# Documentation for AOLS¶

NARMAX Models using the Accelerated Orthogonal Least-Squares algorithm.

## AOLS¶

Bases: Estimators, BaseMSS

Accelerated Orthogonal Least Squares Algorithm.

Build Polynomial NARMAX model using the Accelerated Orthogonal Least-Squares ([1]_). This algorithm is based on the Matlab code available on: https://github.com/realabolfazl/AOLS/

The NARMAX model is described as:

$y_k= F^\ell[y_{k-1}, \dotsc, y_{k-n_y},x_{k-d}, x_{k-d-1}, \dotsc, x_{k-d-n_x}, e_{k-1}, \dotsc, e_{k-n_e}] + e_k$

where $$n_y\in \mathbb{N}^*$$, $$n_x \in \mathbb{N}$$, $$n_e \in \mathbb{N}$$, are the maximum lags for the system output and input respectively; $$x_k \in \mathbb{R}^{n_x}$$ is the system input and $$y_k \in \mathbb{R}^{n_y}$$ is the system output at discrete time $$k \in \mathbb{N}^n$$; $$e_k \in \mathbb{R}^{n_e}$$ stands for uncertainties and possible noise at discrete time $$k$$. In this case, $$\mathcal{F}^\ell$$ is some nonlinear function of the input and output regressors with nonlinearity degree $$\ell \in \mathbb{N}$$ and $$d$$ is a time delay typically set to $$d=1$$.

Parameters:

Name Type Description Default
ylag int

The maximum lag of the output.

2
xlag int

The maximum lag of the input.

2
k int

The sparsity level.

1
L int

Number of selected indices per iteration.

1
threshold float

The desired accuracy.

10e10

Examples:

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from sysidentpy.model_structure_selection import AOLS
>>> from sysidentpy.basis_function._basis_function import Polynomial
>>> from sysidentpy.utils.display_results import results
>>> from sysidentpy.metrics import root_relative_squared_error
>>> from sysidentpy.utils.generate_data import get_miso_data, get_siso_data
>>> x_train, x_valid, y_train, y_valid = get_siso_data(n=1000,
...                                                    colored_noise=True,
...                                                    sigma=0.2,
...                                                    train_percentage=90)
>>> basis_function = Polynomial(degree=2)
>>> model = AOLS(basis_function=basis_function,
...              ylag=2, xlag=2
...              )
>>> model.fit(x_train, y_train)
>>> yhat = model.predict(x_valid, y_valid)
>>> rrse = root_relative_squared_error(y_valid, yhat)
>>> print(rrse)
0.001993603325328823
>>> r = pd.DataFrame(
...     results(
...         model.final_model, model.theta, model.err,
...         model.n_terms, err_precision=8, dtype='sci'
...         ),
...     columns=['Regressors', 'Parameters', 'ERR'])
>>> print(r)
Regressors Parameters         ERR
0        x1(k-2)     0.9000       0.0
1         y(k-1)     0.1999       0.0
2  x1(k-1)y(k-1)     0.1000       0.0

References
Source code in sysidentpy\model_structure_selection\accelerated_orthogonal_least_squares.py
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" \n You'll have to use AOLS(estimator=LeastSquares()) instead. \n The" " only change is that you'll have to define the estimator first instead" " of passing a string like 'least_squares'. \n This change will make" " easier to implement new estimators and it'll improve code" " readability." ), ) class AOLS(Estimators, BaseMSS): r"""Accelerated Orthogonal Least Squares Algorithm. Build Polynomial NARMAX model using the Accelerated Orthogonal Least-Squares ([1]_). This algorithm is based on the Matlab code available on: https://github.com/realabolfazl/AOLS/ The NARMAX model is described as: $$y_k= F^\ell[y_{k-1}, \dotsc, y_{k-n_y},x_{k-d}, x_{k-d-1}, \dotsc, x_{k-d-n_x}, e_{k-1}, \dotsc, e_{k-n_e}] + e_k$$ where $n_y\in \mathbb{N}^*$, $n_x \in \mathbb{N}$, $n_e \in \mathbb{N}$, are the maximum lags for the system output and input respectively; $x_k \in \mathbb{R}^{n_x}$ is the system input and $y_k \in \mathbb{R}^{n_y}$ is the system output at discrete time $k \in \mathbb{N}^n$; $e_k \in \mathbb{R}^{n_e}$ stands for uncertainties and possible noise at discrete time $k$. In this case, $\mathcal{F}^\ell$ is some nonlinear function of the input and output regressors with nonlinearity degree $\ell \in \mathbb{N}$ and $d$ is a time delay typically set to $d=1$. Parameters ---------- ylag : int, default=2 The maximum lag of the output. xlag : int, default=2 The maximum lag of the input. k : int, default=1 The sparsity level. L : int, default=1 Number of selected indices per iteration. threshold : float, default=10e10 The desired accuracy. Examples -------- >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from sysidentpy.model_structure_selection import AOLS >>> from sysidentpy.basis_function._basis_function import Polynomial >>> from sysidentpy.utils.display_results import results >>> from sysidentpy.metrics import root_relative_squared_error >>> from sysidentpy.utils.generate_data import get_miso_data, get_siso_data >>> x_train, x_valid, y_train, y_valid = get_siso_data(n=1000, ... colored_noise=True, ... sigma=0.2, ... train_percentage=90) >>> basis_function = Polynomial(degree=2) >>> model = AOLS(basis_function=basis_function, ... ylag=2, xlag=2 ... ) >>> model.fit(x_train, y_train) >>> yhat = model.predict(x_valid, y_valid) >>> rrse = root_relative_squared_error(y_valid, yhat) >>> print(rrse) 0.001993603325328823 >>> r = pd.DataFrame( ... results( ... model.final_model, model.theta, model.err, ... model.n_terms, err_precision=8, dtype='sci' ... ), ... columns=['Regressors', 'Parameters', 'ERR']) >>> print(r) Regressors Parameters ERR 0 x1(k-2) 0.9000 0.0 1 y(k-1) 0.1999 0.0 2 x1(k-1)y(k-1) 0.1000 0.0 References ---------- - Manuscript: Accelerated Orthogonal Least-Squares for Large-Scale Sparse Reconstruction https://www.sciencedirect.com/science/article/abs/pii/S1051200418305311 - Code: https://github.com/realabolfazl/AOLS/ """ def __init__( self, *, ylag: Union[int, list] = 2, xlag: Union[int, list] = 2, k: int = 1, L: int = 1, threshold: float = 10e-10, model_type: str = "NARMAX", estimator: str = "least_squares", basis_function: Union[Polynomial, Fourier] = Polynomial(), lam: float = 0.98, delta: float = 0.01, offset_covariance: float = 0.2, mu: float = 0.01, eps: np.float64 = np.finfo(np.float64).eps, gama: float = 0.2, weight: float = 0.02, ): self.basis_function = basis_function self.non_degree = basis_function.degree self.model_type = model_type self.build_matrix = self.get_build_io_method(model_type) self.xlag = xlag self.ylag = ylag self.max_lag = self._get_max_lag() self.k = k self.L = L self.estimator = estimator self.threshold = threshold super().__init__( lam=lam, delta=delta, offset_covariance=offset_covariance, mu=mu, eps=eps, gama=gama, weight=weight, basis_function=basis_function, ) self.ensemble = None self.res = None self.n_inputs = None self.theta = None self.regressor_code = None self.pivv = None self.final_model = None self.n_terms = None self.err = None self._validate_params() def _validate_params(self): """Validate input params.""" if isinstance(self.ylag, int) and self.ylag < 1: raise ValueError(f"ylag must be integer and > zero. Got {self.ylag}") if isinstance(self.xlag, int) and self.xlag < 1: raise ValueError(f"xlag must be integer and > zero. Got {self.xlag}") if not isinstance(self.xlag, (int, list)): raise ValueError(f"xlag must be integer and > zero. Got {self.xlag}") if not isinstance(self.ylag, (int, list)): raise ValueError(f"ylag must be integer and > zero. Got {self.ylag}") if not isinstance(self.k, int) or self.k < 1: raise ValueError(f"k must be integer and > zero. Got {self.k}") if not isinstance(self.L, int) or self.L < 1: raise ValueError(f"k must be integer and > zero. Got {self.L}") if not isinstance(self.threshold, (int, float)) or self.threshold < 0: raise ValueError( f"threshold must be integer and > zero. Got {self.threshold}" ) def aols( self, psi: np.ndarray, y: np.ndarray ) -> Tuple[np.ndarray, np.ndarray, np.ndarray]: """Perform the Accelerated Orthogonal Least-Squares algorithm. Parameters ---------- y : array-like of shape = n_samples The target data used in the identification process. psi : ndarray of floats The information matrix of the model. Returns ------- theta : array-like of shape = number_of_model_elements The respective ERR calculated for each regressor. piv : array-like of shape = number_of_model_elements Contains the index to put the regressors in the correct order based on err values. residual_norm : float The final residual norm. References ---------- - Manuscript: Accelerated Orthogonal Least-Squares for Large-Scale Sparse Reconstruction https://www.sciencedirect.com/science/article/abs/pii/S1051200418305311 """ n, m = psi.shape theta = np.zeros([m, 1]) r = y[self.max_lag :].reshape(-1, 1).copy() it = 0 max_iter = int(min(self.k, np.floor(n / self.L))) aols_index = np.zeros(max_iter * self.L) U = np.zeros([n, max_iter * self.L]) T = psi.copy() while LA.norm(r) > self.threshold and it < max_iter: it = it + 1 temp_in = (it - 1) * self.L if it > 1: T = T - U[:, temp_in].reshape(-1, 1) @ ( U[:, temp_in].reshape(-1, 1).T @ psi ) q = ((r.T @ psi) / np.sum(psi * T, axis=0)).ravel() TT = np.sum(T**2, axis=0) * (q**2) sub_ind = list(aols_index[:temp_in].astype(int)) TT[sub_ind] = 0 sorting_indices = np.argsort(TT)[::-1].ravel() aols_index[temp_in : temp_in + self.L] = sorting_indices[: self.L] for i in range(self.L): TEMP = T[:, sorting_indices[i]].reshape(-1, 1) * q[sorting_indices[i]] U[:, temp_in + i] = (TEMP / np.linalg.norm(TEMP, axis=0)).ravel() r = r - TEMP if i == self.L: break T = T - U[:, temp_in + i].reshape(-1, 1) @ ( U[:, temp_in + i].reshape(-1, 1).T @ psi ) q = ((r.T @ psi) / np.sum(psi * T, axis=0)).ravel() aols_index = aols_index[aols_index > 0].ravel().astype(int) residual_norm = LA.norm(r) theta[aols_index] = getattr(self, self.estimator)(psi[:, aols_index], y) if self.L > 1: sorting_indices = np.argsort(np.abs(theta))[::-1] aols_index = sorting_indices[: self.k].ravel().astype(int) theta[aols_index] = getattr(self, self.estimator)(psi[:, aols_index], y) residual_norm = LA.norm( y[self.max_lag :].reshape(-1, 1) - psi[:, aols_index] @ theta[aols_index] ) pivv = np.argwhere(theta.ravel() != 0).ravel() theta = theta[theta != 0] return theta.reshape(-1, 1), pivv, residual_norm def fit(self, *, X: Optional[np.ndarray] = None, y: Optional[np.ndarray] = None): """Fit polynomial NARMAX model using AOLS algorithm. The 'fit' function allows a friendly usage by the user. Given two arguments, X and y, fit training data. Parameters ---------- X : ndarray of floats The input data to be used in the training process. y : ndarray of floats The output data to be used in the training process. Returns ------- model : ndarray of int The model code representation. piv : array-like of shape = number_of_model_elements Contains the index to put the regressors in the correct order based on err values. theta : array-like of shape = number_of_model_elements The estimated parameters of the model. err : array-like of shape = number_of_model_elements The respective ERR calculated for each regressor. info_values : array-like of shape = n_regressor Vector with values of akaike's information criterion for models with N terms (where N is the vector position + 1). """ if y is None: raise ValueError("y cannot be None") self.max_lag = self._get_max_lag() lagged_data = self.build_matrix(X, y) if self.basis_function.__class__.__name__ == "Polynomial": reg_matrix = self.basis_function.fit( lagged_data, self.max_lag, predefined_regressors=None ) else: reg_matrix, self.ensemble = self.basis_function.fit( lagged_data, self.max_lag, predefined_regressors=None ) if X is not None: self.n_inputs = _num_features(X) else: self.n_inputs = 1 # just to create the regressor space base self.regressor_code = self.regressor_space(self.n_inputs) (self.theta, self.pivv, self.res) = self.aols(reg_matrix, y) if self.basis_function.__class__.__name__ == "Polynomial": self.final_model = self.regressor_code[self.pivv, :].copy() elif self.basis_function.__class__.__name__ != "Polynomial" and self.ensemble: basis_code = np.sort( np.tile( self.regressor_code[1:, :], (self.basis_function.repetition, 1) ), axis=0, ) self.regressor_code = np.concatenate([self.regressor_code[1:], basis_code]) self.final_model = self.regressor_code[self.pivv, :].copy() else: self.regressor_code = np.sort( np.tile( self.regressor_code[1:, :], (self.basis_function.repetition, 1) ), axis=0, ) self.final_model = self.regressor_code[self.pivv, :].copy() self.n_terms = len( self.theta ) # the number of terms we selected (necessary in the 'results' methods) self.err = self.n_terms * [ 0 ] # just to use the results method. Will be changed in next update. return self def predict( self, *, X: Optional[np.ndarray] = None, y: Optional[np.ndarray] = None, steps_ahead: Optional[int] = None, forecast_horizon: int = 0, ) -> np.ndarray: """Return the predicted values given an input. The predict function allows a friendly usage by the user. Given a previously trained model, predict values given a new set of data. This method accept y values mainly for prediction n-steps ahead (to be implemented in the future) Parameters ---------- X : ndarray of floats The input data to be used in the prediction process. y : ndarray of floats The output data to be used in the prediction process. steps_ahead : int (default = None) The user can use free run simulation, one-step ahead prediction and n-step ahead prediction. forecast_horizon : int, default=None The number of predictions over the time. Returns ------- yhat : ndarray of floats The predicted values of the model. """ if self.basis_function.__class__.__name__ == "Polynomial": if steps_ahead is None: yhat = self._model_prediction(X, y, forecast_horizon=forecast_horizon) yhat = np.concatenate([y[: self.max_lag], yhat], axis=0) return yhat if steps_ahead == 1: yhat = self._one_step_ahead_prediction(X, y) yhat = np.concatenate([y[: self.max_lag], yhat], axis=0) return yhat _check_positive_int(steps_ahead, "steps_ahead") yhat = self._n_step_ahead_prediction(X, y, steps_ahead=steps_ahead) yhat = np.concatenate([y[: self.max_lag], yhat], axis=0) return yhat if steps_ahead is None: yhat = self._basis_function_predict(X, y, forecast_horizon=forecast_horizon) yhat = np.concatenate([y[: self.max_lag], yhat], axis=0) return yhat if steps_ahead == 1: yhat = self._one_step_ahead_prediction(X, y) yhat = np.concatenate([y[: self.max_lag], yhat], axis=0) return yhat yhat = self._basis_function_n_step_prediction( X, y, steps_ahead=steps_ahead, forecast_horizon=forecast_horizon ) yhat = np.concatenate([y[: self.max_lag], yhat], axis=0) return yhat def _one_step_ahead_prediction( self, X: Optional[np.ndarray], y: Optional[np.ndarray] ) -> np.ndarray: """Perform the 1-step-ahead prediction of a model. Parameters ---------- y : array-like of shape = max_lag Initial conditions values of the model to start recursive process. X : ndarray of floats of shape = n_samples Vector with input values to be used in model simulation. Returns ------- yhat : ndarray of floats The 1-step-ahead predicted values of the model. """ lagged_data = self.build_matrix(X, y) if self.basis_function.__class__.__name__ == "Polynomial": X_base = self.basis_function.transform( lagged_data, self.max_lag, predefined_regressors=self.pivv[: len(self.final_model)], ) else: X_base, _ = self.basis_function.transform( lagged_data, self.max_lag, predefined_regressors=self.pivv[: len(self.final_model)], ) yhat = super()._one_step_ahead_prediction(X_base) return yhat.reshape(-1, 1) def _n_step_ahead_prediction( self, X: Optional[np.ndarray], y: Optional[np.ndarray], steps_ahead: Optional[int], ) -> np.ndarray: """Perform the n-steps-ahead prediction of a model. Parameters ---------- y : array-like of shape = max_lag Initial conditions values of the model to start recursive process. X : ndarray of floats of shape = n_samples Vector with input values to be used in model simulation. steps_ahead : int (default = None) The user can use free run simulation, one-step ahead prediction and n-step ahead prediction. Returns ------- yhat : ndarray of floats The n-steps-ahead predicted values of the model. """ yhat = super()._n_step_ahead_prediction(X, y, steps_ahead) return yhat def _model_prediction( self, X: Optional[np.ndarray], y_initial: Optional[np.ndarray], forecast_horizon: int = 1, ) -> np.ndarray: """Perform the infinity steps-ahead simulation of a model. Parameters ---------- y_initial : array-like of shape = max_lag Number of initial conditions values of output to start recursive process. X : ndarray of floats of shape = n_samples Vector with input values to be used in model simulation. Returns ------- yhat : ndarray of floats The predicted values of the model. """ if self.model_type in ["NARMAX", "NAR"]: return self._narmax_predict(X, y_initial, forecast_horizon) if self.model_type == "NFIR": return self._nfir_predict(X, y_initial) raise ValueError( f"model_type must be NARMAX, NAR or NFIR. Got {self.model_type}" ) def _narmax_predict( self, X: Optional[np.ndarray], y_initial: Optional[np.ndarray], forecast_horizon: int = 1, ) -> np.ndarray: if len(y_initial) < self.max_lag: raise ValueError( "Insufficient initial condition elements! Expected at least" f" {self.max_lag} elements." ) if X is not None: forecast_horizon = X.shape[0] else: forecast_horizon = forecast_horizon + self.max_lag if self.model_type == "NAR": self.n_inputs = 0 y_output = super()._narmax_predict(X, y_initial, forecast_horizon) return y_output def _nfir_predict( self, X: Optional[np.ndarray], y_initial: Optional[np.ndarray] ) -> np.ndarray: y_output = super()._nfir_predict(X, y_initial) return y_output def _basis_function_predict( self, X: Optional[np.ndarray], y_initial: Optional[np.ndarray], forecast_horizon: int = 1, ) -> np.ndarray: if X is not None: forecast_horizon = X.shape[0] else: forecast_horizon = forecast_horizon + self.max_lag if self.model_type == "NAR": self.n_inputs = 0 yhat = super()._basis_function_predict(X, y_initial, forecast_horizon) return yhat.reshape(-1, 1) def _basis_function_n_step_prediction( self, X: Optional[np.ndarray], y: Optional[np.ndarray], steps_ahead: Optional[int], forecast_horizon: int, ) -> np.ndarray: """Perform the n-steps-ahead prediction of a model. Parameters ---------- y : array-like of shape = max_lag Initial conditions values of the model to start recursive process. X : ndarray of floats of shape = n_samples Vector with input values to be used in model simulation. Returns ------- yhat : ndarray of floats The n-steps-ahead predicted values of the model. """ if len(y) < self.max_lag: raise ValueError( "Insufficient initial condition elements! Expected at least" f" {self.max_lag} elements." ) if X is not None: forecast_horizon = X.shape[0] else: forecast_horizon = forecast_horizon + self.max_lag yhat = super()._basis_function_n_step_prediction( X, y, steps_ahead, forecast_horizon ) return yhat.reshape(-1, 1) def _basis_function_n_steps_horizon( self, X: Optional[np.ndarray], y: Optional[np.ndarray], steps_ahead: Optional[int], forecast_horizon: int, ) -> np.ndarray: yhat = super()._basis_function_n_steps_horizon( X, y, steps_ahead, forecast_horizon ) return yhat.reshape(-1, 1) 

### aols(psi, y)¶

Perform the Accelerated Orthogonal Least-Squares algorithm.

Parameters:

Name Type Description Default
y array-like of shape = n_samples

The target data used in the identification process.

required
psi ndarray of floats

The information matrix of the model.

required

Returns:

Name Type Description
theta array-like of shape = number_of_model_elements

The respective ERR calculated for each regressor.

piv array-like of shape = number_of_model_elements

Contains the index to put the regressors in the correct order based on err values.

residual_norm float

The final residual norm.

References
Source code in sysidentpy\model_structure_selection\accelerated_orthogonal_least_squares.py
 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 def aols( self, psi: np.ndarray, y: np.ndarray ) -> Tuple[np.ndarray, np.ndarray, np.ndarray]: """Perform the Accelerated Orthogonal Least-Squares algorithm. Parameters ---------- y : array-like of shape = n_samples The target data used in the identification process. psi : ndarray of floats The information matrix of the model. Returns ------- theta : array-like of shape = number_of_model_elements The respective ERR calculated for each regressor. piv : array-like of shape = number_of_model_elements Contains the index to put the regressors in the correct order based on err values. residual_norm : float The final residual norm. References ---------- - Manuscript: Accelerated Orthogonal Least-Squares for Large-Scale Sparse Reconstruction https://www.sciencedirect.com/science/article/abs/pii/S1051200418305311 """ n, m = psi.shape theta = np.zeros([m, 1]) r = y[self.max_lag :].reshape(-1, 1).copy() it = 0 max_iter = int(min(self.k, np.floor(n / self.L))) aols_index = np.zeros(max_iter * self.L) U = np.zeros([n, max_iter * self.L]) T = psi.copy() while LA.norm(r) > self.threshold and it < max_iter: it = it + 1 temp_in = (it - 1) * self.L if it > 1: T = T - U[:, temp_in].reshape(-1, 1) @ ( U[:, temp_in].reshape(-1, 1).T @ psi ) q = ((r.T @ psi) / np.sum(psi * T, axis=0)).ravel() TT = np.sum(T**2, axis=0) * (q**2) sub_ind = list(aols_index[:temp_in].astype(int)) TT[sub_ind] = 0 sorting_indices = np.argsort(TT)[::-1].ravel() aols_index[temp_in : temp_in + self.L] = sorting_indices[: self.L] for i in range(self.L): TEMP = T[:, sorting_indices[i]].reshape(-1, 1) * q[sorting_indices[i]] U[:, temp_in + i] = (TEMP / np.linalg.norm(TEMP, axis=0)).ravel() r = r - TEMP if i == self.L: break T = T - U[:, temp_in + i].reshape(-1, 1) @ ( U[:, temp_in + i].reshape(-1, 1).T @ psi ) q = ((r.T @ psi) / np.sum(psi * T, axis=0)).ravel() aols_index = aols_index[aols_index > 0].ravel().astype(int) residual_norm = LA.norm(r) theta[aols_index] = getattr(self, self.estimator)(psi[:, aols_index], y) if self.L > 1: sorting_indices = np.argsort(np.abs(theta))[::-1] aols_index = sorting_indices[: self.k].ravel().astype(int) theta[aols_index] = getattr(self, self.estimator)(psi[:, aols_index], y) residual_norm = LA.norm( y[self.max_lag :].reshape(-1, 1) - psi[:, aols_index] @ theta[aols_index] ) pivv = np.argwhere(theta.ravel() != 0).ravel() theta = theta[theta != 0] return theta.reshape(-1, 1), pivv, residual_norm 

### fit(*, X=None, y=None)¶

Fit polynomial NARMAX model using AOLS algorithm.

The 'fit' function allows a friendly usage by the user. Given two arguments, X and y, fit training data.

Parameters:

Name Type Description Default
X ndarray of floats

The input data to be used in the training process.

None
y ndarray of floats

The output data to be used in the training process.

None

Returns:

Name Type Description
model ndarray of int

The model code representation.

piv array-like of shape = number_of_model_elements

Contains the index to put the regressors in the correct order based on err values.

theta array-like of shape = number_of_model_elements

The estimated parameters of the model.

err array-like of shape = number_of_model_elements

The respective ERR calculated for each regressor.

info_values array-like of shape = n_regressor

Vector with values of akaike's information criterion for models with N terms (where N is the vector position + 1).

Source code in sysidentpy\model_structure_selection\accelerated_orthogonal_least_squares.py
 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 def fit(self, *, X: Optional[np.ndarray] = None, y: Optional[np.ndarray] = None): """Fit polynomial NARMAX model using AOLS algorithm. The 'fit' function allows a friendly usage by the user. Given two arguments, X and y, fit training data. Parameters ---------- X : ndarray of floats The input data to be used in the training process. y : ndarray of floats The output data to be used in the training process. Returns ------- model : ndarray of int The model code representation. piv : array-like of shape = number_of_model_elements Contains the index to put the regressors in the correct order based on err values. theta : array-like of shape = number_of_model_elements The estimated parameters of the model. err : array-like of shape = number_of_model_elements The respective ERR calculated for each regressor. info_values : array-like of shape = n_regressor Vector with values of akaike's information criterion for models with N terms (where N is the vector position + 1). """ if y is None: raise ValueError("y cannot be None") self.max_lag = self._get_max_lag() lagged_data = self.build_matrix(X, y) if self.basis_function.__class__.__name__ == "Polynomial": reg_matrix = self.basis_function.fit( lagged_data, self.max_lag, predefined_regressors=None ) else: reg_matrix, self.ensemble = self.basis_function.fit( lagged_data, self.max_lag, predefined_regressors=None ) if X is not None: self.n_inputs = _num_features(X) else: self.n_inputs = 1 # just to create the regressor space base self.regressor_code = self.regressor_space(self.n_inputs) (self.theta, self.pivv, self.res) = self.aols(reg_matrix, y) if self.basis_function.__class__.__name__ == "Polynomial": self.final_model = self.regressor_code[self.pivv, :].copy() elif self.basis_function.__class__.__name__ != "Polynomial" and self.ensemble: basis_code = np.sort( np.tile( self.regressor_code[1:, :], (self.basis_function.repetition, 1) ), axis=0, ) self.regressor_code = np.concatenate([self.regressor_code[1:], basis_code]) self.final_model = self.regressor_code[self.pivv, :].copy() else: self.regressor_code = np.sort( np.tile( self.regressor_code[1:, :], (self.basis_function.repetition, 1) ), axis=0, ) self.final_model = self.regressor_code[self.pivv, :].copy() self.n_terms = len( self.theta ) # the number of terms we selected (necessary in the 'results' methods) self.err = self.n_terms * [ 0 ] # just to use the results method. Will be changed in next update. return self 

### predict(*, X=None, y=None, steps_ahead=None, forecast_horizon=0)¶

Return the predicted values given an input.

The predict function allows a friendly usage by the user. Given a previously trained model, predict values given a new set of data.

This method accept y values mainly for prediction n-steps ahead (to be implemented in the future)

Parameters:

Name Type Description Default
X ndarray of floats

The input data to be used in the prediction process.

None
y ndarray of floats

The output data to be used in the prediction process.

None
steps_ahead int(default=None)

The user can use free run simulation, one-step ahead prediction and n-step ahead prediction.

None
forecast_horizon int

The number of predictions over the time.

None

Returns:

Name Type Description
yhat ndarray of floats

The predicted values of the model.

Source code in sysidentpy\model_structure_selection\accelerated_orthogonal_least_squares.py
 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 def predict( self, *, X: Optional[np.ndarray] = None, y: Optional[np.ndarray] = None, steps_ahead: Optional[int] = None, forecast_horizon: int = 0, ) -> np.ndarray: """Return the predicted values given an input. The predict function allows a friendly usage by the user. Given a previously trained model, predict values given a new set of data. This method accept y values mainly for prediction n-steps ahead (to be implemented in the future) Parameters ---------- X : ndarray of floats The input data to be used in the prediction process. y : ndarray of floats The output data to be used in the prediction process. steps_ahead : int (default = None) The user can use free run simulation, one-step ahead prediction and n-step ahead prediction. forecast_horizon : int, default=None The number of predictions over the time. Returns ------- yhat : ndarray of floats The predicted values of the model. """ if self.basis_function.__class__.__name__ == "Polynomial": if steps_ahead is None: yhat = self._model_prediction(X, y, forecast_horizon=forecast_horizon) yhat = np.concatenate([y[: self.max_lag], yhat], axis=0) return yhat if steps_ahead == 1: yhat = self._one_step_ahead_prediction(X, y) yhat = np.concatenate([y[: self.max_lag], yhat], axis=0) return yhat _check_positive_int(steps_ahead, "steps_ahead") yhat = self._n_step_ahead_prediction(X, y, steps_ahead=steps_ahead) yhat = np.concatenate([y[: self.max_lag], yhat], axis=0) return yhat if steps_ahead is None: yhat = self._basis_function_predict(X, y, forecast_horizon=forecast_horizon) yhat = np.concatenate([y[: self.max_lag], yhat], axis=0) return yhat if steps_ahead == 1: yhat = self._one_step_ahead_prediction(X, y) yhat = np.concatenate([y[: self.max_lag], yhat], axis=0) return yhat yhat = self._basis_function_n_step_prediction( X, y, steps_ahead=steps_ahead, forecast_horizon=forecast_horizon ) yhat = np.concatenate([y[: self.max_lag], yhat], axis=0) return yhat