# Documentation for FROLS¶

Build Polynomial NARMAX Models using FROLS algorithm.

## FROLS¶

Bases: BaseMSS

Forward Regression Orthogonal Least Squares algorithm.

This class uses the FROLS algorithm ([1], [2]) to build NARMAX models. The NARMAX model is described as:

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

### aic(n_theta, n_samples, e_var)¶

Compute the Akaike information criteria value.

##### Parameters¶

n_theta : int Number of parameters of the model. n_samples : int Number of samples given the maximum lag. e_var : float Variance of the residues

##### Returns¶

info_criteria_value : float The computed value given the information criteria selected by the user.

Source code in sysidentpy/model_structure_selection/forward_regression_orthogonal_least_squares.py
 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 def aic(self, n_theta: int, n_samples: int, e_var: float) -> float: """Compute the Akaike information criteria value. Parameters ---------- n_theta : int Number of parameters of the model. n_samples : int Number of samples given the maximum lag. e_var : float Variance of the residues Returns ------- info_criteria_value : float The computed value given the information criteria selected by the user. """ model_factor = 2 * n_theta e_factor = n_samples * np.log(e_var) info_criteria_value = e_factor + model_factor return info_criteria_value 

### aicc(n_theta, n_samples, e_var)¶

Compute the Akaike information Criteria corrected value.

##### Parameters¶

n_theta : int Number of parameters of the model. n_samples : int Number of samples given the maximum lag. e_var : float Variance of the residues

##### Returns¶

aicc : float The computed aicc value.

##### References¶
Source code in sysidentpy/model_structure_selection/forward_regression_orthogonal_least_squares.py
 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 def aicc(self, n_theta: int, n_samples: int, e_var: float) -> float: """Compute the Akaike information Criteria corrected value. Parameters ---------- n_theta : int Number of parameters of the model. n_samples : int Number of samples given the maximum lag. e_var : float Variance of the residues Returns ------- aicc : float The computed aicc value. References ---------- - https://www.mathworks.com/help/ident/ref/idmodel.aic.html """ aic = self.aic(n_theta, n_samples, e_var) aicc = aic + (2 * n_theta * (n_theta + 1) / (n_samples - n_theta - 1)) return aicc 

### bic(n_theta, n_samples, e_var)¶

Compute the Bayesian information criteria value.

##### Parameters¶

n_theta : int Number of parameters of the model. n_samples : int Number of samples given the maximum lag. e_var : float Variance of the residues

##### Returns¶

info_criteria_value : float The computed value given the information criteria selected by the user.

Source code in sysidentpy/model_structure_selection/forward_regression_orthogonal_least_squares.py
 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 def bic(self, n_theta: int, n_samples: int, e_var: float) -> float: """Compute the Bayesian information criteria value. Parameters ---------- n_theta : int Number of parameters of the model. n_samples : int Number of samples given the maximum lag. e_var : float Variance of the residues Returns ------- info_criteria_value : float The computed value given the information criteria selected by the user. """ model_factor = n_theta * np.log(n_samples) e_factor = n_samples * np.log(e_var) info_criteria_value = e_factor + model_factor return info_criteria_value 

### error_reduction_ratio(psi, y, process_term_number)¶

Perform the Error Reduction Ration 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. process_term_number : int Number of Process Terms defined by the user.

##### Returns¶

err : 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. psi_orthogonal : ndarray of floats The updated and orthogonal information matrix.

##### References¶
• Manuscript: Orthogonal least squares methods and their application to non-linear system identification https://eprints.soton.ac.uk/251147/1/778742007_content.pdf
• Manuscript (portuguese): Identificação de Sistemas não Lineares Utilizando Modelos NARMAX Polinomiais - Uma Revisão e Novos Resultados
Source code in sysidentpy/model_structure_selection/forward_regression_orthogonal_least_squares.py
 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 def error_reduction_ratio( self, psi: np.ndarray, y: np.ndarray, process_term_number: int ) -> Tuple[np.ndarray, np.ndarray, np.ndarray]: """Perform the Error Reduction Ration 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. process_term_number : int Number of Process Terms defined by the user. Returns ------- err : 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. psi_orthogonal : ndarray of floats The updated and orthogonal information matrix. References ---------- - Manuscript: Orthogonal least squares methods and their application to non-linear system identification https://eprints.soton.ac.uk/251147/1/778742007_content.pdf - Manuscript (portuguese): Identificação de Sistemas não Lineares Utilizando Modelos NARMAX Polinomiais - Uma Revisão e Novos Resultados """ squared_y = np.dot(y[self.max_lag :].T, y[self.max_lag :]) tmp_psi = psi.copy() y = y[self.max_lag :, 0].reshape(-1, 1) tmp_y = y.copy() dimension = tmp_psi.shape[1] piv = np.arange(dimension) tmp_err = np.zeros(dimension) err = np.zeros(dimension) for i in np.arange(0, dimension): for j in np.arange(i, dimension): # Add eps in the denominator to omit division by zero if # denominator is zero # To implement regularized regression (ridge regression), add # alpha to psi.T @ psi. See S. Chen, Local regularization assisted # orthogonal least squares regression, Neurocomputing 69 (2006) 559-585. # The version implemented below uses the same regularization for every # feature, # What Chen refers to Uniform regularized orthogonal least # squares (UROLS) Set to tiny (self.eps) when you are not regularizing. # alpha = eps is the default. tmp_err[j] = ( (np.dot(tmp_psi[i:, j].T, tmp_y[i:]) ** 2) / ( (np.dot(tmp_psi[i:, j].T, tmp_psi[i:, j]) + self.alpha) * squared_y ) + self.eps )[0, 0] piv_index = np.argmax(tmp_err[i:]) + i err[i] = tmp_err[piv_index] if i == process_term_number: break if (self.err_tol is not None) and (err.cumsum()[i] >= self.err_tol): self.n_terms = i + 1 process_term_number = i + 1 break tmp_psi[:, [piv_index, i]] = tmp_psi[:, [i, piv_index]] piv[[piv_index, i]] = piv[[i, piv_index]] v = Orthogonalization().house(tmp_psi[i:, i]) row_result = Orthogonalization().rowhouse(tmp_psi[i:, i:], v) tmp_y[i:] = Orthogonalization().rowhouse(tmp_y[i:], v) tmp_psi[i:, i:] = np.copy(row_result) tmp_piv = piv[0:process_term_number] psi_orthogonal = psi[:, tmp_piv] return err, tmp_piv, psi_orthogonal 

### fit(*, X=None, y)¶

Fit polynomial NARMAX model.

This is an 'alpha' version of the 'fit' function which 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).

Source code in sysidentpy/model_structure_selection/forward_regression_orthogonal_least_squares.py
 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 def fit(self, *, X: Optional[np.ndarray] = None, y: np.ndarray): """Fit polynomial NARMAX model. This is an 'alpha' version of the 'fit' function which 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) reg_matrix = self.basis_function.fit( lagged_data, self.max_lag, self.ylag, self.xlag, self.model_type, 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) if self.order_selection is True: self.info_values = self.information_criterion(reg_matrix, y) if self.n_terms is None and self.order_selection is True: model_length = self.get_min_info_value(self.info_values) self.n_terms = model_length elif self.n_terms is None and self.order_selection is not True: raise ValueError( "If order_selection is False, you must define n_terms value." ) else: model_length = self.n_terms (self.err, self.pivv, psi) = self.error_reduction_ratio( reg_matrix, y, model_length ) tmp_piv = self.pivv[0:model_length] repetition = len(reg_matrix) if isinstance(self.basis_function, Polynomial): self.final_model = self.regressor_code[tmp_piv, :].copy() else: self.regressor_code = np.sort( np.tile(self.regressor_code[1:, :], (repetition, 1)), axis=0, ) self.final_model = self.regressor_code[tmp_piv, :].copy() self.theta = self.estimator.optimize(psi, y[self.max_lag :, 0].reshape(-1, 1)) if self.estimator.unbiased is True: self.theta = self.estimator.unbiased_estimator( psi, y[self.max_lag :, 0].reshape(-1, 1), self.theta, self.elag, self.max_lag, self.estimator, self.basis_function, self.estimator.uiter, ) return self 

### fpe(n_theta, n_samples, e_var)¶

Compute the Final Error Prediction value.

##### Parameters¶

n_theta : int Number of parameters of the model. n_samples : int Number of samples given the maximum lag. e_var : float Variance of the residues

##### Returns¶

info_criteria_value : float The computed value given the information criteria selected by the user.

Source code in sysidentpy/model_structure_selection/forward_regression_orthogonal_least_squares.py
 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 def fpe(self, n_theta: int, n_samples: int, e_var: float) -> float: """Compute the Final Error Prediction value. Parameters ---------- n_theta : int Number of parameters of the model. n_samples : int Number of samples given the maximum lag. e_var : float Variance of the residues Returns ------- info_criteria_value : float The computed value given the information criteria selected by the user. """ model_factor = n_samples * np.log((n_samples + n_theta) / (n_samples - n_theta)) e_factor = n_samples * np.log(e_var) info_criteria_value = e_factor + model_factor return info_criteria_value 

### get_info_criteria(info_criteria)¶

Get info criteria.

Source code in sysidentpy/model_structure_selection/forward_regression_orthogonal_least_squares.py
 386 387 388 389 390 391 392 393 394 395 def get_info_criteria(self, info_criteria: str): """Get info criteria.""" info_criteria_options = { "aic": self.aic, "aicc": self.aicc, "bic": self.bic, "fpe": self.fpe, "lilc": self.lilc, } return info_criteria_options.get(info_criteria) 

### get_min_info_value(info_values)¶

Find the index of the first increasing value in an array.

##### Parameters¶

info_values : array-like A sequence of numeric values to be analyzed.

##### Returns¶

int The index of the first element where the values start to increase monotonically. If no such element exists, the length of info_values is returned.

##### Notes¶
• The function assumes that info_values is a 1-dimensional array-like structure.
• The function uses np.diff to compute the difference between consecutive elements in the sequence.
• The function checks if any differences are positive, indicating an increase in value.
##### Examples¶

class MyClass: ... def init(self, values): ... self.info_values = values ... def get_min_info_value(self): ... is_monotonique = np.diff(self.info_values) > 0 ... if any(is_monotonique): ... return np.where(is_monotonique)[0][] + 1 ... return len(self.info_values) instance = MyClass([3, 2, 1, 4, 5]) instance.get_min_info_value() 3

Source code in sysidentpy/model_structure_selection/forward_regression_orthogonal_least_squares.py
 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 def get_min_info_value(self, info_values): """Find the index of the first increasing value in an array. Parameters ---------- info_values : array-like A sequence of numeric values to be analyzed. Returns ------- int The index of the first element where the values start to increase monotonically. If no such element exists, the length of info_values is returned. Notes ----- - The function assumes that info_values is a 1-dimensional array-like structure. - The function uses np.diff to compute the difference between consecutive elements in the sequence. - The function checks if any differences are positive, indicating an increase in value. Examples -------- >>> class MyClass: ... def __init__(self, values): ... self.info_values = values ... def get_min_info_value(self): ... is_monotonique = np.diff(self.info_values) > 0 ... if any(is_monotonique): ... return np.where(is_monotonique)[0][0] + 1 ... return len(self.info_values) >>> instance = MyClass([3, 2, 1, 4, 5]) >>> instance.get_min_info_value() 3 """ is_monotonique = np.diff(info_values) > 0 if any(is_monotonique): return np.where(is_monotonique)[0][0] + 1 return len(info_values) 

### information_criterion(X, y)¶

Determine the model order.

This function uses a information criterion to determine the model size. 'Akaike'- Akaike's Information Criterion with critical value 2 (AIC) (default). 'Bayes' - Bayes Information Criterion (BIC). 'FPE' - Final Prediction Error (FPE). 'LILC' - Khundrin's law ofiterated logarithm criterion (LILC).

##### Parameters¶

y : array-like of shape = n_samples Target values of the system. X : array-like of shape = n_samples Input system values measured by the user.

##### Returns¶

output_vector : 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/forward_regression_orthogonal_least_squares.py
 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 def information_criterion(self, X: np.ndarray, y: np.ndarray) -> np.ndarray: """Determine the model order. This function uses a information criterion to determine the model size. 'Akaike'- Akaike's Information Criterion with critical value 2 (AIC) (default). 'Bayes' - Bayes Information Criterion (BIC). 'FPE' - Final Prediction Error (FPE). 'LILC' - Khundrin's law ofiterated logarithm criterion (LILC). Parameters ---------- y : array-like of shape = n_samples Target values of the system. X : array-like of shape = n_samples Input system values measured by the user. Returns ------- output_vector : 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 self.n_info_values is not None and self.n_info_values > X.shape[1]: self.n_info_values = X.shape[1] warnings.warn( "n_info_values is greater than the maximum number of all" " regressors space considering the chosen y_lag, u_lag, and" f" non_degree. We set as {X.shape[1]}", stacklevel=2, ) output_vector = np.zeros(self.n_info_values) output_vector[:] = np.nan n_samples = len(y) - self.max_lag for i in range(self.n_info_values): n_theta = i + 1 regressor_matrix = self.error_reduction_ratio(X, y, n_theta)[2] tmp_theta = self.estimator.optimize( regressor_matrix, y[self.max_lag :, 0].reshape(-1, 1) ) tmp_yhat = np.dot(regressor_matrix, tmp_theta) tmp_residual = y[self.max_lag :] - tmp_yhat e_var = np.var(tmp_residual, ddof=1) output_vector[i] = self.info_criteria_function(n_theta, n_samples, e_var) return output_vector 

### lilc(n_theta, n_samples, e_var)¶

Compute the Lilc information criteria value.

##### Parameters¶

n_theta : int Number of parameters of the model. n_samples : int Number of samples given the maximum lag. e_var : float Variance of the residues

##### Returns¶

info_criteria_value : float The computed value given the information criteria selected by the user.

Source code in sysidentpy/model_structure_selection/forward_regression_orthogonal_least_squares.py
 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 def lilc(self, n_theta: int, n_samples: int, e_var: float) -> float: """Compute the Lilc information criteria value. Parameters ---------- n_theta : int Number of parameters of the model. n_samples : int Number of samples given the maximum lag. e_var : float Variance of the residues Returns ------- info_criteria_value : float The computed value given the information criteria selected by the user. """ model_factor = 2 * n_theta * np.log(np.log(n_samples)) e_factor = n_samples * np.log(e_var) info_criteria_value = e_factor + model_factor return info_criteria_value 

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

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.

Source code in sysidentpy/model_structure_selection/forward_regression_orthogonal_least_squares.py
 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 def predict( self, *, X: Optional[np.ndarray] = None, y: np.ndarray, steps_ahead: Optional[int] = None, forecast_horizon: Optional[int] = None, ) -> 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 isinstance(self.basis_function, 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) 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, forecast_horizon ) yhat = np.concatenate([y[: self.max_lag], yhat], axis=0) return yhat