ARIMA Example

This is a simple example of ARIMA (AutoRegressive Integrated Moving Average) using Python and the statsmodels library.

ARIMA Overview

ARIMA is a popular time series forecasting method that combines autoregression (AR), differencing (I), and moving averages (MA). It is suitable for univariate time series data and is widely used for predicting future values based on past observations.

Key concepts of ARIMA:

• AutoRegressive (AR) Component: Represents the relationship between the current value and its past values.
• Integrated (I) Component: Involves differencing the time series to make it stationary (constant mean and variance).
• Moving Average (MA) Component: Represents the relationship between the current value and the past forecast errors.
• Order (p, d, q): ARIMA is defined by its order (p, d, q), where p is the order of the AR component, d is the degree of differencing, and q is the order of the MA component.

ARIMA models are effective for capturing trends and seasonality in time series data.

Python Source Code:

# Import necessary libraries
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from statsmodels.tsa.arima.model import ARIMA
from statsmodels.graphics.tsaplots import plot_acf, plot_pacf

# Generate synthetic time series data
np.random.seed(42)
time = pd.date_range(start='2022-01-01', end='2023-12-31', freq='D')
values = np.random.randn(len(time)).cumsum()
series = pd.Series(values, index=time)

# Plot the time series data
plt.figure(figsize=(10, 6))
plt.plot(series)
plt.title('Synthetic Time Series Data')
plt.xlabel('Time')
plt.ylabel('Values')
plt.show()

# Plot autocorrelation and partial autocorrelation functions
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 4))
plot_acf(series, ax=ax1, lags=40)
plot_pacf(series, ax=ax2, lags=40)
plt.suptitle('Autocorrelation and Partial Autocorrelation Functions')
plt.show()

# Fit ARIMA model
order = (2, 1, 1)  # Example order (p, d, q)
arima_model = ARIMA(series, order=order)
arima_results = arima_model.fit()

# Generate forecast
forecast_steps = 365
forecast_values = arima_results.get_forecast(steps=forecast_steps).predicted_mean

# Plot the original time series and forecast
plt.figure(figsize=(10, 6))
plt.plot(series, label='Original Time Series')
plt.plot(forecast_values, label='ARIMA Forecast', color='red')
plt.title('ARIMA Forecast')
plt.xlabel('Time')
plt.ylabel('Values')
plt.legend()
plt.show()

Explanation:

• Import Libraries: Import necessary Python libraries, including NumPy for numerical operations, pandas for time series data manipulation, and statsmodels for ARIMA modeling.
• Generate Synthetic Time Series Data: Generate synthetic time series data for demonstration purposes.
• Plot Time Series Data: Visualize the generated time series data.
• Plot Autocorrelation and Partial Autocorrelation: Plot autocorrelation and partial autocorrelation functions to determine the order of the ARIMA model.
• Fit ARIMA Model: Fit an ARIMA model to the time series data using the determined order.
• Generate Forecast: Generate a forecast for a specified number of steps into the future.
• Plot Results: Visualize the original time series data and the ARIMA forecast.