ACGT}} \\ -q_{\texttt{CAGT}}q_{\texttt{ACCA}} + q_{\texttt{CACA}}q_{\texttt{ACGT}} \\ -q_{\texttt{CGCG}}q_{\texttt{ACAC}} + q_{\texttt{CGAT}}q_{\texttt{ACGT}} \\ q_{\texttt{CGTA}}q_{\texttt{ACGT}} - q_{\texttt{CGCG}}q_{\texttt{ACCA}} \\ -q_{\texttt{CGTA}}q_{\texttt{CAAC}} + q_{\texttt{CGAT}}q_{\texttt{CACA}} \\ -q_{\texttt{CGCG}}q_{\texttt{CAAC}} + q_{\texttt{CGAT}}q_{\texttt{CAGT}} \\ q_{\texttt{CGTA}}q_{\texttt{CAGT}} - q_{\texttt{CGCG}}q_{\texttt{CACA}} \\ -q_{\texttt{ACGT}}^2q_{\texttt{AAAA}} + q_{\texttt{ACCA}}q_{\texttt{ACAC}}q_{\texttt{AACC}} \\ -q_{\texttt{CAGT}}q_{\texttt{ACGT}}q_{\texttt{AAAA}} + q_{\texttt{CACA}}q_{\texttt{ACAC}}q_{\texttt{AACC}} \\ q_{\texttt{CGTA}}q_{\texttt{ACAC}}q_{\texttt{AACC}} - q_{\texttt{CGCG}}q_{\texttt{ACGT}}q_{\texttt{AAAA}} \\ -q_{\texttt{CAGT}}^2q_{\texttt{AAAA}} + q_{\texttt{CACA}}q_{\texttt{CAAC}}q_{\texttt{AACC}} \\ q_{\texttt{CGTA}}q_{\texttt{CAAC}}q_{\texttt{AACC}} - q_{\texttt{CGCG}}q_{\texttt{CAGT}}q_{\texttt{AAAA}} \\ q_{\texttt{CGTA}}q_{\texttt{CGAT}}q_{\texttt{AACC}} - q_{\texttt{CGCG}}^2q_{\texttt{AAAA}} \\ -q_{\texttt{CGAT}}^2q_{\texttt{AAAA}} + q_{\texttt{CCAA}}q_{\texttt{CAAC}}q_{\texttt{ACAC}} \\ -q_{\texttt{CGAT}}^2q_{\texttt{AACC}} + q_{\texttt{CCCC}}q_{\texttt{CAAC}}q_{\texttt{ACAC}} \\ -q_{\texttt{CGTA}}q_{\texttt{CGAT}}q_{\texttt{AAAA}} + q_{\texttt{CCAA}}q_{\texttt{CACA}}q_{\texttt{ACAC}} \\ -q_{\texttt{CGCG}}^2q_{\texttt{AAAA}} + q_{\texttt{CCCC}}q_{\texttt{CACA}}q_{\texttt{ACAC}} \\ -q_{\texttt{CGCG}}q_{\texttt{CGAT}}q_{\texttt{AAAA}} + q_{\texttt{CCAA}}q_{\texttt{CAGT}}q_{\texttt{ACAC}} \\ -q_{\texttt{CGCG}}q_{\texttt{CGAT}}q_{\texttt{AACC}} + q_{\texttt{CCCC}}q_{\texttt{CAGT}}q_{\texttt{ACAC}} \\ -q_{\texttt{CGTA}}^2q_{\texttt{AAAA}} + q_{\texttt{CCAA}}q_{\texttt{CACA}}q_{\texttt{ACCA}} \\ -q_{\texttt{CGTA}}^2q_{\texttt{AACC}} + q_{\texttt{CCCC}}q_{\texttt{CACA}}q_{\texttt{ACCA}} \\ -q_{\texttt{CGTA}}q_{\texttt{CGCG}}q_{\texttt{AAAA}} + q_{\texttt{CCAA}}q_{\texttt{CAGT}}q_{\texttt{ACCA}} \\ -q_{\texttt{CGTA}}q_{\texttt{CGCG}}q_{\texttt{AACC}} + q_{\texttt{CCCC}}q_{\texttt{CAGT}}q_{\texttt{ACCA}} \\ -q_{\texttt{CCCC}}q_{\texttt{ACCA}}q_{\texttt{ACAC}} + q_{\texttt{CCAA}}q_{\texttt{ACGT}}^2 \\ -q_{\texttt{CGCG}}^2q_{\texttt{AAAA}} + q_{\texttt{CCAA}}q_{\texttt{CAGT}}q_{\texttt{ACGT}} \\ -q_{\texttt{CGCG}}^2q_{\texttt{AACC}} + q_{\texttt{CCCC}}q_{\texttt{CAGT}}q_{\texttt{ACGT}} \\ -q_{\texttt{CGTA}}q_{\texttt{CCCC}}q_{\texttt{ACAC}} + q_{\texttt{CGCG}}q_{\texttt{CCAA}}q_{\texttt{ACGT}} \\ -q_{\texttt{CCCC}}q_{\texttt{CACA}}q_{\texttt{CAAC}} + q_{\texttt{CCAA}}q_{\texttt{CAGT}}^2 \\ -q_{\texttt{CGTA}}q_{\texttt{CCCC}}q_{\texttt{CAAC}} + q_{\texttt{CGCG}}q_{\texttt{CCAA}}q_{\texttt{CAGT}} \\ q_{\texttt{CGTA}}q_{\texttt{CGAT}}q_{\texttt{CCCC}} - q_{\texttt{CGCG}}^2q_{\texttt{CCAA}} \end{align} $$

Gröbner basis of the vanishing ideal

Gröbner basis with respect to the ordering $ \texttt{degrevlex}\ [ q_{\texttt{CGTA}}, q_{\texttt{CGCG}}, q_{\texttt{CGAT}}, q_{\texttt{CCCC}}, q_{\texttt{CCAA}}, q_{\texttt{CAGT}}, q_{\texttt{CACA}}, q_{\texttt{CAAC}}, q_{\texttt{ACGT}}, 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}} \\ -q_{\texttt{CGTA}}q_{\texttt{ACAC}} + q_{\texttt{CGAT}}q_{\texttt{ACCA}} \\ -q_{\texttt{CAGT}}q_{\texttt{ACAC}} + q_{\texttt{CAAC}}q_{\texttt{ACGT}} \\ -q_{\texttt{CAGT}}q_{\texttt{ACCA}} + q_{\texttt{CACA}}q_{\texttt{ACGT}} \\ -q_{\texttt{CGCG}}q_{\texttt{ACAC}} + q_{\texttt{CGAT}}q_{\texttt{ACGT}} \\ q_{\texttt{CGTA}}q_{\texttt{ACGT}} - q_{\texttt{CGCG}}q_{\texttt{ACCA}} \\ -q_{\texttt{CGTA}}q_{\texttt{CAAC}} + q_{\texttt{CGAT}}q_{\texttt{CACA}} \\ -q_{\texttt{CGCG}}q_{\texttt{CAAC}} + q_{\texttt{CGAT}}q_{\texttt{CAGT}} \\ q_{\texttt{CGTA}}q_{\texttt{CAGT}} - q_{\texttt{CGCG}}q_{\texttt{CACA}} \\ -q_{\texttt{ACGT}}^2q_{\texttt{AAAA}} + q_{\texttt{ACCA}}q_{\texttt{ACAC}}q_{\texttt{AACC}} \\ -q_{\texttt{CAGT}}q_{\texttt{ACGT}}q_{\texttt{AAAA}} + q_{\texttt{CACA}}q_{\texttt{ACAC}}q_{\texttt{AACC}} \\ q_{\texttt{CGTA}}q_{\texttt{ACAC}}q_{\texttt{AACC}} - q_{\texttt{CGCG}}q_{\texttt{ACGT}}q_{\texttt{AAAA}} \\ -q_{\texttt{CAGT}}^2q_{\texttt{AAAA}} + q_{\texttt{CACA}}q_{\texttt{CAAC}}q_{\texttt{AACC}} \\ q_{\texttt{CGTA}}q_{\texttt{CAAC}}q_{\texttt{AACC}} - q_{\texttt{CGCG}}q_{\texttt{CAGT}}q_{\texttt{AAAA}} \\ q_{\texttt{CGTA}}q_{\texttt{CGAT}}q_{\texttt{AACC}} - q_{\texttt{CGCG}}^2q_{\texttt{AAAA}} \\ -q_{\texttt{CGAT}}^2q_{\texttt{AAAA}} + q_{\texttt{CCAA}}q_{\texttt{CAAC}}q_{\texttt{ACAC}} \\ -q_{\texttt{CGAT}}^2q_{\texttt{AACC}} + q_{\texttt{CCCC}}q_{\texttt{CAAC}}q_{\texttt{ACAC}} \\ -q_{\texttt{CGTA}}q_{\texttt{CGAT}}q_{\texttt{AAAA}} + q_{\texttt{CCAA}}q_{\texttt{CACA}}q_{\texttt{ACAC}} \\ -q_{\texttt{CGCG}}^2q_{\texttt{AAAA}} + q_{\texttt{CCCC}}q_{\texttt{CACA}}q_{\texttt{ACAC}} \\ -q_{\texttt{CGCG}}q_{\texttt{CGAT}}q_{\texttt{AAAA}} + q_{\texttt{CCAA}}q_{\texttt{CAGT}}q_{\texttt{ACAC}} \\ -q_{\texttt{CGCG}}q_{\texttt{CGAT}}q_{\texttt{AACC}} + q_{\texttt{CCCC}}q_{\texttt{CAGT}}q_{\texttt{ACAC}} \\ -q_{\texttt{CGTA}}^2q_{\texttt{AAAA}} + q_{\texttt{CCAA}}q_{\texttt{CACA}}q_{\texttt{ACCA}} \\ -q_{\texttt{CGTA}}^2q_{\texttt{AACC}} + q_{\texttt{CCCC}}q_{\texttt{CACA}}q_{\texttt{ACCA}} \\ -q_{\texttt{CGTA}}q_{\texttt{CGCG}}q_{\texttt{AAAA}} + q_{\texttt{CCAA}}q_{\texttt{CAGT}}q_{\texttt{ACCA}} \\ -q_{\texttt{CGTA}}q_{\texttt{CGCG}}q_{\texttt{AACC}} + q_{\texttt{CCCC}}q_{\texttt{CAGT}}q_{\texttt{ACCA}} \\ -q_{\texttt{CCCC}}q_{\texttt{ACCA}}q_{\texttt{ACAC}} + q_{\texttt{CCAA}}q_{\texttt{ACGT}}^2 \\ -q_{\texttt{CGCG}}^2q_{\texttt{AAAA}} + q_{\texttt{CCAA}}q_{\texttt{CAGT}}q_{\texttt{ACGT}} \\ -q_{\texttt{CGCG}}^2q_{\texttt{AACC}} + q_{\texttt{CCCC}}q_{\texttt{CAGT}}q_{\texttt{ACGT}} \\ -q_{\texttt{CGTA}}q_{\texttt{CCCC}}q_{\texttt{ACAC}} + q_{\texttt{CGCG}}q_{\texttt{CCAA}}q_{\texttt{ACGT}} \\ -q_{\texttt{CCCC}}q_{\texttt{CACA}}q_{\texttt{CAAC}} + q_{\texttt{CCAA}}q_{\texttt{CAGT}}^2 \\ -q_{\texttt{CGTA}}q_{\texttt{CCCC}}q_{\texttt{CAAC}} + q_{\texttt{CGCG}}q_{\texttt{CCAA}}q_{\texttt{CAGT}} \\ q_{\texttt{CGTA}}q_{\texttt{CGAT}}q_{\texttt{CCCC}} - q_{\texttt{CGCG}}^2q_{\texttt{CCAA}} \\ -q_{\texttt{CGTA}}^2q_{\texttt{CCCC}}q_{\texttt{ACAC}} + q_{\texttt{CGCG}}^2q_{\texttt{CCAA}}q_{\texttt{ACCA}} \\ -q_{\texttt{CGTA}}^2q_{\texttt{CCCC}}q_{\texttt{CAAC}} + q_{\texttt{CGCG}}^2q_{\texttt{CCAA}}q_{\texttt{CACA}} \end{align} $$

Additional information

Probability $\to$ Fourier

$\begin{align} q_{\texttt{CGTA}} &= \frac{1}{3}p_{\texttt{ACGT}} + \frac{1}{3}p_{\texttt{ACGG}} + \frac{1}{3}p_{\texttt{ACGC}} + \frac{1}{3}p_{\texttt{ACGA}} - \frac{1}{3}p_{\texttt{ACCG}} - \frac{1}{3}p_{\texttt{ACCC}} - \frac{1}{3}p_{\texttt{ACCA}} - \frac{1}{3}p_{\texttt{ACAG}} - \frac{1}{3}p_{\texttt{ACAC}} - \frac{1}{3}p_{\texttt{ACAA}} - \frac{1}{3}p_{\texttt{AACG}} - \frac{1}{3}p_{\texttt{AACC}} - \frac{1}{3}p_{\texttt{AACA}} + p_{\texttt{AAAC}} + p_{\texttt{AAAA}} \\[0.5em] q_{\texttt{CGCG}} &= -\frac{1}{3}p_{\texttt{ACGT}} + \frac{1}{3}p_{\texttt{ACGG}} - \frac{1}{3}p_{\texttt{ACCC}} + \frac{1}{3}p_{\texttt{ACCA}} + \frac{1}{3}p_{\texttt{ACAC}} - \frac{1}{3}p_{\texttt{ACAA}} + \frac{1}{3}p_{\texttt{AACG}} - \frac{1}{3}p_{\texttt{AACC}} - \frac{1}{3}p_{\texttt{AACA}} - \frac{1}{3}p_{\texttt{AAAC}} + p_{\texttt{AAAA}} \\[0.5em] q_{\texttt{CGAT}} &= \frac{1}{3}p_{\texttt{ACGT}} + \frac{1}{3}p_{\texttt{ACGG}} - \frac{1}{3}p_{\texttt{ACGC}} - \frac{1}{3}p_{\texttt{ACGA}} + \frac{1}{3}p_{\texttt{ACCG}} - \frac{1}{3}p_{\texttt{ACCC}} - \frac{1}{3}p_{\texttt{ACCA}} + \frac{1}{3}p_{\texttt{ACAG}} - \frac{1}{3}p_{\texttt{ACAC}} - \frac{1}{3}p_{\texttt{ACAA}} - \frac{1}{3}p_{\texttt{AACG}} - \frac{1}{3}p_{\texttt{AACC}} + p_{\texttt{AACA}} - \frac{1}{3}p_{\texttt{AAAC}} + p_{\texttt{AAAA}} \\[0.5em] q_{\texttt{CCCC}} &= \frac{1}{9}p_{\texttt{ACGT}} - \frac{1}{3}p_{\texttt{ACGG}} + \frac{1}{9}p_{\texttt{ACGC}} + \frac{1}{9}p_{\texttt{ACGA}} + \frac{1}{9}p_{\texttt{ACCG}} - \frac{1}{3}p_{\texttt{ACCC}} + \frac{1}{9}p_{\texttt{ACCA}} + \frac{1}{9}p_{\texttt{ACAG}} + \frac{1}{9}p_{\texttt{ACAC}} - \frac{1}{3}p_{\texttt{ACAA}} - \frac{1}{3}p_{\texttt{AACG}} + p_{\texttt{AACC}} - \frac{1}{3}p_{\texttt{AACA}} - \frac{1}{3}p_{\texttt{AAAC}} + p_{\texttt{AAAA}} \\[0.5em] q_{\texttt{CCAA}} &= -\frac{1}{3}p_{\texttt{ACGT}} - \frac{1}{3}p_{\texttt{ACGG}} - \frac{1}{3}p_{\texttt{ACGC}} - \frac{1}{3}p_{\texttt{ACGA}} - \frac{1}{3}p_{\texttt{ACCG}} - \frac{1}{3}p_{\texttt{ACCC}} - \frac{1}{3}p_{\texttt{ACCA}} - \frac{1}{3}p_{\texttt{ACAG}} - \frac{1}{3}p_{\texttt{ACAC}} - \frac{1}{3}p_{\texttt{ACAA}} + p_{\texttt{AACG}} + p_{\texttt{AACC}} + p_{\texttt{AACA}} + p_{\texttt{AAAC}} + p_{\texttt{AAAA}} \\[0.5em] q_{\texttt{CAGT}} &= \frac{1}{3}p_{\texttt{ACGT}} - \frac{1}{3}p_{\texttt{ACGG}} + \frac{1}{3}p_{\texttt{ACGC}} - \frac{1}{3}p_{\texttt{ACGA}} + \frac{1}{3}p_{\texttt{ACCG}} - \frac{1}{3}p_{\texttt{ACCC}} - \frac{1}{3}p_{\texttt{ACCA}} - \frac{1}{3}p_{\texttt{ACAG}} - \frac{1}{3}p_{\texttt{ACAC}} + p_{\texttt{ACAA}} + \frac{1}{3}p_{\texttt{AACG}} - \frac{1}{3}p_{\texttt{AACC}} - \frac{1}{3}p_{\texttt{AACA}} - \frac{1}{3}p_{\texttt{AAAC}} + p_{\texttt{AAAA}} \\[0.5em] q_{\texttt{CACA}} &= -\frac{1}{3}p_{\texttt{ACGT}} - \frac{1}{3}p_{\texttt{ACGG}} - \frac{1}{3}p_{\texttt{ACGC}} - \frac{1}{3}p_{\texttt{ACGA}} - \frac{1}{3}p_{\texttt{ACCG}} - \frac{1}{3}p_{\texttt{ACCC}} - \frac{1}{3}p_{\texttt{ACCA}} + p_{\texttt{ACAG}} + p_{\texttt{ACAC}} + p_{\texttt{ACAA}} - \frac{1}{3}p_{\texttt{AACG}} - \frac{1}{3}p_{\texttt{AACC}} - \frac{1}{3}p_{\texttt{AACA}} + p_{\texttt{AAAC}} + p_{\texttt{AAAA}} \\[0.5em] q_{\texttt{CAAC}} &= -\frac{1}{3}p_{\texttt{ACGT}} - \frac{1}{3}p_{\texttt{ACGG}} - \frac{1}{3}p_{\texttt{ACGC}} + p_{\texttt{ACGA}} - \frac{1}{3}p_{\texttt{ACCG}} - \frac{1}{3}p_{\texttt{ACCC}} + p_{\texttt{ACCA}} - \frac{1}{3}p_{\texttt{ACAG}} - \frac{1}{3}p_{\texttt{ACAC}} + p_{\texttt{ACAA}} - \frac{1}{3}p_{\texttt{AACG}} - \frac{1}{3}p_{\texttt{AACC}} + p_{\texttt{AACA}} - \frac{1}{3}p_{\texttt{AAAC}} + p_{\texttt{AAAA}} \\[0.5em] q_{\texttt{ACGT}} &= \frac{1}{3}p_{\texttt{ACGT}} - \frac{1}{3}p_{\texttt{ACGG}} - \frac{1}{3}p_{\texttt{ACGC}} + \frac{1}{3}p_{\texttt{ACGA}} - \frac{1}{3}p_{\texttt{ACCG}} + p_{\texttt{ACCC}} - \frac{1}{3}p_{\texttt{ACCA}} + \frac{1}{3}p_{\texttt{ACAG}} - \frac{1}{3}p_{\texttt{ACAC}} - \frac{1}{3}p_{\texttt{ACAA}} + \frac{1}{3}p_{\texttt{AACG}} - \frac{1}{3}p_{\texttt{AACC}} - \frac{1}{3}p_{\texttt{AACA}} - \frac{1}{3}p_{\texttt{AAAC}} + p_{\texttt{AAAA}} \\[0.5em] q_{\texttt{ACCA}} &= -\frac{1}{3}p_{\texttt{ACGT}} - \frac{1}{3}p_{\texttt{ACGG}} - \frac{1}{3}p_{\texttt{ACGC}} - \frac{1}{3}p_{\texttt{ACGA}} + p_{\texttt{ACCG}} + p_{\texttt{ACCC}} + p_{\texttt{ACCA}} - \frac{1}{3}p_{\texttt{ACAG}} - \frac{1}{3}p_{\texttt{ACAC}} - \frac{1}{3}p_{\texttt{ACAA}} - \frac{1}{3}p_{\texttt{AACG}} - \frac{1}{3}p_{\texttt{AACC}} - \frac{1}{3}p_{\texttt{AACA}} + p_{\texttt{AAAC}} + p_{\texttt{AAAA}} \\[0.5em] q_{\texttt{ACAC}} &= -\frac{1}{3}p_{\texttt{ACGT}} - \frac{1}{3}p_{\texttt{ACGG}} + p_{\texttt{ACGC}} - \frac{1}{3}p_{\texttt{ACGA}} - \frac{1}{3}p_{\texttt{ACCG}} + p_{\texttt{ACCC}} - \frac{1}{3}p_{\texttt{ACCA}} - \frac{1}{3}p_{\texttt{ACAG}} + p_{\texttt{ACAC}} - \frac{1}{3}p_{\texttt{ACAA}} - \frac{1}{3}p_{\texttt{AACG}} - \frac{1}{3}p_{\texttt{AACC}} + p_{\texttt{AACA}} - \frac{1}{3}p_{\texttt{AAAC}} + p_{\texttt{AAAA}} \\[0.5em] q_{\texttt{AACC}} &= -\frac{1}{3}p_{\texttt{ACGT}} + p_{\texttt{ACGG}} - \frac{1}{3}p_{\texttt{ACGC}} - \frac{1}{3}p_{\texttt{ACGA}} - \frac{1}{3}p_{\texttt{ACCG}} + p_{\texttt{ACCC}} - \frac{1}{3}p_{\texttt{ACCA}} - \frac{1}{3}p_{\texttt{ACAG}} - \frac{1}{3}p_{\texttt{ACAC}} + p_{\texttt{ACAA}} - \frac{1}{3}p_{\texttt{AACG}} + p_{\texttt{AACC}} - \frac{1}{3}p_{\texttt{AACA}} - \frac{1}{3}p_{\texttt{AAAC}} + p_{\texttt{AAAA}} \\[0.5em] q_{\texttt{AAAA}} &= p_{\texttt{ACGT}} + p_{\texttt{ACGG}} + p_{\texttt{ACGC}} + p_{\texttt{ACGA}} + p_{\texttt{ACCG}} + p_{\texttt{ACCC}} + p_{\texttt{ACCA}} + p_{\texttt{ACAG}} + p_{\texttt{ACAC}} + p_{\texttt{ACAA}} + p_{\texttt{AACG}} + p_{\texttt{AACC}} + p_{\texttt{AACA}} + p_{\texttt{AAAC}} + p_{\texttt{AAAA}} \end{align}$

Fourier $\to$ Probability

$\begin{align} p_{\texttt{ACGT}} &= \frac{3}{16}q_{\texttt{CGTA}} - \frac{3}{8}q_{\texttt{CGCG}} + \frac{3}{16}q_{\texttt{CGAT}} + \frac{3}{32}q_{\texttt{CCCC}} - \frac{3}{32}q_{\texttt{CCAA}} + \frac{3}{16}q_{\texttt{CAGT}} - \frac{3}{32}q_{\texttt{CACA}} - \frac{3}{32}q_{\texttt{CAAC}} + \frac{3}{16}q_{\texttt{ACGT}} - \frac{3}{32}q_{\texttt{ACCA}} - \frac{3}{32}q_{\texttt{ACAC}} - \frac{3}{32}q_{\texttt{AACC}} + \frac{3}{32}q_{\texttt{AAAA}} \\[0.5em] p_{\texttt{ACGG}} &= \frac{3}{16}q_{\texttt{CGTA}} + \frac{3}{8}q_{\texttt{CGCG}} + \frac{3}{16}q_{\texttt{CGAT}} - \frac{9}{32}q_{\texttt{CCCC}} - \frac{3}{32}q_{\texttt{CCAA}} - \frac{3}{16}q_{\texttt{CAGT}} - \frac{3}{32}q_{\texttt{CACA}} - \frac{3}{32}q_{\texttt{CAAC}} - \frac{3}{16}q_{\texttt{ACGT}} - \frac{3}{32}q_{\texttt{ACCA}} - \frac{3}{32}q_{\texttt{ACAC}} + \frac{9}{32}q_{\texttt{AACC}} + \frac{3}{32}q_{\texttt{AAAA}} \\[0.5em] p_{\texttt{ACGC}} &= \frac{3}{16}q_{\texttt{CGTA}} - \frac{3}{16}q_{\texttt{CGAT}} + \frac{3}{32}q_{\texttt{CCCC}} - \frac{3}{32}q_{\texttt{CCAA}} + \frac{3}{16}q_{\texttt{CAGT}} - \frac{3}{32}q_{\texttt{CACA}} - \frac{3}{32}q_{\texttt{CAAC}} - \frac{3}{16}q_{\texttt{ACGT}} - \frac{3}{32}q_{\texttt{ACCA}} + \frac{9}{32}q_{\texttt{ACAC}} - \frac{3}{32}q_{\texttt{AACC}} + \frac{3}{32}q_{\texttt{AAAA}} \\[0.5em] p_{\texttt{ACGA}} &= \frac{3}{16}q_{\texttt{CGTA}} - \frac{3}{16}q_{\texttt{CGAT}} + \frac{3}{32}q_{\texttt{CCCC}} - \frac{3}{32}q_{\texttt{CCAA}} - \frac{3}{16}q_{\texttt{CAGT}} - \frac{3}{32}q_{\texttt{CACA}} + \frac{9}{32}q_{\texttt{CAAC}} + \frac{3}{16}q_{\texttt{ACGT}} - \frac{3}{32}q_{\texttt{ACCA}} - \frac{3}{32}q_{\texttt{ACAC}} - \frac{3}{32}q_{\texttt{AACC}} + \frac{3}{32}q_{\texttt{AAAA}} \\[0.5em] p_{\texttt{ACCG}} &= -\frac{3}{16}q_{\texttt{CGTA}} + \frac{3}{16}q_{\texttt{CGAT}} + \frac{3}{32}q_{\texttt{CCCC}} - \frac{3}{32}q_{\texttt{CCAA}} + \frac{3}{16}q_{\texttt{CAGT}} - \frac{3}{32}q_{\texttt{CACA}} - \frac{3}{32}q_{\texttt{CAAC}} - \frac{3}{16}q_{\texttt{ACGT}} + \frac{9}{32}q_{\texttt{ACCA}} - \frac{3}{32}q_{\texttt{ACAC}} - \frac{3}{32}q_{\texttt{AACC}} + \frac{3}{32}q_{\texttt{AAAA}} \\[0.5em] p_{\texttt{ACCC}} &= -\frac{3}{32}q_{\texttt{CGTA}} - \frac{3}{16}q_{\texttt{CGCG}} - \frac{3}{32}q_{\texttt{CGAT}} - \frac{9}{64}q_{\texttt{CCCC}} - \frac{3}{64}q_{\texttt{CCAA}} - \frac{3}{32}q_{\texttt{CAGT}} - \frac{3}{64}q_{\texttt{CACA}} - \frac{3}{64}q_{\texttt{CAAC}} + \frac{9}{32}q_{\texttt{ACGT}} + \frac{9}{64}q_{\texttt{ACCA}} + \frac{9}{64}q_{\texttt{ACAC}} + \frac{9}{64}q_{\texttt{AACC}} + \frac{3}{64}q_{\texttt{AAAA}} \\[0.5em] p_{\texttt{ACCA}} &= -\frac{3}{32}q_{\texttt{CGTA}} + \frac{3}{16}q_{\texttt{CGCG}} - \frac{3}{32}q_{\texttt{CGAT}} + \frac{3}{64}q_{\texttt{CCCC}} - \frac{3}{64}q_{\texttt{CCAA}} - \frac{3}{32}q_{\texttt{CAGT}} - \frac{3}{64}q_{\texttt{CACA}} + \frac{9}{64}q_{\texttt{CAAC}} - \frac{3}{32}q_{\texttt{ACGT}} + \frac{9}{64}q_{\texttt{ACCA}} - \frac{3}{64}q_{\texttt{ACAC}} - \frac{3}{64}q_{\texttt{AACC}} + \frac{3}{64}q_{\texttt{AAAA}} \\[0.5em] p_{\texttt{ACAG}} &= -\frac{3}{16}q_{\texttt{CGTA}} + \frac{3}{16}q_{\texttt{CGAT}} + \frac{3}{32}q_{\texttt{CCCC}} - \frac{3}{32}q_{\texttt{CCAA}} - \frac{3}{16}q_{\texttt{CAGT}} + \frac{9}{32}q_{\texttt{CACA}} - \frac{3}{32}q_{\texttt{CAAC}} + \frac{3}{16}q_{\texttt{ACGT}} - \frac{3}{32}q_{\texttt{ACCA}} - \frac{3}{32}q_{\texttt{ACAC}} - \frac{3}{32}q_{\texttt{AACC}} + \frac{3}{32}q_{\texttt{AAAA}} \\[0.5em] p_{\texttt{ACAC}} &= -\frac{3}{32}q_{\texttt{CGTA}} + \frac{3}{16}q_{\texttt{CGCG}} - \frac{3}{32}q_{\texttt{CGAT}} + \frac{3}{64}q_{\texttt{CCCC}} - \frac{3}{64}q_{\texttt{CCAA}} - \frac{3}{32}q_{\texttt{CAGT}} + \frac{9}{64}q_{\texttt{CACA}} - \frac{3}{64}q_{\texttt{CAAC}} - \frac{3}{32}q_{\texttt{ACGT}} - \frac{3}{64}q_{\texttt{ACCA}} + \frac{9}{64}q_{\texttt{ACAC}} - \frac{3}{64}q_{\texttt{AACC}} + \frac{3}{64}q_{\texttt{AAAA}} \\[0.5em] p_{\texttt{ACAA}} &= -\frac{3}{32}q_{\texttt{CGTA}} - \frac{3}{16}q_{\texttt{CGCG}} - \frac{3}{32}q_{\texttt{CGAT}} - \frac{9}{64}q_{\texttt{CCCC}} - \frac{3}{64}q_{\texttt{CCAA}} + \frac{9}{32}q_{\texttt{CAGT}} + \frac{9}{64}q_{\texttt{CACA}} + \frac{9}{64}q_{\texttt{CAAC}} - \frac{3}{32}q_{\texttt{ACGT}} - \frac{3}{64}q_{\texttt{ACCA}} - \frac{3}{64}q_{\texttt{ACAC}} + \frac{9}{64}q_{\texttt{AACC}} + \frac{3}{64}q_{\texttt{AAAA}} \\[0.5em] p_{\texttt{AACG}} &= -\frac{3}{16}q_{\texttt{CGTA}} + \frac{3}{8}q_{\texttt{CGCG}} - \frac{3}{16}q_{\texttt{CGAT}} - \frac{9}{32}q_{\texttt{CCCC}} + \frac{9}{32}q_{\texttt{CCAA}} + \frac{3}{16}q_{\texttt{CAGT}} - \frac{3}{32}q_{\texttt{CACA}} - \frac{3}{32}q_{\texttt{CAAC}} + \frac{3}{16}q_{\texttt{ACGT}} - \frac{3}{32}q_{\texttt{ACCA}} - \frac{3}{32}q_{\texttt{ACAC}} - \frac{3}{32}q_{\texttt{AACC}} + \frac{3}{32}q_{\texttt{AAAA}} \\[0.5em] p_{\texttt{AACC}} &= -\frac{3}{32}q_{\texttt{CGTA}} - \frac{3}{16}q_{\texttt{CGCG}} - \frac{3}{32}q_{\texttt{CGAT}} + \frac{27}{64}q_{\texttt{CCCC}} + \frac{9}{64}q_{\texttt{CCAA}} - \frac{3}{32}q_{\texttt{CAGT}} - \frac{3}{64}q_{\texttt{CACA}} - \frac{3}{64}q_{\texttt{CAAC}} - \frac{3}{32}q_{\texttt{ACGT}} - \frac{3}{64}q_{\texttt{ACCA}} - \frac{3}{64}q_{\texttt{ACAC}} + \frac{9}{64}q_{\texttt{AACC}} + \frac{3}{64}q_{\texttt{AAAA}} \\[0.5em] p_{\texttt{AACA}} &= -\frac{3}{32}q_{\texttt{CGTA}} - \frac{3}{16}q_{\texttt{CGCG}} + \frac{9}{32}q_{\texttt{CGAT}} - \frac{9}{64}q_{\texttt{CCCC}} + \frac{9}{64}q_{\texttt{CCAA}} - \frac{3}{32}q_{\texttt{CAGT}} - \frac{3}{64}q_{\texttt{CACA}} + \frac{9}{64}q_{\texttt{CAAC}} - \frac{3}{32}q_{\texttt{ACGT}} - \frac{3}{64}q_{\texttt{ACCA}} + \frac{9}{64}q_{\texttt{ACAC}} - \frac{3}{64}q_{\texttt{AACC}} + \frac{3}{64}q_{\texttt{AAAA}} \\[0.5em] p_{\texttt{AAAC}} &= \frac{9}{32}q_{\texttt{CGTA}} - \frac{3}{16}q_{\texttt{CGCG}} - \frac{3}{32}q_{\texttt{CGAT}} - \frac{9}{64}q_{\texttt{CCCC}} + \frac{9}{64}q_{\texttt{CCAA}} - \frac{3}{32}q_{\texttt{CAGT}} + \frac{9}{64}q_{\texttt{CACA}} - \frac{3}{64}q_{\texttt{CAAC}} - \frac{3}{32}q_{\texttt{ACGT}} + \frac{9}{64}q_{\texttt{ACCA}} - \frac{3}{64}q_{\texttt{ACAC}} - \frac{3}{64}q_{\texttt{AACC}} + \frac{3}{64}q_{\texttt{AAAA}} \\[0.5em] p_{\texttt{AAAA}} &= \frac{3}{32}q_{\texttt{CGTA}} + \frac{3}{16}q_{\texttt{CGCG}} + \frac{3}{32}q_{\texttt{CGAT}} + \frac{9}{64}q_{\texttt{CCCC}} + \frac{3}{64}q_{\texttt{CCAA}} + \frac{3}{32}q_{\texttt{CAGT}} + \frac{3}{64}q_{\texttt{CACA}} + \frac{3}{64}q_{\texttt{CAAC}} + \frac{3}{32}q_{\texttt{ACGT}} + \frac{3}{64}q_{\texttt{ACCA}} + \frac{3}{64}q_{\texttt{ACAC}} + \frac{3}{64}q_{\texttt{AACC}} + \frac{1}{64}q_{\texttt{AAAA}} \end{align}$