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Main Question or Discussion Point
Hello,
I'm working on a CTMC threestate model to obtain timedependent 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 timedependent 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]
My question: 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.
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 threestate model to obtain timedependent 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 timedependent 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]
My question: 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.
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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