Hadamard Matrices and Fast Hadamard Multiplication

A collection of functions to perform Hadamard conjugation. Hadamard matrix H with a vector v using fast Hadamard multiplication.

Usage

hadamard(x)

fhm(v)

h4st(obj, levels = c("a", "c", "g", "t"))

h2st(obj, eps = 0.001)

Arguments

x: a vector of length \(2^n\), where n is an integer.
v: a vector of length \(2^n\), where n is an integer.
obj: a data.frame or character matrix, typical a sequence alignment.
levels: levels of the sequences.
eps: Threshold value for splits.

Value

hadamard returns a Hadamard matrix. fhm returns the fast Hadamard multiplication.

Details

h2st and h4st perform Hadamard conjugation for 2-state (binary, RY-coded) or 4-state (DNA/RNA) data. write.nexus.splits writes splits returned from h2st or distanceHadamard to a nexus file, which can be processed by Spectronet or SplitsTree.

References

Hendy, M.D. (1989). The relationship between simple evolutionary tree models and observable sequence data. Systematic Zoology, 38 310–321.

Hendy, M. D. and Penny, D. (1993). Spectral Analysis of Phylogenetic Data. Journal of Classification, 10, 5–24.

Hendy, M. D. (2005). Hadamard conjugation: an analytical tool for phylogenetics. In O. Gascuel, editor, Mathematics of evolution and phylogeny, Oxford University Press, Oxford

Waddell P. J. (1995). Statistical methods of phylogenetic analysis: Including hadamard conjugation, LogDet transforms, and maximum likelihood. PhD thesis.

Author

Klaus Schliep klaus.schliep@gmail.com

Examples


H <- hadamard(3)
v <- 1:8
H %*% v
#>      [,1]
#> [1,]   36
#> [2,]   -4
#> [3,]   -8
#> [4,]    0
#> [5,]  -16
#> [6,]    0
#> [7,]    0
#> [8,]    0
fhm(v)
#> [1]  36  -4  -8   0 -16   0   0   0

data(yeast)

# RY-coding
dat_ry <- acgt2ry(yeast)
#> Warning: Found unknown characters (not supplied in levels). Deleted sites with unknown states.
fit2 <- h2st(dat_ry)
lento(fit2)


# write.nexus.splits(fit2, file = "test.nxs")
# read this file into Spectronet or SplitsTree to show the network

fit4 <- h4st(yeast)
old.par <- par(no.readonly = TRUE)
par(mfrow=c(3,1))
lento(fit4[[1]], main="Transversion")
lento(fit4[[2]], main="Transition 1")
lento(fit4[[3]], main="Transition 2")

par(old.par)