pyemma.msm.BayesianHMSM

class pyemma.msm.BayesianHMSM(nstates=2, lag=1, stride='effective', p0_prior='mixed', transition_matrix_prior='mixed', nsamples=100, init_hmsm=None, reversible=True, stationary=False, connectivity='largest', mincount_connectivity='1/n', separate=None, observe_nonempty=True, dt_traj='1 step', conf=0.95, store_hidden=False, show_progress=True)

Estimator for a Bayesian Hidden Markov state model

__init__(nstates=2, lag=1, stride='effective', p0_prior='mixed', transition_matrix_prior='mixed', nsamples=100, init_hmsm=None, reversible=True, stationary=False, connectivity='largest', mincount_connectivity='1/n', separate=None, observe_nonempty=True, dt_traj='1 step', conf=0.95, store_hidden=False, show_progress=True)

Estimator for a Bayesian HMSM

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[tau], s[2 tau], ... s[stride], s[stride + tau], s[stride + 2 tau], ...

    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 = None ‘effective’ uses the largest neglected timescale as an estimate for the correlation time and sets the stride accordingly.

  • p0_prior (None, str, float or ndarray(n)) –

    Prior for the initial distribution of the HMM. Will only be active if stationary=False (stationary=True means that p0 is identical to the stationary distribution of the transition matrix). Currently implements different versions of the Dirichlet prior that is conjugate to the Dirichlet distribution of p0. p0 is sampled from: .. math:

    p0 sim prod_i (p0)_i^{a_i + n_i - 1}

    where \(n_i\) are the number of times a hidden trajectory was in state \(i\) at time step 0 and \(a_i\) is the prior count. Following options are available: | ‘mixed’ (default), \(a_i = p_{0,init}\), where \(p_{0,init}\)

    is the initial distribution of initial_model.
    ndarray(n) or float, the given array will be used as A.
    ‘uniform’, \(a_i = 1\)
    None, \(a_i = 0\). This option ensures coincidence between sample mean an MLE. Will sooner or later lead to sampling problems, because as soon as zero trajectories are drawn from a given state, the sampler cannot recover and that state will never serve as a starting state subsequently. Only recommended in the large data regime and when the probability to sample zero trajectories from any state is negligible.
  • transition_matrix_prior (str or ndarray(n, n)) –

    Prior for the HMM transition matrix. Currently implements Dirichlet priors if reversible=False and reversible transition matrix priors as described in [3] if reversible=True. For the nonreversible case the posterior of transition matrix \(P\) is: .. math:

    P sim prod_{i,j} p_{ij}^{b_{ij} + c_{ij} - 1}

    where \(c_{ij}\) are the number of transitions found for hidden trajectories and \(b_{ij}\) are prior counts. | ‘mixed’ (default), \(b_{ij} = p_{ij,init}\), where \(p_{ij,init}\)

    is the transition matrix of initial_model. That means one prior count will be used per row.
    ndarray(n, n) or broadcastable, the given array will be used as B.
    ‘uniform’, \(b_{ij} = 1\)
    None, \(b_ij = 0\). This option ensures coincidence between sample mean an MLE. Will sooner or later lead to sampling problems, because as soon as a transition \(ij\) will not occur in a sample, the sampler cannot recover and that transition will never be sampled again. This option is not recommended unless you have a small HMM and a lot of data.
  • init_hmsm (HMSM, default=None) – Single-point estimate of HMSM object around which errors will be evaluated. If None is give an initial estimate will be automatically generated using the given parameters.
  • store_hidden (bool, optional, default=False) – store hidden trajectories in sampled HMMs
  • show_progress (bool, default=True) – Show progressbars for calculation?

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)
[2]J. D. Chodera Et Al: Bayesian hidden Markov model analysis of single-molecule force spectroscopy: Characterizing kinetics under measurement uncertainty. arXiv:1108.1430 (2011)
[3]Trendelkamp-Schroer, B., H. Wu, F. Paul and F. Noe: Estimation and uncertainty of reversible Markov models. J. Chem. Phys. 143, 174101 (2015).

Methods

__init__([nstates, lag, stride, p0_prior, ...]) Estimator for a Bayesian HMSM
cktest([mlags, conf, err_est, show_progress]) 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])
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)
fingerprint_correlation(a[, b, k, ncv])
fingerprint_relaxation(p0, a[, k, ncv])
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)
propagate(p0, k) Propagates the initial distribution p0 k times
register_progress_callback(call_back[, stage]) Registers the progress reporter.
relaxation(p0, a[, maxtime, k, ncv])
sample_by_observation_probabilities(nsample) Generates samples according to given probability distributions
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
set_model_params([samples, conf, P, pobs, ...])
param samples:sampled MSMs
set_params(**params) Set the parameters of this estimator.
submodel([states, obs, mincount_connectivity])
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

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.
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
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
show_progress
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.
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, 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.
  • 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
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

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
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

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)
propagate(p0, k)

Propagates the initial distribution p0 k times

Computes the product

..1: 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)
register_progress_callback(call_back, stage=0)

Registers the progress reporter.

Parameters:
  • call_back (function) –

    This function will be called with the following arguments:

    1. stage (int)
    2. instance of pyemma.utils.progressbar.ProgressBar
    3. optional *args and named keywords (**kw), for future changes
  • stage (int, optional, default=0) – The stage you want the given call back function to be fired.
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) )
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
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
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
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
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.
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

stationary_distribution

The stationary distribution on the MSM states

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: .. math:

(w_{1,1}, ..., w_{1,T_1}), (w_{N,1}, ..., w_{N,T_N})

that are normalized to one: .. math:

\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: .. math:

\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: .. math:

\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