Description Details Author(s) References See Also Examples

Matrix exponentiation with EXPOKIT in R

Package: | rexpokit |

Type: | Package |

Version: | 0.26.5 |

Date: | 2018-09-30 |

License: | GPL (>= 2) |

LazyLoad: | yes |

This package wraps some of the matrix exponentiation
utilities from EXPOKIT
(http://www.maths.uq.edu.au/expokit/), a FORTRAN
library that is widely recommended for fast matrix
exponentiation (Sidje RB, 1998. "Expokit: A Software
Package for Computing Matrix Exponentials." *ACM
Trans. Math. Softw.* 24(1): 130-156).

The FORTRAN package was developed by Roger B. Sidje, see http://www.maths.uq.edu.au/expokit/. Nicholas J. Matzke adapted the package for use with R and wrote the R interface. Permission to distribute the EXPOKIT source under GPL was obtained from Roger B. Sidje.

EXPOKIT includes functions for exponentiating both small, dense matrices, and large, sparse matrices (in sparse matrices, most of the cells have value 0). Rapid matrix exponentiation is useful in phylogenetics when we have a large number of states (as we do when we are inferring the history of transitions between the possible geographic ranges of a species), but is probably useful in other ways as well.

**Help**

For help with rexpokit or BioGeoBEARS, see (1) the PhyloWiki websites for rexpokit and BioGeoBEARS (http://phylo.wikidot.com/rexpokit, http://phylo.wikidot.com/biogeobears) and (2) the BioGeoBEARS Google Group (https://groups.google.com/forum/#!forum/biogeobears). Minor updates may get posted first to the rexpokit GitHub repository (https://github.com/nmatzke/rexpokit)

**Background**

Various messages on discussion boards have asked whether
or not there is an R package that uses EXPOKIT. There
are only two as of this writing (January 2013) –
`diversitree`

and
`ctarma`

. However, diversitree's usage is nested
deeply in a series of dynamic functions and integrated
with additional libraries (e.g. deSolve) and so is very
difficult to extract for general usage, and `ctarma`

implements only ZEXPM via `ctarma`

::`zexpm`

.
Niels Richard Hansen [email protected] is
also working on an implementation of certain EXPOKIT
functions.

(See the additional notes file in the package "inst" directory, EXPOKIT_For_Dummies_notes_v1.txt for additional notes on wrappers for EXPOKIT in Python etc.)

As it turns out, the EXPOKIT documentation and code is
far from trivial to figure out, since the code as
published does not run "out of the box" – in particular,
the Q transition matrix ("matvec"), which is the major
input into an exponentiation algorithm, is not input
directly, but rather via another function, which requires
the user to put together some FORTRAN code to do this and
make a wrapper for the core function. I couldn't figure
it out in a short amount of time, but Stephen Smith did
for his "LAGRANGE" biogeography package, so I copied and
modified this chunk of his code to get started.

**Installation hints**

Installing
`rexpokit`

from source will require a gfortran
compiler to convert the FORTRAN code files in /src (*.f)
to object files (*.o), and g++ to compile and link the
C++ wrapper. `rexpokit`

was developed on an Intel
Mac running OS X 10.7. I (NJM, 2013) successfully compiled it
using g++ and gfortran from (gcc version 4.2.1). (Update
2017-08-13: repeated with gccgfortran version 7.1.0.)

**Citation**

This code was developed for the following publications. Please cite if used:

Matzke, Nicholas J. (2014). "Model Selection in
Historical Biogeography Reveals that Founder-event
Speciation is a Crucial Process in Island Clades."
*Systematic Biology*, 63(6), 951-970.
http://dx.doi.org/10.1093/sysbio/syu056

Matzke, Nicholas J. (2013). "Probabilistic historical
biogeography: new models for founder-event speciation,
imperfect detection, and fossils allow improved
accuracy and model-testing." *Frontiers of
Biogeography*, 5(4), 242-248.
http://escholarship.org/uc/item/44j7n141

Matzke, Nicholas J. (2012).
"Founder-event speciation in BioGeoBEARS package
dramatically improves likelihoods and alters parameter
inference in Dispersal-Extinction-Cladogenesis (DEC)
analyses." *Frontiers of Biogeography* 4(suppl. 1):
210. Link to abstract and PDF of poster:
http://phylo.wikidot.com/matzke-2013-international-biogeography-society-poster.
(Poster abstract published in the Conference Program and
Abstracts of the International Biogeography Society 6th
Biannual Meeting, Miami, Florida. Poster Session P10:
Historical and Paleo-Biogeography. Poster 129B. January
11, 2013.)

Please also cite Sidje (1998).

**Acknowledgements/sources**

**1.** Niels
Richard Hansen [email protected] helped
greatly with the initial setup of the package. See his
`expoRkit`

for another R
implementation of EXPOKIT routines.

**2.**
EXPOKIT, original FORTRAN package, by Roger B. Sidje
[email protected], Department of Mathematics,
University of Queensland, Brisbane, QLD-4072, Australia,
(c) 1996-2013 All Rights Reserved

Sidje has given
permission to include EXPOKIT code in this R package
under the usual GPL license for CRAN R packages. For the
full EXPOKIT copyright and license, see
`expokit_copyright.txt`

under
`inst/notes`

.

**3.** Revisions
for version 0.26, which fixed many issues with
warnings about obsolescence in F77 code, were
aided by email help/discussions with: Kurt Hornik,
Doug Nychka, Marcello Chiodi, Meabh McCurdy. Also,
thanks to these incredibly helpful websites:
"On-Line Fortran F77 - F90 Converter"
(https://www.fortran.uk/plusfortonline.php),
"Building and checking R source packages for Windows"
(https://win-builder.r-project.org/),
"Modernizing Old Fortran"
(http://fortranwiki.org/fortran/show/Modernizing+Old+Fortran),
"Registering the C++ and FORTRAN calls"
(https://stat.ethz.ch/pipermail/r-devel/2017-February/073755.html)

EXPOKIT was published by Sidje in: Sidje RB (1998).
"Expokit. A Software Package for Computing Matrix
Exponentials." *ACM-Transactions on Mathematical
Software*, 24(1):130-156.
http://tinyurl.com/bwa87rq

**4.** A small
amount of C++ code wrapping EXPOKIT was modified from a
file in LAGRANGE, C++ version by Stephen Smith:

http://code.google.com/p/lagrange/

https://github.com/blackrim/lagrange

Specifically:

` * RateMatrix.cpp`

```
*
```

` * Created on: Aug 14, 2009`

```
*
Author: smitty
```

` *`

...and the my_*.f
wrappers for the EXPOKIT *.f code files.

**5.**
Also copied in part (to get the .h file) from:

Python package "Pyprop":

http://code.google.com/p/pyprop/

(old URL) pyprop.googlecode.com/svn/trunk/core/krylov/expokit/expokitpropagator.cpp

(old URL) www.koders.com/python/fidCA95B5A4B2FB77455A72B8A361CF684FFE48F4DC.aspx?s=fourier+transform

Specifically:

pyprop/core/krylov/expokit/f2c/expokit.h

**6.** The EXPOKIT FORTRAN package is available at:

http://www.maths.uq.edu.au/expokit/

Copyright:

http://www.maths.uq.edu.au/expokit/copyright

...or...

expokit_copyright.txt in this install
(see package "inst" directory)

Nicholas J. Matzke [email protected], Roger B. Sidje [email protected], Drew Schmidt [email protected]

http://www.maths.uq.edu.au/expokit/

http://www.maths.uq.edu.au/expokit/copyright

Matzke, Nicholas J. (2014). "Model Selection in
Historical Biogeography Reveals that Founder-event
Speciation is a Crucial Process in Island Clades."
*Systematic Biology*, 63(6), 951-970.
http://dx.doi.org/10.1093/sysbio/syu056

Matzke, Nicholas J. (2013). "Probabilistic historical
biogeography: new models for founder-event speciation,
imperfect detection, and fossils allow improved
accuracy and model-testing." *Frontiers of
Biogeography*, 5(4), 242-248.
http://escholarship.org/uc/item/44j7n141

Matzke N (2012). "Founder-event speciation in BioGeoBEARS package dramatically improves likelihoods and alters parameter inference in Dispersal-Extinction-Cladogenesis (DEC) analyses." _Frontiers of Biogeography_, *4*(suppl. 1), pp. 210. ISSN 1948-6596, Poster abstract published in the Conference Program and Abstracts of the International Biogeography Society 6th Biannual Meeting, Miami, Florida. Poster Session P10: Historical and Paleo-Biogeography. Poster 129B. January 11, 2013, <URL: http://phylo.wikidot.com/matzke-2013-international-biogeography-society-poster>.

Sidje RB (1998). "Expokit. A Software Package for Computing Matrix Exponentials." _ACM Trans. Math. Softw._, *24*(1), pp. 130-156. <URL: http://dx.doi.org/10.1145/285861.285868>, <URL: http://dl.acm.org/citation.cfm?id=285868>.

Eddelbuettel D and Francois R (2011). "Rcpp: Seamless R and C++ Integration." _Journal of Statistical Software_, *40*(8), pp. 1-18. ISSN 1548-7660, See also: <URL: http://cran.r-project.org/web/packages/Rcpp/vignettes/Rcpp-introduction.pdf>, <URL: http://cran.r-project.org/web/packages/Rcpp/index.html>. , <URL: http://www.jstatsoft.org/v40/i08>.

Moler C and Loan CV (2003). "Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later." _SIAM review_, *45*(1), pp. 3-49. <URL: http://epubs.siam.org/doi/10.1137/S00361445024180>.

Foster PG (2001). "The Idiot's Guide to the Zen of Likelihood in a Nutshell in Seven Days for Dummies, Unleashed." Online PDF, widely copied, <URL: http://www.bioinf.org/molsys/data/idiots.pdf>.

`expoRkit`

`expokit_wrapalldmexpv_tvals`

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 | ```
# Example code
# For background and basic principles, see rexpokit/notes/EXPOKIT_For_Dummies_notes_v1.txt
library(rexpokit)
# Make a square instantaneous rate matrix (Q matrix)
# This matrix is taken from Peter Foster's (2001) "The Idiot's Guide
# to the Zen of Likelihood in a Nutshell in Seven Days for Dummies,
# Unleashed" at:
# \url{http://www.bioinf.org/molsys/data/idiots.pdf}
#
# The Q matrix includes the stationary base freqencies, which Pmat
# converges to as t becomes large.
Qmat = matrix(c(-1.218, 0.504, 0.336, 0.378, 0.126, -0.882, 0.252, 0.504,
0.168, 0.504, -1.05, 0.378, 0.126, 0.672, 0.252, -1.05), nrow=4, byrow=TRUE)
# Make a series of t values
tvals = c(0.001, 0.005, 0.01, 0.05, 0.1, 0.5, 1, 2, 5, 14)
# Exponentiate each with EXPOKIT's dgpadm (good for small dense matrices)
for (t in tvals)
{
Pmat = expokit_dgpadm_Qmat(Qmat=Qmat, t=t, transpose_needed=TRUE)
cat("\n\nTime=", t, "\n", sep="")
print(Pmat)
}
# Exponentiate each with EXPOKIT's dmexpv (should be fast for large sparse matrices)
for (t in tvals)
{
Pmat = expokit_dmexpv_Qmat(Qmat=Qmat, t=t, transpose_needed=TRUE)
cat("\n\nTime=", t, "\n", sep="")
print(Pmat)
}
# DMEXPV and DGEXPV are designed for large, sparse Q matrices (sparse = lots of zeros).
# DMEXPV is specifically designed for Markov chains and so may be slower, but more accurate.
# DMEXPV, single t-value
expokit_wrapalldmexpv_tvals(Qmat=Qmat, tvals=tvals[1], transpose_needed=TRUE)
expokit_wrapalldmexpv_tvals(Qmat=Qmat, tvals=2)
# DGEXPV, single t-value
expokit_wrapalldgexpv_tvals(Qmat=Qmat, tvals=tvals[1], transpose_needed=TRUE)
expokit_wrapalldgexpv_tvals(Qmat=Qmat, tvals=2)
# These functions runs the for-loop itself (sadly, we could not get mapply() to work
# on a function that calls dmexpv/dgexpv), returning a list of probability matrices.
# DMEXPV functions
list_of_P_matrices_dmexpv = expokit_wrapalldmexpv_tvals(Qmat=Qmat,
tvals=tvals, transpose_needed=TRUE)
list_of_P_matrices_dmexpv
# DGEXPV functions
list_of_P_matrices_dgexpv = expokit_wrapalldgexpv_tvals(Qmat=Qmat,
tvals=tvals, transpose_needed=TRUE)
list_of_P_matrices_dgexpv
# Check if there are differences in the results (might only happen for large problems)
cat("\n")
cat("Differences between dmexpv and dgexpv\n")
for (i in 1:length(list_of_P_matrices_dmexpv))
{
diffs = list_of_P_matrices_dmexpv[[i]] - list_of_P_matrices_dgexpv[[i]]
print(diffs)
cat("\n")
}
``` |

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