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October 2003 A note on parameter differentiation of matrix exponentials, with applications to continuous-time modelling
Henghsiu Tsai, K.S. Chan
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Bernoulli 9(5): 895-919 (October 2003). DOI: 10.3150/bj/1066418883


We propose a new analytic formula for evaluating the derivatives of a matrix exponential. In contrast to the diagonalization method, eigenvalues and eigenvectors do not appear explicitly in the derivation, although we show that a necessary and sufficient condition for the validity of the formula is that the matrix has distinct eigenvalues. The new formula expresses the derivatives of a matrix exponential in terms of minors, polynomials, the exponential of the matrix as well as matrix inversion, and hence is algebraically more manageable. For sparse matrices, the formula can be further simplified. Two examples are discussed in some detail. For the companion matrix of a continuous-time autoregress\-ive moving average process, the derivatives of the exponential of the companion matrix can be computed recursively. The second example concerns the exponential of the tri\-diagonal transition intensity matrix of a finite-state-space continuous-time Markov chain whose instantaneous transitions must be between adjacent states. We present a numerical study to show that the new method may yield numerically more accurate results than the diagonalization method, at the expense of a slight increase in computation.


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Henghsiu Tsai. K.S. Chan. "A note on parameter differentiation of matrix exponentials, with applications to continuous-time modelling." Bernoulli 9 (5) 895 - 919, October 2003.


Published: October 2003
First available in Project Euclid: 17 October 2003

zbMATH: 1053.62009
MathSciNet: MR2047691
Digital Object Identifier: 10.3150/bj/1066418883

Keywords: CARMA models , Cayley-Hamilton theorem , finite-state-space continuous-time Markov chain , maximum likelihood estimation , transition intensity matrix

Rights: Copyright © 2003 Bernoulli Society for Mathematical Statistics and Probability

Vol.9 • No. 5 • October 2003
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