# Documentation for Entropic Regression¶

Build Polynomial NARMAX Models using the Entropic Regression algorithm.

## ER¶

Bases: Estimators, BaseMSS

Entropic Regression Algorithm.

Build Polynomial NARMAX model using the Entropic Regression Algorithm ([1]_). This algorithm is based on the Matlab package available on: https://github.com/almomaa/ERFit-Package

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 kth nearest neighbor to be used in estimation.

2
q float

Quantile to compute, which must be between 0 and 1 inclusive.

0.99
p default=inf,

Lp Measure of the distance in Knn estimator.

inf
n_perm int

Number of permutation to be used in shuffle test

200
estimator str

The parameter estimation method.

"least_squares"
skip_forward bool

To be used for difficult and highly uncertain problems. Skipping the forward selection results in more accurate solution, but comes with higher computational cost.

False
lam float

Forgetting factor of the Recursive Least Squares method.

0.98
delta float

Normalization factor of the P matrix.

0.01
offset_covariance float

The offset covariance factor of the affine least mean squares filter.

0.2
mu float

The convergence coefficient (learning rate) of the filter.

0.01
eps float

Normalization factor of the normalized filters.

eps
gama float

The leakage factor of the Leaky LMS method.

0.2
weight float

Weight factor to control the proportions of the error norms and offers an extra degree of freedom within the adaptation of the LMS mixed norm method.

0.02
model_type str

The user can choose "NARMAX", "NAR" and "NFIR" models

'NARMAX'

Examples:

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from sysidentpy.model_structure_selection import ER
>>> 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 = ER(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
• Abd AlRahman R. AlMomani, Jie Sun, and Erik Bollt. How Entropic Regression Beats the Outliers Problem in Nonlinear System Identification. Chaos 30, 013107 (2020).
• Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004
• Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004
• Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004
Source code in sysidentpy\model_structure_selection\entropic_regression.py
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" \n You'll have to use ER(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 ER(Estimators, BaseMSS): r"""Entropic Regression Algorithm. Build Polynomial NARMAX model using the Entropic Regression Algorithm ([1]_). This algorithm is based on the Matlab package available on: https://github.com/almomaa/ERFit-Package 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=2 The kth nearest neighbor to be used in estimation. q : float, default=0.99 Quantile to compute, which must be between 0 and 1 inclusive. p : default=inf, Lp Measure of the distance in Knn estimator. n_perm: int, default=200 Number of permutation to be used in shuffle test estimator : str, default="least_squares" The parameter estimation method. skip_forward = bool, default=False To be used for difficult and highly uncertain problems. Skipping the forward selection results in more accurate solution, but comes with higher computational cost. lam : float, default=0.98 Forgetting factor of the Recursive Least Squares method. delta : float, default=0.01 Normalization factor of the P matrix. offset_covariance : float, default=0.2 The offset covariance factor of the affine least mean squares filter. mu : float, default=0.01 The convergence coefficient (learning rate) of the filter. eps : float Normalization factor of the normalized filters. gama : float, default=0.2 The leakage factor of the Leaky LMS method. weight : float, default=0.02 Weight factor to control the proportions of the error norms and offers an extra degree of freedom within the adaptation of the LMS mixed norm method. model_type: str, default="NARMAX" The user can choose "NARMAX", "NAR" and "NFIR" models Examples -------- >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from sysidentpy.model_structure_selection import ER >>> 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 = ER(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 ---------- - Abd AlRahman R. AlMomani, Jie Sun, and Erik Bollt. How Entropic Regression Beats the Outliers Problem in Nonlinear System Identification. Chaos 30, 013107 (2020). - Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004 - Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004 - Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004 """ def __init__( self, *, ylag: Union[int, list] = 1, xlag: Union[int, list] = 1, q: float = 0.99, estimator: str = "least_squares", extended_least_squares: bool = False, h: float = 0.01, k: int = 2, mutual_information_estimator: str = "mutual_information_knn", n_perm: int = 200, p: float = np.inf, skip_forward: bool = False, lam: float = 0.98, delta: float = 0.01, offset_covariance: float = 0.2, mu: float = 0.01, eps: float = np.finfo(np.float64).eps, gama: float = 0.2, weight: float = 0.02, model_type: str = "NARMAX", basis_function: Union[Polynomial, Fourier] = Polynomial(), random_state: Optional[int] = None, ): self.basis_function = basis_function self.model_type = model_type self.build_matrix = self.get_build_io_method(model_type) self.xlag = xlag self.ylag = ylag self.non_degree = basis_function.degree self.max_lag = self._get_max_lag() self.k = k self.estimator = estimator self.extended_least_squares = extended_least_squares self.q = q self.h = h self.mutual_information_estimator = mutual_information_estimator self.n_perm = n_perm self.p = p self.skip_forward = skip_forward self.random_state = random_state self.rng = check_random_state(random_state) self.tol = None self.ensemble = None self.n_inputs = None self.estimated_tolerance = None self.regressor_code = None self.final_model = None self.theta = None self.n_terms = None self.err = None self.pivv = None self._validate_params() super().__init__( lam=lam, delta=delta, offset_covariance=offset_covariance, mu=mu, eps=eps, gama=gama, weight=weight, basis_function=basis_function, ) 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.n_perm, int) or self.n_perm < 1: raise ValueError(f"n_perm must be integer and > zero. Got {self.n_perm}") if not isinstance(self.q, float) or self.q > 1 or self.q <= 0: raise ValueError( f"q must be float and must be between 0 and 1 inclusive. Got {self.q}" ) if not isinstance(self.skip_forward, bool): raise TypeError( f"skip_forward must be False or True. Got {self.skip_forward}" ) if not isinstance(self.extended_least_squares, bool): raise TypeError( "extended_least_squares must be False or True. Got" f" {self.extended_least_squares}" ) if self.model_type not in ["NARMAX", "NAR", "NFIR"]: raise ValueError( f"model_type must be NARMAX, NAR or NFIR. Got {self.model_type}" ) def mutual_information_knn(self, y, y_perm): """Find the mutual information. Finds the mutual information between $x$ and $y$ given $z$. This code is based on Matlab Entropic Regression package. Parameters ---------- y : ndarray of floats The source signal. y_perm : ndarray of floats The destination signal. Returns ------- ksg_estimation : float The conditioned mutual information. References ---------- - Abd AlRahman R. AlMomani, Jie Sun, and Erik Bollt. How Entropic Regression Beats the Outliers Problem in Nonlinear System Identification. Chaos 30, 013107 (2020). - Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004 - Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004 - Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004 """ joint_space = np.concatenate([y, y_perm], axis=1) smallest_distance = np.sort( cdist(joint_space, joint_space, "minkowski", p=self.p).T ) idx = np.argpartition(smallest_distance[-1, :], self.k + 1)[: self.k + 1] smallest_distance = smallest_distance[:, idx] epsilon = smallest_distance[:, -1].reshape(-1, 1) smallest_distance_y = cdist(y, y, "minkowski", p=self.p) less_than_array_nx = np.array((smallest_distance_y < epsilon)).astype(int) nx = (np.sum(less_than_array_nx, axis=1) - 1).reshape(-1, 1) smallest_distance_y_perm = cdist(y_perm, y_perm, "minkowski", p=self.p) less_than_array_ny = np.array((smallest_distance_y_perm < epsilon)).astype(int) ny = (np.sum(less_than_array_ny, axis=1) - 1).reshape(-1, 1) arr = psi(nx + 1) + psi(ny + 1) ksg_estimation = ( psi(self.k) + psi(y.shape[0]) - np.nanmean(arr[np.isfinite(arr)]) ) return ksg_estimation def entropic_regression_backward(self, reg_matrix, y, piv): """Entropic Regression Backward Greedy Feature Elimination. This algorithm is based on the Matlab package available on: https://github.com/almomaa/ERFit-Package Parameters ---------- reg_matrix : 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. piv : ndarray of ints The set of indices to investigate Returns ------- piv : ndarray of ints The set of remaining indices after the Backward Greedy Feature Elimination. """ min_value = -np.inf piv = np.array(piv) ix = [] while (min_value <= self.tol) and (len(piv) > 1): initial_array = np.full((1, len(piv)), np.inf) for i in range(initial_array.shape[1]): if piv[i] not in []: # if you want to keep any regressor rem = np.setdiff1d(piv, piv[i]) f1 = reg_matrix[:, piv] @ LA.pinv(reg_matrix[:, piv]) @ y f2 = reg_matrix[:, rem] @ LA.pinv(reg_matrix[:, rem]) @ y initial_array[0, i] = self.conditional_mutual_information(y, f1, f2) ix = np.argmin(initial_array) min_value = initial_array[0, ix] piv = np.delete(piv, ix) return piv def entropic_regression_forward(self, reg_matrix, y): """Entropic Regression Forward Greedy Feature Selection. This algorithm is based on the Matlab package available on: https://github.com/almomaa/ERFit-Package Parameters ---------- reg_matrix : 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. Returns ------- selected_terms : ndarray of ints The set of selected regressors after the Forward Greedy Feature Selection. success : boolean Indicate if the forward selection succeed. If high degree of uncertainty is detected, and many parameters are selected, the success flag will be set to false. Then, the backward elimination will be applied for all indices. """ success = True ix = [] selected_terms = [] reg_matrix_columns = np.array(list(range(reg_matrix.shape[1]))) self.tol = self.tolerance_estimator(y) ksg_max = getattr(self, self.mutual_information_estimator)( y, reg_matrix @ LA.pinv(reg_matrix) @ y ) stop_criteria = False while stop_criteria is False: selected_terms = np.ravel( [*selected_terms, *np.array([reg_matrix_columns[ix]])] ) if len(selected_terms) != 0: ksg_local = getattr(self, self.mutual_information_estimator)( y, reg_matrix[:, selected_terms] @ LA.pinv(reg_matrix[:, selected_terms]) @ y, ) else: ksg_local = getattr(self, self.mutual_information_estimator)( y, np.zeros_like(y) ) initial_vector = np.full((1, reg_matrix.shape[1]), -np.inf) for i in range(reg_matrix.shape[1]): if reg_matrix_columns[i] not in selected_terms: f1 = ( reg_matrix[:, [*selected_terms, reg_matrix_columns[i]]] @ LA.pinv( reg_matrix[:, [*selected_terms, reg_matrix_columns[i]]] ) @ y ) if len(selected_terms) != 0: f2 = ( reg_matrix[:, selected_terms] @ LA.pinv(reg_matrix[:, selected_terms]) @ y ) else: f2 = np.zeros_like(y) vp_estimation = self.conditional_mutual_information(y, f1, f2) initial_vector[0, i] = vp_estimation else: continue ix = np.nanargmax(initial_vector) max_value = initial_vector[0, ix] if (ksg_max - ksg_local <= self.tol) or (max_value <= self.tol): stop_criteria = True elif len(selected_terms) > np.max([8, reg_matrix.shape[1] / 2]): success = False stop_criteria = True return selected_terms, success def conditional_mutual_information(self, y, f1, f2): """Find the conditional mutual information. Finds the conditioned mutual information between $y$ and $f1$ given $f2$. This code is based on Matlab Entropic Regression package. https://github.com/almomaa/ERFit-Package Parameters ---------- y : ndarray of floats The source signal. f1 : ndarray of floats The destination signal. f2 : ndarray of floats The condition set. Returns ------- vp_estimation : float The conditioned mutual information. References ---------- - Abd AlRahman R. AlMomani, Jie Sun, and Erik Bollt. How Entropic Regression Beats the Outliers Problem in Nonlinear System Identification. Chaos 30, 013107 (2020). - Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004 - Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004 - Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004 """ joint_space = np.concatenate([y, f1, f2], axis=1) smallest_distance = np.sort( cdist(joint_space, joint_space, "minkowski", p=self.p).T ) idx = np.argpartition(smallest_distance[-1, :], self.k + 1)[: self.k + 1] smallest_distance = smallest_distance[:, idx] epsilon = smallest_distance[:, -1].reshape(-1, 1) # Find number of points from (y,f2), (f1,f2), and (f2,f2) that lies withing the # k^{th} nearest neighbor distance from each point of themselves. smallest_distance_y_f2 = cdist( np.concatenate([y, f2], axis=1), np.concatenate([y, f2], axis=1), "minkowski", p=self.p, ) less_than_array_y_f2 = np.array((smallest_distance_y_f2 < epsilon)).astype(int) y_f2 = (np.sum(less_than_array_y_f2, axis=1) - 1).reshape(-1, 1) smallest_distance_f1_f2 = cdist( np.concatenate([f1, f2], axis=1), np.concatenate([f1, f2], axis=1), "minkowski", p=self.p, ) less_than_array_f1_f2 = np.array((smallest_distance_f1_f2 < epsilon)).astype( int ) f1_f2 = (np.sum(less_than_array_f1_f2, axis=1) - 1).reshape(-1, 1) smallest_distance_f2 = cdist(f2, f2, "minkowski", p=self.p) less_than_array_f2 = np.array((smallest_distance_f2 < epsilon)).astype(int) f2_f2 = (np.sum(less_than_array_f2, axis=1) - 1).reshape(-1, 1) arr = psi(y_f2 + 1) + psi(f1_f2 + 1) - psi(f2_f2 + 1) vp_estimation = psi(self.k) - np.nanmean(arr[np.isfinite(arr)]) return vp_estimation def tolerance_estimator(self, y): """Tolerance Estimation for mutual independence test. Finds the conditioned mutual information between $y$ and $f1$ given $f2$. This code is based on Matlab Entropic Regression package. https://github.com/almomaa/ERFit-Package Parameters ---------- y : ndarray of floats The source signal. Returns ------- tol : float The tolerance value given q. References ---------- - Abd AlRahman R. AlMomani, Jie Sun, and Erik Bollt. How Entropic Regression Beats the Outliers Problem in Nonlinear System Identification. Chaos 30, 013107 (2020). - Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004 - Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004 - Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004 """ ksg_estimation = [] for _ in range(self.n_perm): mutual_information_output = getattr( self, self.mutual_information_estimator )(y, self.rng.permutation(y)) ksg_estimation.append(mutual_information_output) ksg_estimation = np.array(ksg_estimation) tol = np.quantile(ksg_estimation, self.q) return tol def fit(self, *, 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. The Entropic Regression algorithm is based on the Matlab package available on: https://github.com/almomaa/ERFit-Package 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. theta : array-like of shape = number_of_model_elements The estimated parameters of the model. References ---------- - Abd AlRahman R. AlMomani, Jie Sun, and Erik Bollt. How Entropic Regression Beats the Outliers Problem in Nonlinear System Identification. Chaos 30, 013107 (2020). - Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004 - Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004 - Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004 """ 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) if self.regressor_code.shape[0] > 90: warnings.warn( "Given the higher number of possible regressors" f" ({self.regressor_code.shape[0]}), the Entropic Regression" " algorithm may take long time to run. Consider reducing the" " number of regressors ", stacklevel=2, ) y_full = y.copy() y = y[self.max_lag :].reshape(-1, 1) self.tol = 0 ksg_estimation = [] for _ in range(self.n_perm): mutual_information_output = getattr( self, self.mutual_information_estimator )(y, self.rng.permutation(y)) ksg_estimation.append(mutual_information_output) ksg_estimation = np.array(ksg_estimation).reshape(-1, 1) self.tol = np.quantile(ksg_estimation, self.q) self.estimated_tolerance = self.tol success = False if not self.skip_forward: selected_terms, success = self.entropic_regression_forward(reg_matrix, y) if not success or self.skip_forward: selected_terms = np.array(list(range(reg_matrix.shape[1]))) selected_terms_backward = self.entropic_regression_backward( reg_matrix[:, selected_terms], y, list(range(len(selected_terms))) ) final_model = selected_terms[selected_terms_backward] # re-check for the constant term (add it to the estimated indices) if 0 not in final_model: final_model = np.array([0, *final_model]) if self.basis_function.__class__.__name__ == "Polynomial": self.final_model = self.regressor_code[final_model, :].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[final_model, :].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[final_model, :].copy() self.theta = getattr(self, self.estimator)(reg_matrix[:, final_model], y_full) if (np.abs(self.theta[0]) < self.h) and ( np.sum((self.theta != 0).astype(int)) > 1 ): self.theta = self.theta[1:].reshape(-1, 1) self.final_model = self.final_model[1:, :] final_model = final_model[1:] 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. self.pivv = final_model return self def predict(self, *, X=None, y=None, 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. 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, y): """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, y, steps_ahead): """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. """ yhat = super()._n_step_ahead_prediction(X, y, steps_ahead) return yhat def _model_prediction(self, X, y_initial, forecast_horizon=None): """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) elif self.model_type == "NFIR": return self._nfir_predict(X, y_initial) else: raise ValueError( f"model_type must be NARMAX, NAR or NFIR. Got {self.model_type}" ) def _narmax_predict(self, X, y_initial, forecast_horizon): 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, y_initial): y_output = super()._nfir_predict(X, y_initial) return y_output def _basis_function_predict(self, X, y_initial, forecast_horizon=None): 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, y, steps_ahead, forecast_horizon): """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, y, steps_ahead, forecast_horizon): yhat = super()._basis_function_n_steps_horizon( X, y, steps_ahead, forecast_horizon ) return yhat.reshape(-1, 1) 

### conditional_mutual_information(y, f1, f2)¶

Find the conditional mutual information.

Finds the conditioned mutual information between $$y$$ and $$f1$$ given $$f2$$.

This code is based on Matlab Entropic Regression package. https://github.com/almomaa/ERFit-Package

Parameters:

Name Type Description Default
y ndarray of floats

The source signal.

required
f1 ndarray of floats

The destination signal.

required
f2 ndarray of floats

The condition set.

required

Returns:

Name Type Description
vp_estimation float

The conditioned mutual information.

References
• Abd AlRahman R. AlMomani, Jie Sun, and Erik Bollt. How Entropic Regression Beats the Outliers Problem in Nonlinear System Identification. Chaos 30, 013107 (2020).
• Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004
• Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004
• Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004
Source code in sysidentpy\model_structure_selection\entropic_regression.py
 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 def conditional_mutual_information(self, y, f1, f2): """Find the conditional mutual information. Finds the conditioned mutual information between $y$ and $f1$ given $f2$. This code is based on Matlab Entropic Regression package. https://github.com/almomaa/ERFit-Package Parameters ---------- y : ndarray of floats The source signal. f1 : ndarray of floats The destination signal. f2 : ndarray of floats The condition set. Returns ------- vp_estimation : float The conditioned mutual information. References ---------- - Abd AlRahman R. AlMomani, Jie Sun, and Erik Bollt. How Entropic Regression Beats the Outliers Problem in Nonlinear System Identification. Chaos 30, 013107 (2020). - Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004 - Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004 - Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004 """ joint_space = np.concatenate([y, f1, f2], axis=1) smallest_distance = np.sort( cdist(joint_space, joint_space, "minkowski", p=self.p).T ) idx = np.argpartition(smallest_distance[-1, :], self.k + 1)[: self.k + 1] smallest_distance = smallest_distance[:, idx] epsilon = smallest_distance[:, -1].reshape(-1, 1) # Find number of points from (y,f2), (f1,f2), and (f2,f2) that lies withing the # k^{th} nearest neighbor distance from each point of themselves. smallest_distance_y_f2 = cdist( np.concatenate([y, f2], axis=1), np.concatenate([y, f2], axis=1), "minkowski", p=self.p, ) less_than_array_y_f2 = np.array((smallest_distance_y_f2 < epsilon)).astype(int) y_f2 = (np.sum(less_than_array_y_f2, axis=1) - 1).reshape(-1, 1) smallest_distance_f1_f2 = cdist( np.concatenate([f1, f2], axis=1), np.concatenate([f1, f2], axis=1), "minkowski", p=self.p, ) less_than_array_f1_f2 = np.array((smallest_distance_f1_f2 < epsilon)).astype( int ) f1_f2 = (np.sum(less_than_array_f1_f2, axis=1) - 1).reshape(-1, 1) smallest_distance_f2 = cdist(f2, f2, "minkowski", p=self.p) less_than_array_f2 = np.array((smallest_distance_f2 < epsilon)).astype(int) f2_f2 = (np.sum(less_than_array_f2, axis=1) - 1).reshape(-1, 1) arr = psi(y_f2 + 1) + psi(f1_f2 + 1) - psi(f2_f2 + 1) vp_estimation = psi(self.k) - np.nanmean(arr[np.isfinite(arr)]) return vp_estimation 

### entropic_regression_backward(reg_matrix, y, piv)¶

Entropic Regression Backward Greedy Feature Elimination.

This algorithm is based on the Matlab package available on: https://github.com/almomaa/ERFit-Package

Parameters:

Name Type Description Default
reg_matrix ndarray of floats

The input data to be used in the prediction process.

required
y ndarray of floats

The output data to be used in the prediction process.

required
piv ndarray of ints

The set of indices to investigate

required

Returns:

Name Type Description
piv ndarray of ints

The set of remaining indices after the Backward Greedy Feature Elimination.

Source code in sysidentpy\model_structure_selection\entropic_regression.py
 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 def entropic_regression_backward(self, reg_matrix, y, piv): """Entropic Regression Backward Greedy Feature Elimination. This algorithm is based on the Matlab package available on: https://github.com/almomaa/ERFit-Package Parameters ---------- reg_matrix : 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. piv : ndarray of ints The set of indices to investigate Returns ------- piv : ndarray of ints The set of remaining indices after the Backward Greedy Feature Elimination. """ min_value = -np.inf piv = np.array(piv) ix = [] while (min_value <= self.tol) and (len(piv) > 1): initial_array = np.full((1, len(piv)), np.inf) for i in range(initial_array.shape[1]): if piv[i] not in []: # if you want to keep any regressor rem = np.setdiff1d(piv, piv[i]) f1 = reg_matrix[:, piv] @ LA.pinv(reg_matrix[:, piv]) @ y f2 = reg_matrix[:, rem] @ LA.pinv(reg_matrix[:, rem]) @ y initial_array[0, i] = self.conditional_mutual_information(y, f1, f2) ix = np.argmin(initial_array) min_value = initial_array[0, ix] piv = np.delete(piv, ix) return piv 

### entropic_regression_forward(reg_matrix, y)¶

Entropic Regression Forward Greedy Feature Selection.

This algorithm is based on the Matlab package available on: https://github.com/almomaa/ERFit-Package

Parameters:

Name Type Description Default
reg_matrix ndarray of floats

The input data to be used in the prediction process.

required
y ndarray of floats

The output data to be used in the prediction process.

required

Returns:

Name Type Description
selected_terms ndarray of ints

The set of selected regressors after the Forward Greedy Feature Selection.

success boolean

Indicate if the forward selection succeed. If high degree of uncertainty is detected, and many parameters are selected, the success flag will be set to false. Then, the backward elimination will be applied for all indices.

Source code in sysidentpy\model_structure_selection\entropic_regression.py
 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 def entropic_regression_forward(self, reg_matrix, y): """Entropic Regression Forward Greedy Feature Selection. This algorithm is based on the Matlab package available on: https://github.com/almomaa/ERFit-Package Parameters ---------- reg_matrix : 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. Returns ------- selected_terms : ndarray of ints The set of selected regressors after the Forward Greedy Feature Selection. success : boolean Indicate if the forward selection succeed. If high degree of uncertainty is detected, and many parameters are selected, the success flag will be set to false. Then, the backward elimination will be applied for all indices. """ success = True ix = [] selected_terms = [] reg_matrix_columns = np.array(list(range(reg_matrix.shape[1]))) self.tol = self.tolerance_estimator(y) ksg_max = getattr(self, self.mutual_information_estimator)( y, reg_matrix @ LA.pinv(reg_matrix) @ y ) stop_criteria = False while stop_criteria is False: selected_terms = np.ravel( [*selected_terms, *np.array([reg_matrix_columns[ix]])] ) if len(selected_terms) != 0: ksg_local = getattr(self, self.mutual_information_estimator)( y, reg_matrix[:, selected_terms] @ LA.pinv(reg_matrix[:, selected_terms]) @ y, ) else: ksg_local = getattr(self, self.mutual_information_estimator)( y, np.zeros_like(y) ) initial_vector = np.full((1, reg_matrix.shape[1]), -np.inf) for i in range(reg_matrix.shape[1]): if reg_matrix_columns[i] not in selected_terms: f1 = ( reg_matrix[:, [*selected_terms, reg_matrix_columns[i]]] @ LA.pinv( reg_matrix[:, [*selected_terms, reg_matrix_columns[i]]] ) @ y ) if len(selected_terms) != 0: f2 = ( reg_matrix[:, selected_terms] @ LA.pinv(reg_matrix[:, selected_terms]) @ y ) else: f2 = np.zeros_like(y) vp_estimation = self.conditional_mutual_information(y, f1, f2) initial_vector[0, i] = vp_estimation else: continue ix = np.nanargmax(initial_vector) max_value = initial_vector[0, ix] if (ksg_max - ksg_local <= self.tol) or (max_value <= self.tol): stop_criteria = True elif len(selected_terms) > np.max([8, reg_matrix.shape[1] / 2]): success = False stop_criteria = True return selected_terms, success 

### 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.

The Entropic Regression algorithm is based on the Matlab package available on: https://github.com/almomaa/ERFit-Package

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.

theta array-like of shape = number_of_model_elements

The estimated parameters of the model.

References
• Abd AlRahman R. AlMomani, Jie Sun, and Erik Bollt. How Entropic Regression Beats the Outliers Problem in Nonlinear System Identification. Chaos 30, 013107 (2020).
• Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004
• Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004
• Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004
Source code in sysidentpy\model_structure_selection\entropic_regression.py
 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 def fit(self, *, 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. The Entropic Regression algorithm is based on the Matlab package available on: https://github.com/almomaa/ERFit-Package 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. theta : array-like of shape = number_of_model_elements The estimated parameters of the model. References ---------- - Abd AlRahman R. AlMomani, Jie Sun, and Erik Bollt. How Entropic Regression Beats the Outliers Problem in Nonlinear System Identification. Chaos 30, 013107 (2020). - Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004 - Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004 - Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004 """ 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) if self.regressor_code.shape[0] > 90: warnings.warn( "Given the higher number of possible regressors" f" ({self.regressor_code.shape[0]}), the Entropic Regression" " algorithm may take long time to run. Consider reducing the" " number of regressors ", stacklevel=2, ) y_full = y.copy() y = y[self.max_lag :].reshape(-1, 1) self.tol = 0 ksg_estimation = [] for _ in range(self.n_perm): mutual_information_output = getattr( self, self.mutual_information_estimator )(y, self.rng.permutation(y)) ksg_estimation.append(mutual_information_output) ksg_estimation = np.array(ksg_estimation).reshape(-1, 1) self.tol = np.quantile(ksg_estimation, self.q) self.estimated_tolerance = self.tol success = False if not self.skip_forward: selected_terms, success = self.entropic_regression_forward(reg_matrix, y) if not success or self.skip_forward: selected_terms = np.array(list(range(reg_matrix.shape[1]))) selected_terms_backward = self.entropic_regression_backward( reg_matrix[:, selected_terms], y, list(range(len(selected_terms))) ) final_model = selected_terms[selected_terms_backward] # re-check for the constant term (add it to the estimated indices) if 0 not in final_model: final_model = np.array([0, *final_model]) if self.basis_function.__class__.__name__ == "Polynomial": self.final_model = self.regressor_code[final_model, :].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[final_model, :].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[final_model, :].copy() self.theta = getattr(self, self.estimator)(reg_matrix[:, final_model], y_full) if (np.abs(self.theta[0]) < self.h) and ( np.sum((self.theta != 0).astype(int)) > 1 ): self.theta = self.theta[1:].reshape(-1, 1) self.final_model = self.final_model[1:, :] final_model = final_model[1:] 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. self.pivv = final_model return self 

### mutual_information_knn(y, y_perm)¶

Find the mutual information.

Finds the mutual information between $$x$$ and $$y$$ given $$z$$.

This code is based on Matlab Entropic Regression package.

Parameters:

Name Type Description Default
y ndarray of floats

The source signal.

required
y_perm ndarray of floats

The destination signal.

required

Returns:

Name Type Description
ksg_estimation float

The conditioned mutual information.

References
• Abd AlRahman R. AlMomani, Jie Sun, and Erik Bollt. How Entropic Regression Beats the Outliers Problem in Nonlinear System Identification. Chaos 30, 013107 (2020).
• Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004
• Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004
• Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004
Source code in sysidentpy\model_structure_selection\entropic_regression.py
 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 def mutual_information_knn(self, y, y_perm): """Find the mutual information. Finds the mutual information between $x$ and $y$ given $z$. This code is based on Matlab Entropic Regression package. Parameters ---------- y : ndarray of floats The source signal. y_perm : ndarray of floats The destination signal. Returns ------- ksg_estimation : float The conditioned mutual information. References ---------- - Abd AlRahman R. AlMomani, Jie Sun, and Erik Bollt. How Entropic Regression Beats the Outliers Problem in Nonlinear System Identification. Chaos 30, 013107 (2020). - Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004 - Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004 - Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004 """ joint_space = np.concatenate([y, y_perm], axis=1) smallest_distance = np.sort( cdist(joint_space, joint_space, "minkowski", p=self.p).T ) idx = np.argpartition(smallest_distance[-1, :], self.k + 1)[: self.k + 1] smallest_distance = smallest_distance[:, idx] epsilon = smallest_distance[:, -1].reshape(-1, 1) smallest_distance_y = cdist(y, y, "minkowski", p=self.p) less_than_array_nx = np.array((smallest_distance_y < epsilon)).astype(int) nx = (np.sum(less_than_array_nx, axis=1) - 1).reshape(-1, 1) smallest_distance_y_perm = cdist(y_perm, y_perm, "minkowski", p=self.p) less_than_array_ny = np.array((smallest_distance_y_perm < epsilon)).astype(int) ny = (np.sum(less_than_array_ny, axis=1) - 1).reshape(-1, 1) arr = psi(nx + 1) + psi(ny + 1) ksg_estimation = ( psi(self.k) + psi(y.shape[0]) - np.nanmean(arr[np.isfinite(arr)]) ) return ksg_estimation 

### predict(*, X=None, y=None, 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.

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\entropic_regression.py
 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 def predict(self, *, X=None, y=None, 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. 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 

### tolerance_estimator(y)¶

Tolerance Estimation for mutual independence test.

Finds the conditioned mutual information between $$y$$ and $$f1$$ given $$f2$$.

This code is based on Matlab Entropic Regression package. https://github.com/almomaa/ERFit-Package

Parameters:

Name Type Description Default
y ndarray of floats

The source signal.

required

Returns:

Name Type Description
tol float

The tolerance value given q.

References
• Abd AlRahman R. AlMomani, Jie Sun, and Erik Bollt. How Entropic Regression Beats the Outliers Problem in Nonlinear System Identification. Chaos 30, 013107 (2020).
• Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004
• Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004
• Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004
Source code in sysidentpy\model_structure_selection\entropic_regression.py
 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 def tolerance_estimator(self, y): """Tolerance Estimation for mutual independence test. Finds the conditioned mutual information between $y$ and $f1$ given $f2$. This code is based on Matlab Entropic Regression package. https://github.com/almomaa/ERFit-Package Parameters ---------- y : ndarray of floats The source signal. Returns ------- tol : float The tolerance value given q. References ---------- - Abd AlRahman R. AlMomani, Jie Sun, and Erik Bollt. How Entropic Regression Beats the Outliers Problem in Nonlinear System Identification. Chaos 30, 013107 (2020). - Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004 - Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004 - Alexander Kraskov, Harald St¨ogbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066-138,2004 """ ksg_estimation = [] for _ in range(self.n_perm): mutual_information_output = getattr( self, self.mutual_information_estimator )(y, self.rng.permutation(y)) ksg_estimation.append(mutual_information_output) ksg_estimation = np.array(ksg_estimation) tol = np.quantile(ksg_estimation, self.q) return tol