def test_XX_weighted_meanconst(self):
est = OnlineCovariance(compute_c0t=False, bessel=False)
cc = est.fit(self.data - self.mean_const, weights=self.data_weights).fetch_model()
np.testing.assert_allclose(cc.mean_0, self.mx_c_wobj_lag0)
np.testing.assert_allclose(cc.cov_00, self.Mxx_c_wobj_lag0)
cc = est.fit(self.data - self.mean_const, weights=self.data_weights, column_selection=self.cols_2).fetch_model()
np.testing.assert_allclose(cc.cov_00, self.Mxx_c_wobj_lag0[:, self.cols_2])
def test_XX_with_mean(self):
# many passes
est = OnlineCovariance(compute_c0t=False, remove_data_mean=False, bessel=False)
cc = est.fit(self.data).fetch_model()
np.testing.assert_allclose(cc.mean_0, self.mx_lag0)
np.testing.assert_allclose(cc.cov_00, self.Mxx_lag0)
cc = est.fit(self.data, column_selection=self.cols_2).fetch_model()
np.testing.assert_allclose(cc.cov_00, self.Mxx_lag0[:, self.cols_2])
def test_XX_weightobj_meanfree(self):
# many passes
est = OnlineCovariance(compute_c0t=False, remove_data_mean=True, bessel=False)
cc = est.fit(self.data, weights=self.data_weights, n_splits=10).fetch_model()
np.testing.assert_allclose(cc.mean_0, self.mx_wobj_lag0)
np.testing.assert_allclose(cc.cov_00, self.Mxx0_wobj_lag0)
cc = est.fit(self.data, column_selection=self.cols_2, weights=self.data_weights).fetch_model()
np.testing.assert_allclose(cc.cov_00, self.Mxx0_wobj_lag0[:, self.cols_2])
def test_XY_sym_meanconst(self):
est = OnlineCovariance(compute_c0t=True, reversible=True, bessel=False)
cc = est.fit((self.X - self.mean_const, self.Y - self.mean_const)).fetch_model()
np.testing.assert_allclose(cc.mean_0, self.m_c_sym)
np.testing.assert_allclose(cc.cov_00, self.Mxx_c_sym)
np.testing.assert_allclose(cc.cov_0t, self.Mxy_c_sym)
cc = est.fit((self.X - self.mean_const, self.Y - self.mean_const), column_selection=self.cols_2).fetch_model()
np.testing.assert_allclose(cc.cov_00, self.Mxx_c_sym[:, self.cols_2])
np.testing.assert_allclose(cc.cov_0t, self.Mxy_c_sym[:, self.cols_2])
def test_XXXY_weightobj_sym_meanfree(self):
# many passes
est = OnlineCovariance(remove_data_mean=True, compute_c0t=True, reversible=True, bessel=False)
cc = est.fit((self.X, self.Y), weights=self.data_weights_lagged).fetch_model()
np.testing.assert_allclose(cc.mean_0, self.m_sym_wobj)
np.testing.assert_allclose(cc.cov_00, self.Mxx0_sym_wobj)
np.testing.assert_allclose(cc.cov_0t, self.Mxy0_sym_wobj)
cc = est.fit((self.X, self.Y), weights=self.data_weights_lagged, column_selection=self.cols_2).fetch_model()
np.testing.assert_allclose(cc.cov_00, self.Mxx0_sym_wobj[:, self.cols_2])
np.testing.assert_allclose(cc.cov_0t, self.Mxy0_sym_wobj[:, self.cols_2])
def test_XXXY_sym_withmean(self):
# many passes
est = OnlineCovariance(remove_data_mean=False, compute_c0t=True, reversible=True, bessel=False)
cc = est.fit((self.X, self.Y)).fetch_model()
np.testing.assert_allclose(cc.mean_0, self.m_sym)
np.testing.assert_allclose(cc.cov_00, self.Mxx_sym)
np.testing.assert_allclose(cc.cov_0t, self.Mxy_sym)
cc = est.fit((self.X, self.Y), column_selection=self.cols_2).fetch_model()
np.testing.assert_allclose(cc.cov_00, self.Mxx_sym[:, self.cols_2])
np.testing.assert_allclose(cc.cov_0t, self.Mxy_sym[:, self.cols_2])
def test_XX_meanconst(self):
est = OnlineCovariance(lagtime=self.lag,
compute_c0t=False,
bessels_correction=False)
cc = est.fit(self.data - self.mean_const).fetch_model()
np.testing.assert_allclose(cc.mean_0, self.mx_c_lag0)
np.testing.assert_allclose(cc.cov_00, self.Mxx_c_lag0)
cc = est.fit(self.data - self.mean_const,
column_selection=self.cols_2).fetch_model()
np.testing.assert_allclose(cc.cov_00, self.Mxx_c_lag0[:, self.cols_2])
def test_XXXY_meanfree(self):
# many passes
est = OnlineCovariance(remove_data_mean=True, compute_c0t=True, bessel=False)
cc = est.fit((self.X, self.Y)).fetch_model()
np.testing.assert_allclose(cc.mean_0, self.mx)
np.testing.assert_allclose(cc.mean_t, self.my)
np.testing.assert_allclose(cc.cov_00, self.Mxx0)
np.testing.assert_allclose(cc.cov_0t, self.Mxy0)
cc = est.fit((self.X, self.Y), column_selection=self.cols_2).fetch_model()
np.testing.assert_allclose(cc.cov_00, self.Mxx0[:, self.cols_2])
np.testing.assert_allclose(cc.cov_0t, self.Mxy0[:, self.cols_2])
def test_non_matching_length(self):
n = 100
data = [np.random.random(size=(n, 2)) for n in range(3)]
data = (data[:-1], data[1:])
weights = [np.random.random(n) for _ in range(3)]
weights[0] = weights[0][:-3]
with self.assertRaises(ValueError):
OnlineCovariance(compute_c0t=True).fit(data, weights=weights)
with self.assertRaises(ValueError):
OnlineCovariance(compute_c0t=True).fit(data, weights=weights[:10])
def test_XY_weighted_meanconst(self):
est = OnlineCovariance(compute_c0t=True, bessel=False)
cc = est.fit((self.X - self.mean_const, self.Y - self.mean_const),
weights=self.data_weights_lagged).fetch_model()
np.testing.assert_allclose(cc.mean_0, self.mx_c_wobj)
np.testing.assert_allclose(cc.mean_t, self.my_c_wobj)
np.testing.assert_allclose(cc.cov_00, self.Mxx_c_wobj)
np.testing.assert_allclose(cc.cov_0t, self.Mxy_c_wobj)
cc = est.fit((self.X - self.mean_const, self.Y - self.mean_const),
weights=self.data_weights_lagged, column_selection=self.cols_2).fetch_model()
np.testing.assert_allclose(cc.cov_00, self.Mxx_c_wobj[:, self.cols_2])
np.testing.assert_allclose(cc.cov_0t, self.Mxy_c_wobj[:, self.cols_2])
def test_XY_meanconst(self):
est = OnlineCovariance(lagtime=self.lag,
compute_c0t=True,
bessels_correction=False)
cc = est.fit(self.Xc_lag0).fetch_model()
np.testing.assert_allclose(cc.mean_0, self.mx_c)
np.testing.assert_allclose(cc.mean_t, self.my_c)
np.testing.assert_allclose(cc.cov_00, self.Mxx_c)
np.testing.assert_allclose(cc.cov_0t, self.Mxy_c)
cc = est.fit(self.Xc_lag0, column_selection=self.cols_2).fetch_model()
np.testing.assert_allclose(cc.cov_00, self.Mxx_c[:, self.cols_2])
np.testing.assert_allclose(cc.cov_0t, self.Mxy_c[:, self.cols_2])
def __init__(self, lagtime, epsilon=1e-6, reversible=True, dim=0.95,
scaling='kinetic_map', ncov=5):
# tica parameters
self.epsilon = epsilon
self.dim = dim
self.scaling = scaling
# online cov parameters
self.reversible = reversible
self._covar = OnlineCovariance(lagtime=lagtime, compute_c00=True, compute_c0t=True, compute_ctt=False, remove_data_mean=True,
reversible=self.reversible, bessels_correction=False, ncov=ncov)
super(TICA, self).__init__()
def test_XXXY_sym_meanfree(self):
# many passes
est = OnlineCovariance(lagtime=self.lag,
remove_data_mean=True,
compute_c0t=True,
reversible=True,
bessels_correction=False)
cc = est.fit(self.data, lagtime=self.lag).fetch_model()
np.testing.assert_allclose(cc.mean_0, self.m_sym)
np.testing.assert_allclose(cc.cov_00, self.Mxx0_sym)
np.testing.assert_allclose(cc.cov_0t, self.Mxy0_sym)
cc = est.fit(self.data, column_selection=self.cols_2).fetch_model()
np.testing.assert_allclose(cc.cov_00, self.Mxx0_sym[:, self.cols_2])
np.testing.assert_allclose(cc.cov_0t, self.Mxy0_sym[:, self.cols_2])
def test_re_estimate_weight_types(self):
# check different types are allowed and re-estimation works
x = np.random.random((100, 2))
c = OnlineCovariance(lagtime=1, compute_c0t=True)
c.fit(x, weights=np.ones((len(x), ))).fetch_model()
c.fit(x, weights=np.ones((len(x), ))).fetch_model()
c.fit(x, weights=None).fetch_model()
c.fit(x, weights=x[:, 0]).fetch_model()
def test_XXXY_weightobj_withmean(self):
# many passes
est = OnlineCovariance(lagtime=self.lag,
remove_data_mean=False,
compute_c0t=True,
bessels_correction=False)
cc = est.fit(self.data, weights=self.data_weights).fetch_model()
np.testing.assert_allclose(cc.mean_0, self.mx_wobj)
np.testing.assert_allclose(cc.mean_t, self.my_wobj)
np.testing.assert_allclose(cc.cov_00, self.Mxx_wobj)
np.testing.assert_allclose(cc.cov_0t, self.Mxy_wobj)
cc = est.fit(self.data,
weights=self.data_weights,
column_selection=self.cols_2).fetch_model()
np.testing.assert_allclose(cc.cov_00, self.Mxx_wobj[:, self.cols_2])
np.testing.assert_allclose(cc.cov_0t, self.Mxy_wobj[:, self.cols_2])
def test_XXXY_withmean(self):
# many passes
est = OnlineCovariance(lagtime=self.lag,
remove_data_mean=False,
compute_c0t=True,
bessels_correction=False)
cc = est.fit(self.data, n_splits=1).fetch_model()
assert not cc.bessels_correction
np.testing.assert_allclose(cc.mean_0, self.mx)
np.testing.assert_allclose(cc.mean_t, self.my)
np.testing.assert_allclose(cc.cov_00, self.Mxx)
np.testing.assert_allclose(cc.cov_0t, self.Mxy)
cc = est.fit(self.data, n_splits=1,
column_selection=self.cols_2).fetch_model()
np.testing.assert_allclose(cc.cov_00, self.Mxx[:, self.cols_2])
np.testing.assert_allclose(cc.cov_0t, self.Mxy[:, self.cols_2])
def test_XY_sym_weighted_meanconst(self):
est = OnlineCovariance(lagtime=self.lag,
compute_c0t=True,
reversible=True,
bessels_correction=False)
cc = est.fit(self.Xc_lag0, n_splits=1,
weights=self.data_weights).fetch_model()
np.testing.assert_allclose(cc.mean_0, self.m_c_sym_wobj)
np.testing.assert_allclose(cc.cov_00, self.Mxx_c_sym_wobj)
np.testing.assert_allclose(cc.cov_0t, self.Mxy_c_sym_wobj)
cc = est.fit(self.Xc_lag0,
weights=self.data_weights,
n_splits=1,
column_selection=self.cols_2).fetch_model()
np.testing.assert_allclose(cc.cov_00, self.Mxx_c_sym_wobj[:,
self.cols_2])
np.testing.assert_allclose(cc.cov_0t, self.Mxy_c_sym_wobj[:,
self.cols_2])
def test_XXXY_weightobj_meanfree(self):
#TODO: tests do not pass for n_splits > 1!
# many passes
est = OnlineCovariance(lagtime=self.lag,
remove_data_mean=True,
compute_c0t=True,
bessels_correction=False)
cc = est.fit(self.data, weights=self.data_weights,
n_splits=1).fetch_model()
np.testing.assert_allclose(cc.mean_0, self.mx_wobj)
np.testing.assert_allclose(cc.mean_t, self.my_wobj)
np.testing.assert_allclose(cc.cov_00, self.Mxx0_wobj)
np.testing.assert_allclose(cc.cov_0t, self.Mxy0_wobj)
cc = est.fit(self.data,
weights=self.data_weights,
column_selection=self.cols_2,
n_splits=1).fetch_model()
np.testing.assert_allclose(cc.cov_00, self.Mxx0_wobj[:, self.cols_2])
np.testing.assert_allclose(cc.cov_0t, self.Mxy0_wobj[:, self.cols_2])
def test_weights_equal_to_zero(self):
n = 1000
data = [np.random.random(size=(n, 2)) for _ in range(5)]
# create some artificial correlations
data[0][:, 0] *= np.random.randint(n)
weights = [np.ones(n, dtype=np.float32) for _ in range(5)]
# omit the first trajectory by setting a weight close to zero.
weights[0][:] = 0
weights[0][800:850] = 1
est = OnlineCovariance(compute_c0t=True)
for x, w in zip(data, weights):
est.partial_fit((x[:-3], x[3:]), w[:-3])
cov = est.fetch_model()
zeros = sum((sum(w == 0) for w in weights))
assert np.all(cov.cov_00 < 1), cov.cov_00
assert np.all(cov.cov_00 > 0), cov.cov_00
def test_weights_close_to_zero(self):
n = 1000
data = [np.random.random(size=(n, 2)) for _ in range(5)]
# create some artificial correlations
data[0][:, 0] *= np.random.randint(n)
data = np.asarray(data)
weights = [np.ones(n, dtype=np.float32) for _ in range(5)]
# omit the first trajectory by setting a weight close to zero.
weights[0][:] = 1E-44
weights = np.asarray(weights)
est = OnlineCovariance(compute_c0t=True)
for data_traj, weights_traj in zip(data, weights):
est.partial_fit((data_traj[:-3], data_traj[3:]), weights=weights_traj[:-3])
cov = est.fetch_model()
# cov = covariance_lagged(data, lag=3, weights=weights, chunksize=10)
assert np.all(cov.cov_00 < 1)
import numpy as np
from sktime.covariance.online_covariance import OnlineCovariance
if __name__ == '__main__':
data = np.random.normal(size=(500000, 10))
################################################################################################
# compute covariance matrix C00
################################################################################################
# configure estimator with estimator-global parameters
estimator = OnlineCovariance(compute_c00=True, remove_data_mean=True)
for batch in np.array_split(data, 100):
# during fit or partial fit parameters can be entered that are relevant for that batch only
estimator.partial_fit(batch, weights=None, column_selection=None)
# this finalizes the partial estimation (ie extracts means & covariance matrices from running covar)
# and returns the current model
model = estimator.fetch_model()
print(model.mean_0)
# retrieves copy of current model
model_copy = model.copy()
assert np.all(
model_copy.mean_0 == model.mean_0) and model_copy is not model
################################################################################################
# compute covariance matrix C0t
################################################################################################
class TICA(Estimator, Transformer):
r""" Time-lagged independent component analysis (TICA) [1]_, [2]_, [3]_.
Parameters
----------
lagtime : int
the time of the lag
dim : int or float, optional, default 0.95
Number of dimensions (independent components) to project onto.
* if dim is not set (None) all available ranks are kept:
`n_components == min(n_samples, n_uncorrelated_features)`
* if dim is an integer >= 1, this number specifies the number
of dimensions to keep.
* if dim is a float with ``0 < dim <= 1``, select the number
of dimensions such that the amount of kinetic variance
that needs to be explained is greater than the percentage
specified by dim.
epsilon : float
eigenvalue norm cutoff. Eigenvalues of C0 with norms <= epsilon will be
cut off. The remaining number of eigenvalues define the size of the output.
reversible: bool, default=True
symmetrize correlation matrices C_0, C_{\tau}.
scaling: str or None, default='kinetic_map'
* None: unscaled.
* 'kinetic_map': Eigenvectors will be scaled by eigenvalues. As a result, Euclidean
distances in the transformed data approximate kinetic distances [4]_.
This is a good choice when the data is further processed by clustering.
* 'commute_map': Eigenvector_i will be scaled by sqrt(timescale_i / 2). As a result,
Euclidean distances in the transformed data will approximate commute distances [5]_.
Notes
-----
Given a sequence of multivariate data :math:`X_t`, computes the mean-free
covariance and time-lagged covariance matrix:
.. math::
C_0 &= (X_t - \mu)^T (X_t - \mu) \\
C_{\tau} &= (X_t - \mu)^T (X_{t + \tau} - \mu)
and solves the eigenvalue problem
.. math:: C_{\tau} r_i = C_0 \lambda_i(tau) r_i,
where :math:`r_i` are the independent components and :math:`\lambda_i(tau)` are
their respective normalized time-autocorrelations. The eigenvalues are
related to the relaxation timescale by
.. math:: t_i(tau) = -\tau / \ln |\lambda_i|.
When used as a dimension reduction method, the input data is projected
onto the dominant independent components.
References
----------
.. [1] Perez-Hernandez G, F Paul, T Giorgino, G De Fabritiis and F Noe. 2013.
Identification of slow molecular order parameters for Markov model construction
J. Chem. Phys. 139, 015102. doi:10.1063/1.4811489
.. [2] Schwantes C, V S Pande. 2013.
Improvements in Markov State Model Construction Reveal Many Non-Native Interactions in the Folding of NTL9
J. Chem. Theory. Comput. 9, 2000-2009. doi:10.1021/ct300878a
.. [3] L. Molgedey and H. G. Schuster. 1994.
Separation of a mixture of independent signals using time delayed correlations
Phys. Rev. Lett. 72, 3634.
.. [4] Noe, F. and Clementi, C. 2015. Kinetic distance and kinetic maps from molecular dynamics simulation.
J. Chem. Theory. Comput. doi:10.1021/acs.jctc.5b00553
.. [5] Noe, F., Banisch, R., Clementi, C. 2016. Commute maps: separating slowly-mixing molecular configurations
for kinetic modeling. J. Chem. Theory. Comput. doi:10.1021/acs.jctc.6b00762
"""
def __init__(self,
lagtime: int,
epsilon=1e-6,
reversible=True,
dim=0.95,
scaling='kinetic_map',
ncov=5):
super(TICA, self).__init__()
# tica parameters
self._model.epsilon = epsilon
self._model.dim = dim
self._model.scaling = scaling
self._model.lagtime = lagtime
# online cov parameters
self.reversible = reversible
self._covar = OnlineCovariance(compute_c00=True,
compute_c0t=True,
compute_ctt=False,
remove_data_mean=True,
reversible=self.reversible,
bessel=False,
ncov=ncov)
def _create_model(self) -> TICAModel:
return TICAModel()
def transform(self, data):
r"""Projects the data onto the dominant independent components.
Parameters
----------
data : ndarray(n, m)
the input data
Returns
-------
Y : ndarray(n,)
the projected data
"""
return self.fetch_model().transform(data)
def partial_fit(self, X):
""" incrementally update the covariances and mean.
Parameters
----------
X: array, list of arrays
input data.
"""
self._covar.partial_fit(X)
return self
def fit(self, X, **kw):
self._covar.fit(X, **kw)
return self
def fetch_model(self) -> TICAModel:
covar_model = self._covar.fetch_model()
self._model.cov_00 = covar_model.cov_00
self._model.cov_0t = covar_model.cov_0t
self._model.mean_0 = covar_model.mean_0
return self._model
def __init__(self,
lagtime=1,
dim=None,
scaling=None,
right=False,
epsilon=1e-6,
ncov=float('inf')):
r""" Variational approach for Markov processes (VAMP) [1]_.
Parameters
----------
dim : float or int, default=None
Number of dimensions to keep:
* if dim is not set (None) all available ranks are kept:
`n_components == min(n_samples, n_uncorrelated_features)`
* if dim is an integer >= 1, this number specifies the number
of dimensions to keep.
* if dim is a float with ``0 < dim < 1``, select the number
of dimensions such that the amount of kinetic variance
that needs to be explained is greater than the percentage
specified by dim.
scaling : None or string
Scaling to be applied to the VAMP order parameters upon transformation
* None: no scaling will be applied, variance of the order parameters is 1
* 'kinetic map' or 'km': order parameters are scaled by singular value.
Only the left singular functions induce a kinetic map wrt the
conventional forward propagator. The right singular functions induce
a kinetic map wrt the backward propagator.
right : boolean
Whether to compute the right singular functions.
If `right==True`, `get_output()` will return the right singular
functions. Otherwise, `get_output()` will return the left singular
functions.
Beware that only `frames[tau:, :]` of each trajectory returned
by `get_output()` contain valid values of the right singular
functions. Conversely, only `frames[0:-tau, :]` of each
trajectory returned by `get_output()` contain valid values of
the left singular functions. The remaining frames might
possibly be interpreted as some extrapolation.
epsilon : float
eigenvalue cutoff. Eigenvalues of :math:`C_{00}` and :math:`C_{11}`
with norms <= epsilon will be cut off. The remaining number of
eigenvalues together with the value of `dim` define the size of the output.
ncov : int, default=infinity
limit the memory usage of the algorithm from [3]_ to an amount that corresponds
to ncov additional copies of each correlation matrix
Notes
-----
VAMP is a method for dimensionality reduction of Markov processes.
The Koopman operator :math:`\mathcal{K}` is an integral operator
that describes conditional future expectation values. Let
:math:`p(\mathbf{x},\,\mathbf{y})` be the conditional probability
density of visiting an infinitesimal phase space volume around
point :math:`\mathbf{y}` at time :math:`t+\tau` given that the phase
space point :math:`\mathbf{x}` was visited at the earlier time
:math:`t`. Then the action of the Koopman operator on a function
:math:`f` can be written as follows:
.. math::
\mathcal{K}f=\int p(\mathbf{x},\,\mathbf{y})f(\mathbf{y})\,\mathrm{dy}=\mathbb{E}\left[f(\mathbf{x}_{t+\tau}\mid\mathbf{x}_{t}=\mathbf{x})\right]
The Koopman operator is defined without any reference to an
equilibrium distribution. Therefore it is well-defined in
situations where the dynamics is irreversible or/and non-stationary
such that no equilibrium distribution exists.
If we approximate :math:`f` by a linear superposition of ansatz
functions :math:`\boldsymbol{\chi}` of the conformational
degrees of freedom (features), the operator :math:`\mathcal{K}`
can be approximated by a (finite-dimensional) matrix :math:`\mathbf{K}`.
The approximation is computed as follows: From the time-dependent
input features :math:`\boldsymbol{\chi}(t)`, we compute the mean
:math:`\boldsymbol{\mu}_{0}` (:math:`\boldsymbol{\mu}_{1}`) from
all data excluding the last (first) :math:`\tau` steps of every
trajectory as follows:
.. math::
\boldsymbol{\mu}_{0} :=\frac{1}{T-\tau}\sum_{t=0}^{T-\tau}\boldsymbol{\chi}(t)
\boldsymbol{\mu}_{1} :=\frac{1}{T-\tau}\sum_{t=\tau}^{T}\boldsymbol{\chi}(t)
Next, we compute the instantaneous covariance matrices
:math:`\mathbf{C}_{00}` and :math:`\mathbf{C}_{11}` and the
time-lagged covariance matrix :math:`\mathbf{C}_{01}` as follows:
.. math::
\mathbf{C}_{00} :=\frac{1}{T-\tau}\sum_{t=0}^{T-\tau}\left[\boldsymbol{\chi}(t)-\boldsymbol{\mu}_{0}\right]\left[\boldsymbol{\chi}(t)-\boldsymbol{\mu}_{0}\right]
\mathbf{C}_{11} :=\frac{1}{T-\tau}\sum_{t=\tau}^{T}\left[\boldsymbol{\chi}(t)-\boldsymbol{\mu}_{1}\right]\left[\boldsymbol{\chi}(t)-\boldsymbol{\mu}_{1}\right]
\mathbf{C}_{01} :=\frac{1}{T-\tau}\sum_{t=0}^{T-\tau}\left[\boldsymbol{\chi}(t)-\boldsymbol{\mu}_{0}\right]\left[\boldsymbol{\chi}(t+\tau)-\boldsymbol{\mu}_{1}\right]
The Koopman matrix is then computed as follows:
.. math::
\mathbf{K}=\mathbf{C}_{00}^{-1}\mathbf{C}_{01}
It can be shown [1]_ that the leading singular functions of the
half-weighted Koopman matrix
.. math::
\bar{\mathbf{K}}:=\mathbf{C}_{00}^{-\frac{1}{2}}\mathbf{C}_{01}\mathbf{C}_{11}^{-\frac{1}{2}}
encode the best reduced dynamical model for the time series.
The singular functions can be computed by first performing the
singular value decomposition
.. math::
\bar{\mathbf{K}}=\mathbf{U}^{\prime}\mathbf{S}\mathbf{V}^{\prime}
and then mapping the input conformation to the left singular
functions :math:`\boldsymbol{\psi}` and right singular
functions :math:`\boldsymbol{\phi}` as follows:
.. math::
\boldsymbol{\psi}(t):=\mathbf{U}^{\prime\top}\mathbf{C}_{00}^{-\frac{1}{2}}\left[\boldsymbol{\chi}(t)-\boldsymbol{\mu}_{0}\right]
\boldsymbol{\phi}(t):=\mathbf{V}^{\prime\top}\mathbf{C}_{11}^{-\frac{1}{2}}\left[\boldsymbol{\chi}(t)-\boldsymbol{\mu}_{1}\right]
References
----------
.. [1] Wu, H. and Noe, F. 2017. Variational approach for learning Markov processes from time series data.
arXiv:1707.04659v1
.. [2] Noe, F. and Clementi, C. 2015. Kinetic distance and kinetic maps from molecular dynamics simulation.
J. Chem. Theory. Comput. doi:10.1021/acs.jctc.5b00553
.. [3] Chan, T. F., Golub G. H., LeVeque R. J. 1979. Updating formulae and pairwise algorithms for
computing sample variances. Technical Report STAN-CS-79-773, Department of Computer Science, Stanford University.
"""
self.dim = dim
self.scaling = scaling
self.right = right
self.epsilon = epsilon
self.ncov = ncov
self._covar = OnlineCovariance(lagtime=lagtime,
compute_c00=True,
compute_c0t=True,
compute_ctt=True,
remove_data_mean=True,
reversible=False,
bessels_correction=False,
ncov=self.ncov)
self.lagtime = lagtime
super(VAMP, self).__init__()
