ARIMA Models

AutoRegressive Integrated Moving Average (ARIMA) combines three components to model non-seasonal time series: autoregression (AR), differencing (I), and moving average (MA).

The Three Components

Component	Parameter	What It Does
AR (AutoRegressive)	\(p\)	Uses past values to predict the current value
I (Integrated)	\(d\)	Number of times the series is differenced to achieve stationarity
MA (Moving Average)	\(q\)	Uses past forecast errors to predict the current value

The model is specified as ARIMA(\(p, d, q\)).

Choosing \(p, d, q\)

1. \(d\): Difference the series until it is stationary (ADF test p-value < 0.05). Usually \(d = 0, 1,\) or \(2\).
2. \(p\): Look at the PACF plot — the lag where it cuts off.
3. \(q\): Look at the ACF plot — the lag where it cuts off.

Implementation

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

# Generate synthetic non-stationary data
rng = np.random.default_rng(42)
dates = pd.date_range("2020-01-01", periods=120, freq="MS")
ts = pd.Series(rng.normal(0, 1, 120).cumsum() + np.linspace(0, 20, 120), index=dates)

# 1. Check stationarity
adf_result = adfuller(ts)
print(f"ADF p-value: {adf_result[1]:.4f}")

# 2. Fit ARIMA(1,1,1)
model = ARIMA(ts, order=(1, 1, 1))
fitted = model.fit()
print(fitted.summary())

# 3. Forecast 12 months ahead
forecast = fitted.forecast(steps=12)

plt.figure(figsize=(10, 4))
ts.plot(label="Observed")
forecast.plot(label="Forecast", color="red", linestyle="--")
plt.legend()
plt.title("ARIMA(1,1,1) Forecast")
plt.tight_layout()
plt.show()

Model Diagnostics

# Check residuals — they should look like white noise
fitted.plot_diagnostics(figsize=(10, 6))
plt.tight_layout()
plt.show()

Workplace Tip

Use pmdarima.auto_arima() to automatically search for the best (\(p, d, q\)) combination using AIC. This saves manual inspection of ACF/PACF plots.

KSB Mapping

KSB	Description	How This Addresses It
K4.1	Statistical models and methods	ARIMA, SARIMA, and exponential smoothing foundations
K4.2	Predictive analytics and ML techniques	Time series forecasting and model comparison
K5.3	Common patterns in real-world data	Identifying trends, seasonality, and stationarity
S1	Scientific methods and hypothesis testing	Stationarity testing, model diagnostics, forecast validation
S4	Analysis and models to inform outcomes	Building forecasts to support business planning
B5	Impartial, hypothesis-driven approach	Honest evaluation of forecast accuracy and limitations