Calculate the Chapman-Kolmogorov equation.
This method evaluates both sides of the Chapman-Kolmogorov equation
\[T(\tau n) = T^n(\tau)\;.\]
So to compare the transition probability estimated based on the lag time
\(n\tau\) (referred as "MD") with the transition probability estimated based
on the lag time \(\tau\) and propagated \(n\) times (referred as "MSM"), we can
use the Chapman-Kolmogorov test. If the model is Markovian, both sides are
identical, and the deviation indicates how Markovian the model is. The
Chapman-Kolmogorov test is commonly projected onto the diagonal (so
limiting to \(T_{ii}\)). For more details, see the review by Prinz et al.
.
The returned dictionary can be visualized using
msmhelper.plot.plot_ck_test. An example can be found in the
tutorial.
Parameters:
-
trajs
(StateTraj or list or ndarray or list of ndarray)
–
State trajectory/trajectories. The states should start from zero and
need to be integers.
-
lagtimes
(list or ndarray int)
–
Lagtimes for estimating the markov model given in [frames].
-
tmax
(int)
–
Longest time to evaluate the CK equation given in [frames].
Returns:
-
cktest ( dict
) –
Dictionary holding for each lagtime the CK equation and with 'md' the
reference.
Source code in src/msmhelper/msm/tests.py
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87 | @decorit.alias('ck_test')
def chapman_kolmogorov_test(trajs, lagtimes, tmax):
r"""Calculate the Chapman-Kolmogorov equation.
This method evaluates both sides of the Chapman-Kolmogorov equation
$$T(\tau n) = T^n(\tau)\;.$$
So to compare the transition probability estimated based on the lag time
$n\tau$ (referred as "MD") with the transition probability estimated based
on the lag time $\tau$ and propagated $n$ times (referred as "MSM"), we can
use the Chapman-Kolmogorov test. If the model is Markovian, both sides are
identical, and the deviation indicates how Markovian the model is. The
Chapman-Kolmogorov test is commonly projected onto the diagonal (so
limiting to $T_{ii}$). For more details, see the review by Prinz et al.
[^1].
The returned dictionary can be visualized using
[msmhelper.plot.plot_ck_test][]. An example can be found in the
[tutorial](/msmhelper/tutorials/msm/#chapman-kolmogorov-test).
[^1]: Prinz et al., **Markov models of molecular kinetics: Generation and
validation**, *J. Chem. Phys.*, 134, 174105 (2011),
doi:[10.1063/1.3565032](https://doi.org/10.1063/1.3565032)
Parameters
----------
trajs : StateTraj or list or ndarray or list of ndarray
State trajectory/trajectories. The states should start from zero and
need to be integers.
lagtimes : list or ndarray int
Lagtimes for estimating the markov model given in [frames].
tmax : int
Longest time to evaluate the CK equation given in [frames].
Returns
-------
cktest : dict
Dictionary holding for each lagtime the CK equation and with 'md' the
reference.
"""
# format input
trajs = StateTraj(trajs)
lagtimes = np.atleast_1d(lagtimes)
lagtimes = np.sort(lagtimes)
# check that lag times are array of integers
if not np.issubdtype(lagtimes.dtype, np.integer):
raise TypeError(
'Lagtimes needs to be integers but are {0}'.format(lagtimes.dtype),
)
if not (lagtimes > 0).all():
raise TypeError('Lagtimes needs to be positive integers')
if lagtimes.ndim != 1:
raise TypeError(
'Lagtimes needs to be maximal 1d, but {0}'.format(lagtimes),
)
if not isinstance(tmax, int) or tmax < 0:
raise TypeError('tmax needs to be a positive integer')
ckeqs = {}
for lagtime in lagtimes:
ckeqs[lagtime] = _chapman_kolmogorov_test(trajs, lagtime, tmax)
ckeqs['md'] = _chapman_kolmogorov_test_md(
trajs, tmin=lagtimes[0], tmax=tmax,
)
return ckeqs
|