pyemma.msm.MaximumLikelihoodHMSM¶

class pyemma.msm.MaximumLikelihoodHMSM(nstates=2, lag=1, stride=1, msm_init='largest-strong', reversible=True, stationary=False, connectivity=None, mincount_connectivity='1/n', observe_nonempty=True, separate=None, dt_traj='1 step', accuracy=0.001, maxit=1000)¶

Maximum likelihood estimator for a Hidden MSM given a MSM

__init__(nstates=2, lag=1, stride=1, msm_init='largest-strong', reversible=True, stationary=False, connectivity=None, mincount_connectivity='1/n', observe_nonempty=True, separate=None, dt_traj='1 step', accuracy=0.001, maxit=1000)¶

Maximum likelihood estimator for a Hidden MSM given a MSM

Parameters:

nstates (int, optional, default=2) – number of hidden states
lag (int, optional, default=1) – lagtime to estimate the HMSM at
stride (str or int, default=1) –
stride between two lagged trajectories extracted from the input trajectories. Given trajectory s[t], stride and lag will result in trajectories

s[0], s[lag], s[2 lag], ... s[stride], s[stride + lag], s[stride + 2 lag], ...

Setting stride = 1 will result in using all data (useful for maximum likelihood estimator), while a Bayesian estimator requires a longer stride in order to have statistically uncorrelated trajectories. Setting stride = ‘effective’ uses the largest neglected timescale as an estimate for the correlation time and sets the stride accordingly
msm_init (str or MSM) –
MSM object to initialize the estimation, or one of following keywords:
- ‘largest-strong’ or None (default) : Estimate MSM on the largest
  
  strongly connected set and use spectral clustering to generate an initial HMM
- ‘all’ : Estimate MSM(s) on the full state space to initialize the
  
  HMM. This estimate maybe weakly connected or disconnected.
reversible (bool, optional, default = True) – If true compute reversible MSM, else non-reversible MSM
stationary (bool, optional, default=False) – If True, the initial distribution of hidden states is self-consistently computed as the stationary distribution of the transition matrix. If False, it will be estimated from the starting states. Only set this to true if you’re sure that the observation trajectories are initiated from a global equilibrium distribution.
connectivity (str, optional, default = None) –
Defines if the resulting HMM will be defined on all hidden states or on a connected subset. Connectivity is defined by counting only transitions with at least mincount_connectivity counts. If a subset of states is used, all estimated quantities (transition matrix, stationary distribution, etc) are only defined on this subset and are correspondingly smaller than nstates. Following modes are available:
- None or ‘all’ : The active set is the full set of states. Estimation is done on all weakly connected subsets separately. The resulting transition matrix may be disconnected.
- ‘largest’ : The active set is the largest reversibly connected set.
- ‘populous’ : The active set is the reversibly connected set with most counts.
mincount_connectivity (float or '1/n') – minimum number of counts to consider a connection between two states. Counts lower than that will count zero in the connectivity check and may thus separate the resulting transition matrix. The default evaluates to 1/nstates.
separate (None or iterable of int) – Force the given set of observed states to stay in a separate hidden state. The remaining nstates-1 states will be assigned by a metastable decomposition.
observe_nonempty (bool) – If True, will restricted the observed states to the states that have at least one observation in the lagged input trajectories. If an initial MSM is given, this option is ignored and the observed subset is always identical to the active set of that MSM.
dt_traj (str, optional, default='1 step') –
Description of the physical time corresponding to the trajectory time step. May be used by analysis algorithms such as plotting tools to pretty-print the axes. By default ‘1 step’, i.e. there is no physical time unit. Specify by a number, whitespace and unit. Permitted units are (* is an arbitrary string):

‘fs’, ‘femtosecond*’

‘ps’, ‘picosecond*’

‘ns’, ‘nanosecond*’

‘us’, ‘microsecond*’

‘ms’, ‘millisecond*’

‘s’, ‘second*’
accuracy (float, optional, default = 1e-3) – convergence threshold for EM iteration. When two the likelihood does not increase by more than accuracy, the iteration is stopped successfully.
maxit (int, optional, default = 1000) – stopping criterion for EM iteration. When so many iterations are performed without reaching the requested accuracy, the iteration is stopped without convergence (a warning is given)

Methods

__init__([nstates, lag, stride, msm_init, ...]) Maximum likelihood estimator for a Hidden MSM given a MSM

cktest([mlags, conf, err_est, n_jobs, ...]) Conducts a Chapman-Kolmogorow test.

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

estimate(X, **params) Estimates the model given the data X

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.

fit(X) Estimates parameters - for compatibility with sklearn.

get_model_params([deep]) Get parameters for this model.

get_params([deep]) Get parameters for this estimator.

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_by_observation_probabilities(nsample) Generates samples according to given probability distributions

set_model_params([P, pobs, pi, reversible, ...])

param P:	coarse-grained or hidden transition matrix

set_params(**params) Set the parameters of this estimator.

simulate(N[, start, stop, dt]) Generates a realization of the Hidden Markov Model

submodel([states, obs, mincount_connectivity]) Returns a HMM with restricted state space

submodel_disconnect([mincount_connectivity]) Disconnects sets of hidden states that are barely connected

submodel_largest([strong, mincount_connectivity]) Returns the largest connected sub-HMM (convenience function)

submodel_populous([strong, ...]) Returns the most populous connected sub-HMM (convenience function)

timescales([k]) The relaxation timescales corresponding to the eigenvalues

trajectory_weights() Uses the HMSM to assign a probability weight to each trajectory frame.

transition_matrix_obs([k]) Computes the transition matrix between observed states

update_model_params(**params) Update given model parameter if they are set to specific values

Attributes

`P`	The transition matrix on the active set.
`active_set`	The active set of hidden states on which all hidden state computations are done
`discrete_trajectories_full`	A list of integer arrays with the original trajectories.
`discrete_trajectories_lagged`	Transformed original trajectories that are used as an input into the HMM estimation
`discrete_trajectories_obs`	A list of integer arrays with the discrete trajectories mapped to the observation mode used.
`dt_model`	Description of the physical time corresponding to the lag.
`dtrajs_full`	A list of integer arrays with the original trajectories.
`dtrajs_lagged`	Transformed original trajectories that are used as an input into the HMM estimation
`dtrajs_obs`	A list of integer arrays with the discrete trajectories mapped to the observation mode used.
`eigenvectors_left_obs`
`eigenvectors_right_obs`
`is_reversible`	Returns whether the MSM is reversible
`is_sparse`	Returns whether the MSM is sparse
`lagtime`	The lag time in steps
`lifetimes`	Lifetimes of states of the hidden transition matrix
`logger`	The logger for this class instance
`metastable_assignments`	Computes the assignment to metastable sets for observable states
`metastable_distributions`	Returns the output probability distributions.
`metastable_memberships`	Computes the memberships of observable states to metastable sets by Bayesian inversion as described in [1]_.
`metastable_sets`	Computes the metastable sets of observable states within each
`model`	The model estimated by this Estimator
`name`	The name of this instance
`neig`	number of eigenvalues to compute.
`nstates`	Number of active states on which all computations and estimations are done
`nstates_obs`	Number of states in discrete trajectories
`observable_set`	The active set of states on which all computations and estimations will be done
`observable_state_indexes`	Ensures that the observable states are indexed and returns the indices
`observation_probabilities`	returns the output probability matrix
`pi`	The stationary distribution on the MSM states
`reversible`	Returns whether the MSM is reversible
`sparse`	Returns whether the MSM is sparse
`stationary_distribution`	The stationary distribution on the MSM states
`stationary_distribution_obs`
`timestep_model`	Physical time corresponding to one transition matrix step, e.g.
`transition_matrix`	The transition matrix on the active set.

P¶: The transition matrix on the active set.

active_set¶: The active set of hidden states on which all hidden state computations are done

cktest(mlags=10, conf=0.95, err_est=False, n_jobs=1, show_progress=True)¶

Conducts a Chapman-Kolmogorow test.

Parameters:	mlags (int or int-array, default=10) – multiples of lag times for testing the Model, e.g. range(10). A single int will trigger a range, i.e. mlags=10 maps to mlags=range(10). The setting None will choose mlags automatically according to the longest available trajectory conf (float, optional, default = 0.95) – confidence interval err_est (bool, default=False) – compute errors also for all estimations (computationally expensive) If False, only the prediction will get error bars, which is often sufficient to validate a model. n_jobs (int, default=1) – how many jobs to use during calculation show_progress (bool, default=True) – Show progressbars for calculation?
Returns:	cktest
Return type:	`ChapmanKolmogorovValidator`

References

This is an adaption of the Chapman-Kolmogorov Test described in detail in [1]_ to Hidden MSMs as described in [2].

[1]	Prinz, J H, H Wu, M Sarich, B Keller, M Senne, M Held, J D Chodera, C Schuette and F Noe. 2011. Markov models of molecular kinetics: Generation and validation. J Chem Phys 134: 174105

[2]	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)

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

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

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)  

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.

discrete_trajectories_full¶: A list of integer arrays with the original trajectories.

discrete_trajectories_lagged¶: Transformed original trajectories that are used as an input into the HMM estimation

discrete_trajectories_obs¶: A list of integer arrays with the discrete trajectories mapped to the observation mode used. When using observe_active = True, the indexes will be given on the MSM active set. Frames that are not in the observation set will be -1. When observe_active = False, this attribute is identical to discrete_trajectories_full

dt_model¶: Description of the physical time corresponding to the lag.

dtrajs_full¶: A list of integer arrays with the original trajectories.

dtrajs_lagged¶: Transformed original trajectories that are used as an input into the HMM estimation

dtrajs_obs¶: A list of integer arrays with the discrete trajectories mapped to the observation mode used. When using observe_active = True, the indexes will be given on the MSM active set. Frames that are not in the observation set will be -1. When observe_active = False, this attribute is identical to discrete_trajectories_full

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,)

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)

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)

estimate(X, **params)¶

Estimates the model given the data X

Parameters:	X (object) – A reference to the data from which the model will be estimated params (dict) – New estimation parameter values. The parameters must that have been announced in the __init__ method of this estimator. The present settings will overwrite the settings of parameters given in the __init__ method, i.e. the parameter values after this call will be those that have been used for this estimation. Use this option if only one or a few parameters change with respect to the __init__ settings for this run, and if you don’t need to remember the original settings of these changed parameters.
Returns:	estimator – The estimated estimator with the model being available.
Return type:	object

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 \(P\).

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.

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.

fit(X)¶

Estimates parameters - for compatibility with sklearn.

Parameters:	X (object) – A reference to the data from which the model will be estimated
Returns:	estimator – The estimator (self) with estimated model.
Return type:	object

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

get_params(deep=True)¶

Get parameters for this estimator.

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

is_reversible¶: Returns whether the MSM is reversible

is_sparse¶: Returns whether the MSM is sparse

lagtime¶: The lag time in steps

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)

logger¶: The logger for this class instance

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)

metastable_distributions¶

Returns the output probability distributions. Identical to: observation_probabilities()

Returns:	Pout – output probability matrix from hidden to observable discrete states
Return type:	ndarray (m,n)

See also

observation_probabilities

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)

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

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

model¶: The model estimated by this Estimator

name¶: The name of this instance

neig¶: number of eigenvalues to compute.

nstates¶: Number of active states on which all computations and estimations are done

nstates_obs¶: Number of states in discrete trajectories

observable_set¶: The active set of states on which all computations and estimations will be done

observable_state_indexes¶: Ensures that the observable states are indexed and returns the indices

observation_probabilities¶

returns the output probability matrix

Returns:	Pout – output probability matrix from hidden to observable discrete states
Return type:	ndarray (m,n)

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() and metastable_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()` and `metastable_assignments()`.
Return type:	`PCCA`

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

pi¶: The stationary distribution on the MSM states

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)

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

reversible¶: Returns whether the MSM is reversible

sample_by_observation_probabilities(nsample)¶

Generates samples according to given probability distributions

Parameters:	distributions (list or array of ndarray ( (n) )) – m distributions over states. Each distribution must be of length n and must sum up to 1.0 nsample (int) – Number of samples per distribution. If replace = False, the number of returned samples per state could be smaller if less than nsample indexes are available for a state.
Returns:	indexes – List of the sampled indices by distribution. Each element is an index array with a number of rows equal to nsample, with rows consisting of a tuple (i, t), where i is the index of the trajectory and t is the time index within the trajectory.
Return type:	length m list of ndarray( (nsample, 2) )

set_model_params(P=None, pobs=None, pi=None, reversible=None, dt_model='1 step', neig=None)¶

Parameters:

P (ndarray(m,m)) – coarse-grained or hidden transition matrix
pobs (ndarray (m,n)) – observation probability matrix from hidden to observable discrete states
pi (ndarray(m), optional, default=None) – stationary or distribution. Can be optionally given in case if it was already computed, e.g. by the estimator.
dt_model (str, optional, default='1 step') – time step of the model
neig (int or None) – The number of eigenvalues / eigenvectors to be kept. If set to None, all eigenvalues will be used.

set_params(**params)¶: Set the parameters of this estimator. The method works on simple estimators as well as on nested objects (such as pipelines). The former have parameters of the form <component>__<parameter> so that it’s possible to update each component of a nested object. :returns: :rtype: self

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, mincount_connectivity='1/n')¶

Returns a HMM with restricted state space

Parameters:	states (None, str or int-array) – Hidden states to restrict the model to. In addition to specifying the subset, possible options are: * None : all states - don’t restrict * ‘populous-strong’ : strongly connected subset with maximum counts * ‘populous-weak’ : weakly connected subset with maximum counts * ‘largest-strong’ : strongly connected subset with maximum size * ‘largest-weak’ : weakly connected subset with maximum size obs (None, str or int-array) – Observed states to restrict the model to. In addition to specifying an array with the state labels to be observed, possible options are: * None : all states - don’t restrict * ‘nonempty’ : all states with at least one observation in the estimator mincount_connectivity (float or '1/n') – minimum number of counts to consider a connection between two states. Counts lower than that will count zero in the connectivity check and may thus separate the resulting transition matrix. Default value: 1/nstates.
Returns:	hmm – The restricted HMM.
Return type:	HMM

submodel_disconnect(mincount_connectivity='1/n')¶

Disconnects sets of hidden states that are barely connected

Runs a connectivity check excluding all transition counts below mincount_connectivity. The transition matrix and stationary distribution will be re-estimated. Note that the resulting transition matrix may have both strongly and weakly connected subsets.

Parameters:	mincount_connectivity (float or '1/n') – minimum number of counts to consider a connection between two states. Counts lower than that will count zero in the connectivity check and may thus separate the resulting transition matrix. The default evaluates to 1/nstates.
Returns:	hmm – The restricted HMM.
Return type:	HMM

submodel_largest(strong=True, mincount_connectivity='1/n')¶

Returns the largest connected sub-HMM (convenience function)

Returns:	hmm – The restricted HMM.
Return type:	HMM

submodel_populous(strong=True, mincount_connectivity='1/n')¶

Returns the most populous connected sub-HMM (convenience function)

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)

timestep_model¶: Physical time corresponding to one transition matrix step, e.g. ‘10 ps’

trajectory_weights()¶

Uses the HMSM to assign a probability weight to each trajectory frame.

This is a powerful function for the calculation of arbitrary observables in the trajectories one has started the analysis with. The stationary probability of the MSM will be used to reweigh all states. Returns a list of weight arrays, one for each trajectory, and with a number of elements equal to trajectory frames. Given \(N\) trajectories of lengths \(T_1\) to \(T_N\), this function returns corresponding weights:

\[(w_{1,1}, ..., w_{1,T_1}), (w_{N,1}, ..., w_{N,T_N})\]

that are normalized to one:

\[\sum_{i=1}^N \sum_{t=1}^{T_i} w_{i,t} = 1\]

Suppose you are interested in computing the expectation value of a function \(a(x)\), where \(x\) are your input configurations. Use this function to compute the weights of all input configurations and obtain the estimated expectation by:

\[\langle a \rangle = \sum_{i=1}^N \sum_{t=1}^{T_i} w_{i,t} a(x_{i,t})\]

Or if you are interested in computing the time-lagged correlation between functions \(a(x)\) and \(b(x)\) you could do:

\[\langle a(t) b(t+\tau) \rangle_t = \sum_{i=1}^N \sum_{t=1}^{T_i} w_{i,t} a(x_{i,t}) a(x_{i,t+\tau})\]

Returns:	The normalized trajectory weights. Given \(N\) trajectories of lengths \(T_1\) to \(T_N\), returns the corresponding weights .. math:: – (w_{1,1}, ..., w_{1,T_1}), (w_{N,1}, ..., w_{N,T_N})

transition_matrix¶: The transition matrix on the active set.

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,

update_model_params(**params)¶: Update given model parameter if they are set to specific values