Welcome to SysIdentPy’s documentation!

SysIdentPy is a Python module for System Identification using NARMAX models built on top of numpy and is distributed under the 3-Clause BSD license.

The NARMAX model is described as:

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

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, \(\mathcal{F}\) is some nonlinear function of the input and output regressors and \(d\) is a time delay typically set to \(d=1\).

Note

The update v0.2.0 has been released with major changes and additional features.

There are several API modifications and you will need to change your code to have the new (and upcoming) features.

Check the examples of how to use the new version in the documentation page

For more details, please see the changelog

See also

The examples directory has several Jupyter notebooks presenting basic tutorials of how to use the package and some specific applications of SysIdentPy. Try it out!

Tip

SysIdentPy now support NARX Neural Network and General estimators, e.g., sklearn estimators and Catboost. Check it out!

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from sysidentpy.metrics import mean_squared_error
from sysidentpy.utils.generate_data import get_siso_data


# Generate a dataset of a simulated dynamical system
x_train, x_valid, y_train, y_valid = get_siso_data(
        n=1000,
        colored_noise=False,
sigma=0.001,
train_percentage=80
)

Polynomial NARX

from sysidentpy.model_structure_selection import FROLS
from sysidentpy.basis_function._basis_function import Polynomial
from sysidentpy.utils.display_results import results
from sysidentpy.utils.plotting import plot_residues_correlation, plot_results
from sysidentpy.residues.residues_correlation import compute_residues_autocorrelation, compute_cross_correlation

basis_function = Polynomial(degree=2)
model = FROLS(
        order_selection=True,
        n_info_values=10,
        extended_least_squares=False,
        ylag=2,
        xlag=2,
        info_criteria='aic',
        estimator='least_squares',
        basis_function=basis_function
)
model.fit(X=x_train, y=y_train)
yhat = model.predict(X=x_valid, y=y_valid)
rrse = root_relative_squared_error(y_valid, yhat)
print(rrse)
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  0.95556574
1         y(k-1)     0.1999  0.04107943
2  x1(k-1)y(k-1)     0.1000  0.00335113

plot_results(y=y_valid, yhat=yhat, n=1000)
ee = compute_residues_autocorrelation(y_valid, yhat)
plot_residues_correlation(data=ee, title="Residues", ylabel="$e^2$")
x1e = compute_cross_correlation(y_valid, yhat, x2_val)
plot_residues_correlation(data=x1e, title="Residues", ylabel="$x_1e$")
_images/polynomial_narmax.png

NARX Neural Network

from torch import nn
from sysidentpy.neural_network import NARXNN
from sysidentpy.basis_function._basis_function import Polynomial
from sysidentpy.utils.plotting import plot_residues_correlation, plot_results
from sysidentpy.residues.residues_correlation import compute_residues_autocorrelation, compute_cross_correlation

class NARX(nn.Module):
        def __init__(self):
                super().__init__()
                self.lin = nn.Linear(4, 10)
                self.lin2 = nn.Linear(10, 10)
                self.lin3 = nn.Linear(10, 1)
                self.tanh = nn.Tanh()

        def forward(self, xb):
                z = self.lin(xb)
                z = self.tanh(z)
                z = self.lin2(z)
                z = self.tanh(z)
                z = self.lin3(z)
                return z

basis_function=Polynomial(degree=1)

narx_net = NARXNN(
        net=NARX(),
        ylag=2,
        xlag=2,
        basis_function=basis_function,
        model_type="NARMAX",
        loss_func='mse_loss',
        optimizer='Adam',
        epochs=200,
        verbose=False,
        optim_params={'betas': (0.9, 0.999), 'eps': 1e-05} # optional parameters of the optimizer
)

narx_net.fit(X=x_train, y=y_train)
yhat = narx_net.predict(X=x_valid, y=y_valid)
plot_results(y=y_valid, yhat=yhat, n=200)
ee = compute_residues_autocorrelation(y_valid, yhat)
plot_residues_correlation(data=ee, title="Residues", ylabel="$e^2$")
x1e = compute_cross_correlation(y_valid, yhat, x_valid)
plot_residues_correlation(data=x1e, title="Residues", ylabel="$x_1e$")
_images/narx_network.png

Catboost-narx

from sysidentpy.general_estimators import NARX
from catboost import CatBoostRegressor
from sysidentpy.basis_function._basis_function import Polynomial
from sysidentpy.utils.plotting import plot_residues_correlation, plot_results
from sysidentpy.residues.residues_correlation import compute_residues_autocorrelation, compute_cross_correlation

catboost_narx = NARX(
        base_estimator=CatBoostRegressor(
                iterations=300,
                learning_rate=0.1,
                depth=6),
        xlag=2,
        ylag=2,
        basis_function=basis_function,
        model_type="NARMAX",
        fit_params={'verbose': False}
)

catboost_narx.fit(X=x_train, y=y_train)
yhat = catboost_narx.predict(X=x_valid, y=y_valid)
plot_results(y=y_valid, yhat=yhat, n=200)
ee = compute_residues_autocorrelation(y_valid, yhat)
plot_residues_correlation(data=ee, title="Residues", ylabel="$e^2$")
x1e = compute_cross_correlation(y_valid, yhat, x_valid)
plot_residues_correlation(data=x1e, title="Residues", ylabel="$x_1e$")
_images/catboost_narx.png

Catboost without NARX configuration

The following is the Catboost performance without the NARX configuration.

def plot_results_tmp(y_valid, yhat):
        _, ax = plt.subplots(figsize=(14, 8))
        ax.plot(y_valid[:200], label='Data', marker='o')
        ax.plot(yhat[:200], label='Prediction', marker='*')
        ax.set_xlabel("$n$", fontsize=18)
        ax.set_ylabel("$y[n]$", fontsize=18)
        ax.grid()
        ax.legend(fontsize=18)
        plt.show()

catboost = CatBoostRegressor(
        iterations=300,
        learning_rate=0.1,
        depth=6
)

catboost.fit(x_train, y_train, verbose=False)
plot_results(y_valid, catboost.predict(x_valid))
_images/catboost.png

Changelog

See the changelog for a history of notable changes to SysIdentPy.

Development

We welcome new contributors of all experience levels. The SysIdentPy community goals are to be helpful, welcoming, and effective.

Note

We use the pytest package for testing. The test functions are located in tests subdirectories at each folder inside SysIdentPy, which check the validity of the algorithms.

Run the pytest in the respective folder to perform all the tests of the corresponding sub-packages.

Currently, we have around 81% of code coverage.

You can install pytest using

pip install -U pytest

Example of how to run the tests:

Open a terminal emulator of your choice and go to a subdirectory, e.g,

\sysidentpy\metrics\

Just type pytest and you get a result like

========== test session starts ==========

platform linux -- Python 3.7.6, pytest-5.4.2, py-1.8.1, pluggy-0.13.1

rootdir: ~/sysidentpy

plugins: cov-2.8.1

collected 12 items

tests/test_regression.py ............ [100%]

========== 12 passed in 2.45s ==================

You can also see the code coverage using the pytest-cov package. First, install pytest-cov using

pip install pytest-cov

Run the command below in the SysIdentPy root directory, to generate the report.

pytest --cov=.

Source code

You can check the latest sources with the command:

git clone https://github.com/wilsonrljr/sysidentpy.git

Project History

The project was started by Wilson R. L. Junior, Luan Pascoal and Samir A. M. Martins as a project for System Identification discipline. Samuel joined early in 2019.

The project is actively maintained by Wilson R. L. Junior and looking for contributors.

Communication

Citation

If you use SysIdentPy on your project, please drop me a line.

If you use SysIdentPy on your scientific publication, we would appreciate citations to the following paper:

  • Lacerda et al., (2020). SysIdentPy: A Python package for System Identification using NARMAX models. Journal of Open Source Software, 5(54), 2384, https://doi.org/10.21105/joss.02384

    @article{Lacerda2020,
      doi = {10.21105/joss.02384},
      url = {https://doi.org/10.21105/joss.02384},
      year = {2020},
      publisher = {The Open Journal},
      volume = {5},
      number = {54},
      pages = {2384},
      author = {Wilson Rocha Lacerda Junior and Luan Pascoal Costa da Andrade and Samuel Carlos Pessoa Oliveira and Samir Angelo Milani Martins},
      title = {SysIdentPy: A Python package for System Identification using NARMAX models},
      journal = {Journal of Open Source Software}
    }
    

Inspiration

The documentation and structure (even this section) is openly inspired by sklearn, einsteinpy, and many others as we used (and keep using) them to learn.