# Documentation for MetaMSS¶

Meta Model Structure Selection.

## MetaMSS¶

Bases: SimulateNARMAX, BPSOGSA

Meta-Model Structure Selection: Building Polynomial NARMAX model.

This class uses the MetaMSS ([1], [2], [3]_) algorithm to build NARMAX models. The NARMAX model is described as:

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

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
loss_func str

The loss function to be minimized.

"metamss_loss"
estimator str

The parameter estimation method.

"least_squares"
estimate_parameter bool

Whether to estimate the model parameters.

True
extended_least_squares bool

Whether to use extended least squares method for parameter estimation. Note that we define a specific set of noise regressors.

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
maxiter int

The maximum number of iterations.

30
alpha int

The descending coefficient of the gravitational constant.

23
g_zero int

The initial value of the gravitational constant.

100
k_agents_percent int

Percent of agents applying force to the others in the last iteration.

2
norm int

The information criteria method to be used.

-2
power int

The number of the model terms to be selected. Note that n_terms overwrite the information criteria values.

2
n_agents int

The number of agents to search the optimal solution.

10
p_zeros float

The probability of getting ones in the construction of the population.

0.5
p_zeros float

The probability of getting zeros in the construction of the population.

0.5

Examples:

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from sysidentpy.model_structure_selection import MetaMSS
>>> from sysidentpy.metrics import root_relative_squared_error
>>> from sysidentpy.basis_function._basis_function import Polynomial
>>> from sysidentpy.utils.display_results import results
>>> from sysidentpy.utils.generate_data import get_siso_data
>>> x_train, x_valid, y_train, y_valid = get_siso_data(n=400,
...                                                    colored_noise=False,
...                                                    sigma=0.001,
...                                                    train_percentage=80)
>>> basis_function = Polynomial(degree=2)
>>> model = MetaMSS(
...     basis_function=basis_function,
...     norm=-2,
...     xlag=7,
...     ylag=7,
...     estimator="least_squares",
...     k_agents_percent=2,
...     estimate_parameter=True,
...     maxiter=30,
...     n_agents=10,
...     p_value=0.05,
...     loss_func='metamss_loss'
... )
>>> model.fit(x_train, y_train, x_valid, y_valid)
>>> 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: Meta-Model Structure Selection: Building Polynomial NARX Model for Regression and Classification https://arxiv.org/pdf/2109.09917.pdf
• Manuscript (Portuguese): Identificação de Sistemas Não Lineares Utilizando o Algoritmo Híbrido e Binário de Otimização por Enxame de Partículas e Busca Gravitacional DOI: 10.17648/sbai-2019-111317
• Master thesis: Meta model structure selection: an algorithm for building polynomial NARX models for regression and classification
Source code in sysidentpy\model_structure_selection\meta_model_structure_selection.py
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" \n You'll have to use MetaMSS(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 MetaMSS(SimulateNARMAX, BPSOGSA): r"""Meta-Model Structure Selection: Building Polynomial NARMAX model. This class uses the MetaMSS ([1]_, [2]_, [3]_) algorithm to build NARMAX models. The NARMAX model is described as: $$y_k= F^\ell[y_{k-1}, \dotsc, y_{k-n_y},x_{k-d}, x_{k-d-1}, \dotsc, x_{k-d-n_x}, e_{k-1}, \dotsc, e_{k-n_e}] + e_k$$ 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. loss_func : str, default="metamss_loss" The loss function to be minimized. estimator : str, default="least_squares" The parameter estimation method. estimate_parameter : bool, default=True Whether to estimate the model parameters. extended_least_squares : bool, default=False Whether to use extended least squares method for parameter estimation. Note that we define a specific set of noise regressors. 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. maxiter : int, default=30 The maximum number of iterations. alpha : int, default=23 The descending coefficient of the gravitational constant. g_zero : int, default=100 The initial value of the gravitational constant. k_agents_percent: int, default=2 Percent of agents applying force to the others in the last iteration. norm : int, default=-2 The information criteria method to be used. power : int, default=2 The number of the model terms to be selected. Note that n_terms overwrite the information criteria values. n_agents : int, default=10 The number of agents to search the optimal solution. p_zeros : float, default=0.5 The probability of getting ones in the construction of the population. p_zeros : float, default=0.5 The probability of getting zeros in the construction of the population. Examples -------- >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from sysidentpy.model_structure_selection import MetaMSS >>> from sysidentpy.metrics import root_relative_squared_error >>> from sysidentpy.basis_function._basis_function import Polynomial >>> from sysidentpy.utils.display_results import results >>> from sysidentpy.utils.generate_data import get_siso_data >>> x_train, x_valid, y_train, y_valid = get_siso_data(n=400, ... colored_noise=False, ... sigma=0.001, ... train_percentage=80) >>> basis_function = Polynomial(degree=2) >>> model = MetaMSS( ... basis_function=basis_function, ... norm=-2, ... xlag=7, ... ylag=7, ... estimator="least_squares", ... k_agents_percent=2, ... estimate_parameter=True, ... maxiter=30, ... n_agents=10, ... p_value=0.05, ... loss_func='metamss_loss' ... ) >>> model.fit(x_train, y_train, x_valid, y_valid) >>> 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: Meta-Model Structure Selection: Building Polynomial NARX Model for Regression and Classification https://arxiv.org/pdf/2109.09917.pdf - Manuscript (Portuguese): Identificação de Sistemas Não Lineares Utilizando o Algoritmo Híbrido e Binário de Otimização por Enxame de Partículas e Busca Gravitacional DOI: 10.17648/sbai-2019-111317 - Master thesis: Meta model structure selection: an algorithm for building polynomial NARX models for regression and classification """ def __init__( self, *, maxiter: int = 30, alpha: int = 23, g_zero: int = 100, k_agents_percent: int = 2, norm: float = -2, power: int = 2, n_agents: int = 10, p_zeros: float = 0.5, p_ones: float = 0.5, p_value: float = 0.05, xlag: Union[int, list] = 1, ylag: Union[int, list] = 1, elag: Union[int, list] = 1, estimator: str = "least_squares", extended_least_squares: bool = False, 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, estimate_parameter: bool = True, loss_func: str = "metamss_loss", model_type: str = "NARMAX", basis_function: Polynomial = Polynomial(), steps_ahead: Optional[int] = None, random_state: Optional[int] = None, ): super().__init__( estimator=estimator, extended_least_squares=extended_least_squares, lam=lam, delta=delta, offset_covariance=offset_covariance, mu=mu, eps=eps, gama=gama, weight=weight, estimate_parameter=estimate_parameter, model_type=model_type, basis_function=basis_function, ) BPSOGSA.__init__( self, n_agents=n_agents, maxiter=maxiter, g_zero=g_zero, alpha=alpha, k_agents_percent=k_agents_percent, norm=norm, power=power, p_zeros=p_zeros, p_ones=p_ones, ) self.xlag = xlag self.ylag = ylag self.elag = elag self.non_degree = basis_function.degree self.p_value = p_value self.estimator = estimator self.estimate_parameter = estimate_parameter self.loss_func = loss_func self.steps_ahead = steps_ahead self.random_state = random_state self.build_matrix = self.get_build_io_method(model_type) self.n_inputs = None self.regressor_code = None self.best_model_history = None self.tested_models = None self.final_model = None self._validate_metamss_params() def _validate_metamss_params(self): 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}") @deprecated( version="v0.3.0", future_version="v0.4.0", message=( "You will not need to pass X_test and y_test in v0.4.0." " \n You'll have to use MetaMSS(test_size=0.25) instead. \n This" " change will make easier to use the MetaMSS model and will" " follow the same structure of the other methods." ), ) def fit( self, *, X: Optional[np.ndarray] = None, y: Optional[np.ndarray] = None, X_test: Optional[np.ndarray] = None, y_test: Optional[np.ndarray] = None, ): """Fit the polynomial NARMAX model. 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. X_test : ndarray of floats The input data to be used in the prediction process. y_test : ndarray of floats The output data (initial conditions) to be used in the prediction process. Returns ------- self : returns an instance of self. """ if self.basis_function.__class__.__name__ != "Polynomial": raise NotImplementedError( "Currently MetaMSS only supports polynomial models." ) if y is None: raise ValueError("y cannot be None") if X is not None: check_X_y(X, y) self.n_inputs = _num_features(X) else: self.n_inputs = 1 # just to create the regressor space base # self.n_inputs = _num_features(X_train) self.max_lag = self._get_max_lag() self.regressor_code = self.regressor_space(self.n_inputs) self.dimension = self.regressor_code.shape[0] velocity = np.zeros([self.dimension, self.n_agents]) self.random_state = check_random_state(self.random_state) population = self.generate_random_population(self.random_state) self.best_by_iter = [] self.mean_by_iter = [] self.optimal_fitness_value = np.inf self.optimal_model = None self.best_model_history = [] self.tested_models = [] for i in range(self.maxiter): fitness = self.evaluate_objective_function(X, y, X_test, y_test, population) column_of_best_solution = np.nanargmin(fitness) current_best_fitness = fitness[column_of_best_solution] if current_best_fitness < self.optimal_fitness_value: self.optimal_fitness_value = current_best_fitness self.optimal_model = population[:, column_of_best_solution].copy() self.best_model_history.append(self.optimal_model) self.best_by_iter.append(self.optimal_fitness_value) self.mean_by_iter.append(np.mean(fitness)) agent_mass = self.mass_calculation(fitness) gravitational_constant = self.calculate_gravitational_constant(i) acceleration = self.calculate_acceleration( population, agent_mass, gravitational_constant, i ) velocity, population = self.update_velocity_position( population, acceleration, velocity, i, ) self.final_model = self.regressor_code[self.optimal_model == 1].copy() _ = self.simulate( X_train=X, y_train=y, X_test=X_test, y_test=y_test, model_code=self.final_model, steps_ahead=self.steps_ahead, ) self.max_lag = self._get_max_lag() return self def evaluate_objective_function( self, X_train: Optional[np.ndarray], y_train: Optional[np.ndarray], X_test: Optional[np.ndarray], y_test: Optional[np.ndarray], population: np.ndarray, ): """Fit the polynomial NARMAX model. Parameters ---------- X_train : ndarray of floats The input data to be used in the training process. y_train : ndarray of floats The output data to be used in the training process. X_test : ndarray of floats The input data to be used in the prediction process. y_test : ndarray of floats The output data (initial conditions) to be used in the prediction process. population : ndarray of zeros and ones The initial population of agents. Returns ------- fitness_value : ndarray The fitness value of each agent. """ fitness = [] for agent in population.T: if np.all(agent == 0): fitness.append(30) # penalty for cases where there is no terms continue m = self.regressor_code[agent == 1].copy() yhat = self.simulate( X_train=X_train, y_train=y_train, X_test=X_test, y_test=y_test, model_code=m, steps_ahead=self.steps_ahead, ) residues = y_test - yhat self.max_lag = self._get_max_lag() lagged_data = self.build_matrix(X_train, y_train) psi = self.basis_function.fit( lagged_data, self.max_lag, predefined_regressors=self.pivv ) pos_insignificant_terms, _, _ = self.perform_t_test( psi, self.theta, residues ) pos_aux = np.where(agent == 1)[0] pos_aux = pos_aux[pos_insignificant_terms] agent[pos_aux] = 0 m = self.regressor_code[agent == 1].copy() if np.all(agent == 0): fitness.append(1000) # just a big number as penalty continue yhat = self.simulate( X_train=X_train, y_train=y_train, X_test=X_test, y_test=y_test, model_code=m, steps_ahead=self.steps_ahead, ) self.final_model = m.copy() self.tested_models.append(m) if len(self.theta) == 0: print(m) d = getattr(self, self.loss_func)(y_test, yhat, len(self.theta)) fitness.append(d) return fitness def perform_t_test( self, psi: np.ndarray, theta: np.ndarray, residues: np.ndarray ) -> Tuple[np.ndarray, np.ndarray, np.ndarray]: """Perform the t-test given the p-value defined by the user. Parameters ---------- psi : array the data matrix of regressors theta : array the parameters estimated via least squares algorithm residues : array the identification residues of the solution Returns ------- pos_insignificant_terms : array these regressors in the actual candidate solution are removed from the population since they are insignificant t_test : array the values of the p_value of each regressor of the model tail2P: array The calculated two-tailed p-value. """ sum_of_squared_residues = np.sum(residues**2) variance_of_residues = (sum_of_squared_residues) / ( len(residues) - psi.shape[1] ) if np.isnan(variance_of_residues): variance_of_residues = 4.3645e05 skk = np.linalg.pinv(psi.T.dot(psi)) skk_diag = np.diag(skk) var_e = variance_of_residues * skk_diag se_theta = np.sqrt(var_e) se_theta = se_theta.reshape(-1, 1) t_test = theta / se_theta degree_of_freedom = psi.shape[0] - psi.shape[1] tail2P = 2 * t.cdf(-np.abs(t_test), degree_of_freedom) pos_insignificant_terms = np.where(tail2P > self.p_value)[0] pos_insignificant_terms = pos_insignificant_terms.reshape(-1, 1).T if pos_insignificant_terms.shape == 0: return np.array([]), t_test, tail2P # t_test and tail2P will be returned in future updates return pos_insignificant_terms, t_test, tail2P def aic(self, y_test: np.ndarray, yhat: np.ndarray, n_theta: int) -> float: """Calculate the Akaike Information Criterion. Parameters ---------- y_test : ndarray of floats The output data (initial conditions) to be used in the prediction process. yhat : ndarray of floats The n-steps-ahead predicted values of the model. n_theta : ndarray of floats The number of model parameters. Returns ------- aic : float The Akaike Information Criterion """ mse = mean_squared_error(y_test, yhat) n = y_test.shape[0] return n * np.log(mse) + 2 * n_theta def bic(self, y_test: np.ndarray, yhat: np.ndarray, n_theta: int) -> float: """Calculate the Bayesian Information Criterion. Parameters ---------- y_test : ndarray of floats The output data (initial conditions) to be used in the prediction process. yhat : ndarray of floats The n-steps-ahead predicted values of the model. n_theta : ndarray of floats The number of model parameters. Returns ------- bic : float The Bayesian Information Criterion """ mse = mean_squared_error(y_test, yhat) n = y_test.shape[0] return n * np.log(mse) + n_theta + np.log(n) def metamss_loss(self, y_test: np.ndarray, yhat: np.ndarray, n_terms: int) -> float: """Calculate the MetaMSS loss function. Parameters ---------- y_test : ndarray of floats The output data (initial conditions) to be used in the prediction process. yhat : ndarray of floats The n-steps-ahead predicted values of the model. n_terms : ndarray of floats The number of model parameters. Returns ------- metamss_loss : float The MetaMSS loss function """ penalty_count = np.arange(0, self.dimension) penalty_distribution = (np.log(n_terms + 1) ** (-1)) / self.dimension penalty = self.sigmoid_linear_unit_derivative( penalty_count, self.dimension / 2, penalty_distribution ) penalty = penalty - np.min(penalty) rmse = root_relative_squared_error(y_test, yhat) fitness = rmse * penalty[n_terms] if np.isnan(fitness): fitness = 30 return fitness def sigmoid_linear_unit_derivative(self, x, c, a): """Calculate the derivative of the Sigmoid Linear Unit function. The derivative of Sigmoid Linear Unit (dSiLU) function can be viewed as a overshooting version of the sigmoid function. Parameters ---------- x : ndarray The range of the regressors space. a : float The rate of change. c : int Corresponds to the x value where y = 0.5. Returns ------- penalty : ndarray of floats The values of the penalty function """ return ( 1 / (1 + np.exp(-a * (x - c))) * (1 + (a * (x - c)) * (1 - 1 / (1 + np.exp(-a * (x - c))))) ) def predict( self, *, X: Optional[np.ndarray] = None, y: Optional[np.ndarray] = None, steps_ahead: Optional[int] = None, forecast_horizon: int = 1, ) -> 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 raise NotImplementedError( "MetaMSS doesn't support basis functions other than polynomial yet.", ) 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. """ yhat = super()._one_step_ahead_prediction(X, y) 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. 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, ): """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: 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, y_initial, forecast_horizon=None): """Not implemented.""" raise NotImplementedError( "You can only use Polynomial Basis Function in MetaMSS for now." ) def _basis_function_n_step_prediction(self, X, y, steps_ahead, forecast_horizon): """Not implemented.""" raise NotImplementedError( "You can only use Polynomial Basis Function in MetaMSS for now." ) def _basis_function_n_steps_horizon(self, X, y, steps_ahead, forecast_horizon): """Not implemented.""" raise NotImplementedError( "You can only use Polynomial Basis Function in MetaMSS for now." ) 

### aic(y_test, yhat, n_theta)¶

Calculate the Akaike Information Criterion.

Parameters:

Name Type Description Default
y_test ndarray of floats

The output data (initial conditions) to be used in the prediction process.

required
yhat ndarray of floats

The n-steps-ahead predicted values of the model.

required
n_theta ndarray of floats

The number of model parameters.

required

Returns:

Name Type Description
aic float

The Akaike Information Criterion

Source code in sysidentpy\model_structure_selection\meta_model_structure_selection.py
 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 def aic(self, y_test: np.ndarray, yhat: np.ndarray, n_theta: int) -> float: """Calculate the Akaike Information Criterion. Parameters ---------- y_test : ndarray of floats The output data (initial conditions) to be used in the prediction process. yhat : ndarray of floats The n-steps-ahead predicted values of the model. n_theta : ndarray of floats The number of model parameters. Returns ------- aic : float The Akaike Information Criterion """ mse = mean_squared_error(y_test, yhat) n = y_test.shape[0] return n * np.log(mse) + 2 * n_theta 

### bic(y_test, yhat, n_theta)¶

Calculate the Bayesian Information Criterion.

Parameters:

Name Type Description Default
y_test ndarray of floats

The output data (initial conditions) to be used in the prediction process.

required
yhat ndarray of floats

The n-steps-ahead predicted values of the model.

required
n_theta ndarray of floats

The number of model parameters.

required

Returns:

Name Type Description
bic float

The Bayesian Information Criterion

Source code in sysidentpy\model_structure_selection\meta_model_structure_selection.py
 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 def bic(self, y_test: np.ndarray, yhat: np.ndarray, n_theta: int) -> float: """Calculate the Bayesian Information Criterion. Parameters ---------- y_test : ndarray of floats The output data (initial conditions) to be used in the prediction process. yhat : ndarray of floats The n-steps-ahead predicted values of the model. n_theta : ndarray of floats The number of model parameters. Returns ------- bic : float The Bayesian Information Criterion """ mse = mean_squared_error(y_test, yhat) n = y_test.shape[0] return n * np.log(mse) + n_theta + np.log(n) 

### evaluate_objective_function(X_train, y_train, X_test, y_test, population)¶

Fit the polynomial NARMAX model.

Parameters:

Name Type Description Default
X_train ndarray of floats

The input data to be used in the training process.

required
y_train ndarray of floats

The output data to be used in the training process.

required
X_test ndarray of floats

The input data to be used in the prediction process.

required
y_test ndarray of floats

The output data (initial conditions) to be used in the prediction process.

required
population ndarray of zeros and ones

The initial population of agents.

required

Returns:

Name Type Description
fitness_value ndarray

The fitness value of each agent.

Source code in sysidentpy\model_structure_selection\meta_model_structure_selection.py
 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 def evaluate_objective_function( self, X_train: Optional[np.ndarray], y_train: Optional[np.ndarray], X_test: Optional[np.ndarray], y_test: Optional[np.ndarray], population: np.ndarray, ): """Fit the polynomial NARMAX model. Parameters ---------- X_train : ndarray of floats The input data to be used in the training process. y_train : ndarray of floats The output data to be used in the training process. X_test : ndarray of floats The input data to be used in the prediction process. y_test : ndarray of floats The output data (initial conditions) to be used in the prediction process. population : ndarray of zeros and ones The initial population of agents. Returns ------- fitness_value : ndarray The fitness value of each agent. """ fitness = [] for agent in population.T: if np.all(agent == 0): fitness.append(30) # penalty for cases where there is no terms continue m = self.regressor_code[agent == 1].copy() yhat = self.simulate( X_train=X_train, y_train=y_train, X_test=X_test, y_test=y_test, model_code=m, steps_ahead=self.steps_ahead, ) residues = y_test - yhat self.max_lag = self._get_max_lag() lagged_data = self.build_matrix(X_train, y_train) psi = self.basis_function.fit( lagged_data, self.max_lag, predefined_regressors=self.pivv ) pos_insignificant_terms, _, _ = self.perform_t_test( psi, self.theta, residues ) pos_aux = np.where(agent == 1)[0] pos_aux = pos_aux[pos_insignificant_terms] agent[pos_aux] = 0 m = self.regressor_code[agent == 1].copy() if np.all(agent == 0): fitness.append(1000) # just a big number as penalty continue yhat = self.simulate( X_train=X_train, y_train=y_train, X_test=X_test, y_test=y_test, model_code=m, steps_ahead=self.steps_ahead, ) self.final_model = m.copy() self.tested_models.append(m) if len(self.theta) == 0: print(m) d = getattr(self, self.loss_func)(y_test, yhat, len(self.theta)) fitness.append(d) return fitness 

### fit(*, X=None, y=None, X_test=None, y_test=None)¶

Fit the polynomial NARMAX model.

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
X_test ndarray of floats

The input data to be used in the prediction process.

None
y_test ndarray of floats

The output data (initial conditions) to be used in the prediction process.

None

Returns:

Name Type Description
self returns an instance of self.
Source code in sysidentpy\model_structure_selection\meta_model_structure_selection.py
 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 @deprecated( version="v0.3.0", future_version="v0.4.0", message=( "You will not need to pass X_test and y_test in v0.4.0." " \n You'll have to use MetaMSS(test_size=0.25) instead. \n This" " change will make easier to use the MetaMSS model and will" " follow the same structure of the other methods." ), ) def fit( self, *, X: Optional[np.ndarray] = None, y: Optional[np.ndarray] = None, X_test: Optional[np.ndarray] = None, y_test: Optional[np.ndarray] = None, ): """Fit the polynomial NARMAX model. 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. X_test : ndarray of floats The input data to be used in the prediction process. y_test : ndarray of floats The output data (initial conditions) to be used in the prediction process. Returns ------- self : returns an instance of self. """ if self.basis_function.__class__.__name__ != "Polynomial": raise NotImplementedError( "Currently MetaMSS only supports polynomial models." ) if y is None: raise ValueError("y cannot be None") if X is not None: check_X_y(X, y) self.n_inputs = _num_features(X) else: self.n_inputs = 1 # just to create the regressor space base # self.n_inputs = _num_features(X_train) self.max_lag = self._get_max_lag() self.regressor_code = self.regressor_space(self.n_inputs) self.dimension = self.regressor_code.shape[0] velocity = np.zeros([self.dimension, self.n_agents]) self.random_state = check_random_state(self.random_state) population = self.generate_random_population(self.random_state) self.best_by_iter = [] self.mean_by_iter = [] self.optimal_fitness_value = np.inf self.optimal_model = None self.best_model_history = [] self.tested_models = [] for i in range(self.maxiter): fitness = self.evaluate_objective_function(X, y, X_test, y_test, population) column_of_best_solution = np.nanargmin(fitness) current_best_fitness = fitness[column_of_best_solution] if current_best_fitness < self.optimal_fitness_value: self.optimal_fitness_value = current_best_fitness self.optimal_model = population[:, column_of_best_solution].copy() self.best_model_history.append(self.optimal_model) self.best_by_iter.append(self.optimal_fitness_value) self.mean_by_iter.append(np.mean(fitness)) agent_mass = self.mass_calculation(fitness) gravitational_constant = self.calculate_gravitational_constant(i) acceleration = self.calculate_acceleration( population, agent_mass, gravitational_constant, i ) velocity, population = self.update_velocity_position( population, acceleration, velocity, i, ) self.final_model = self.regressor_code[self.optimal_model == 1].copy() _ = self.simulate( X_train=X, y_train=y, X_test=X_test, y_test=y_test, model_code=self.final_model, steps_ahead=self.steps_ahead, ) self.max_lag = self._get_max_lag() return self 

### metamss_loss(y_test, yhat, n_terms)¶

Calculate the MetaMSS loss function.

Parameters:

Name Type Description Default
y_test ndarray of floats

The output data (initial conditions) to be used in the prediction process.

required
yhat ndarray of floats

The n-steps-ahead predicted values of the model.

required
n_terms ndarray of floats

The number of model parameters.

required

Returns:

Name Type Description
metamss_loss float

The MetaMSS loss function

Source code in sysidentpy\model_structure_selection\meta_model_structure_selection.py
 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 def metamss_loss(self, y_test: np.ndarray, yhat: np.ndarray, n_terms: int) -> float: """Calculate the MetaMSS loss function. Parameters ---------- y_test : ndarray of floats The output data (initial conditions) to be used in the prediction process. yhat : ndarray of floats The n-steps-ahead predicted values of the model. n_terms : ndarray of floats The number of model parameters. Returns ------- metamss_loss : float The MetaMSS loss function """ penalty_count = np.arange(0, self.dimension) penalty_distribution = (np.log(n_terms + 1) ** (-1)) / self.dimension penalty = self.sigmoid_linear_unit_derivative( penalty_count, self.dimension / 2, penalty_distribution ) penalty = penalty - np.min(penalty) rmse = root_relative_squared_error(y_test, yhat) fitness = rmse * penalty[n_terms] if np.isnan(fitness): fitness = 30 return fitness 

### perform_t_test(psi, theta, residues)¶

Perform the t-test given the p-value defined by the user.

Parameters:

Name Type Description Default
psi array

the data matrix of regressors

required
theta array

the parameters estimated via least squares algorithm

required
residues array

the identification residues of the solution

required

Returns:

Name Type Description
pos_insignificant_terms array

these regressors in the actual candidate solution are removed from the population since they are insignificant

t_test array

the values of the p_value of each regressor of the model

tail2P array

The calculated two-tailed p-value.

Source code in sysidentpy\model_structure_selection\meta_model_structure_selection.py
 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 def perform_t_test( self, psi: np.ndarray, theta: np.ndarray, residues: np.ndarray ) -> Tuple[np.ndarray, np.ndarray, np.ndarray]: """Perform the t-test given the p-value defined by the user. Parameters ---------- psi : array the data matrix of regressors theta : array the parameters estimated via least squares algorithm residues : array the identification residues of the solution Returns ------- pos_insignificant_terms : array these regressors in the actual candidate solution are removed from the population since they are insignificant t_test : array the values of the p_value of each regressor of the model tail2P: array The calculated two-tailed p-value. """ sum_of_squared_residues = np.sum(residues**2) variance_of_residues = (sum_of_squared_residues) / ( len(residues) - psi.shape[1] ) if np.isnan(variance_of_residues): variance_of_residues = 4.3645e05 skk = np.linalg.pinv(psi.T.dot(psi)) skk_diag = np.diag(skk) var_e = variance_of_residues * skk_diag se_theta = np.sqrt(var_e) se_theta = se_theta.reshape(-1, 1) t_test = theta / se_theta degree_of_freedom = psi.shape[0] - psi.shape[1] tail2P = 2 * t.cdf(-np.abs(t_test), degree_of_freedom) pos_insignificant_terms = np.where(tail2P > self.p_value)[0] pos_insignificant_terms = pos_insignificant_terms.reshape(-1, 1).T if pos_insignificant_terms.shape == 0: return np.array([]), t_test, tail2P # t_test and tail2P will be returned in future updates return pos_insignificant_terms, t_test, tail2P 

### predict(*, X=None, y=None, steps_ahead=None, forecast_horizon=1)¶

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\meta_model_structure_selection.py
 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 def predict( self, *, X: Optional[np.ndarray] = None, y: Optional[np.ndarray] = None, steps_ahead: Optional[int] = None, forecast_horizon: int = 1, ) -> 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 raise NotImplementedError( "MetaMSS doesn't support basis functions other than polynomial yet.", ) 

### sigmoid_linear_unit_derivative(x, c, a)¶

Calculate the derivative of the Sigmoid Linear Unit function.

The derivative of Sigmoid Linear Unit (dSiLU) function can be viewed as a overshooting version of the sigmoid function.

Parameters:

Name Type Description Default
x ndarray

The range of the regressors space.

required
a float

The rate of change.

required
c int

Corresponds to the x value where y = 0.5.

required

Returns:

Name Type Description
penalty ndarray of floats

The values of the penalty function

Source code in sysidentpy\model_structure_selection\meta_model_structure_selection.py
 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 def sigmoid_linear_unit_derivative(self, x, c, a): """Calculate the derivative of the Sigmoid Linear Unit function. The derivative of Sigmoid Linear Unit (dSiLU) function can be viewed as a overshooting version of the sigmoid function. Parameters ---------- x : ndarray The range of the regressors space. a : float The rate of change. c : int Corresponds to the x value where y = 0.5. Returns ------- penalty : ndarray of floats The values of the penalty function """ return ( 1 / (1 + np.exp(-a * (x - c))) * (1 + (a * (x - c)) * (1 - 1 / (1 + np.exp(-a * (x - c))))) )