Matrix exponential

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Description

Calculates the exponential of a square matrix.

Usage

1
MatrixExp(mat, t = 1, method=NULL, ...)

Arguments

mat

A square matrix

t

An optional scaling factor for mat.

method

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.

Details

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 (http://mcmc-jags.sourceforge.net).

Value

The exponentiated matrix exp(mat). Or, if t is a vector of length 2 or more, an array of exponentiated matrices.

References

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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