MatrixExp | R Documentation |
Calculates the exponential of a square matrix.
MatrixExp(mat, t = 1, method = NULL, ...)
mat |
A square matrix |
t |
An optional scaling factor for |
method |
Under the default of Otherwise, for backwards compatibility, the following options, which use
code in the msm package, are available: |
... |
Arguments to pass to |
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).
The exponentiated matrix \exp(mat)
. Or, if t
is a vector of length 2 or more, an array of exponentiated matrices.
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.
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