Computes the LogDet (a.k.a. log-determinant) nucleotide distance between two sequences from a 4×4 A/C/G/T joint count table. The method normalizes the joint frequency matrix by the marginal base compositions of the two sequences and then applies the log-determinant transform.
Arguments
- m
A numeric matrix of pairwise counts with row/column names including
A,C,G,T. Ifmis a list, the field$probability_matrixis used (compatible with your other helpers).- pseudocount
A non-negative number added to each A/C/G/T cell before normalization (default
0). Use a small value (e.g.0.5) to reduce the chance of a singular matrix when some joint cells are zero. 2
Value
A single numeric: the LogDet distance (substitutions/site, unitless).
Returns Inf if any required determinant/sign is non‑positive.
A single numeric: the HKY85 distance (substitutions/site).
Details
Let \(M\) be the 4×4 joint count matrix over A,C,G,T and
\(F = M / \sum M\) the joint frequency (divergence) matrix. Let
\(\boldsymbol{r} = F \mathbf{1}\) (row sums, seq1 base frequencies) and
\(\boldsymbol{c} = F^\top \mathbf{1}\) (column sums, seq2 base frequencies).
Define diagonal matrices \(D_R = \mathrm{diag}(\boldsymbol{r})\) and
\(D_C = \mathrm{diag}(\boldsymbol{c})\). The LogDet normalization uses
$$
S \;=\; D_R^{-1/2}\, F \, D_C^{-1/2},
$$
and the distance is
$$
d_{\mathrm{LogDet}} \;=\; -\,\ln \!\big( \det S \big)
\;=\; -\,\ln \!\big( \det F \big) \;+\; \tfrac{1}{2}\,\big[ \ln \!\big( \det D_R \big)
\;+\; \ln \!\big( \det D_C \big) \big].
$$
This formulation follows the LogDet/paralinear family where determinants
commute (real scalars), enabling robustness to composition heterogeneity; see
Lockhart et al. (1994) for the transform and discussion, and MEGA’s note on
non-computability when log terms approach zero. 12
References
Lockhart PJ, Steel MA, Hendy MD, Penny D (1994). Recovering evolutionary trees under a more realistic model of sequence evolution. Mol Biol Evol 11:605–612. https://evoluscope.fr/phylographe/biblio/Lockhartetal1994.pdf 1
MEGA help: “LogDet Distance Could Not Be Computed” (notes on log terms). https://www.megasoftware.net/web_help_11/logDet_distance_Could_Not_Be_Computed.htm 2
Gu X, Li WH (1996). Bias-corrected paralinear and LogDet distances and tests under nonstationary nucleotide frequencies. Mol Biol Evol 13:1375–1383. https://paperity.org/p/54207439/bias-corrected-paralinear-and-logdet-distances-and-tests-of-molecular-clocks-and 3
See also
Other Alignment statistics:
F81_distance(),
GCequilibrium(),
GCpressure(),
GCproportion(),
HKY85_distance(),
JC69_distance(),
K80_distance(),
P_distance(),
T92_distance(),
TN93_distance(),
exampleSubstitutionMatrix,
gapProportion()
Other Similarity indexes:
F81_distance(),
GOC(),
HKY85_distance(),
JC69_distance(),
K80_distance(),
P_distance(),
T92_distance(),
TN93_distance(),
correlation_index(),
karyotype_index(),
slidingWindow(),
strand_randomisation_index(),
synteny_index(),
tau_index()