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Hello,

I'm working on a CTMC three-state model to obtain time-dependent populations of each state.

[itex]S <=> E <=> G[/itex]

I have built a rate matrix for this (diffusion) process.

[itex]

K =

\begin{pmatrix}

K_{SS} & K_{SE} & K_{SG}\\

K_{ES} & K_{EE} & K_{EG}\\

K_{GS} & K_{GE} & K_{GG}

\end{pmatrix}

=

\begin{pmatrix}

-3.13E+06 & 4.29E+07 & 0\\

3.13E+06 &-4.29E+07 & 3.33E+09\\

0 & 2.26E+06 & -3.33E+09

\end{pmatrix}

[/itex]

The time-dependent population of state G (final state), is given as the product of (I) sum of the (elements??) of the matrix exponential of the transition matrix multiplied by time, and (II) the initial population of state S, P_S(0)=1.

[itex]

P_G(t) = \left( \sum_{S}(exp(tK))\right)*P_S(0)=\left(\sum_{S}(\frac{t^n*K^n}{n!} )\right)*P_S(0)

[/itex]

In this handout, they estimate the population but for the initial state (here, denoted S) instead of final (G, what I seek). They do so by obtaining by an eigenvalue decomposition of the rate matrix, to obtain a final expression for P(t) in terms of t.

http://www.stats.ox.ac.uk/~laws/AppliedProb/handout5.pdf [Broken]

I'm working on a CTMC three-state model to obtain time-dependent populations of each state.

[itex]S <=> E <=> G[/itex]

I have built a rate matrix for this (diffusion) process.

[itex]

K =

\begin{pmatrix}

K_{SS} & K_{SE} & K_{SG}\\

K_{ES} & K_{EE} & K_{EG}\\

K_{GS} & K_{GE} & K_{GG}

\end{pmatrix}

=

\begin{pmatrix}

-3.13E+06 & 4.29E+07 & 0\\

3.13E+06 &-4.29E+07 & 3.33E+09\\

0 & 2.26E+06 & -3.33E+09

\end{pmatrix}

[/itex]

The time-dependent population of state G (final state), is given as the product of (I) sum of the (elements??) of the matrix exponential of the transition matrix multiplied by time, and (II) the initial population of state S, P_S(0)=1.

[itex]

P_G(t) = \left( \sum_{S}(exp(tK))\right)*P_S(0)=\left(\sum_{S}(\frac{t^n*K^n}{n!} )\right)*P_S(0)

[/itex]

**I have many difficulties understanding how I could solve this matrix exponential to obtain the population of state G in practice, given the above transition matrix.***My question:*In this handout, they estimate the population but for the initial state (here, denoted S) instead of final (G, what I seek). They do so by obtaining by an eigenvalue decomposition of the rate matrix, to obtain a final expression for P(t) in terms of t.

http://www.stats.ox.ac.uk/~laws/AppliedProb/handout5.pdf [Broken]

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