General NARX Models¶

Example created by Wilson Rocha Lacerda Junior

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In this example we will create NARX models using different estimator like GradientBoostingRegressor, Bayesian Regression, Automatic Relevance Determination (ARD) Regression and Catboost

pip install sysidentpy

import matplotlib.pyplot as plt
from sysidentpy.metrics import mean_squared_error
from sysidentpy.utils.generate_data import get_siso_data
from sysidentpy.general_estimators import NARX
from sklearn.linear_model import BayesianRidge, ARDRegression
from sklearn.ensemble import GradientBoostingRegressor
from catboost import CatBoostRegressor

from sysidentpy.basis_function import Polynomial, Fourier
from sysidentpy.utils.plotting import plot_residues_correlation, plot_results
from sysidentpy.residues.residues_correlation import (
    compute_residues_autocorrelation,
    compute_cross_correlation,
)

# simulated dataset
x_train, x_valid, y_train, y_valid = get_siso_data(
    n=10000, colored_noise=False, sigma=0.01, train_percentage=80
)

Importance of the NARX architecture¶

To get an idea of the importance of the NARX architecture, lets take a look in the performance of the models without the NARX configuration.

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

gb = GradientBoostingRegressor(
    loss="quantile",
    alpha=0.90,
    n_estimators=250,
    max_depth=10,
    learning_rate=0.1,
    min_samples_leaf=9,
    min_samples_split=9,
)

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.fit(x_train, y_train, verbose=False)
plot_results_tmp(y_valid, catboost.predict(x_valid))

gb.fit(x_train, y_train.ravel())
plot_results_tmp(y_valid, gb.predict(x_valid))

Introducing the NARX configuration using SysIdentPy¶

As you can see, you just need to pass the base estimator you want to the NARX class from SysIdentPy do build the NARX model! You can choose the lags of the input and output variables to build the regressor matrix.

We keep the fit/predict method to make the process straightforward.

NARX with Catboost¶

basis_function = Fourier(degree=1)

catboost_narx = NARX(
    base_estimator=CatBoostRegressor(iterations=300, learning_rate=0.1, depth=8),
    xlag=10,
    ylag=10,
    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, steps_ahead=1)
print("MSE: ", mean_squared_error(y_valid, yhat))
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="$x_1e$")

MSE:  0.00024145290395678653

NARX with Gradient Boosting¶

basis_function = Fourier(degree=1)

gb_narx = NARX(
    base_estimator=GradientBoostingRegressor(
        loss="quantile",
        alpha=0.90,
        n_estimators=250,
        max_depth=10,
        learning_rate=0.1,
        min_samples_leaf=9,
        min_samples_split=9,
    ),
    xlag=2,
    ylag=2,
    basis_function=basis_function,
    model_type="NARMAX",
)

gb_narx.fit(X=x_train, y=y_train)
yhat = gb_narx.predict(X=x_valid, y=y_valid)
print(mean_squared_error(y_valid, yhat))

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="$x_1e$")

0.0011824693986863938

NARX with ARD¶

from sysidentpy.general_estimators import NARX

ARD_narx = NARX(
    base_estimator=ARDRegression(),
    xlag=2,
    ylag=2,
    basis_function=basis_function,
    model_type="NARMAX",
)

ARD_narx.fit(X=x_train, y=y_train)
yhat = ARD_narx.predict(X=x_valid, y=y_valid)
print(mean_squared_error(y_valid, yhat))

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="$x_1e$")

0.0011058934497373794

NARX with Bayesian Ridge¶

from sysidentpy.general_estimators import NARX

BayesianRidge_narx = NARX(
    base_estimator=BayesianRidge(),
    xlag=2,
    ylag=2,
    basis_function=basis_function,
    model_type="NARMAX",
)

BayesianRidge_narx.fit(X=x_train, y=y_train)
yhat = BayesianRidge_narx.predict(X=x_valid, y=y_valid)
print(mean_squared_error(y_valid, yhat))

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="$x_1e$")

0.0011077874945734536

Note¶

Remember you can use n-steps-ahead prediction and NAR and NFIR models now. Check how to use it in their respective examples.