pyemma.msm.SampledHMSM¶
-
class
pyemma.msm.
SampledHMSM
(P, pobs, pi=None, dt_model='1 step')¶ Sampled Hidden Markov state model
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__init__
(P, pobs, pi=None, dt_model='1 step')¶ - Parameters
Pcoarse (ndarray (m,m)) – coarse-grained or hidden transition matrix
Pobs (ndarray (m,n)) – observation probability matrix from hidden to observable discrete states
dt_model (str, optional, default='1 step') – time step of the model
Methods
__init__
(P, pobs[, pi, dt_model])- param Pcoarse
coarse-grained or hidden transition matrix
committor_backward
(A, B)Backward committor from set A to set B
committor_forward
(A, B)Forward committor (also known as p_fold or splitting probability) from set A to set B
correlation
(a[, b, maxtime, k, ncv])Time-correlation for equilibrium experiment.
eigenvalues
([k])Compute the transition matrix eigenvalues
eigenvectors_left
([k])Compute the left transition matrix eigenvectors
eigenvectors_right
([k])Compute the right transition matrix eigenvectors
expectation
(a)Equilibrium expectation value of a given observable.
fingerprint_correlation
(a[, b, k, ncv])Dynamical fingerprint for equilibrium time-correlation experiment.
fingerprint_relaxation
(p0, a[, k, ncv])Dynamical fingerprint for perturbation/relaxation experiment.
get_model_params
([deep])Get parameters for this model.
load
(file_name[, model_name])Loads a previously saved PyEMMA object from disk.
mfpt
(A, B)Mean first passage times from set A to set B, in units of the input trajectory time step
pcca
(m)Runs PCCA++ [1]_ to compute a metastable decomposition of MSM states
propagate
(p0, k)Propagates the initial distribution p0 k times
relaxation
(p0, a[, maxtime, k, ncv])Simulates a perturbation-relaxation experiment.
sample_conf
(f, *args, **kwargs)Sample confidence interval of numerical method f over all samples
sample_f
(f, *args, **kwargs)Evaluated method f for all samples
sample_mean
(f, *args, **kwargs)Sample mean of numerical method f over all samples
sample_std
(f, *args, **kwargs)Sample standard deviation of numerical method f over all samples
save
(file_name[, model_name, overwrite, …])saves the current state of this object to given file and name.
set_model_params
([samples, conf, P, pobs, …])- param samples
sampled MSMs
simulate
(N[, start, stop, dt])Generates a realization of the Hidden Markov Model
submodel
([states, obs])Returns a HMM with restricted state space
timescales
([k])The relaxation timescales corresponding to the eigenvalues
Computes the transition matrix between observed states
update_model_params
(**params)Update given model parameter if they are set to specific values
Attributes
The transition matrix on the active set.
Description of the physical time corresponding to the lag.
eigenvectors_left_obs
eigenvectors_right_obs
Returns whether the MSM is reversible
Returns whether the MSM is sparse
Lifetimes of states of the hidden transition matrix
Computes the assignment to metastable sets for observable states
Returns the output probability distributions. Identical to
Computes the memberships of observable states to metastable sets by
Computes the metastable sets of observable states within each
Number of states chosen for PCCA++ computation.
number of eigenvalues to compute.
Number of active states on which all computations and estimations are done
nstates_obs
returns the output probability matrix
The stationary distribution on the MSM states
Returns whether the MSM is reversible
samples
Returns whether the MSM is sparse
The stationary distribution on the MSM states
stationary_distribution_obs
Physical time corresponding to one transition matrix step, e.g.
The transition matrix on the active set.
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P
¶ The transition matrix on the active set.
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committor_backward
(A, B)¶ Backward committor from set A to set B
- Parameters
A (int or int array) – set of starting states
B (int or int array) – set of target states
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committor_forward
(A, B)¶ Forward committor (also known as p_fold or splitting probability) from set A to set B
- Parameters
A (int or int array) – set of starting states
B (int or int array) – set of target states
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correlation
(a, b=None, maxtime=None, k=None, ncv=None)¶ Time-correlation for equilibrium experiment.
In order to simulate a time-correlation experiment (e.g. fluorescence correlation spectroscopy [NDD11], dynamical neutron scattering [LYP13], …), first compute the mean values of your experimental observable \(a\) by MSM state:
\[a_i = \frac{1}{N_i} \sum_{x_t \in S_i} f(x_t)\]where \(S_i\) is the set of configurations belonging to MSM state \(i\) and \(f()\) is a function that computes the experimental observable of interest for configuration \(x_t\). If a cross-correlation function is wanted, also apply the above computation to a second experimental observable \(b\).
Then the accurate (i.e. without statistical error) autocorrelation function of \(f(x_t)\) given the Markov model is computed by correlation(a), and the accurate cross-correlation function is computed by correlation(a,b). This is done by evaluating the equation
\[\begin{split}acf_a(k\tau) &= \mathbf{a}^\top \mathrm{diag}(\boldsymbol{\pi}) \mathbf{P(\tau)}^k \mathbf{a} \\ ccf_{a,b}(k\tau) &= \mathbf{a}^\top \mathrm{diag}(\boldsymbol{\pi}) \mathbf{P(\tau)}^k \mathbf{b}\end{split}\]where \(acf\) stands for autocorrelation function and \(ccf\) stands for cross-correlation function, \(\mathbf{P(\tau)}\) is the transition matrix at lag time \(\tau\), \(\boldsymbol{\pi}\) is the equilibrium distribution of \(\mathbf{P}\), and \(k\) is the time index.
Note that instead of using this method you could generate a long synthetic trajectory from the MSM and then estimating the time-correlation of your observable(s) directly from this trajectory. However, there is no reason to do this because the present method does that calculation without any sampling, and only in the limit of an infinitely long synthetic trajectory the two results will agree exactly. The correlation function computed by the present method still has statistical uncertainty from the fact that the underlying MSM transition matrix has statistical uncertainty when being estimated from data, but there is no additional (and unnecessary) uncertainty due to synthetic trajectory generation.
- Parameters
a ((n,) ndarray) – Observable, represented as vector on state space
maxtime (int or float) – Maximum time (in units of the input trajectory time step) until which the correlation function will be evaluated. Internally, the correlation function can only be computed in integer multiples of the Markov model lag time, and therefore the actual last time point will be computed at \(\mathrm{ceil}(\mathrm{maxtime} / \tau)\) By default (None), the maxtime will be set equal to the 5 times the slowest relaxation time of the MSM, because after this time the signal is almost constant.
b ((n,) ndarray (optional)) – Second observable, for cross-correlations
k (int (optional)) – Number of eigenvalues and eigenvectors to use for computation. This option is only relevant for sparse matrices and long times for which an eigenvalue decomposition will be done instead of using the matrix power.
ncv (int (optional)) – Only relevant for sparse matrices and large lag times where the relaxation will be computed using an eigenvalue decomposition. The number of Lanczos vectors generated, ncv must be greater than k; it is recommended that ncv > 2*k.
- Returns
times (ndarray (N)) – Time points (in units of the input trajectory time step) at which the correlation has been computed
correlations (ndarray (N)) – Correlation values at given times
Examples
This example computes the autocorrelation function of a simple observable on a three-state Markov model and plots the result using matplotlib:
>>> import numpy as np >>> import pyemma.msm as msm >>> >>> P = np.array([[0.99, 0.01, 0], [0.01, 0.9, 0.09], [0, 0.1, 0.9]]) >>> a = np.array([0.0, 0.5, 1.0]) >>> M = msm.markov_model(P) >>> times, acf = M.correlation(a) >>> >>> import matplotlib.pylab as plt >>> plt.plot(times, acf) # doctest: +SKIP
References
- NDD11
Noe, F., S. Doose, I. Daidone, M. Loellmann, J. D. Chodera, M. Sauer and J. C. Smith. 2011 Dynamical fingerprints for probing individual relaxation processes in biomolecular dynamics with simulations and kinetic experiments. Proc. Natl. Acad. Sci. USA 108, 4822-4827.
- LYP13
Lindner, B., Z. Yi, J.-H. Prinz, J. C. Smith and F. Noe. 2013. Dynamic Neutron Scattering from Conformational Dynamics I: Theory and Markov models. J. Chem. Phys. 139, 175101.
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dt_model
¶ Description of the physical time corresponding to the lag.
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eigenvalues
(k=None)¶ Compute the transition matrix eigenvalues
- Parameters
k (int) – number of eigenvalues to be returned. By default will return all available eigenvalues
- Returns
ts – transition matrix eigenvalues \(\lambda_i, i = 1, ..., k\)., sorted by descending norm.
- Return type
ndarray(k,)
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eigenvectors_left
(k=None)¶ Compute the left transition matrix eigenvectors
- Parameters
k (int) – number of eigenvectors to be returned. By default all available eigenvectors.
- Returns
L – left eigenvectors in a row matrix. l_ij is the j’th component of the i’th left eigenvector
- Return type
ndarray(k,n)
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eigenvectors_right
(k=None)¶ Compute the right transition matrix eigenvectors
- Parameters
k (int) – number of eigenvectors to be computed. By default all available eigenvectors.
- Returns
R – right eigenvectors in a column matrix. r_ij is the i’th component of the j’th right eigenvector
- Return type
ndarray(n,k)
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expectation
(a)¶ Equilibrium expectation value of a given observable.
- Parameters
a ((n,) ndarray) – Observable vector on the MSM state space
- Returns
val – Equilibrium expectation value fo the given observable
- Return type
float
Notes
The equilibrium expectation value of an observable \(a\) is defined as follows
\[\mathbb{E}_{\mu}[a] = \sum_i \pi_i a_i\]\(\pi=(\pi_i)\) is the stationary vector of the transition matrix \(P\).
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fingerprint_correlation
(a, b=None, k=None, ncv=None)¶ Dynamical fingerprint for equilibrium time-correlation experiment.
- Parameters
a ((n,) ndarray) – Observable, represented as vector on MSM state space
b ((n,) ndarray, optional) – Second observable, for cross-correlations
k (int, optional) – Number of eigenvalues and eigenvectors to use for computation. This option is only relevant for sparse matrices and long times for which an eigenvalue decomposition will be done instead of using the matrix power
ncv (int, optional) – Only relevant for sparse matrices and large lag times, where the relaxation will be computed using an eigenvalue decomposition. The number of Lanczos vectors generated, ncv must be greater than k; it is recommended that ncv > 2*k
- Returns
timescales ((k,) ndarray) – Time-scales (in units of the input trajectory time step) of the transition matrix
amplitudes ((k,) ndarray) – Amplitudes for the correlation experiment
References
Spectral densities are commonly used in spectroscopy. Dynamical fingerprints are a useful representation for computational spectroscopy results and have been introduced in [NDD11].
- NDD11
Noe, F, S Doose, I Daidone, M Loellmann, M Sauer, J D Chodera and J Smith. 2010. Dynamical fingerprints for probing individual relaxation processes in biomolecular dynamics with simulations and kinetic experiments. PNAS 108 (12): 4822-4827.
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fingerprint_relaxation
(p0, a, k=None, ncv=None)¶ Dynamical fingerprint for perturbation/relaxation experiment.
- Parameters
p0 ((n,) ndarray) – Initial distribution for a relaxation experiment
a ((n,) ndarray) – Observable, represented as vector on state space
k (int, optional) – Number of eigenvalues and eigenvectors to use for computation
ncv (int, optional) – Only relevant for sparse matrices and large lag times, where the relaxation will be computes using an eigenvalue decomposition. The number of Lanczos vectors generated, ncv must be greater than k; it is recommended that ncv > 2*k
- Returns
timescales ((k,) ndarray) – Time-scales (in units of the input trajectory time step) of the transition matrix
amplitudes ((k,) ndarray) – Amplitudes for the relaxation experiment
References
Spectral densities are commonly used in spectroscopy. Dynamical fingerprints are a useful representation for computational spectroscopy results and have been introduced in [NDD11].
- NDD11
Noe, F, S Doose, I Daidone, M Loellmann, M Sauer, J D Chodera and J Smith. 2010. Dynamical fingerprints for probing individual relaxation processes in biomolecular dynamics with simulations and kinetic experiments. PNAS 108 (12): 4822-4827.
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get_model_params
(deep=True)¶ Get parameters for this model.
- Parameters
deep (boolean, optional) – If True, will return the parameters for this estimator and contained subobjects that are estimators.
- Returns
params – Parameter names mapped to their values.
- Return type
mapping of string to any
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is_reversible
¶ Returns whether the MSM is reversible
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is_sparse
¶ Returns whether the MSM is sparse
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lifetimes
¶ Lifetimes of states of the hidden transition matrix
- Returns
l – state lifetimes in units of the input trajectory time step, defined by \(-\tau / ln \mid p_{ii} \mid, i = 1,...,nstates\), where \(p_{ii}\) are the diagonal entries of the hidden transition matrix.
- Return type
ndarray(nstates)
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classmethod
load
(file_name, model_name='default')¶ Loads a previously saved PyEMMA object from disk.
- Parameters
file_name (str or file like object (has to provide read method)) – The file like object tried to be read for a serialized object.
model_name (str, default='default') – if multiple models are contained in the file, these can be accessed by their name. Use
pyemma.list_models()
to get a representation of all stored models.
- Returns
obj
- Return type
the de-serialized object
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metastable_assignments
¶ Computes the assignment to metastable sets for observable states
Notes
This is only recommended for visualization purposes. You cannot compute any actual quantity of the coarse-grained kinetics without employing the fuzzy memberships!
- Returns
For each observable state, the metastable state it is located in.
- Return type
ndarray((n) ,dtype=int)
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metastable_distributions
¶ - Returns the output probability distributions. Identical to
- Returns
Pout – output probability matrix from hidden to observable discrete states
- Return type
ndarray (m,n)
See also
-
metastable_memberships
¶ - Computes the memberships of observable states to metastable sets by
Bayesian inversion as described in [1]_.
- Returns
M – A matrix containing the probability or membership of each observable state to be assigned to each metastable or hidden state. The row sums of M are 1.
- Return type
ndarray((n,m))
References
- 1
F. Noe, H. Wu, J.-H. Prinz and N. Plattner: Projected and hidden Markov models for calculating kinetics and metastable states of complex molecules. J. Chem. Phys. 139, 184114 (2013)
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metastable_sets
¶ - Computes the metastable sets of observable states within each
metastable set
Notes
This is only recommended for visualization purposes. You cannot compute any actual quantity of the coarse-grained kinetics without employing the fuzzy memberships!
- Returns
sets – A list of length equal to metastable states. Each element is an array with observable state indexes contained in it
- Return type
list of int-arrays
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mfpt
(A, B)¶ Mean first passage times from set A to set B, in units of the input trajectory time step
- Parameters
A (int or int array) – set of starting states
B (int or int array) – set of target states
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n_metastable
¶ Number of states chosen for PCCA++ computation.
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neig
¶ number of eigenvalues to compute.
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nstates
¶ Number of active states on which all computations and estimations are done
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observation_probabilities
¶ returns the output probability matrix
- Returns
Pout – output probability matrix from hidden to observable discrete states
- Return type
ndarray (m,n)
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pcca
(m)¶ Runs PCCA++ [1]_ to compute a metastable decomposition of MSM states
After calling this method you can access
metastable_memberships()
,metastable_distributions()
,metastable_sets()
andmetastable_assignments()
.- Parameters
m (int) – Number of metastable sets
- Returns
pcca_obj – An object containing all PCCA quantities. However, you can also ignore this return value and instead retrieve the quantities of your interest with the following MSM functions:
metastable_memberships()
,metastable_distributions()
,metastable_sets()
andmetastable_assignments()
.- Return type
Notes
If you coarse grain with PCCA++, the order of the obtained memberships might not be preserved. This also applies for
metastable_memberships()
,metastable_distributions()
,metastable_sets()
,metastable_assignments()
References
- 1
Roeblitz, S and M Weber. 2013. Fuzzy spectral clustering by PCCA+: application to Markov state models and data classification. Advances in Data Analysis and Classification 7 (2): 147-179
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pi
¶ The stationary distribution on the MSM states
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propagate
(p0, k)¶ Propagates the initial distribution p0 k times
Computes the product
\[p_k = p_0^T P^k\]If the lag time of transition matrix \(P\) is \(\tau\), this will provide the probability distribution at time \(k \tau\).
- Parameters
p0 (ndarray(n)) – Initial distribution. Vector of size of the active set.
k (int) – Number of time steps
- Returns
pk – Distribution after k steps. Vector of size of the active set.
- Return type
ndarray(n)
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relaxation
(p0, a, maxtime=None, k=None, ncv=None)¶ Simulates a perturbation-relaxation experiment.
In perturbation-relaxation experiments such as temperature-jump, pH-jump, pressure jump or rapid mixing experiments, an ensemble of molecules is initially prepared in an off-equilibrium distribution and the expectation value of some experimental observable is then followed over time as the ensemble relaxes towards equilibrium.
In order to simulate such an experiment, first determine the distribution of states at which the experiment is started, \(p_0\) and compute the mean values of your experimental observable \(a\) by MSM state:
\[a_i = \frac{1}{N_i} \sum_{x_t \in S_i} f(x_t)\]where \(S_i\) is the set of configurations belonging to MSM state \(i\) and \(f()\) is a function that computes the experimental observable of interest for configuration \(x_t\).
Then the accurate (i.e. without statistical error) time-dependent expectation value of \(f(x_t)\) given the Markov model is computed by relaxation(p0, a). This is done by evaluating the equation
\[E_a(k\tau) = \mathbf{p_0}^{\top} \mathbf{P(\tau)}^k \mathbf{a}\]where \(E\) stands for the expectation value that relaxes to its equilibrium value that is identical to expectation(a), \(\mathbf{P(\tau)}\) is the transition matrix at lag time \(\tau\), \(\boldsymbol{\pi}\) is the equilibrium distribution of \(\mathbf{P}\), and \(k\) is the time index.
Note that instead of using this method you could generate many synthetic trajectory from the MSM with starting points drawn from the initial distribution and then estimating the time-dependent expectation value by an ensemble average. However, there is no reason to do this because the present method does that calculation without any sampling, and only in the limit of an infinitely many trajectories the two results will agree exactly. The relaxation function computed by the present method still has statistical uncertainty from the fact that the underlying MSM transition matrix has statistical uncertainty when being estimated from data, but there is no additional (and unnecessary) uncertainty due to synthetic trajectory generation.
- Parameters
p0 ((n,) ndarray) – Initial distribution for a relaxation experiment
a ((n,) ndarray) – Observable, represented as vector on state space
maxtime (int or float, optional) – Maximum time (in units of the input trajectory time step) until which the correlation function will be evaluated. Internally, the correlation function can only be computed in integer multiples of the Markov model lag time, and therefore the actual last time point will be computed at \(\mathrm{ceil}(\mathrm{maxtime} / \tau)\). By default (None), the maxtime will be set equal to the 5 times the slowest relaxation time of the MSM, because after this time the signal is constant.
k (int (optional)) – Number of eigenvalues and eigenvectors to use for computation
ncv (int (optional)) – Only relevant for sparse matrices and large lag times, where the relaxation will be computes using an eigenvalue decomposition. The number of Lanczos vectors generated, ncv must be greater than k; it is recommended that ncv > 2*k
- Returns
times (ndarray (N)) – Time points (in units of the input trajectory time step) at which the relaxation has been computed
res (ndarray) – Array of expectation value at given times
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reversible
¶ Returns whether the MSM is reversible
-
sample_conf
(f, *args, **kwargs)¶ Sample confidence interval of numerical method f over all samples
Calls f(*args, **kwargs) on all samples and computes the confidence interval. Size of confidence interval is given in the construction of the SampledModel. f must return a numerical value or an ndarray.
- Parameters
f (method reference or name (str)) – Model method to be evaluated for each model sample
args (arguments) – Non-keyword arguments to be passed to the method in each call
kwargs (keyword-argments) – Keyword arguments to be passed to the method in each call
- Returns
L (float or ndarray) – lower value or array of confidence interval
R (float or ndarray) – upper value or array of confidence interval
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sample_f
(f, *args, **kwargs)¶ Evaluated method f for all samples
Calls f(*args, **kwargs) on all samples.
- Parameters
f (method reference or name (str)) – Model method to be evaluated for each model sample
args (arguments) – Non-keyword arguments to be passed to the method in each call
kwargs (keyword-argments) – Keyword arguments to be passed to the method in each call
- Returns
vals – list of results of the method calls
- Return type
list
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sample_mean
(f, *args, **kwargs)¶ Sample mean of numerical method f over all samples
Calls f(*args, **kwargs) on all samples and computes the mean. f must return a numerical value or an ndarray.
- Parameters
f (method reference or name (str)) – Model method to be evaluated for each model sample
args (arguments) – Non-keyword arguments to be passed to the method in each call
kwargs (keyword-argments) – Keyword arguments to be passed to the method in each call
- Returns
mean – mean value or mean array
- Return type
float or ndarray
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sample_std
(f, *args, **kwargs)¶ Sample standard deviation of numerical method f over all samples
Calls f(*args, **kwargs) on all samples and computes the standard deviation. f must return a numerical value or an ndarray.
- Parameters
f (method reference or name (str)) – Model method to be evaluated for each model sample
args (arguments) – Non-keyword arguments to be passed to the method in each call
kwargs (keyword-argments) – Keyword arguments to be passed to the method in each call
- Returns
std – standard deviation or array of standard deviations
- Return type
float or ndarray
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save
(file_name, model_name='default', overwrite=False, save_streaming_chain=False)¶ saves the current state of this object to given file and name.
- Parameters
file_name (str) – path to desired output file
model_name (str, default='default') – creates a group named ‘model_name’ in the given file, which will contain all of the data. If the name already exists, and overwrite is False (default) will raise a RuntimeError.
overwrite (bool, default=False) – Should overwrite existing model names?
save_streaming_chain (boolean, default=False) – if True, the data_producer(s) of this object will also be saved in the given file.
Examples
>>> import pyemma, numpy as np >>> from pyemma.util.contexts import named_temporary_file >>> m = pyemma.msm.MSM(P=np.array([[0.1, 0.9], [0.9, 0.1]]))
>>> with named_temporary_file() as file: # doctest: +SKIP ... m.save(file, 'simple') # doctest: +SKIP ... inst_restored = pyemma.load(file, 'simple') # doctest: +SKIP >>> np.testing.assert_equal(m.P, inst_restored.P) # doctest: +SKIP
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set_model_params
(samples=None, conf=0.95, P=None, pobs=None, pi=None, reversible=None, dt_model='1 step', neig=None)¶ - Parameters
samples (list of MSM objects) – sampled MSMs
conf (float, optional, default=0.68) – Confidence interval. By default one-sigma (68.3%) is used. Use 95.4% for two sigma or 99.7% for three sigma.
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simulate
(N, start=None, stop=None, dt=1)¶ Generates a realization of the Hidden Markov Model
- Parameters
N (int) – trajectory length in steps of the lag time
start (int, optional, default = None) – starting hidden state. If not given, will sample from the stationary distribution of the hidden transition matrix.
stop (int or int-array-like, optional, default = None) – stopping hidden set. If given, the trajectory will be stopped before N steps once a hidden state of the stop set is reached
dt (int) – trajectory will be saved every dt time steps. Internally, the dt’th power of P is taken to ensure a more efficient simulation.
- Returns
htraj ((N/dt, ) ndarray) – The hidden state trajectory with length N/dt
otraj ((N/dt, ) ndarray) – The observable state discrete trajectory with length N/dt
-
sparse
¶ Returns whether the MSM is sparse
-
stationary_distribution
¶ The stationary distribution on the MSM states
-
submodel
(states=None, obs=None)¶ Returns a HMM with restricted state space
- Parameters
states (None or int-array) – Hidden states to restrict the model to (if not None).
obs (None, str or int-array) – Observed states to restrict the model to (if not None).
- Returns
hmm – The restricted HMM.
- Return type
HMM
-
timescales
(k=None)¶ The relaxation timescales corresponding to the eigenvalues
- Parameters
k (int) – number of timescales to be returned. By default all available eigenvalues, minus 1.
- Returns
ts – relaxation timescales in units of the input trajectory time step, defined by \(-\tau / ln | \lambda_i |, i = 2,...,k+1\).
- Return type
ndarray(m)
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timestep_model
¶ Physical time corresponding to one transition matrix step, e.g. ‘10 ps’
-
transition_matrix
¶ The transition matrix on the active set.
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transition_matrix_obs
(k=1)¶ Computes the transition matrix between observed states
Transition matrices for longer lag times than the one used to parametrize this HMSM can be obtained by setting the k option. Note that a HMSM is not Markovian, thus we cannot compute transition matrices at longer lag times using the Chapman-Kolmogorow equality. I.e.:
\[P (k \tau) \neq P^k (\tau)\]This function computes the correct transition matrix using the metastable (coarse) transition matrix \(P_c\) as:
\[P (k \tau) = {\Pi}^-1 \chi^{\top} ({\Pi}_c) P_c^k (\tau) \chi\]where \(\chi\) is the output probability matrix, \(\Pi_c\) is a diagonal matrix with the metastable-state (coarse) stationary distribution and \(\Pi\) is a diagonal matrix with the observable-state stationary distribution.
- Parameters
k (int, optional, default=1) – Multiple of the lag time for which the By default (k=1), the transition matrix at the lag time used to construct this HMSM will be returned. If a higher power is given,
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update_model_params
(**params)¶ Update given model parameter if they are set to specific values
-