### BibTeX

@MISC{Jones08ratematrix,

author = {Graham Jones},

title = {Rate Matrix Prior (Draft)},

year = {2008}

}

### OpenURL

### Abstract

My convention is to have row vectors (of state frequencies) on the left acted on by transition matrices on the right. This seems to be the convention for Markov chains, although the opposite convention is generally more common. Rate matrices have rows summing to zero; transition matrices have rows summing to one. It is usual to impose the condition that the non-diagonal elements of a rate matrix sum to one, but I will work with unnormalised rate matrices. For nucleotides, an arbitrary 12-parameter rate matrix, which I will call a non-time reversible, or NTR rate matrix, can be written as follows. Note that the numbering has wi diagonally opposite to wi+6. The diagonal entries follow from the fact that rows sum to zero. To: A G C T From A- w1 w2 w3 G w7- w4 w5 C w8 w10- w6 T w9 w11 w12-Let vi = log(wi) for 1 ≤ i ≤ 12. The GTR rate matrix can be written as follows ([1] p205). To: A G C T From A- πGa πCb πT c G πAa- πCd πTe C πAb πGd- πT f T πAc πGe πCf-where a,...f are arbitrary positive numbers. Note that there are 10 parameters but there is also a redundancy, since if all of πA, πG, πC, πT are multiplied by x and all of a,...f are multiplied by x −1 the same rate matrix is produced. The dimensionality of the unnormalised GTR rate matrix is therefore 9. Usually πA, πG, πC, πT are normalised to sum to one, so that they can be interpreted as state frequencies at equilibrium, but only the form of the above GTR rate matrix is of concern here, not the meaning of the parameters. By comparing the NTR and GTR rate matrices, three conditions can be found relating ratios of the wi, which become sums and differences of the vi. v1 − v7 + v8 − v2 + v4 − v10 = 0 (1) v7 − v1 + v2 − v8 + v11 − v5 + v6 − v12 = 0 (2) v7 − v1 + v3 − v9 + v10 − v4 + v12 − v6 = 0 (3) 1 The HKY rate matrix can be written as follows ([1] p201). To: A G C T From A- πG(a + b) πCb πT b G πA(a + b)- πCb πT b C πAb πGb- πT(a + b) T πAb πGb πC(a + b)-where a = αR/πR = αY /πY in Felsenstein’s notation. From this, four further conditions can be found, namely v8 = v9, v10 = v11, and v2 = v4 (4) v7 − v8 + v12 − v4 = 0 (5) The two-parameter Kimura rate matrix ([1] p196), which I’ll denote as KIM, is

### Keyphrases

rate matrix prior gtr rate matrix transition matrix state frequency rate matrix ga cb opposite convention diagonal entry unnormalised gtr rate matrix markov chain v2 v4 aa cd te ab gd arbitrary 12-parameter rate matrix non-diagonal element felsenstein notation hky rate matrix cb ab gb v8 v9 ab gb unnormalised rate matrix row vector v7 v8 v12 v4 rate matrix sum ntr rate matrix ac ge cf-where w12-let vi log arbitrary positive number two-parameter kimura rate matrix