NARMAX Methods In The Time Domain

From Classical Approaches to Neural Networks

SysIdentPy offers State of the Art techniques to build your NARMAX models, including it's variants NARX, NARMA, NAR, NFIR, ARMAX, ARX, ARMA and others. It also includes tons of interesting examples to help you build nonlinear forecasting models using SysIdentPy.

What is SysIdentPy?

SysIdentPy is an open-source Python module for System Identification using NARMAX models built on top of numpy and is distributed under the 3-Clause BSD license. SysIdentPy provides an easy-to-use and flexible framework for building Dynamical Nonlinear Models for time series and dynamic systems.

Check our documentation!

How do I install SysIdentPy?

The easiest way to get SysIdentPy running is to install it using pip

pip install sysidentpy

Requirements

SysIdentPy requires:

• Python (>= 3.7)
• NumPy (>= 1.9.2) for all numerical algorithms
• Matplotlib >= 3.3.2 for static plotting and visualizations
• Pytorch (>=1.7.1) for building feed-forward neural networks

Platform	Status
Linux	ok
Windows	ok
macOS	ok

A few examples require pandas >= 0.18.0. However, it is not required to use SysIdentPy.

What are the main features of SysIdentPy?

Feature	What is this?
NARMAX philosophy	You can build variations of NARMAX models like NARX, NAR, NARMA, NFIR, ARMA, ARX, AR, and others.
Model Structure Selection	Easy-to-use methods to select the best terms to build your models, including FROLS and MetaMSS and several combinations with parameter estimation techniques to select the model terms.
Basis Function	You can use different basis functions to build your models. You can set linear and nonlinear basis functions and ensemble them to get custom NARMAX models.
Parameter Estimation	More than 15 methods to estimate the model parameters and test different structure selection scenarios.
Multiobjective Parameter Estimation	You can use affine information to estimate the model parameters minimizing different objective functions.
Model Simulation	You can reproduce results from papers easily with SimulateNARMAX class. Moreover, you can test published models with different parameter estimation methods and compare the performance.
Neural NARX	You can use SysIdentPy with Pytorch to create custom neural NARX models architectures which support all the optimizers and loss functions from Pytorch.
General Estimators	You can use estimators from packages like scikit-learn, Catboost, and many other compatible interfaces and composition tools, to create NARMAX models.

Why does SysIdentPy exist?

SysIdentPy aims to be a free and open source package to help the community to design NARMAX models. More than that, be a free and robust alternative to one of the most used tools to build NARMAX models, which is the Matlab's System Identification Toolbox.

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

How do I use sysIdentPy?

The SysIdentPy documentation includes more than 20 examples to help get you started:

• Typical "Hello World" example, for an entry-level description of the main SysIdentPy concepts
• A dedicated section focusing on SysIdentPy features, like model structure selection algorithms, basis functions, parameter estimation, and more.
• A dedicated section focusing on use cases using SysIdentPy with real world datasets. Besides, there is some brief comparisons and benchmarks against other time series tools, like Prophet, Neural Prophet, ARIMA, and more.

Examples

from torch import nn
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from sysidentpy.metrics import root_relative_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
)

Building Polynomial NARX models with FROLS algorithm

from sysidentpy.model_structure_selection import FROLS
from sysidentpy.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
from sysidentpy.residues.residues_correlation import 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="$x_1e$")

NARX Neural Network

from sysidentpy.neural_network import NARXNN
from sysidentpy.basis_function import Polynomial
from sysidentpy.utils.plotting import plot_residues_correlation, plot_results
from sysidentpy.residues.residues_correlation import compute_residues_autocorrelation
from sysidentpy.residues.residues_correlation import 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=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, x_valid)
plot_residues_correlation(data=x1e, title="Residues", ylabel="$x_1e$")

Catboost-narx

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


basis_function=Polynomial(degree=1)

catboost_narx = NARX(
  base_estimator=CatBoostRegressor(
    iterations=300,
    learning_rate=0.1,
    depth=6),
  xlag=2,
  ylag=2,
  basis_function=basis_function,
  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=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, x_valid)
plot_residues_correlation(data=x1e, title="Residues", ylabel="$x_1e$")

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_tmp(y_valid, catboost.predict(x_valid))

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

Communication

• Discord server: https://discord.gg/8eGE3PQ

  

• Website: http://sysidentpy.org

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.

Contributors

Sponsors

Special thanks to our sponsors

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Gold Sponsors

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