Hankel DMD¶
Derived module from dmdbase.py for hankel dmd.
Reference:  H. Arbabi, I. Mezic, Ergodic theory, dynamic mode decomposition, and computation of spectral properties of the Koopman operator. SIAM Journal on Applied Dynamical Systems, 2017, 16.4: 20962126.

class
HankelDMD
(svd_rank=0, tlsq_rank=0, exact=False, opt=False, rescale_mode=None, forward_backward=False, d=1, sorted_eigs=False, reconstruction_method='first', tikhonov_regularization=None)[source] Bases:
pydmd.dmdbase.DMDBase
Hankel Dynamic Mode Decomposition
 Parameters
svd_rank (int or float) – the rank for the truncation; If 0, the method computes the optimal rank and uses it for truncation; if positive interger, the method uses the argument for the truncation; if float between 0 and 1, the rank is the number of the biggest singular values that are needed to reach the ‘energy’ specified by svd_rank; if 1, the method does not compute truncation.
tlsq_rank (int) – rank truncation computing Total Least Square. Default is 0, that means no truncation.
exact (bool) – flag to compute either exact DMD or projected DMD. Default is False.
opt (bool or int) – argument to control the computation of DMD modes amplitudes. See
DMDBase
. Default is False.rescale_mode ({'auto'} or None or numpy.ndarray) – Scale Atilde as shown in 10.1016/j.jneumeth.2015.10.010 (section 2.4) before computing its eigendecomposition. None means no rescaling, ‘auto’ means automatic rescaling using singular values, otherwise the scaling factors.
forward_backward (bool) – If True, the lowrank operator is computed like in fbDMD (reference: https://arxiv.org/abs/1507.02264). Default is False.
d (int) – the new order for spatial dimension of the input snapshots. Default is 1.
sorted_eigs ({'real', 'abs'} or False) – Sort eigenvalues (and modes/dynamics accordingly) by magnitude if sorted_eigs=’abs’, by real part (and then by imaginary part to break ties) if sorted_eigs=’real’. Default: False.
reconstruction_method ({'first', 'mean'} or arraylike) – Method used to reconstruct the snapshots of the dynamical system from the multiple versions available due to how HankelDMD is conceived. If ‘first’ (default) the first version available is selected (i.e. the nearest to the 0th row in the augmented matrix). If ‘mean’ we compute the elementwise mean. If reconstruction_method is an array of float values we compute the weighted average (for each snapshots) using the given values as weights (the number of weights must be equal to d).

_first_reconstructions
(reconstructions)[source] Return the first occurrence of each snapshot available in the given matrix (which must be the result of self._sub_dmd.reconstructed_data, or have the same shape).
 Parameters
reconstructions (np.ndarray) – A matrix of (higherorder) snapshots having shape (space*self.d, time_instants)
 Returns
The first snapshot that occurs in reconstructions for each available time instant.
 Return type
np.ndarray

_hankel_first_occurrence
(time)[source] For a given t such that there is k \in \mathbb{N} such that t = t_0 + k dt, return the index of the first column in Hankel pseudo matrix (see also
_pseudo_hankel_matrix()
) which contains the snapshot corresponding to t. Parameters
time – The time corresponding to the requested snapshot.
 Returns
The index of the first appeareance of time in the columns of Hankel pseudo matrix.
 Return type

_pseudo_hankel_matrix
(X)[source] Arrange the snapshot in the matrix X into the (pseudo) Hankel matrix. The attribute d controls the number of snapshot from X in each snapshot of the Hankel matrix.
 Example
>>> from pydmd import HankelDMD >>> import numpy as np
>>> dmd = HankelDMD(d=2) >>> a = np.array([[1, 2, 3, 4, 5]]) >>> dmd._pseudo_hankel_matrix(a) array([[1, 2, 3, 4], [2, 3, 4, 5]]) >>> dmd = HankelDMD(d=4) >>> dmd._pseudo_hankel_matrix(a) array([[1, 2], [2, 3], [3, 4], [4, 5]])
>>> dmd = HankelDMD(d=2) >>> a = np.array([1,2,3,4,5,6]).reshape(2,3) >>> a array([[1, 2, 3], [4, 5, 6]]) >>> dmd._pseudo_hankel_matrix(a) array([[1, 2], [4, 5], [2, 3], [5, 6]])

_update_sub_dmd_time
()[source] Update the time dictionaries (dmd_time and original_time) of the auxiliary DMD instance HankelDMD._sub_dmd after an update of the time dictionaries of the time dictionaries of this instance of the higher level instance of HankelDMD.

property
amplitudes
Get the coefficients that minimize the error between the original system and the reconstructed one. For futher information, see dmdbase._compute_amplitudes.
 Returns
the array that contains the amplitudes coefficient.
 Return type

property
d
The new order for spatial dimension of the input snapshots.

property
eigs
Get the eigenvalues of A tilde.
 Returns
the eigenvalues from the eigendecomposition of atilde.
 Return type

fit
(X)[source] Compute the Dynamic Modes Decomposition to the input data.
 Parameters
X (numpy.ndarray or iterable) – the input snapshots.

property
modes
Get the matrix containing the DMD modes, stored by column.
 Returns
the matrix containing the DMD modes.
 Return type

property
modes_activation_bitmask
Get the bitmask which controls which DMD modes are enabled at the moment in this DMD instance.
The DMD instance must be fitted before this property becomes valid. After
fit()
is called, the defalt value of modes_activation_bitmask is an array of True values of the same shape ofamplitudes()
.The array returned is readonly (this allow us to react appropriately to changes in the bitmask). In order to modify the bitmask you need to set the field to a brandnew value (see example below).
Example:
>>> # this is an error >>> dmd.modes_activation_bitmask[[1,2]] = False ValueError: assignment destination is readonly >>> tmp = np.array(dmd.modes_activation_bitmask) >>> tmp[[1,2]] = False >>> dmd.modes_activation_bitmask = tmp
 Returns
The DMD modes activation bitmask.
 Return type

property
operator
Get the instance of DMDOperator.
 Returns
the instance of DMDOperator
 Return type
DMDOperator

property
reconstructed_data
Get the reconstructed data.
 Returns
the matrix that contains the reconstructed snapshots.
 Return type

reconstructions_of_timeindex
(timeindex=None)[source] Build a collection of all the available versions of the given timeindex. The indexing of time instants is the same used for
reconstructed_data()
. For each time instant there are at least one and at most d versions. If timeindex is None the function returns the whole collection, for all the time instants. Parameters
timeindex (int) – The index of the time snapshot.
 Returns
a collection of all the available versions for the given time snapshot, or for all the time snapshots if timeindex is None (in the second case, time varies along the first dimension of the array returned).
 Return type
numpy.ndarray or list

property
svd_rank