Calculates the exponential of a square matrix.
MatrixExp(mat, t = 1, method=NULL, ...)The exponentiated matrix \(\exp(mat)\). Or, if t is
a vector of length 2 or more, an array of exponentiated matrices.
A square matrix
An optional scaling factor for mat.
Under the default of NULL, this simply wraps
the expm function from the expm package.
This is recommended. Options to expm can be supplied to
MatrixExp, including method.
Otherwise, for backwards compatibility, the following options, which
use code in the msm package, are available:
"pade" for a Pade approximation method, "series" for the
power series approximation, or "analytic" for the analytic
formulae for simpler Markov model intensity matrices (see below).
These options are only used if mat has repeated
eigenvalues, thus the usual eigen-decomposition method cannot be used.
Arguments to pass to expm.
See the expm documentation for details of the
algorithms it uses.
Generally the exponential \(E\) of a square matrix \(M\) can often be calculated as
$$E = U \exp(D) U^{-1}$$
where \(D\) is a diagonal matrix with the eigenvalues of \(M\) on the diagonal, \(\exp(D)\) is a diagonal matrix with the exponentiated eigenvalues of \(M\) on the diagonal, and \(U\) is a matrix whose columns are the eigenvectors of \(M\).
This method of calculation is used if "pade" or "series"
is supplied but \(M\) has distinct
eigenvalues. I If \(M\) has repeated eigenvalues, then its
eigenvector matrix may be non-invertible. In this case, the matrix
exponential is calculated using the Pade approximation defined by
Moler and van Loan (2003), or the less robust power series
approximation,
$$\exp(M) = I + M + M^2/2 + M^3 / 3! + M^4 / 4! + ...$$
For a continuous-time homogeneous Markov process with transition intensity matrix \(Q\), the probability of occupying state \(s\) at time \(u + t\) conditional on occupying state \(r\) at time \(u\) is given by the \((r,s)\) entry of the matrix \(\exp(tQ)\).
If mat is a valid transition intensity matrix for a
continuous-time Markov model (i.e. diagonal entries non-positive,
off-diagonal entries non-negative, rows sum to zero), then for certain
simpler model structures, there are
analytic formulae for the individual entries of the exponential
of mat. These structures are
listed in the PDF manual and the formulae are coded in the msm
source file src/analyticp.c. These formulae are only used if
method="analytic". This is more efficient, but it is not the
default in MatrixExp because the code is not robust to extreme
values. However it is the default when calculating likelihoods for models fitted
by msm.
The implementation of the Pade approximation used by
method="pade" was taken from
JAGS by Martyn Plummer (https://mcmc-jags.sourceforge.io).
Cox, D. R. and Miller, H. D. The theory of stochastic processes, Chapman and Hall, London (1965)
Moler, C and van Loan, C (2003). Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Review 45, 3--49.