Codes

sysidentpy base

Base classes for NARX estimator.

class sysidentpy.base.GenerateRegressors[source]

Polynomial NARMAX model

Methods

regressor_space(self, non_degree, xlag, …)

Create the code representation of the regressors.
regressor_space(self, non_degree, xlag, ylag, n_inputs)[source]

Create the code representation of the regressors.

This function generates a codification from all possibles regressors given the maximum lag of the input and output. This is used to write the final terms of the model in a readable form. [1001] -> y(k-1). This code format was based on a dissertation from UFMG. See reference below.

Parameters

  • non_degree (int) – The desired maximum nonlinearity degree.

  • ylag (int) – The maximum lag of output regressors.

  • xlag (int) – The maximum lag of input regressors.

Returns

  • max_lag (int) – This value can be used by another functions.

  • regressor_code (ndarray of int) – Matrix codification of all possible regressors.

Examples

The codification is defined as:

>>> 100n = y(k-n)
>>> 200n = u(k-n)
>>> [100n 100n] = y(k-n)y(k-n)
>>> [200n 200n] = u(k-n)u(k-n)

References

[1]`Master Thesis: Barbosa, Alípio Monteiro. “Técnicas de otimizaçao bi-objetivo para a determinaçao

da estrutura de modelos NARX.” (2010).

<https://repositorio.ufmg.br/bitstream/1843/BUOS-8EXJV3/1/7m.pdf>`_

class sysidentpy.base.HouseHolder[source]

Householder reflection and transformation.

class sysidentpy.base.InformationMatrix[source]

Class for methods regarding preprocessing of columns

Methods

build_information_matrix(self, X, y, xlag, …)

Build the information matrix.

initial_lagged_matrix(self, X, y, xlag, ylag)

Build a lagged matrix concerning each lag for each column.

shift_column(self, col_to_shift, lag)

Shift values based on a lag.
shift_column(self, col_to_shift, lag)[source]

Shift values based on a lag.

Parameters

  • col_to_shift (array-like of shape = n_samples) – The samples of the input or output.

  • lag (int) – The respective lag of the regressor.

Returns

tmp_column – The shifted array of the input or output.

Return type

array-like of shape = n_samples

Examples

>>> y = [1, 2, 3, 4, 5]
>>> shift_column(y, 1)
[0, 1, 2, 3, 4]
initial_lagged_matrix(self, X, y, xlag, ylag)[source]

Build a lagged matrix concerning each lag for each column.

Parameters

  • model (ndarray of int) – The model code representation.

  • y (array-like) – Target data used on training phase.

  • X (array-like) – Input data used on training phase.

  • ylag (int) – The maximum lag of output regressors.

  • xlag (int) – The maximum lag of input regressors.

Returns

lagged_data – The lagged matrix built in respect with each lag and column.

Return type

ndarray of floats

Examples

Let X and y be the input and output values of shape Nx1. If the chosen lags are 2 for both input and output the initial lagged matrix will be formed by Y[k-1], Y[k-2], X[k-1], and X[k-2].

build_information_matrix(self, X, y, xlag, ylag, non_degree)[source]

Build the information matrix.

Each columns of the information matrix represents a candidate regressor. The set of candidate regressors are based on xlag, ylag, and non_degree entered by the user.

Parameters

  • model (ndarray of int) – The model code representation.

  • y (array-like) – Target data used on training phase.

  • X (array-like) – Input data used on training phase.

  • ylag (int) – The maximum lag of output regressors.

  • xlag (int) – The maximum lag of input regressors.

  • non_degree (int) – The desired maximum nonlinearity degree.

Returns

The lagged matrix built in respect with each lag and column.

Return type

lagged_data = ndarray of floats

sysidentpy main

sysidentpy residues

class sysidentpy.residues.residues_correlation.ResiduesAnalysis[source]

Bases: object

Residues analysis for Polynomial NARX model.

Methods

plot_result(self, y, yhat, e_acf, xe_ccf[, …])

Plot the free run simulation and residues analysis.

residuals(self, X, y, yhat)

Performs the residual analysis of output to validate model.
residuals(self, X, y, yhat)[source]

Performs the residual analysis of output to validate model.

Parameters

  • y (array-like of shape = n_samples) – The target data used in the identification process.

  • yhat (array-like of shape = n_samples) – The prediction values of the identification process.

  • X (ndarray of floats) – The input data.

Returns

  • output_autocorr (ndarray of floats:) – 1st column - Residuals normalized autocorrelation. 2nd/3rd columns - Superior and inferior limits of a 95% confidence interval.

  • output_crosscorr (ndarray of floats:) – 1st column - Correlation between residuals and input. 2nd/3rd columns - Superior and inferior limits of a 95% confidence interval.

References

[1] `Wikipedia entry on the Autocorrelation

<https://en.wikipedia.org/wiki/Autocorrelation>`_

Examples

>>> y = [3, -0.5, 2, 7]
>>> autocorr(y)
[62.25 11.5   2.5  21.  ]
_input_ccf(self, X, e, len_confidence)[source]
_residuals_acf(self, y, yhat)[source]
_normalized_correlation(self, signal1, signal2)[source]

Compute the normalized correlation between two signals.

Parameters

  • signal1 (array-like of shape = n_samples.) –

  • signal2 (array-like of shape = n_samples.) –

Returns

ruy – The normalized cross correlation between the two signals.

Return type

ndarray of floats:

plot_result(self, y, yhat, e_acf, xe_ccf, figsize=10, 8, n=100)[source]

Plot the free run simulation and residues analysis.

Parameters

  • y (array-like of shape = n_samples) – The target data used in the identification process.

  • yhat (array-like of shape = n_samples) – The prediction values of the identification process.

  • e_acf (ndarray of floats:) – 1st column - Residuals normalized autocorrelation. 2nd/3rd columns - Superior and inferior limits of a 95% confidence interval.

  • xe_ccf (ndarray of floats:) – 1st column - Correlation between residuals and input. 2nd/3rd columns - Superior and inferior limits of a 95% confidence interval.

__dict__ = mappingproxy({'__module__': 'sysidentpy.residues.residues_correlation', '__doc__': 'Residues analysis for Polynomial NARX model.', 'residuals': <function ResiduesAnalysis.residuals>, '_input_ccf': <function ResiduesAnalysis._input_ccf>, '_residuals_acf': <function ResiduesAnalysis._residuals_acf>, '_normalized_correlation': <function ResiduesAnalysis._normalized_correlation>, 'plot_result': <function ResiduesAnalysis.plot_result>, '__dict__': <attribute '__dict__' of 'ResiduesAnalysis' objects>, '__weakref__': <attribute '__weakref__' of 'ResiduesAnalysis' objects>})
__module__ = 'sysidentpy.residues.residues_correlation'
__weakref__

list of weak references to the object (if defined)

sysidentpy metrics

Common metrics to assess performance on NARX models.

sysidentpy.metrics._regression.forecast_error(y, y_predicted)[source]

Calculate the forecast error in a regression model.

Parameters

  • y (array-like of shape = number_of_outputs) – Represent the target values.

  • y_predicted (array-like of shape = number_of_outputs) – Target values predicted by the model.

Returns

loss – The difference between the true target values and the predicted or forecast value in regression or any other phenomenon.

Return type

ndarray of floats

References

[1] `Wikipedia entry on the Forecast error

<https://en.wikipedia.org/wiki/Forecast_error>`_

Examples

>>> y = [3, -0.5, 2, 7]
>>> y_predicted = [2.5, 0.0, 2, 8]
>>> forecast_error(y, y_predicted)
[0.5, -0.5, 0, -1]
sysidentpy.metrics._regression.mean_forecast_error(y, y_predicted)[source]

Calculate the mean of forecast error of a regression model.

Parameters

  • y (array-like of shape = number_of_outputs) – Represent the target values.

  • y_predicted (array-like of shape = number_of_outputs) – Target values predicted by the model.

Returns

loss – The mean value of the difference between the true target values and the predicted or forecast value in regression or any other phenomenon.

Return type

float

References

[1] `Wikipedia entry on the Forecast error

<https://en.wikipedia.org/wiki/Forecast_error>`_

Examples

>>> y = [3, -0.5, 2, 7]
>>> y_predicted = [2.5, 0.0, 2, 8]
>>> mean_forecast_error(y, y_predicted)
-0.25
sysidentpy.metrics._regression.mean_squared_error(y, y_predicted)[source]

Calculate the Mean Squared Error.

Parameters

  • y (array-like of shape = number_of_outputs) – Represent the target values.

  • y_predicted (array-like of shape = number_of_outputs) – Target values predicted by the model.

Returns

loss – MSE output is non-negative values. Becoming 0.0 means your model outputs are exactly matched by true target values.

Return type

float

References

[1] `Wikipedia entry on the Mean Squared Error

<https://en.wikipedia.org/wiki/Mean_squared_error>`_

Examples

>>> y = [3, -0.5, 2, 7]
>>> y_predicted = [2.5, 0.0, 2, 8]
>>> mean_squared_error(y, y_predicted)
0.375
sysidentpy.metrics._regression.root_mean_squared_error(y, y_predicted)[source]

Calculate the Root Mean Squared Error.

Parameters

  • y (array-like of shape = number_of_outputs) – Represent the target values.

  • y_predicted (array-like of shape = number_of_outputs) – Target values predicted by the model.

Returns

loss – RMSE output is non-negative values. Becoming 0.0 means your model outputs are exactly matched by true target values.

Return type

float

References

[1] `Wikipedia entry on the Root Mean Squared Error

<https://en.wikipedia.org/wiki/Root-mean-square_deviation>`_

Examples

>>> y = [3, -0.5, 2, 7]
>>> y_predicted = [2.5, 0.0, 2, 8]
>>> root_mean_squared_error(y, y_predicted)
0.612
sysidentpy.metrics._regression.normalized_root_mean_squared_error(y, y_predicted)[source]

Calculate the normalized Root Mean Squared Error.

Parameters

  • y (array-like of shape = number_of_outputs) – Represent the target values.

  • y_predicted (array-like of shape = number_of_outputs) – Target values predicted by the model.

Returns

loss – nRMSE output is non-negative values. Becoming 0.0 means your model outputs are exactly matched by true target values.

Return type

float

References

[1] `Wikipedia entry on the normalized Root Mean Squared Error

<https://en.wikipedia.org/wiki/Root-mean-square_deviation>`_

Examples

>>> y = [3, -0.5, 2, 7]
>>> y_predicted = [2.5, 0.0, 2, 8]
>>> normalized_root_mean_squared_error(y, y_predicted)
0.081
sysidentpy.metrics._regression.root_relative_squared_error(y, y_predicted)[source]

Calculate the Root Relative Mean Squared Error.

Parameters

  • y (array-like of shape = number_of_outputs) – Represent the target values.

  • y_predicted (array-like of shape = number_of_outputs) – Target values predicted by the model.

Returns

loss – RRSE output is non-negative values. Becoming 0.0 means your model outputs are exactly matched by true target values.

Return type

float

Examples

>>> y = [3, -0.5, 2, 7]
>>> y_predicted = [2.5, 0.0, 2, 8]
>>> root_relative_mean_squared_error(y, y_predicted)
0.206
sysidentpy.metrics._regression.mean_absolute_error(y, y_predicted)[source]

Calculate the Mean absolute error.

Parameters

  • y (array-like of shape = number_of_outputs) – Represent the target values.

  • y_predicted (array-like of shape = number_of_outputs) – Target values predicted by the model.

Returns

loss – MAE output is non-negative values. Becoming 0.0 means your model outputs are exactly matched by true target values.

Return type

float or ndarray of floats

References

[1] `Wikipedia entry on the Mean absolute error

<https://en.wikipedia.org/wiki/Mean_absolute_error>`_

Examples

>>> y = [3, -0.5, 2, 7]
>>> y_predicted = [2.5, 0.0, 2, 8]
>>> mean_absolute_error(y, y_predicted)
0.5
sysidentpy.metrics._regression.mean_squared_log_error(y, y_predicted)[source]

Calculate the Mean Squared Logarithmic Error.

Parameters

  • y (array-like of shape = number_of_outputs) – Represent the target values.

  • y_predicted (array-like of shape = number_of_outputs) – Target values predicted by the model.

Returns

loss – MSLE output is non-negative values. Becoming 0.0 means your model outputs are exactly matched by true target values.

Return type

float

Examples

>>> y = [3, 5, 2.5, 7]
>>> y_predicted = [2.5, 5, 4, 8]
>>> mean_squared_log_error(y, y_predicted)
0.039
sysidentpy.metrics._regression.median_absolute_error(y, y_predicted)[source]

Calculate the Median Absolute Error.

Parameters

  • y (array-like of shape = number_of_outputs) – Represent the target values.

  • y_predicted (array-like of shape = number_of_outputs) – Target values predicted by the model.

Returns

loss – MdAE output is non-negative values. Becoming 0.0 means your model outputs are exactly matched by true target values.

Return type

float

References

[1] `Wikipedia entry on the Median absolute deviation

<https://en.wikipedia.org/wiki/Median_absolute_deviation>`_

Examples

>>> y = [3, -0.5, 2, 7]
>>> y_predicted = [2.5, 0.0, 2, 8]
>>> median_absolute_error(y, y_predicted)
0.5
sysidentpy.metrics._regression.explained_variance_score(y, y_predicted)[source]

Calculate the Explained Variance Score.

Parameters

  • y (array-like of shape = number_of_outputs) – Represent the target values.

  • y_predicted (array-like of shape = number_of_outputs) – Target values predicted by the model.

Returns

loss – EVS output is non-negative values. Becoming 1.0 means your model outputs are exactly matched by true target values. Lower values means worse results.

Return type

float

References

[1] `Wikipedia entry on the Explained Variance

<https://en.wikipedia.org/wiki/Explained_variation>`_

Examples

>>> y = [3, -0.5, 2, 7]
>>> y_predicted = [2.5, 0.0, 2, 8]
>>> explained_variance_score(y, y_predicted)
0.957
sysidentpy.metrics._regression.r2_score(y, y_predicted)[source]

Calculate the R2 score.

Parameters

  • y (array-like of shape = number_of_outputs) – Represent the target values.

  • y_predicted (array-like of shape = number_of_outputs) – Target values predicted by the model.

Returns

loss – R2 output can be non-negative values or negative value. Becoming 1.0 means your model outputs are exactly matched by true target values. Lower values means worse results.

Return type

float

Notes

This is not a symmetric function.

References

[1] `Wikipedia entry on the Coefficient of determination

<https://en.wikipedia.org/wiki/Coefficient_of_determination>`_

Examples

>>> y = [3, -0.5, 2, 7]
>>> y_predicted = [2.5, 0.0, 2, 8]
>>> explained_variance_score(y, y_predicted)
0.948
sysidentpy.metrics._regression.symmetric_mean_absolute_percentage_error(y, y_predicted)[source]

Calculate the SMAPE score.

Parameters

  • y (array-like of shape = number_of_outputs) – Represent the target values.

  • y_predicted (array-like of shape = number_of_outputs) – Target values predicted by the model.

Returns

loss – SMAPE output is a non-negative value. The results are percentages values.

Return type

float

Notes

One supposed problem with SMAPE is that it is not symmetric since over-forecasts and under-forecasts are not treated equally.

References

[1] `Wikipedia entry on the Symmetric mean absolute percentage error

<https://en.wikipedia.org/wiki/Symmetric_mean_absolute_percentage_error>`_

Examples

>>> y = [3, -0.5, 2, 7]
>>> y_predicted = [2.5, 0.0, 2, 8]
>>> symmetric_mean_absolute_percentage_error(y, y_predicted)
57.87

sysidentpy estimators

Least Squares Methodos for parameter estimation

class sysidentpy.parameter_estimation.estimators.Estimators(aux_lag=1, lam=0.98, delta=0.01, offset_covariance=0.2, mu=0.01, eps=2.220446049250313e-16, gama=0.2, weight=0.02)[source]

Oridanry Least squares for linear parameter estimation

Methods

affine_least_mean_squares(self, psi, y)

Estimate the model parameters using the Affine Least Mean Squares.

least_mean_squares(self, psi, y)

Estimate the model parameters using the Least Mean Squares filter.

least_mean_squares_fourth(self, psi, y)

Parameter estimation using the LMS Fourth filter.

least_mean_squares_leaky(self, psi, y)

Parameter estimation using the Leaky LMS filter.

least_mean_squares_mixed_norm(self, psi, y)

Parameter estimation using the Mixed-norm LMS filter.

least_mean_squares_normalized_leaky(self, psi, y)

Parameter estimation using the Normalized Leaky LMS filter.

least_mean_squares_normalized_sign_error(…)

Parameter estimation using the Normalized Sign-Error LMS filter.

least_mean_squares_normalized_sign_regressor(…)

Parameter estimation using the Normalized Sign-Regressor LMS filter.

least_mean_squares_normalized_sign_sign(…)

Parameter estimation using the Normalized Sign-Sign LMS filter.

least_mean_squares_sign_error(self, psi, y)

Parameter estimation using the Sign-Error Least Mean Squares filter.

least_mean_squares_sign_regressor(self, psi, y)

Parameter estimation using the Sign-Regressor LMS filter.

least_mean_squares_sign_sign(self, psi, y)

Parameter estimation using the Sign-Sign LMS filter.

least_squares(self, psi, y)

Estimate the model parameters using Least Squares method.

normalized_least_mean_squares(self, psi, y)

Parameter estimation using the Normalized Least Mean Squares filter.

recursive_least_squares(self, psi, y)

Estimate the model parameters using the Recursive Least Squares method.

total_least_squares(self, psi, y)

Estimate the model parameters using Total Least Squares method.
least_squares(self, psi, y)[source]

Estimate the model parameters using Least Squares method.

Parameters

  • psi (ndarray of floats) – The information matrix of the model.

  • y_train (array-like of shape = y_training) – The data used to training the model.

Returns

theta – The estimated parameters of the model.

Return type

array-like of shape = number_of_model_elements

References

[1]`Manuscript: Sorenson, H. W. (1970). Least-squares estimation:

from Gauss to Kalman. IEEE spectrum, 7(7), 63-68. <http://pzs.dstu.dp.ua/DataMining/mls/bibl/Gauss2Kalman.pdf>`_

[2]`Book (Portuguese): Aguirre, L. A. (2007). Introduçaoa identificaçao

de sistemas: técnicas lineares enao-lineares aplicadas a sistemas reais. Editora da UFMG. 3a ediçao. <https://books.google.com.br/books?hl=pt-BR&lr=&id=f9IwE7Ph0fYC&oi=fnd&pg=PA2&dq=Introdu%C3%A7%C3%A3o+%C3%A0+identifica%C3%A7%C3%A3o+de+sistemas+-+T%C3%A9cnicas+lineares+e+n%C3%A3o-lineares+aplicadas+a+sistemas+reais&ots=Qiyc4VsMdt&sig=6gumj1AEWh_b0tUGR4quI5oETUA#v=onepage&q=Introdu%C3%A7%C3%A3o%20%C3%A0%20identifica%C3%A7%C3%A3o%20de%20sistemas%20-%20T%C3%A9cnicas%20lineares%20e%20n%C3%A3o-lineares%20aplicadas%20a%20sistemas%20reais&f=false>`_

[3]`Manuscript: Markovsky, I., & Van Huffel, S. (2007).

Overview of total least-squares methods. Signal processing, 87(10), 2283-2302. <https://eprints.soton.ac.uk/263855/1/tls_overview.pdf>`_

[4]`Wikipedia entry on Least Squares

<https://en.wikipedia.org/wiki/Least_squares>`_

total_least_squares(self, psi, y)[source]

Estimate the model parameters using Total Least Squares method.

Parameters

  • psi (ndarray of floats) – The information matrix of the model.

  • y_train (array-like of shape = y_training) – The data used to training the model.

Returns

theta – The estimated parameters of the model.

Return type

array-like of shape = number_of_model_elements

References

[1]`Manuscript: Golub, G. H., & Van Loan, C. F. (1980).

An analysis of the total least squares problem. SIAM journal on numerical analysis, 17(6), 883-893. <https://epubs.siam.org/doi/pdf/10.1137/0717073?casa_token=218O16LygKkAAAAA:GyssnBnNEWzVg2Wvbmu5K1pj-XwkzpTSknUsddVTZfEJafpKANUstMuRDyJjIdcTgO-tFuQYb4Y>`_

[2]`Manuscript: Markovsky, I., & Van Huffel, S. (2007).

Overview of total least-squares methods. Signal processing, 87(10), 2283-2302. <https://eprints.soton.ac.uk/263855/1/tls_overview.pdf>`_

[3]`Wikipedia entry on Total Least Squares

<https://en.wikipedia.org/wiki/Total_least_squares>`_

recursive_least_squares(self, psi, y)[source]

Estimate the model parameters using the Recursive Least Squares method.

The implementation consider the forgeting factor. :param psi: The information matrix of the model. :type psi: ndarray of floats :param y_train: The data used to training the model. :type y_train: array-like of shape = y_training

Returns

theta – The estimated parameters of the model.

Return type

array-like of shape = number_of_model_elements

Notes

A more in-depth documentation of all methods for parameters estimation will be available soon. For now, please refer to the mentioned references.

References

[1]`Book (Portuguese): Aguirre, L. A. (2007). Introduçaoa identificaçao

de sistemas: técnicas lineares enao-lineares aplicadas a sistemas reais. Editora da UFMG. 3a ediçao. <https://books.google.com.br/books?hl=pt-BR&lr=&id=f9IwE7Ph0fYC&oi=fnd&pg=PA2&dq=Introdu%C3%A7%C3%A3o+%C3%A0+identifica%C3%A7%C3%A3o+de+sistemas+-+T%C3%A9cnicas+lineares+e+n%C3%A3o-lineares+aplicadas+a+sistemas+reais&ots=Qiyc4VsMdt&sig=6gumj1AEWh_b0tUGR4quI5oETUA#v=onepage&q=Introdu%C3%A7%C3%A3o%20%C3%A0%20identifica%C3%A7%C3%A3o%20de%20sistemas%20-%20T%C3%A9cnicas%20lineares%20e%20n%C3%A3o-lineares%20aplicadas%20a%20sistemas%20reais&f=false>`_

affine_least_mean_squares(self, psi, y)[source]

Estimate the model parameters using the Affine Least Mean Squares.

Parameters

  • psi (ndarray of floats) – The information matrix of the model.

  • y_train (array-like of shape = y_training) – The data used to training the model.

Returns

theta – The estimated parameters of the model.

Return type

array-like of shape = number_of_model_elements

Notes

A more in-depth documentation of all methods for parameters estimation will be available soon. For now, please refer to the mentioned references.

References

[1]`Book: Poularikas, A. D. (2017). Adaptive filtering: Fundamentals

of least mean squares with MATLAB®. CRC Press. <https://books.google.com.br/books?hl=pt-BR&lr=&id=OJPSBQAAQBAJ&oi=fnd&pg=PP1&dq=adaptive+filtering+fundamentals+of+least+mean+squares+with+matlab&ots=dMNzB_2erC&sig=7l0VvIm9-GwUDgj0xuy1m0c0Gdo#v=onepage&q=adaptive%20filtering%20fundamentals%20of%20least%20mean%20squares%20with%20matlab&f=false>`_

least_mean_squares(self, psi, y)[source]

Estimate the model parameters using the Least Mean Squares filter.

Parameters

  • psi (ndarray of floats) – The information matrix of the model.

  • y_train (array-like of shape = y_training) – The data used to training the model.

Returns

theta – The estimated parameters of the model.

Return type

array-like of shape = number_of_model_elements

Notes

A more in-depth documentation of all methods for parameters estimation will be available soon. For now, please refer to the mentioned references.

References

[1]`Book: Haykin, S., & Widrow, B. (Eds.). (2003). Least-mean-square

adaptive filters (Vol. 31). John Wiley & Sons. <https://books.google.com.br/books?hl=pt-BR&lr=&id=U8X3mJtawUkC&oi=fnd&pg=PR9&dq=%22least+mean+square%22&ots=Bzp42ZklVe&sig=ZilhP9bYuuagpi30hrJk53sWj_8&redir_esc=y#v=onepage&q=%22least%20mean%20square%22&f=false>`_

[2]`Dissertation (Portuguese): Zipf, J. G. F. (2011). Classificação,

análise estatística e novas estratégias de algoritmos LMS de passo variável. <https://repositorio.ufsc.br/bitstream/handle/123456789/94953/296734.pdf?sequence=1>`_

[3]`Wikipedia entry on Least Mean Squares

<https://en.wikipedia.org/wiki/Least_mean_squares_filter>`_

least_mean_squares_sign_error(self, psi, y)[source]

Parameter estimation using the Sign-Error Least Mean Squares filter.

The sign-error LMS algorithm uses the sign of the error vector to change the filter coefficients.

Parameters

  • psi (ndarray of floats) – The information matrix of the model.

  • y_train (array-like of shape = y_training) – The data used to training the model.

Returns

theta – The estimated parameters of the model.

Return type

array-like of shape = number_of_model_elements

Notes

A more in-depth documentation of all methods for parameters estimation will be available soon. For now, please refer to the mentioned references.

References

[1]`Book: Hayes, M. H. (2009). Statistical digital signal processing

and modeling. John Wiley & Sons.

[2]`Dissertation (Portuguese): Zipf, J. G. F. (2011). Classificação,

análise estatística e novas estratégias de algoritmos LMS de passo variável. <https://repositorio.ufsc.br/bitstream/handle/123456789/94953/296734.pdf?sequence=1>`_

[3]`Wikipedia entry on Least Mean Squares

<https://en.wikipedia.org/wiki/Least_mean_squares_filter>`_

normalized_least_mean_squares(self, psi, y)[source]

Parameter estimation using the Normalized Least Mean Squares filter.

The normalization is used to avoid numerical instability when updating the estimated parameters.

Parameters

  • psi (ndarray of floats) – The information matrix of the model.

  • y_train (array-like of shape = y_training) – The data used to training the model.

Returns

theta – The estimated parameters of the model.

Return type

array-like of shape = number_of_model_elements

Notes

A more in-depth documentation of all methods for parameters estimation will be available soon. For now, please refer to the mentioned references.

References

[1]`Book: Hayes, M. H. (2009). Statistical digital signal processing

and modeling. John Wiley & Sons.

[2]`Dissertation (Portuguese): Zipf, J. G. F. (2011). Classificação,

análise estatística e novas estratégias de algoritmos LMS de passo variável. <https://repositorio.ufsc.br/bitstream/handle/123456789/94953/296734.pdf?sequence=1>`_

[3]`Wikipedia entry on Least Mean Squares

<https://en.wikipedia.org/wiki/Least_mean_squares_filter>`_

least_mean_squares_normalized_sign_error(self, psi, y)[source]

Parameter estimation using the Normalized Sign-Error LMS filter.

The normalization is used to avoid numerical instability when updating the estimated parameters and the sign of the error vector is used to to change the filter coefficients.

Parameters

  • psi (ndarray of floats) – The information matrix of the model.

  • y_train (array-like of shape = y_training) – The data used to training the model.

Returns

theta – The estimated parameters of the model.

Return type

array-like of shape = number_of_model_elements

Notes

A more in-depth documentation of all methods for parameters estimation will be available soon. For now, please refer to the mentioned references.

References

[1]`Book: Hayes, M. H. (2009). Statistical digital signal processing

and modeling. John Wiley & Sons.

[2]`Dissertation (Portuguese): Zipf, J. G. F. (2011). Classificação,

análise estatística e novas estratégias de algoritmos LMS de passo variável. <https://repositorio.ufsc.br/bitstream/handle/123456789/94953/296734.pdf?sequence=1>`_

[3]`Wikipedia entry on Least Mean Squares

<https://en.wikipedia.org/wiki/Least_mean_squares_filter>`_

least_mean_squares_sign_regressor(self, psi, y)[source]

Parameter estimation using the Sign-Regressor LMS filter.

The sign-regressor LMS algorithm uses the sign of the matrix information to change the filter coefficients.

Parameters

  • psi (ndarray of floats) – The information matrix of the model.

  • y_train (array-like of shape = y_training) – The data used to training the model.

Returns

theta – The estimated parameters of the model.

Return type

array-like of shape = number_of_model_elements

Notes

A more in-depth documentation of all methods for parameters estimation will be available soon. For now, please refer to the mentioned references.

References

[1]`Book: Hayes, M. H. (2009). Statistical digital signal processing

and modeling. John Wiley & Sons.

[2]`Dissertation (Portuguese): Zipf, J. G. F. (2011). Classificação,

análise estatística e novas estratégias de algoritmos LMS de passo variável. <https://repositorio.ufsc.br/bitstream/handle/123456789/94953/296734.pdf?sequence=1>`_

[3]`Wikipedia entry on Least Mean Squares

<https://en.wikipedia.org/wiki/Least_mean_squares_filter>`_

least_mean_squares_normalized_sign_regressor(self, psi, y)[source]

Parameter estimation using the Normalized Sign-Regressor LMS filter.

The normalization is used to avoid numerical instability when updating the estimated parameters and the sign of the information matrix is used to change the filter coefficients.

Parameters

  • psi (ndarray of floats) – The information matrix of the model.

  • y_train (array-like of shape = y_training) – The data used to training the model.

Returns

theta – The estimated parameters of the model.

Return type

array-like of shape = number_of_model_elements

Notes

A more in-depth documentation of all methods for parameters estimation will be available soon. For now, please refer to the mentioned references.

References

[1]`Book: Hayes, M. H. (2009). Statistical digital signal processing

and modeling. John Wiley & Sons.

[2]`Dissertation (Portuguese): Zipf, J. G. F. (2011). Classificação,

análise estatística e novas estratégias de algoritmos LMS de passo variável. <https://repositorio.ufsc.br/bitstream/handle/123456789/94953/296734.pdf?sequence=1>`_

[3]`Wikipedia entry on Least Mean Squares

<https://en.wikipedia.org/wiki/Least_mean_squares_filter>`_

least_mean_squares_sign_sign(self, psi, y)[source]

Parameter estimation using the Sign-Sign LMS filter.

The sign-regressor LMS algorithm uses both the sign of the matrix information and the sign of the error vector to change the filter coefficients.

Parameters

  • psi (ndarray of floats) – The information matrix of the model.

  • y_train (array-like of shape = y_training) – The data used to training the model.

Returns

theta – The estimated parameters of the model.

Return type

array-like of shape = number_of_model_elements

Notes

A more in-depth documentation of all methods for parameters estimation will be available soon. For now, please refer to the mentioned references.

References

[1]`Book: Hayes, M. H. (2009). Statistical digital signal processing

and modeling. John Wiley & Sons.

[2]`Dissertation (Portuguese): Zipf, J. G. F. (2011). Classificação,

análise estatística e novas estratégias de algoritmos LMS de passo variável. <https://repositorio.ufsc.br/bitstream/handle/123456789/94953/296734.pdf?sequence=1>`_

[3]`Wikipedia entry on Least Mean Squares

<https://en.wikipedia.org/wiki/Least_mean_squares_filter>`_

least_mean_squares_normalized_sign_sign(self, psi, y)[source]

Parameter estimation using the Normalized Sign-Sign LMS filter.

The normalization is used to avoid numerical instability when updating the estimated parameters and both the sign of the information matrix and the sign of the error vector are used to change the filter coefficients.

Parameters

  • psi (ndarray of floats) – The information matrix of the model.

  • y_train (array-like of shape = y_training) – The data used to training the model.

Returns

theta – The estimated parameters of the model.

Return type

array-like of shape = number_of_model_elements

Notes

A more in-depth documentation of all methods for parameters estimation will be available soon. For now, please refer to the mentioned references.

References

[1]`Book: Hayes, M. H. (2009). Statistical digital signal processing

and modeling. John Wiley & Sons.

[2]`Dissertation (Portuguese): Zipf, J. G. F. (2011). Classificação,

análise estatística e novas estratégias de algoritmos LMS de passo variável. <https://repositorio.ufsc.br/bitstream/handle/123456789/94953/296734.pdf?sequence=1>`_

[3]`Wikipedia entry on Least Mean Squares

<https://en.wikipedia.org/wiki/Least_mean_squares_filter>`_

least_mean_squares_normalized_leaky(self, psi, y)[source]

Parameter estimation using the Normalized Leaky LMS filter.

When the leakage factor, gama, is set to 0 then there is no leakage in the estimation process.

Parameters

  • psi (ndarray of floats) – The information matrix of the model.

  • y_train (array-like of shape = y_training) – The data used to training the model.

Returns

theta – The estimated parameters of the model.

Return type

array-like of shape = number_of_model_elements

Notes

A more in-depth documentation of all methods for parameters estimation will be available soon. For now, please refer to the mentioned references.

References

[1]`Book: Hayes, M. H. (2009). Statistical digital signal processing

and modeling. John Wiley & Sons.

[2]`Dissertation (Portuguese): Zipf, J. G. F. (2011). Classificação,

análise estatística e novas estratégias de algoritmos LMS de passo variável. <https://repositorio.ufsc.br/bitstream/handle/123456789/94953/296734.pdf?sequence=1>`_

[3]`Wikipedia entry on Least Mean Squares

<https://en.wikipedia.org/wiki/Least_mean_squares_filter>`_

least_mean_squares_leaky(self, psi, y)[source]

Parameter estimation using the Leaky LMS filter.

When the leakage factor, gama, is set to 0 then there is no leakage in the estimation process.

Parameters

  • psi (ndarray of floats) – The information matrix of the model.

  • y_train (array-like of shape = y_training) – The data used to training the model.

Returns

theta – The estimated parameters of the model.

Return type

array-like of shape = number_of_model_elements

Notes

A more in-depth documentation of all methods for parameters estimation will be available soon. For now, please refer to the mentioned references.

References

[1]`Book: Hayes, M. H. (2009). Statistical digital signal processing

and modeling. John Wiley & Sons.

[2]`Dissertation (Portuguese): Zipf, J. G. F. (2011). Classificação,

análise estatística e novas estratégias de algoritmos LMS de passo variável. <https://repositorio.ufsc.br/bitstream/handle/123456789/94953/296734.pdf?sequence=1>`_

[3]`Wikipedia entry on Least Mean Squares

<https://en.wikipedia.org/wiki/Least_mean_squares_filter>`_

least_mean_squares_fourth(self, psi, y)[source]

Parameter estimation using the LMS Fourth filter.

When the leakage factor, gama, is set to 0 then there is no leakage in the estimation process.

Parameters

  • psi (ndarray of floats) – The information matrix of the model.

  • y_train (array-like of shape = y_training) – The data used to training the model.

Returns

theta – The estimated parameters of the model.

Return type

array-like of shape = number_of_model_elements

Notes

A more in-depth documentation of all methods for parameters estimation will be available soon. For now, please refer to the mentioned references.

References

[1]`Book: Hayes, M. H. (2009). Statistical digital signal processing

and modeling. John Wiley & Sons.

[2]`Dissertation (Portuguese): Zipf, J. G. F. (2011). Classificação,

análise estatística e novas estratégias de algoritmos LMS de passo variável. <https://repositorio.ufsc.br/bitstream/handle/123456789/94953/296734.pdf?sequence=1>`_

[3]`Manuscript:Gui, G., Mehbodniya, A., & Adachi, F. (2013).

Least mean square/fourth algorithm with application to sparse channel estimation. arXiv preprint arXiv:1304.3911. <https://arxiv.org/pdf/1304.3911.pdf>`_

[4]`Manuscript: Nascimento, V. H., & Bermudez, J. C. M. (2005, March).

When is the least-mean fourth algorithm mean-square stable? In Proceedings.(ICASSP’05). IEEE International Conference on Acoustics, Speech, and Signal Processing, 2005. (Vol. 4, pp. iv-341). IEEE. <http://www.lps.usp.br/vitor/artigos/icassp05.pdf>`_

[5]`Wikipedia entry on Least Mean Squares

<https://en.wikipedia.org/wiki/Least_mean_squares_filter>`_

least_mean_squares_mixed_norm(self, psi, y)[source]

Parameter estimation using the Mixed-norm LMS filter.

The weight factor controls the proportions of the error norms and offers an extra degree of freedom within the adaptation.

Parameters

  • psi (ndarray of floats) – The information matrix of the model.

  • y_train (array-like of shape = y_training) – The data used to training the model.

Returns

theta – The estimated parameters of the model.

Return type

array-like of shape = number_of_model_elements

Notes

A more in-depth documentation of all methods for parameters estimation will be available soon. For now, please refer to the mentioned references.

References

[1]`Chambers, J. A., Tanrikulu, O., & Constantinides, A. G. (1994).

Least mean mixed-norm adaptive filtering. Electronics letters, 30(19), 1574-1575. <https://ieeexplore.ieee.org/document/326382>`_

[2]`Dissertation (Portuguese): Zipf, J. G. F. (2011). Classificação,

análise estatística e novas estratégias de algoritmos LMS de passo variável. <https://repositorio.ufsc.br/bitstream/handle/123456789/94953/296734.pdf?sequence=1>`_

[3]`Wikipedia entry on Least Mean Squares

<https://en.wikipedia.org/wiki/Least_mean_squares_filter>`_

sysidentpy utils

Utilities fo data validation

sysidentpy.utils._check_arrays.check_infinity(X, y)[source]

Check that X and y have no NaN or Inf samples.

If there is any NaN or Inf samples a ValueError is raised.

Parameters

  • X (ndarray of floats) – The input data.

  • y (ndarray of floats) – The output data.

sysidentpy.utils._check_arrays.check_nan(X, y)[source]

Check that X and y have no NaN or Inf samples.

If there is any NaN or Inf samples a ValueError is raised.

Parameters

  • X (ndarray of floats) – The input data.

  • y (ndarray of floats) – The output data.

sysidentpy.utils._check_arrays.check_length(X, y)[source]

Check that X and y have the same number of samples.

If the length of X and y are different a ValueError is raised.

Parameters

  • X (ndarray of floats) – The input data.

  • y (ndarray of floats) – The output data.

sysidentpy.utils._check_arrays.check_dimension(X, y)[source]

Check if X and y have only real values.

If there is any string or object samples a ValueError is raised.

Parameters

  • X (ndarray of floats) – The input data.

  • y (ndarray of floats) – The output data.

sysidentpy.utils._check_arrays.check_X_y(X, y)[source]

Validate input and output data using some crucial tests.

Parameters

  • X (ndarray of floats) – The input data.

  • y (ndarray of floats) – The output data.

sysidentpy generate data

Utilities for data generation

sysidentpy.utils.generate_data.get_siso_data(n=5000, colored_noise=False, sigma=0.05, train_percentage=90)[source]

Perform the Error Reduction Ration algorithm.

Parameters

  • n (int) – The number of samples.

  • colored_noise (bool) – Select white noise or colored noise (autoregressive noise).

  • sigma (float) – The standard deviation of the random distribution to generate the noise.

  • train_percentage (int) – The percentage of the data to be used as train data.

Returns

  • x_train, x_valid (array-like) – The input data to be used in identification and validation, respectively.

  • y_train, y_valid (array-like) – The output data to be used in identification and validation, respectively.

sysidentpy.utils.generate_data.get_miso_data(n=5000, colored_noise=False, sigma=0.05, train_percentage=90)[source]

Perform the Error Reduction Ration algorithm.

Parameters

  • n (int) – The number of samples.

  • colored_noise (bool) – Select white noise or colored noise (autoregressive noise).

  • sigma (float) – The standard deviation of the random distribution to generate the noise.

  • train_percentage (int) – The percentage of the data to be used as train data.

Returns

  • x_train, x_valid (array-like) – The input data to be used in identification and validation, respectively.

  • y_train, y_valid (array-like) – The output data to be used in identification and validation, respectively.

Indices and tables