Code Implementation of ARIMAX Model in Python
We take an example of Python code that generates synthetic data and fits an ARIMAX model using the statsmodels library.
Pre requisite:
pip install statsmodels
It is important to note that the statsmodels library does not provide a distinct ARIMAX class separate from SARIMAX because SARIMAX is used for a variety of state space models including ARIMAX. Therefore, when using statsmodels, SARIMAX is the appropriate choice even when referring to ARIMAX models.
Step 1: Data Generation Function
This function generates synthetic time series data
import numpy as np
import pandas as pd
def generate_data(n=100, seed=42):
"""
Generate synthetic time series data with exogenous variables.
Parameters:
n (int): Number of data points.
seed (int): Seed for reproducibility.
Returns:
pd.DataFrame: DataFrame containing the endogenous and exogenous variables.
"""
np.random.seed(seed)
Y = np.cumsum(np.random.randn(n)) # Random walk
X1 = np.random.randn(n)
X2 = np.random.randn(n)
data = pd.DataFrame({'Y': Y, 'X1': X1, 'X2': X2}, index=pd.date_range(start='2020-01-01', periods=n))
return data
Step 2: ARIMAX Model Fitting and Forecasting Function
This function fits an ARIMAX model to the provided time series data and makes future forecasts.
from statsmodels.tsa.statespace.sarimax import SARIMAX
def fit_arimax(data, order=(1, 1, 1), exog_cols=['X1', 'X2'], forecast_steps=10):
"""
Fit an ARIMAX model to the data and make forecasts.
Parameters:
data (pd.DataFrame): DataFrame containing the time series and exogenous variables.
order (tuple): The (p,d,q) order of the ARIMA model.
exog_cols (list): List of column names for the exogenous variables.
forecast_steps (int): Number of steps to forecast.
Returns:
pd.DataFrame: DataFrame containing the observed, forecasted values, and confidence intervals.
"""
exog = data[exog_cols]
model = SARIMAX(data['Y'], exog=exog, order=order)
results = model.fit()
print(results.summary())
forecast = results.get_forecast(steps=forecast_steps, exog=np.random.randn(forecast_steps, len(exog_cols)))
forecast_index = pd.date_range(start=data.index[-1] + pd.Timedelta(days=1), periods=forecast_steps)
forecast_mean = forecast.predicted_mean
forecast_ci = forecast.conf_int()
forecast_df = pd.DataFrame({'Forecast': forecast_mean}, index=forecast_index)
forecast_df['Lower CI'] = forecast_ci.iloc[:, 0]
forecast_df['Upper CI'] = forecast_ci.iloc[:, 1]
return forecast_df
Step 3: Plotting Function
This function plots the observed data along with the forecasted values and their confidence intervals.
import matplotlib.pyplot as plt
def plot_results(data, forecast_df):
"""
Plot the observed data and forecasted values.
Parameters:
data (pd.DataFrame): DataFrame containing the observed time series.
forecast_df (pd.DataFrame): DataFrame containing the forecasted values and confidence intervals.
"""
plt.figure(figsize=(12, 6))
plt.plot(data.index, data['Y'], label='Observed')
plt.plot(forecast_df.index, forecast_df['Forecast'], label='Forecast')
plt.fill_between(forecast_df.index, forecast_df['Lower CI'], forecast_df['Upper CI'], color='pink', alpha=0.3)
plt.legend()
plt.title('ARIMAX Model Forecast')
plt.show()
Step 4: Plotting the forecast values
The main script coordinates the execution of the functions to generate data, fit the ARIMAX model, and plot the results.
# Generate synthetic data
data = generate_data()
# Fit ARIMAX model and forecast
forecast_df = fit_arimax(data, order=(1, 1, 1), exog_cols=['X1', 'X2'], forecast_steps=10)
# Plot the results
plot_results(data, forecast_df)
Output:
What Is an ARIMAX Model?
In the world of time series analysis and forecasting, various models help us understand and predict future values based on past data. Among these models, the ARIMAX model stands out due to its ability to incorporate external variables, providing a more robust and accurate forecasting mechanism. This article delves into the intricacies of the ARIMAX model, exploring its components, mathematical formulation, applications, and key points to understand its functionality better.