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NARMAX Methods For System Identification and TimeSeries Forecasting

From Classical Approaches to Neural Networks

**SysIdentPy** offers State-of-the-Art techniques to build your NARMAX models, including its 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.

Table of Contents

• What is SysIdentPy?
• How do I install SysIdentPy?
• Features
• Why does SysIdentPy exist?
• How do I use SysIdentPy?
• Examples
• Communication
• Citation
• Inspiration
• Contributors
• Sponsors

Introduction

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.

With SysIdentPy, you can:

• Build and customize nonlinear forecasting models.
• Utilize state-of-the-art techniques for model structure selection and parameter estimation.
• Experiment with neural NARX models and other advanced algorithms.

Check our documentation!

For an in depth documentation, check our companion book:

How do I install SysIdentPy?

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

Requirements

SysIdentPy requires:

• Python (>= 3.8) (3.7 works for everything except for 3 parameter estimation methods)
• NumPy (>= 1.9.2) for numerical algorithms
• Matplotlib >= 3.3.2 for static plotting and visualizations
• Pytorch (>=1.7.1) for building NARX neural networks
• scipy (>= 1.8.0) for numerical and optimization algorithms

The library is compatible with Linux, Windows, and macOS. Some examples may also require additional packages like pandas.

For more details, check our installation guide

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 up to 8 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 for System Identification and TimeSeries Forecasting. 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: - Quickstart guide, 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

```python from torch import nn import numpy as np import pandas as pd import matplotlib.pyplot as plt from sysidentpy.metrics import rootrelativesquarederror from sysidentpy.utils.generatedata import getsisodata

Generate a dataset of a simulated dynamical system

xtrain, xvalid, ytrain, yvalid = getsisodata( n=1000, colorednoise=False, sigma=0.001, trainpercentage=80 ) ```

Building Polynomial NARX models with FROLS algorithm

```python from sysidentpy.modelstructureselection import FROLS from sysidentpy.basisfunction import Polynomial from sysidentpy.parameterestimation import LeastSquares from sysidentpy.metrics import rootrelativesquarederror from sysidentpy.utils.generatedata import getsisodata from sysidentpy.utils.displayresults import results from sysidentpy.utils.plotting import plotresiduescorrelation, plotresults from sysidentpy.residues.residuescorrelation import ( computeresiduesautocorrelation, computecross_correlation, )

basisfunction = Polynomial(degree=2) estimator = LeastSquares() model = FROLS( orderselection=True, ninfovalues=3, ylag=2, xlag=2, infocriteria="aic", estimator=estimator, errtol=None, basisfunction=basisfunction, ) model.fit(X=xtrain, y=ytrain) yhat = model.predict(X=xvalid, y=yvalid) rrse = rootrelativesquarederror(yvalid, yhat) print(rrse) r = pd.DataFrame( results( model.finalmodel, model.theta, model.err, model.nterms, err_precision=8, dtype='sci' ), columns=['Regressors', 'Parameters', 'ERR']) print(r)

console 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 ` python plotresults(y=yvalid, yhat=yhat, n=100, figsize=(14, 3), linewidth=1.5) ```

NARX Neural Network

```python from sysidentpy.neuralnetwork import NARXNN from sysidentpy.basisfunction import Polynomial from sysidentpy.utils.plotting import plotresiduescorrelation, plotresults from sysidentpy.residues.residuescorrelation import computeresiduesautocorrelation from sysidentpy.residues.residuescorrelation import computecross_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)

narxnet = NARXNN( net=NARX(), ylag=2, xlag=2, basisfunction=basisfunction, modeltype="NARMAX", lossfunc='mseloss', optimizer='Adam', epochs=200, verbose=False, optim_params={'betas': (0.9, 0.999), 'eps': 1e-05} # optional parameters of the optimizer )

narxnet.fit(X=xtrain, y=ytrain) yhat = narxnet.predict(X=xvalid, y=yvalid) plotresults(y=yvalid, yhat=yhat, n=100, figsize=(14, 3), linewidth=1.5) ```

Catboost-narx

```python from catboost import CatBoostRegressor from sysidentpy.generalestimators import NARX from sysidentpy.basisfunction import Polynomial from sysidentpy.utils.plotting import plotresiduescorrelation, plotresults from sysidentpy.residues.residuescorrelation import computeresiduesautocorrelation from sysidentpy.residues.residuescorrelation import computecross_correlation

basis_function=Polynomial(degree=1)

catboostnarx = NARX( baseestimator=CatBoostRegressor( iterations=300, learningrate=0.1, depth=6), xlag=2, ylag=2, basisfunction=basisfunction, fitparams={'verbose': False} )

catboostnarx.fit(X=xtrain, y=ytrain) yhat = catboostnarx.predict(X=xvalid, y=yvalid) plotresults(y=yvalid, yhat=yhat, n=100, figsize=(14, 3), linewidth=1.5) ```

Catboost without NARX configuration

The following is the Catboost performance without the NARX configuration.

python catboost = CatBoostRegressor( iterations=300, learning_rate=0.1, depth=6 ) catboost.fit(x_train, y_train, verbose=False) plot_results(y=y_valid, yhat=catboost.predict(x_valid), n=100, figsize=(14, 3), linewidth=1.5)

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 Scikit-learn, EinsteinPy, and many others as we used (and keep using) them to learn.

Contributors

Sponsors

Special thanks to our sponsors

Monthly Sponsors

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