ShiftScaleTransformer

ShiftScaleTransformer#

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

Process snapshots by vector centering and/or affine 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. Transformation parameters are learned from a training data set, not provided explicitly by the user as in ShiftTransformer or ScaleTransformer.

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).

All scaling options multiply \(\Q'\) by a constant; others (symmetric scalings, 'standard' and those ending in 'sym') shift the entries of \(\Q'\) by a constant (the mean entry) as well. This is different from setting centering=True, which shifts each column of \(\Q\) by a vector; however, when centering=True symmetric scaling options are equivalent to their non-symmetric counterparts because in that case the mean of \(\Q'\) is zero.

Options:

byrowbool

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

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.

A cleaner alternative is to use a ShiftTransformer, which takes a custom shifting vector, and/or a ScaleTransformer, which takes a custom scaling. These can be joined with a TransformerPipeline.

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. None unless centering = True.

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()`.

Formula	\[\Q'' = \frac{\Q' - \mean(\Q')}{\std(\Q')}\]
`byrow=False`	\(\mean(\Q'') = 0\) and \(\std(\Q'') = 1\)
`byrow=True`	\(\mean_{j}(\Q_{i,j}'') = 0\) and \(\std_j(\Q_{i,j}'') = 1\) for each row index \(i\)

Formula	\[\Q'' = \frac{\Q'-\min(\Q')}{\max(\Q')-\min(\Q')}\]
`byrow=False`	\(\min(\Q'') = 0\) and \(\max(\Q'') = 1\)
`byrow=True`	\(\min_{j}(\Q_{i,j}'') = 0\) and \(\max_{j}(\Q_{i,j}'') = 1\) for each row index \(i\)

Formula	\[\Q'' = 2\frac{\Q'-\min(\Q')}{\max(\Q')-\min(\Q')}-1\]
`byrow=False`	\(\min(\Q'') = -1\) and \(\max(\Q'') = 1\)
`byrow=True`	\(\min_{j}(\Q_{i,j}'') = -1\) and \(\max_{j}(\Q_{i,j}'') = 1\) for each row index \(i\)

Formula	\[\Q'' = \frac{1}{\max(\text{abs}(\Q'))}\Q'\]
`byrow=False`	\(\mean(\Q'')=\frac{\mean(\Q')}{\max(\text{abs}(\Q'))}\) and \(\max(\text{abs}(\Q'')) = 1\)
`byrow=True`	\(\mean_{j}(\Q_{i,j}'') = \frac{\mean_j(\Q_{i,j}')}{\max_j(\text{abs}(\Q_{i,j}'))}\) and \(\max_{j}(\text{abs}(\Q_{i,j}'')) = 1\) for each row index \(i\)

Formula	\[\Q'' = \frac{\Q' - \mean(\Q')}{ \max(\text{abs}(\Q' - \mean(\Q')))}\]
`byrow=False`	\(\mean(\Q'')=0\) and \(\max(\text{abs}(\Q''))=1\)
`byrow=True`	\(\mean_j(\Q_{i,j}'') = 0\) and \(\max_j(\text{abs}(\Q_{i,j}'')) = 1\) for each row index \(i\)

Formula	\[\Q'' = \frac{1}{\max_j(\\|\Q'_{:,j}\\|_2)}\Q'\]
`byrow=False`	\(\mean(\Q'')=\frac{\mean(\Q')}{\max_j(\\|\Q'_{:,j}\\|)}\) and \(\max_j(\\|\Q''_{:,j}\\|) = 1\)
`byrow=True`	`ValueError`: use `'maxabs'` instead

Formula	\[\Q'' = \frac{\Q' - \text{mean}(\Q')}{ \max_j(\\|\Q'_{:,j} - \text{mean}(\Q')\\|_2)}\]
`byrow=False`	\(\mean(\Q'')=0\) and \(\max_j(\\|\Q''_{:,j}\\|) = 1\)
`byrow=True`	`ValueError`: use `'maxabssym'` instead

ShiftScaleTransformer

Contents

ShiftScaleTransformer#