ShiftScaleTransformer

class ShiftScaleTransformer(centering: bool = False, scaling: str = None, byrow: bool = False, name: str = None, verbose: bool = False)

Process snapshots by centering and/or scaling (in that order).

Transformations with this class are notated below as

\[\Q \mapsto \Q' ~\text{(centered)}~ \mapsto \Q'' ~\text{(centered/scaled)},\]

where \(\Q\in\RR^{n \times k}\) is the snapshot matrix to be transformed and \(\Q''\in\RR^{n \times k}\) is the transformed snapshot matrix.

All transformations with this class are affine and hence can be written componentwise as \(\Q_{i,j}'' = \alpha_{i,j} \Q_{i,j} + \beta_{i,j}\) for some choice of \(\alpha_{i,j},\beta_{i,j}\in\RR\).

Parameters

centeringbool

If True, shift the snapshots by the mean training snapshot, i.e.,

\[\Q'_{:,j} = \Q_{:,j} - \frac{1}{k}\sum_{j=0}^{k-1}\Q_{:,j}.\]

Otherwise, \(\Q' = \Q\) (default).

scalingstr or None

If given, scale (non-dimensionalize) the centered snapshot entries. Otherwise, \(\Q'' = \Q'\) (default).

Options:

• 'standard': standardize to zero mean and unit standard deviation.

  \[\Q'' = \Q' - \frac{\mean(\Q')}{\std(\Q')}.\]

  Guarantees \(\mean(\Q'') = 0\) and \(\std(\Q'') = 1\).

  If byrow=True, then \(\mean_{j}(\Q_{i,j}'') = 0\) and \(\std_j(\Q_{i,j}'') = 1\) for each row index \(i\).

• 'minmax': minmax scaling to \([0, 1]\).

  \[\Q'' = \frac{\Q' - \min(\Q')}{\max(\Q') - \min(\Q')}.\]

  Guarantees \(\min(\Q'') = 0\) and \(\max(\Q'') = 1\).

  If byrow=True, then \(\min_{j}(\Q_{i,j}'') = 0\) and \(\max_{j}(\Q_{i,j}'') = 1\) for each row index \(i\).

• 'minmaxsym': minmax scaling to \([-1, 1]\).

  \[\Q'' = 2\frac{\Q' - \min(\Q')}{\max(\Q') - \min(\Q')} - 1.\]

  Guarantees \(\min(\Q'') = -1\) and \(\max(\Q'') = 1\).

  If byrow=True, then \(\min_{j}(\Q_{i,j}'') = -1\) and \(\max_{j}(\Q_{i,j}'') = 1\) for each row index \(i\).

• 'maxabs': maximum absolute scaling to \([-1, 1]\) without scalar mean shift.

  \[\Q'' = \frac{1}{\max(\text{abs}(\Q'))}\Q'.\]

  Guarantees \(\mean(\Q'') = \frac{\mean(\Q')}{\max(\text{abs}(\Q'))}\) and \(\max(\text{abs}(\Q'')) = 1\).

  If byrow=True, then \(\mean_{j}(\Q_{i,j}'') = \frac{\mean_j(\Q_{i,j}')}{\max_j(\text{abs}(\Q_{i,j}'))}\) and \(\max_{j}(\text{abs}(\Q_{i,j}'')) = 1\) for each row index \(i\).

• 'maxabssym': maximum absolute scaling to \([-1, 1]\) with scalar mean shift.

  \[\Q'' = \frac{\Q' - \mean(\Q')}{\max(\text{abs}(\Q' - \mean(\Q')))}.\]

  Guarantees \(\mean(\Q'') = 0\) and \(\max(\text{abs}(\Q'')) = 1\).

  If byrow=True, then \(\mean_j(\Q_{i,j}'') = 0\) and \(\max_j(\text{abs}(\Q_{i,j}'')) = 1\) for each row index \(i\).

byrowbool

If True, scale each row of the snapshot matrix separately when a scaling is specified. Otherwise, scale the entire matrix at once.

verbosebool

If True, print information upon learning a transformation.

Notes

A custom shifting vector (i.e., the mean snapshot) can be specified by setting the mean_ attribute. Similarly, the scaling \(q'\mapsto q'' = \alpha q' + \beta\) can be adjusted by setting the scale_ (\(\alpha\)) and shift_ (\(\beta\)) attributes. However, calling fit() or fit_transform() will overwrite all three attributes.

Properties

byrow	If True, scale each row of the snapshot matrix separately.
centering	If True, center the snapshots by the mean training snapshot.
mean_	Mean training snapshot.
name	Label for the state variable that this transformer acts on.
scale_	Multiplicative factor of the scaling, the \(\alpha\) of \(q'' = \alpha q' + \beta\).
scaling	Type of scaling (non-dimensionalization).
shift_	Additive factor of the scaling, the \(\beta\) of \(q'' = \alpha q' + \beta\).
state_dimension	Dimension \(n\) of the state.
verbose	If True, print information upon learning a transformation.

Methods

fit	Learn (but do not apply) the transformation.
fit_transform	Learn and apply the transformation.
inverse_transform	Apply the inverse of the learned transformation.
load	Load a previously saved transformer from an HDF5 file.
save	Save the transformer to an HDF5 file.
transform	Apply the learned transformation.
transform_ddts	Apply the learned transformation to snapshot time derivatives.
verify	Verify that transform() and inverse_transform() are consistent and that transform_ddts(), if implemented, is consistent with transform().