def get_type_1_info(P):
"""
The expected time for a type 1 event is computed as follows.
It is the sum of two expected times.
The first time is the expected number of steps from AB to either Ab or aB
conditional on not entering the states AB or ab.
The second time is the expected number of steps from Ab to ab
conditional on not entering AB.
Note that this formulation depends on the assumption
that the process associated with the first
step is equally likely to end up in Ab as in aB.
It should also include a bit of exponential delay that it takes
to leave the final fixed AB state before embark.
@param P: a huge transition matrix which is not modified
@return: expectation and variance of compensatory substitution time
"""
MatrixUtil.assert_transition_matrix(P)
nstates = len(P)
# get the expected time for the first stage
plain = range(4, nstates)
forbidden = [0, 3]
target = [1, 2]
H = hittingtime.get_conditional_transition_matrix(
P, plain, forbidden, target)
t = hittingtime.get_absorption_time(
H, plain+forbidden, target)
v = hittingtime.get_absorption_variance(
H, plain+forbidden, target)
t_first = t[0]
v_first = v[0]
# get the expected time for the second stage
plain = [1, 2] + range(4, nstates)
forbidden = [0]
target = [3]
H = hittingtime.get_conditional_transition_matrix(
P, plain, forbidden, target)
t = hittingtime.get_absorption_time(
H, plain+forbidden, target)
v = hittingtime.get_absorption_variance(
H, plain+forbidden, target)
t_second = t[1]
v_second = v[1]
# add a geometric rv that depends on probability of leaving fixed AB
p = 1 - P[0, 0]
t_third = (1 - p) / p
v_third = (1 - p) / (p*p)
# return the moments of the distribution accounting for both stages
return t_first + t_second + t_third, v_first + v_second + v_third
def get_type_2_info(P):
"""
The expected time for a type 2 event is computed as follows.
It is the expected number of steps from AB to ab
conditional on not entering the states AB, Ab, or aB.
It should also include a bit of exponential delay that it takes
to leave the final fixed AB state before embark.
@param P: a huge transition matrix which is not modified
@return: expectation and variance of compensatory substitution time
"""
MatrixUtil.assert_transition_matrix(P)
nstates = len(P)
# define index sequences
plain = range(4, nstates)
forbidden = [0, 1, 2]
target = [3]
#
H = hittingtime.get_conditional_transition_matrix(
P, plain, forbidden, target)
t = hittingtime.get_absorption_time(
H, plain+forbidden, target)
v = hittingtime.get_absorption_variance(
H, plain+forbidden, target)
#
t0 = t[0]
v0 = v[0]
# add a geometric rv that depends on probability of leaving fixed AB
p = 1 - P[0, 0]
t0 += (1 - p) / p
v0 += (1 - p) / (p*p)
#
return t0, v0
def get_type_2_info(P):
"""
The expected time for a type 2 event is computed as follows.
It is the expected number of steps from AB to ab
conditional on not entering the states AB, Ab, or aB.
It should also include a bit of exponential delay that it takes
to leave the final fixed AB state before embark.
@param P: a huge transition matrix which is not modified
@return: expectation and variance of compensatory substitution time
"""
MatrixUtil.assert_transition_matrix(P)
nstates = len(P)
# define index sequences
plain = range(4, nstates)
forbidden = [0, 1, 2]
target = [3]
#
H = hittingtime.get_conditional_transition_matrix(P, plain, forbidden,
target)
t = hittingtime.get_absorption_time(H, plain + forbidden, target)
v = hittingtime.get_absorption_variance(H, plain + forbidden, target)
#
t0 = t[0]
v0 = v[0]
# add a geometric rv that depends on probability of leaving fixed AB
p = 1 - P[0, 0]
t0 += (1 - p) / p
v0 += (1 - p) / (p * p)
#
return t0, v0
def get_type_1_info(P):
"""
The expected time for a type 1 event is computed as follows.
It is the sum of two expected times.
The first time is the expected number of steps from AB to either Ab or aB
conditional on not entering the states AB or ab.
The second time is the expected number of steps from Ab to ab
conditional on not entering AB.
Note that this formulation depends on the assumption
that the process associated with the first
step is equally likely to end up in Ab as in aB.
It should also include a bit of exponential delay that it takes
to leave the final fixed AB state before embark.
@param P: a huge transition matrix which is not modified
@return: expectation and variance of compensatory substitution time
"""
MatrixUtil.assert_transition_matrix(P)
nstates = len(P)
# get the expected time for the first stage
plain = range(4, nstates)
forbidden = [0, 3]
target = [1, 2]
H = hittingtime.get_conditional_transition_matrix(P, plain, forbidden,
target)
t = hittingtime.get_absorption_time(H, plain + forbidden, target)
v = hittingtime.get_absorption_variance(H, plain + forbidden, target)
t_first = t[0]
v_first = v[0]
# get the expected time for the second stage
plain = [1, 2] + range(4, nstates)
forbidden = [0]
target = [3]
H = hittingtime.get_conditional_transition_matrix(P, plain, forbidden,
target)
t = hittingtime.get_absorption_time(H, plain + forbidden, target)
v = hittingtime.get_absorption_variance(H, plain + forbidden, target)
t_second = t[1]
v_second = v[1]
# add a geometric rv that depends on probability of leaving fixed AB
p = 1 - P[0, 0]
t_third = (1 - p) / p
v_third = (1 - p) / (p * p)
# return the moments of the distribution accounting for both stages
return t_first + t_second + t_third, v_first + v_second + v_third