ARMA

The Auto-Regressive Moving Average (ARMA) model is designed to represent stationary time series data, where statistical properties remain consistent over time.

It combines the following two components:

• Auto-Regressive (AR) Model: Predicts future values using relationships with past observations.
• Moving Average (MA) Model: Models relationships with past errors to correct forecasts and improve accuracy.

Installation

To run the ARMA model, the statsmodels and pandas libraries need to be installed using pip:

pip install statsmodels pandas

Syntax

The syntaxes for performing different operations regarding the ARMA model are listed below.

ARIMA Model

To implement ARMA in Python, the ARIMA (Auto-Regressive Integrated Moving Average) model is used:

model = ARIMA(endog, order=(p, d, q))

• endog: The time series data (y-axis).
• order: A tuple (p, d, q) where:
  • p: Order of the AR (Auto-Regressive) component.
  • d: Differencing order (set to 0 for ARMA).
  • q: Order of the MA (Moving Average) component.

Fitting the Model

Once the ARIMA model is initialized, the .fit() method trains it on the data:

model_fit = model.fit()

The result, model_fit, contains:

• Parameters of the fitted ARMA model.
• Methods for forecasting and evaluating the model.

Forecasting

The .forecast() method generates predictions for a specified number of future steps:

forecast = model_fit.forecast(steps=n)

• steps: Number of future time steps to predict.

Visualizing

Matplotlib is used to visualize the historical data and the forecast:

plt.plot(data.index, data["temperature"], label="Historical Data")
plt.plot(future_dates, forecast, label="Forecast", color="red")

• data.index: Dates for the historical data.
• forecast: Forecasted values for the next steps number of days.
• future_dates: Generated dates for the forecast period.

Example

In this example, static data is generated randomly and forecasted using the ARMA model:

import pandas as pd
import numpy as np
from statsmodels.tsa.arima.model import ARIMA
import matplotlib.pyplot as plt

# Generate synthetic time series data
np.random.seed(42)  # Set seed for reproducibility
dates = pd.date_range(start="2023-01-01", periods=100)
temperatures = 20 + np.sin(np.linspace(0, 3 * np.pi, 100)) + np.random.normal(0, 0.5, 100)
data = pd.DataFrame({"date": dates, "temperature": temperatures})
data.set_index("date", inplace=True)

# Fit the ARMA model (ARIMA with no differencing)
model = ARIMA(data["temperature"], order=(2, 0, 2))
model_fit = model.fit()

# Forecast for the next 10 days
forecast = model_fit.forecast(steps=10)

# Generate future dates for the forecast
future_dates = pd.date_range(start=data.index[-1] + pd.Timedelta(days=1), periods=10)

# Plot historical data and forecast
plt.figure(figsize=(10, 6))
plt.plot(data.index, data["temperature"], label="Historical Data")
plt.plot(future_dates, forecast, label="Forecast", color="red")
plt.title("ARMA Forecast")
plt.xlabel("Date")
plt.ylabel("Temperature (°C)")
plt.legend()
plt.grid()
plt.show()

Here is the output for the above code: