class BirthDeathChain¶
- class deeptime.data.BirthDeathChain(q, p, sparse: bool = False)¶
Birth and death chain.
A general birth and death chain on a d-dimensional state space has the following transition matrix
\[p_{ij} = \begin{cases} q_i &\text{, if } j=i-1 \text{ and } i>0,\\ r_i &\text{, if } j=i,\\ p_i &\text{, if } j=i+1 \text{ and } i < d-1 \end{cases}\]- Parameters:
q (array_like) – Annihilation probabilities for transition from i to i-1.
p (array-like) – Creation probabilities for transition from i to i+1.
sparse (bool, optional, default=False) – Whether sparse matrices are used.
Attributes
MarkovStateModel for this birth death chain
The stationary distribution of the birth-death chain.
Transition matrix for birth and death chain with given creation and anhilation probabilities.
Methods
committor_backward(a, b)Backward committor for birth-and-death-chain.
committor_forward(a, b)Forward committor for birth-and-death-chain.
flux(a, b)The flux network for the reaction from A=[0,...,a] => B=[b,...,M].
netflux(a, b)The netflux network for the reaction from A=[0,...,a] => B=[b,...,M].
rate(a, b)Yields the total transition rate between state sets A=[0,...,a] and B=[0,...,b].
totalflux(a, b)The tiotal flux for the reaction A=[0,...,a] => B=[b,...,M].
- committor_backward(a, b)¶
Backward committor for birth-and-death-chain.
The backward committor is the probability for a chain in state x chain to originate from state a instead of coming from state b \(w_x=P_x(t_a<t_b)\), \(t_i\) is the last exit time of the chain from state i, \(t_i = \inf ( t>0 \mid X_{-t} = i )\).
- Parameters:
a (int) – State index
b (int) – State index
- Returns:
w – Vector of committor probabilities.
- Return type:
(M,) ndarray
Notes
The birth-death chain is time-reversible
\[P(t_a < t_b) = P(T_a < T_b) = 1-P(T_btherefore we can express the backward comittor probabilities in terms of the forward committor probabilities \(w=1-u\).
- committor_forward(a, b)¶
Forward committor for birth-and-death-chain.
The forward committor is the probability to hit state b before hitting state a starting in state x,
\[u_x = P_x(T_b\(T_i\) is the first arrival time of the chain to state i,
\[T_i = \inf ( t>0 \mid X_t=i ).\]- Parameters:
a (int) – State index
b (int) – State index
- Returns:
u – Vector of committor probabilities.
- Return type:
(M,) ndarray
- flux(a, b)¶
The flux network for the reaction from A=[0,…,a] => B=[b,…,M].
- Parameters:
a (int) – State index
b (int) – State index
- Returns:
flux – Matrix of flux values between pairs of states.
- Return type:
(M, M) ndarray
- netflux(a, b)¶
The netflux network for the reaction from A=[0,…,a] => B=[b,…,M].
- Parameters:
a (int) – State index
b (int) – State index
- Returns:
netflux – Matrix of flux values between pairs of states.
- Return type:
(M, M) ndarray
- rate(a, b)¶
Yields the total transition rate between state sets A=[0,…,a] and B=[0,…,b].
- Parameters:
a (int) – State index.
b (int) – State index.
- Returns:
kAB – Total transition rate.
- Return type:
float
- totalflux(a, b)¶
The tiotal flux for the reaction A=[0,…,a] => B=[b,…,M].
- Parameters:
a (int) – State index
b (int) – State index
- Returns:
F – The total flux between reactant and product
- Return type:
float
- property msm¶
MarkovStateModel for this birth death chain
- Getter:
Yields the MSM.
- Type:
- property stationary_distribution¶
The stationary distribution of the birth-death chain.
- property transition_matrix¶
Transition matrix for birth and death chain with given creation and anhilation probabilities.
- Getter:
Yields the transition matrix.
- Type:
(N,N) ndarray