NARX#

class fastcan.narx.NARX(*, feat_ids=None, delay_ids=None, output_ids=None, fit_intercept=True)#

The Nonlinear Autoregressive eXogenous (NARX) model class. For example, a (polynomial) NARX model is like y(k) = y(k-1)*u(k-1) + u(k-1)^2 + u(k-2) + 1.5 where y(k) is the system output at the k-th time step, u(k) is the system input at the k-th time step, u and y is called features, u(k-1) is called a (time shift) variable, u(k-1)^2 is called a (polynomial) term, and 1.5 is called an intercept.

Parameters:

  • feat_ids (array-like of shape (n_terms, degree), default=None) – The unique id numbers of features to form polynomial terms. The id -1 stands for the constant 1. The id 0 to n_features_in_-1 are the input features. The id n_features_in_ to n_features_in_ + n_outputs_-1 are the output features. E.g., for a 2-input 1-output system, the feat_ids [[-1, 0], [-1, 1], [0, 2]] may represent the polynomial terms 1*u(k-1, 0), 1*u(k, 1), and u(k-1, 0)*y(k-2, 0).

  • delay_ids (array-like of shape (n_terms, degree), default=None) – The delays of each feature in polynomial terms. The id -1 stands for empty. The id 0 stands for 0 delay. The positive integer id k stands for k-th delay. E.g., for the polynomial terms 1*u(k-1, 0), 1*u(k, 1), and u(k-1, 0)*y(k-2, 0), the delay_ids [[-1, 1], [-1, 0], [1, 2]].

  • output_ids (array-like of shape (n_terms,), default=None) – The id numbers indicate which output the polynomial term belongs to. It is useful in multi-output case.

  • fit_intercept (bool, default=True) – Whether to fit the intercept. If set to False, intercept will be zeros.

coef_#

Estimated coefficients for the linear regression problem.

Type:

array of shape (n_terms,)

intercept_#

Independent term in the linear model.

Type:

array of shape (n_outputs_,)

n_features_in_#

Number of features seen during fit.

Type:

int

n_outputs_#

Number of outputs seen during fit.

Type:

int

feature_names_in_#

Names of features seen during fit. Defined only when X has feature names that are all strings.

Type:

ndarray of shape (n_features_in_,)

max_delay_#

The maximum time delay of the time shift variables.

Type:

int

feat_ids_#

The unique id numbers of features to form polynomial terms. The id -1 stands for the constant 1.

Type:

array-like of shape (n_terms, degree)

delay_ids_#

The delays of each feature in polynomial terms. The id -1 stands for empty.

Type:

array-like of shape (n_terms, degree)

References

  • Billings, Stephen A. (2013).

    Nonlinear System Identification: Narmax Methods in the Time, Frequency, and Spatio-Temporal Domains.

Examples

>>> import numpy as np
>>> from fastcan.narx import NARX, print_narx
>>> rng = np.random.default_rng(12345)
>>> n_samples = 1000
>>> max_delay = 3
>>> e = rng.normal(0, 0.1, n_samples)
>>> u = rng.uniform(0, 1, n_samples+max_delay) # input
>>> y = np.zeros(n_samples+max_delay) # output
>>> for i in range(max_delay, n_samples+max_delay):
...     y[i] = 0.5*y[i-1] + 0.7*u[i-2] + 1.5*u[i-1]*u[i-3] + 1
>>> y = y[max_delay:]+e
>>> X = u[max_delay:].reshape(-1, 1)
>>> feat_ids = [[-1, 1], # 1*y
...             [-1, 0], # 1*u
...             [0, 0]]  # u^2
>>> delay_ids = [[-1, 1], # 1*y(k-1)
...              [-1, 2], # 1*u(k-2)
...              [1, 3]]  # u(k-1)*u(k-3)
>>> narx = NARX(feat_ids=feat_ids,
...             delay_ids=delay_ids).fit(X, y, coef_init="one_step_ahead")
>>> print_narx(narx)
| yid |        Term        |   Coef   |
|-----|--------------------|----------|
|  0  |     Intercept      |  1.008   |
|  0  |    y_hat[k-1,0]    |  0.498   |
|  0  |      X[k-2,0]      |  0.701   |
|  0  | X[k-1,0]*X[k-3,0]  |  1.496   |
fit(X, y, sample_weight=None, coef_init=None, session_sizes=None, solver='least_squares', **params)#

Fit narx model.

Parameters:

  • X (array-like of shape (n_samples, n_features_in_) or None) – Training data.

  • y (array-like of shape (n_samples,) or (n_samples, n_outputs_)) – Target values. Will be cast to X’s dtype if necessary.

  • sample_weight (array-like of shape (n_samples,), default=None) – Individual weights for each sample.

  • coef_init (array-like of shape (n_terms,), default=None) –

    The initial values of coefficients and intercept for optimization. When coef_init is None, the model will be a One-Step-Ahead NARX. When coef_init is one_step_ahead, the model will be a Multi-Step-Ahead NARX whose coefficients and intercept are initialized by the a One-Step-Ahead NARX. When coef_init is an array, the model will be a Multi-Step-Ahead NARX whose coefficients and intercept are initialized by the array.

    Note

    When coef_init is one_step_ahead, the model will be trained as a Multi-Step-Ahead NARX, rather than a One-Step-Ahead NARX.

  • session_sizes (array-like of shape (n_sessions,), default=None) –

    The sizes of measurement sessions for time-series. The sum of session_sizes should be equal to n_samples. If None, the whole data is treated as one session.

    Added in version 0.5.0.

  • solver ({'least_squares', 'minimize'}, default='least_squares') –

    The SciPy solver for optimization.

    Added in version 0.5.1.

  • **params (dict) – Keyword arguments passed to scipy.optimize.least_squares() or scipy.optimize.minimize().

Returns:

self – Fitted Estimator.

Return type:

object

get_metadata_routing()#

Get metadata routing of this object.

Please check User Guide on how the routing mechanism works.

Returns:

routing – A MetadataRequest encapsulating routing information.

Return type:

MetadataRequest

get_params(deep=True)#

Get parameters for this estimator.

Parameters:

deep (bool, default=True) – If True, will return the parameters for this estimator and contained subobjects that are estimators.

Returns:

params – Parameter names mapped to their values.

Return type:

dict

predict(X, y_init=None)#

Predict using the linear model.

Parameters:

  • X (array-like of shape (n_samples, n_features_in_) or int) – When X is an array, it is input data. When (nonlinear) AR model is adopted, where n_features_in_ is 0, X can be an integer, which indicates the total steps to predict.

  • y_init (array-like of shape (n_init,) or (n_init, n_outputs_), default=None) – The initial values for the prediction of y. It should at least have one sample.

Returns:

y_hat – Returns predicted values. The number of dimensions is the same as that of y in fit.

Return type:

array-like of shape (n_samples,) or (n_samples, n_outputs_)

score(X, y, sample_weight=None)#

Return coefficient of determination on test data.

The coefficient of determination, \(R^2\), is defined as \((1 - \frac{u}{v})\), where \(u\) is the residual sum of squares ((y_true - y_pred)** 2).sum() and \(v\) is the total sum of squares ((y_true - y_true.mean()) ** 2).sum(). The best possible score is 1.0 and it can be negative (because the model can be arbitrarily worse). A constant model that always predicts the expected value of y, disregarding the input features, would get a \(R^2\) score of 0.0.

Parameters:

  • X (array-like of shape (n_samples, n_features)) – Test samples. For some estimators this may be a precomputed kernel matrix or a list of generic objects instead with shape (n_samples, n_samples_fitted), where n_samples_fitted is the number of samples used in the fitting for the estimator.

  • y (array-like of shape (n_samples,) or (n_samples, n_outputs)) – True values for X.

  • sample_weight (array-like of shape (n_samples,), default=None) – Sample weights.

Returns:

score – \(R^2\) of self.predict(X) w.r.t. y.

Return type:

float

Notes

The \(R^2\) score used when calling score on a regressor uses multioutput='uniform_average' from version 0.23 to keep consistent with default value of r2_score(). This influences the score method of all the multioutput regressors (except for MultiOutputRegressor).

set_fit_request(*, coef_init: bool | None | str = '$UNCHANGED$', sample_weight: bool | None | str = '$UNCHANGED$', session_sizes: bool | None | str = '$UNCHANGED$', solver: bool | None | str = '$UNCHANGED$') → NARX#

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:

  • coef_init (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for coef_init parameter in fit.

  • sample_weight (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for sample_weight parameter in fit.

  • session_sizes (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for session_sizes parameter in fit.

  • solver (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for solver parameter in fit.

Returns:

self – The updated object.

Return type:

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:

**params (dict) – Estimator parameters.

Returns:

self – Estimator instance.

Return type:

estimator instance

set_predict_request(*, y_init: bool | None | str = '$UNCHANGED$') → NARX#

Configure whether metadata should be requested to be passed to the predict 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 predict 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 predict.

  • 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:

y_init (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for y_init parameter in predict.

Returns:

self – The updated object.

Return type:

object

set_score_request(*, sample_weight: bool | None | str = '$UNCHANGED$') → NARX#

Configure whether metadata should be requested to be passed to the score 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 score 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 score.

  • 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:

sample_weight (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for sample_weight parameter in score.

Returns:

self – The updated object.

Return type:

object