msm

Markov State Modeling

This submodule contains methods related to Markov state modeling, a powerful technique for analyzing complex systems. It provides a set of functions for constructing and analyzing Markov models, including methods for calculating transition probabilities and estimating various time scales.

The submodule is structured into the following submodules:

• msm: This submodule contains all methods related to estimate the Markov state model.
• tests: This submodule holds methods for validating Markov state models.
• timescales: This submodule contains methods for estimating various timescales based on a Markov model.
• utils: This submodule provides some useful linear algebra methods.

chapman_kolmogorov_test(trajs, lagtimes, tmax)

Calculate the Chapman-Kolmogorov equation.

This method evaluates both sides of the Chapman-Kolmogorov equation

\[T(\tau n) = T^n(\tau)\;.\]

So to compare the transition probability estimated based on the lag time \(n\tau\) (referred as "MD") with the transition probability estimated based on the lag time \(\tau\) and propagated \(n\) times (referred as "MSM"), we can use the Chapman-Kolmogorov test. If the model is Markovian, both sides are identical, and the deviation indicates how Markovian the model is. The Chapman-Kolmogorov test is commonly projected onto the diagonal (so limiting to \(T_{ii}\)). For more details, see the review by Prinz et al. ¹.

The returned dictionary can be visualized using msmhelper.plot.plot_ck_test. An example can be found in the tutorial.

---

1. Prinz et al., Markov models of molecular kinetics: Generation and validation, J. Chem. Phys., 134, 174105 (2011), doi:10.1063/1.3565032 ↩

Parameters:

• trajs (StateTraj or list or ndarray or list of ndarray) –

  State trajectory/trajectories. The states should start from zero and need to be integers.

• lagtimes (list or ndarray int) –

  Lagtimes for estimating the markov model given in [frames].

• tmax (int) –

  Longest time to evaluate the CK equation given in [frames].

Returns:

• cktest ( dict ) –

  Dictionary holding for each lagtime the CK equation and with 'md' the reference.

Source code in src/msmhelper/msm/tests.py

@decorit.alias('ck_test')
def chapman_kolmogorov_test(trajs, lagtimes, tmax):
    r"""Calculate the Chapman-Kolmogorov equation.

    This method evaluates both sides of the Chapman-Kolmogorov equation

    $$T(\tau n) = T^n(\tau)\;.$$

    So to compare the transition probability estimated based on the lag time
    $n\tau$ (referred as "MD") with the transition probability estimated based
    on the lag time $\tau$ and propagated $n$ times (referred as "MSM"), we can
    use the Chapman-Kolmogorov test. If the model is Markovian, both sides are
    identical, and the deviation indicates how Markovian the model is. The
    Chapman-Kolmogorov test is commonly projected onto the diagonal (so
    limiting to $T_{ii}$). For more details, see the review by Prinz et al.
    [^1].

    The returned dictionary can be visualized using
    [msmhelper.plot.plot_ck_test][]. An example can be found in the
    [tutorial](/msmhelper/tutorials/msm/#chapman-kolmogorov-test).

    [^1]: Prinz et al., **Markov models of molecular kinetics: Generation and
        validation**, *J. Chem. Phys.*, 134, 174105 (2011),
        doi:[10.1063/1.3565032](https://doi.org/10.1063/1.3565032)

    Parameters
    ----------
    trajs : StateTraj or list or ndarray or list of ndarray
        State trajectory/trajectories. The states should start from zero and
        need to be integers.
    lagtimes : list or ndarray int
        Lagtimes for estimating the markov model given in [frames].
    tmax : int
        Longest time to evaluate the CK equation given in [frames].

    Returns
    -------
    cktest : dict
        Dictionary holding for each lagtime the CK equation and with 'md' the
        reference.

    """
    # format input
    trajs = StateTraj(trajs)
    lagtimes = np.atleast_1d(lagtimes)
    lagtimes = np.sort(lagtimes)

    # check that lag times are array of integers
    if not np.issubdtype(lagtimes.dtype, np.integer):
        raise TypeError(
            'Lagtimes needs to be integers but are {0}'.format(lagtimes.dtype),
        )
    if not (lagtimes > 0).all():
        raise TypeError('Lagtimes needs to be positive integers')

    if lagtimes.ndim != 1:
        raise TypeError(
            'Lagtimes needs to be maximal 1d, but {0}'.format(lagtimes),
        )

    if not isinstance(tmax, int) or tmax < 0:
        raise TypeError('tmax needs to be a positive integer')

    ckeqs = {}
    for lagtime in lagtimes:
        ckeqs[lagtime] = _chapman_kolmogorov_test(trajs, lagtime, tmax)
    ckeqs['md'] = _chapman_kolmogorov_test_md(
        trajs, tmin=lagtimes[0], tmax=tmax,
    )

    return ckeqs

estimate_markov_model(trajs, lagtime)

Estimates Markov State Model.

This method estimates the MSM based on the transition count matrix.

Parameters:

• trajs (StateTraj or list or ndarray or list of ndarray) –

  State trajectory/trajectories used to estimate the MSM.

• lagtime (int) –

  Lag time for estimating the markov model given in [frames].

Returns:

• T ( ndarray ) –

  Transition probability matrix \(T_{ij}\), containing the transition probability transition from state \(i o j\).

• states ( ndarray ) –

  Array holding states corresponding to the columns of \(T_{ij}\).

Source code in src/msmhelper/msm/msm.py

def estimate_markov_model(trajs, lagtime):
    """Estimates Markov State Model.

    This method estimates the MSM based on the transition count matrix.

    Parameters
    ----------
    trajs : StateTraj or list or ndarray or list of ndarray
        State trajectory/trajectories used to estimate the MSM.
    lagtime : int
        Lag time for estimating the markov model given in [frames].

    Returns
    -------
    T : ndarray
        Transition probability matrix $T_{ij}$, containing the transition
        probability transition from state $i\to j$.
    states : ndarray
        Array holding states corresponding to the columns of $T_{ij}$.

    """
    trajs = StateTraj(trajs)
    return trajs.estimate_markov_model(lagtime)

equilibrium_population(tmat, allow_non_ergodic=True)

Calculate equilibirum population.

If there are non ergodic states, their population is set to zero.

Parameters:

• tmat (ndarray) –

  Quadratic transition matrix, needs to be ergodic.

• allow_non_ergodic (bool, default: True ) –

  If True only the largest ergodic subset will be used. Otherwise it will throw an error if not ergodic.

Returns:

• peq ( ndarray ) –

  Equilibrium population of input matrix.

Source code in src/msmhelper/msm/msm.py

@decorit.alias('peq')
def equilibrium_population(tmat, allow_non_ergodic=True):
    """Calculate equilibirum population.

    If there are non ergodic states, their population is set to zero.

    Parameters
    ----------
    tmat : ndarray
        Quadratic transition matrix, needs to be ergodic.
    allow_non_ergodic : bool
        If True only the largest ergodic subset will be used. Otherwise it will
        throw an error if not ergodic.

    Returns
    -------
    peq : ndarray
        Equilibrium population of input matrix.

    """
    tmat = np.asarray(tmat)
    is_ergodic = tests.is_ergodic(tmat)
    if not allow_non_ergodic and not is_ergodic:
        raise ValueError('tmat needs to be ergodic transition matrix.')

    # calculate ev for ergodic subset
    if is_ergodic:
        _, eigenvectors = linalg.left_eigenvectors(tmat, nvals=1)
        eigenvectors = eigenvectors[0]
    else:
        mask = tests.ergodic_mask(tmat)
        _, evs_mask = linalg.left_eigenvectors(
            row_normalize_matrix(
                tmat[np.ix_(mask, mask)],
            ),
            nvals=1,
        )

        eigenvectors = np.zeros(len(tmat), dtype=tmat.dtype)
        eigenvectors[mask] = evs_mask[0]

    return eigenvectors / np.sum(eigenvectors)

row_normalize_matrix(mat)

Row normalize the given 2d matrix.

Parameters:

• mat (ndarray) –

  Matrix to be row normalized.

Returns:

• mat ( ndarray ) –

  Normalized matrix.

Source code in src/msmhelper/msm/msm.py

@numba.njit
def row_normalize_matrix(mat):
    """Row normalize the given 2d matrix.

    Parameters
    ----------
    mat : ndarray
        Matrix to be row normalized.

    Returns
    -------
    mat : ndarray
        Normalized matrix.

    """
    row_sum = np.sum(mat, axis=1)
    if not row_sum.all():
        row_sum[row_sum == 0] = 1

    # due to missing np.newaxis row_sum[:, np.newaxis] becomes # noqa: SC100
    return mat / row_sum.reshape(mat.shape[0], 1)

implied_timescales(trajs, lagtimes, ntimescales=None, reversible=False)

Calculate the implied timescales.

Calculate the implied timescales, which are defined by

\[t_i = - \frac{t_\text{lag}}{\log\lambda_i}\]

the \(i\)-th eigenvalue \(\lambda_i\).

Note

It is not checked if for higher lagtimes the dimensionality changes.

Parameters:

• trajs (StateTraj or list or ndarray or list of ndarray) –

  State trajectory/trajectories. The states should start from zero and need to be integers.

• lagtimes (list or ndarray int) –

  Lagtimes for estimating the markov model given in [frames]. This is not implemented yet!

• ntimescales (int, default: None ) –

  Number of returned lagtimes.

• reversible (bool, default: False ) –

  If reversibility should be enforced for the markov state model.

Returns:

• ts ( ndarray ) –

  Matrix containing the implied Timescales.

Source code in src/msmhelper/msm/timescales.py

def implied_timescales(trajs, lagtimes, ntimescales=None, reversible=False):
    r"""Calculate the implied timescales.

    Calculate the implied timescales, which are defined by

    $$t_i = - \frac{t_\text{lag}}{\log\lambda_i}$$

    the $i$-th eigenvalue $\lambda_i$.

    !!! note
        It is not checked if for higher lagtimes the dimensionality changes.

    Parameters
    ----------
    trajs : StateTraj or list or ndarray or list of ndarray
        State trajectory/trajectories. The states should start from zero and
        need to be integers.
    lagtimes : list or ndarray int
        Lagtimes for estimating the markov model given in [frames].
        This is not implemented yet!
    ntimescales : int, optional
        Number of returned lagtimes.
    reversible : bool
        If reversibility should be enforced for the markov state model.

    Returns
    -------
    ts : ndarray
        Matrix containing the implied Timescales.

    """
    # format input
    trajs = StateTraj(trajs)
    lagtimes = np.atleast_1d(lagtimes)

    # check that lag times are array of integers
    if not np.issubdtype(lagtimes.dtype, np.integer):
        raise TypeError(
            'Lagtimes needs to be integers but are {0}'.format(lagtimes.dtype),
        )
    if not (lagtimes > 0).all():
        raise TypeError('Lagtimes needs to be positive integers')
    if reversible:
        raise NotImplementedError(
            'Reversible matrices are not anymore supported.'
        )

    if ntimescales is None:
        ntimescales = trajs.nstates - 1

    # initialize result
    impl_timescales = np.zeros((len(lagtimes), ntimescales))

    for idx, lagtime in enumerate(lagtimes):
        transmat, _ = trajs.estimate_markov_model(lagtime)
        impl_timescales[idx] = _implied_timescales(
            transmat, lagtime, ntimescales=ntimescales,
        )

    return impl_timescales

estimate_waiting_times(*, trajs, lagtime, start, final, steps, return_list=False)

Estimates waiting times between stated states.

The stated states (from/to) will be treated as a basin. The function calculates all transitions from first entering the start-basin until first reaching the final-basin.

Parameters:

• trajs (statetraj or list or ndarray or list of ndarray) –

  State trajectory/trajectories. The states should start from zero and need to be integers.

• lagtime (int) –

  Lag time for estimating the markov model given in [frames].

• start (int or list of) –

  States to start counting.

• final (int or list of) –

  States to start counting.

• steps (int) –

  Number of MCMC propagation steps of MCMC run.

• return_list (bool, default: False ) –

  If true a list of all events is returned, else the probability density together with the edges is returned.

Returns:

• ts ( ndarray ) –

  Density probability of the time distribution. If return_list=True, return a sorted (!) list containing all times.

• edges ( ndarray ) –

  Array containing the edges corresponding to the probability, given in frames. Only for return_list=False.

Source code in src/msmhelper/msm/timescales.py

@decorit.alias('estimate_wt')
def estimate_waiting_times(
    *,
    trajs,
    lagtime,
    start,
    final,
    steps,
    return_list=False,
):
    """Estimates waiting times between stated states.

    The stated states (from/to) will be treated as a basin. The function
    calculates all transitions from first entering the start-basin until first
    reaching the final-basin.

    Parameters
    ----------
    trajs : statetraj or list or ndarray or list of ndarray
        State trajectory/trajectories. The states should start from zero and
        need to be integers.
    lagtime : int
        Lag time for estimating the markov model given in [frames].
    start : int or list of
        States to start counting.
    final : int or list of
        States to start counting.
    steps : int
        Number of MCMC propagation steps of MCMC run.
    return_list : bool
        If true a list of all events is returned, else the probability density
        together with the edges is returned.

    Returns
    -------
    ts : ndarray
        Density probability of the time distribution. If `return_list=True`,
        return a sorted (!) list containing all times.
    edges : ndarray
        Array containing the edges corresponding to the probability, given in
        frames. Only for `return_list=False`.

    """
    return _estimate_times(
        trajs=trajs,
        lagtime=lagtime,
        start=start,
        final=final,
        steps=steps,
        estimator=_estimate_waiting_times,
        return_list=return_list,
    )

estimate_waiting_time_dist(trajs, max_lagtime, start, final, steps, n_lagtimes=50)

Estimate waiting time distribution.

Parameters:

• trajs (statetraj or list or ndarray or list of ndarray) –

  State trajectory/trajectories. The states should start from zero and need to be integers.

• max_lagtime (int) –

  Maximal lag time for estimating the markov model given in [frames].

• start (int or list of) –

  States to start counting.

• final (int or list of) –

  States to start counting.

• steps (int) –

  Number of MCMC propagation steps of MCMC run.

Returns:

• wtd ( dict ) –

  Dictionary containing waiting time distribution.

Source code in src/msmhelper/msm/timescales.py

@decorit.alias('estimate_wtd')
def estimate_waiting_time_dist(
    trajs,
    max_lagtime,
    start,
    final,
    steps,
    n_lagtimes=50,
):
    """Estimate waiting time distribution.

    Parameters
    ----------
    trajs : statetraj or list or ndarray or list of ndarray
        State trajectory/trajectories. The states should start from zero and
        need to be integers.
    max_lagtime : int
        Maximal lag time for estimating the markov model given in [frames].
    start : int or list of
        States to start counting.
    final : int or list of
        States to start counting.
    steps : int
        Number of MCMC propagation steps of MCMC run.

    Returns
    -------
    wtd : dict
        Dictionary containing waiting time distribution.

    """
    lagtimes = np.unique(
        np.linspace(1, max_lagtime, num=n_lagtimes, dtype=int),
    )

    # get stats
    wtd = {
        lagtime: boxplot_stats(
            estimate_waiting_times(
                trajs=trajs,
                lagtime=lagtime,
                start=start,
                final=final,
                steps=steps,
                return_list=True,
            ),
        )[0]
        for lagtime in lagtimes
    }

    # include MD
    wtd['MD'] = boxplot_stats(
        md_estimate_wt(
            trajs=trajs,
            start=start,
            final=final,
        ),
    )
    return wtd

estimate_paths(*, trajs, lagtime, start, final, steps)

Estimates paths and waiting times between stated states.

The stated states (from/to) will be treated as a basin. The function estimates transitions from first entering the start-basin until first reaching the final-basin. The results will be listed by the corresponding pathways, where loops are removed occuring first.

Note

This function is a simple wrapper and in contrast to estimate_wt it stores the whole MCMC trajectory in memory. Hence, it memory-hungry.

Parameters:

• trajs (statetraj or list or ndarray or list of ndarray) –

  State trajectory/trajectories. The states should start from zero and need to be integers.

• lagtime (int) –

  Lag time for estimating the markov model given in [frames].

• start (int or list of) –

  States to start counting.

• final (int or list of) –

  States to start counting.

• steps (int) –

  Number of MCMC propagation steps of MCMC run.

Returns:

• paths ( dict ) –

  Dictionary containing the the paths as keys and and an array holding the times of all paths as value.

Source code in src/msmhelper/msm/timescales.py

def estimate_paths(
    *,
    trajs,
    lagtime,
    start,
    final,
    steps,
):
    """Estimates paths and waiting times between stated states.

    The stated states (from/to) will be treated as a basin. The function
    estimates transitions from first entering the start-basin until first
    reaching the final-basin. The results will be listed by the corresponding
    pathways, where loops are removed occuring first.

    !!! note
        This function is a simple wrapper and in contrast to
        [estimate_wt][msmhelper.msm.estimate_waiting_times] it stores the whole
        MCMC trajectory in memory. Hence, it memory-hungry.

    Parameters
    ----------
    trajs : statetraj or list or ndarray or list of ndarray
        State trajectory/trajectories. The states should start from zero and
        need to be integers.
    lagtime : int
        Lag time for estimating the markov model given in [frames].
    start : int or list of
        States to start counting.
    final : int or list of
        States to start counting.
    steps : int
        Number of MCMC propagation steps of MCMC run.

    Returns
    -------
    paths : dict
        Dictionary containing the the paths as keys and and an array holding
        the times of all paths as value.

    """
    # check correct input format
    trajs = StateTraj(trajs)

    states_start, states_final = np.unique(start), np.unique(final)

    if intersect(states_start, states_final):
        raise ValueError('States `start` and `final` do overlap.')

    # check that all states exist in trajectory
    for states in (states_start, states_final):
        if intersect(states, trajs.states) != len(states):
            raise ValueError(
                'Selected states does not exist in state trajectory.',
            )

    return md_estimate_paths(
        propagate_MCMC(trajs, lagtime, steps),
        start,
        final,
    )