bcam.indar.ema.AAA¶

class bcam.indar.ema.AAA(*, order: int | None = None, compute_r: bool = True, prune_tol: float = 0.0, d: bool = False, lapack_driver: str | None = None, cond: float | None = None)¶

Rational approximation using the AAA algorithm.

This is an extension of the AAA algorithm to vector-valued functions using a common-denominator representation. Since this class was created for subsequent use in spectral estimation, it is assumed that the data to be approximated is the DFT of a real signal. Hence, if the data $y$ has length $n$, then the rational function $R(z)$ satisfies

\[R\big(e^{2\pi i k / n}\big) \approx y_k,\quad\text{for } k = 0, \ldots, n-1,\]

where $y_k = y_{n-k}^*$, and $R(z) = R(z^*)^*$. Because of the symmetry of the data, instead of $\{y_k\}_{k=0}^{n-1}$ we work with $\{y_k\}_{k=0}^{N-1}$ and the parity $\sigma := n \mod 2$ so that $n = 2(N-1) + \sigma$.

This class is a sklearn estimator.

Parameters:

orderint: Number of poles of the rational function; recall that complex poles come in conjugate pairs. If order > max_order, where max_order $= \lfloor \frac{r\cdot n}{r+1} \rfloor$, and $r$ is the rank of the data, that is, $y_k \in \mathbb{C}^r$, then order is set to max_order and a warning is raised.
compute_rbool, default True: Whether to compute residues and constant term after fitting poles.
prune_tolfloat, default 0.: Tolerance for pruning poles based on the norm of their residues. If compute_r=False, then residues are computed for pruning, but not updated or stored.
dbool, default False: Whether to include a constant term in the rational function.
lapack_driverstr, default None: LAPACK driver to use for least-squares problems. Should be one of “gelsd”, “gelss”, or “gelsy”. If None, the default driver is used.
condfloat, default None: Condition number threshold for rank estimation in least-squares problems. If None, the default threshold is used.

Attributes:

poles_Poles: Poles of the fitted rational function. To access the real or complex poles, use the attributes poles_.real and poles_.cx, respectively. Only complex poles with positive imaginary part are returned.
r_HCoeffs: Residues of the fitted rational function; only computed if compute_r=True. To access the real or complex residues, use the attributes r_.real and r_.cx, respectively. Only complex residues associated with complex poles with positive imaginary part are returned.
d_np.ndarray: Constant term of the fitted rational function; only computed if compute_r=True.

Methods

`fit`(y[, parity])	Fit rational function.
`get_metadata_routing`()	Get metadata routing of this object.
`get_params`([deep])	Get parameters for this estimator.
`predict`(X)	Predict using the rational function.
`set_fit_request`(*[, parity])	Configure whether metadata should be requested to be passed to the `fit` method.
`set_params`(**params)	Set the parameters of this estimator.

Notes

A rational function $R(z) = p(z)/q(z)$ is of type $(l, m)$ if $p(z)$ is a vector-valued polynomial of degree $l$, and $q(z)$ is a scalar polynomial of degree $m$. When $l = m$, the $R$ can be written as

\[R(z) = d + \frac{p_0(z)}{q(z)},\]

where $d$ is a constant vector, and $p_0(z)$ has degree at most $m-1$; by symmetry, $d$ is real. For spectral estimation $d = 0$, however, if the parameter d is set to True, then $d$ is not forced to vanish.

In the computation of the poles, it is assumed that the (generic) condition that all the roots of $q(z)$ are simple is satisfied.

The AAA algorithm is iterative in the number of poles, and at each step it adds a new support point, so this class stores the indices of the support points for further reuse if the order is increased in a subsequent fit. This property is exploited in the construction of stabilization diagrams.

fit(y: array - like, parity: bool | None = None) → AAA¶

Fit rational function.

Parameters:

yarray-like, shape (N, n_channels): The data to fit, where for each frequency $k = 0, \ldots, N-1$, the value $y_k$ is a vector of length n_channels.
paritybool, optional: Parity of the data as explained in the class docstring. If not given, it is deduced from the imaginary part of the last frequency: if it is negligible, then parity is set to 0, otherwise it is set to 1.

Returns:

selfAAA

get_metadata_routing()¶

Get metadata routing of this object.

Please check User Guide on how the routing mechanism works.

Returns:

routingMetadataRequest: A MetadataRequest encapsulating routing information.

get_params(deep=True)¶

Get parameters for this estimator.

Parameters:

deepbool, default=True: If True, will return the parameters for this estimator and contained subobjects that are estimators.

Returns:

paramsdict: Parameter names mapped to their values.

predict(X: array - like) → ndarray¶

Predict using the rational function.

Parameters:

Xarray-like, shape (n_eval_points,): Complex points at which to evaluate the rational function.

Returns:

Rnp.ndarray, shape (n_eval_points, n_channels): Predicted values.

set_fit_request(*, parity: bool | None | str = '$UNCHANGED$') → AAA¶

Configure whether metadata should be requested to be passed to the fit method.

Note that this method is only relevant when this estimator is used as a sub-estimator within a meta-estimator and metadata routing is enabled with enable_metadata_routing=True (see sklearn.set_config()). Please check the User Guide on how the routing mechanism works.

The options for each parameter are:

True: metadata is requested, and passed to fit if provided. The request is ignored if metadata is not provided.
False: metadata is not requested and the meta-estimator will not pass it to fit.
None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.
str: metadata should be passed to the meta-estimator with this given alias instead of the original name.

The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.

Added in version 1.3.

Parameters:

paritystr, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED: Metadata routing for parity parameter in fit.

Returns:

selfobject: The updated object.

set_params(**params)¶

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as Pipeline). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each component of a nested object.

Parameters:

**paramsdict: Estimator parameters.

Returns:

selfestimator instance: Estimator instance.