Autoregressive Integrated Moving Average (ARIMA) Models in Python
Many real-world decisions depend on how a metric evolves over time: demand, temperature, sales, traffic, or usage. **ARIMA** models are a foundational forecasting method because they combine trend handling, autoregressive behavior, and moving-average error structure in one interpretable framework. In this tutorial, you will build an ARIMA forecasting workflow in Python using `statsmodels`. ## What ARIMA means ARIMA stands for: - **AR (Autoregressive)**: use past values of the series - **I (Integrated)**: difference the series to make it more stationary - **MA (Moving Average)**: use past forecast errors An ARIMA model is written as `ARIMA(p, d, q)`: - `p`: number of autoregressive lags - `d`: number of differences - `q`: number of moving-average lags ## Creating the dataset
import requests
import pandas as pd
# Download source file once so later blocks can focus on modeling steps
url = "https://raw.githubusercontent.com/jbrownlee/Datasets/master/daily-min-temperatures.csv"
response = requests.get(url, timeout=30)
response.raise_for_status()
with open("daily-min-temperatures.csv", "w", encoding="utf-8") as f:
f.write(response.text)
# Parse dates and set a daily frequency-aware index
df = pd.read_csv("daily-min-temperatures.csv")
df["Date"] = pd.to_datetime(df["Date"])
df = df.set_index("Date")
series = df["Temp"].asfreq("D")
print(series.head())
print(series.index.freq)Date 1981-01-01 20.7 1981-01-02 17.9 1981-01-03 18.8 1981-01-04 14.6 1981-01-05 15.8 Freq: D, Name: Temp, dtype: float64 <Day>
This block downloads the dataset once, stores it locally, parses the date index, and creates a daily time series used by the remaining examples. Keeping a local copy avoids repeated network calls and makes the tutorial easier to rerun. ## Visualizing the time series
import pandas as pd
import matplotlib.pyplot as plt
# Load prepared series for a quick visual inspection
df = pd.read_csv("daily-min-temperatures.csv")
df["Date"] = pd.to_datetime(df["Date"])
df = df.set_index("Date")
series = df["Temp"].asfreq("D")
plt.figure(figsize=(10, 4))
plt.plot(series.index, series.values, linewidth=1)
plt.title("Daily Minimum Temperatures")
plt.xlabel("Date")
plt.ylabel("Temperature")
plt.tight_layout()
plt.show()
Plotting first helps you quickly inspect trend, variability, and potential seasonality before selecting ARIMA parameters. This initial check reduces guesswork when deciding whether differencing is needed. ## Differencing to reduce non-stationarity
import pandas as pd
import matplotlib.pyplot as plt
# Create first-differenced series (x_t - x_{t-1})
df = pd.read_csv("daily-min-temperatures.csv")
df["Date"] = pd.to_datetime(df["Date"])
df = df.set_index("Date")
series = df["Temp"].asfreq("D")
diff_series = series.diff().dropna()
plt.figure(figsize=(10, 4))
plt.plot(diff_series.index, diff_series.values, linewidth=1, color="tab:orange")
plt.title("First-Differenced Temperature Series")
plt.xlabel("Date")
plt.ylabel("Differenced Temperature")
plt.tight_layout()
plt.show()
First differencing (`d=1`) is a common way to stabilize the mean level so ARIMA assumptions are more reasonable. A more stable series generally leads to better-behaved parameter estimates. ## Fitting an ARIMA model
import pandas as pd
from statsmodels.tsa.arima.model import ARIMA
# Fit ARIMA with chosen (p, d, q) values
df = pd.read_csv("daily-min-temperatures.csv")
df["Date"] = pd.to_datetime(df["Date"])
df = df.set_index("Date")
series = df["Temp"].asfreq("D")
model = ARIMA(series, order=(2, 1, 2))
results = model.fit()
print(results.summary()) SARIMAX Results
==============================================================================
Dep. Variable: Temp No. Observations: 3652
Model: ARIMA(2, 1, 2) Log Likelihood -8386.458
Date: Wed, 11 Mar 2026 AIC 16782.915
Time: 16:47:59 BIC 16813.929
Sample: 01-01-1981 HQIC 16793.960
- 12-31-1990
Covariance Type: opg
==============================================================================
coef std err z P>|z| [0.025 0.975]
------------------------------------------------------------------------------
ar.L1 0.4852 0.134 3.620 0.000 0.222 0.748
ar.L2 -0.1243 0.067 -1.865 0.062 -0.255 0.006
ma.L1 -0.8895 0.136 -6.548 0.000 -1.156 -0.623
ma.L2 -0.0105 0.126 -0.083 0.934 -0.258 0.237
sigma2 5.8022 0.128 45.191 0.000 5.551 6.054
===================================================================================
Ljung-Box (L1) (Q): 0.00 Jarque-Bera (JB): 14.49
Prob(Q): 0.99 Prob(JB): 0.00
Heteroskedasticity (H): 0.86 Skew: 0.08
Prob(H) (two-sided): 0.01 Kurtosis: 3.27
===================================================================================
Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).
This fits an `ARIMA(2, 1, 2)` model and prints parameter estimates and diagnostic statistics to assess model behavior. The summary is where you inspect coefficient significance and residual diagnostics before trusting forecasts. ## Train-test forecasting
import pandas as pd
from statsmodels.tsa.arima.model import ARIMA
# Hold out the last year to evaluate forecasting performance
df = pd.read_csv("daily-min-temperatures.csv")
df["Date"] = pd.to_datetime(df["Date"])
df = df.set_index("Date")
series = df["Temp"].asfreq("D")
train = series.iloc[:-365]
test = series.iloc[-365:]
model = ARIMA(train, order=(2, 1, 2))
results = model.fit()
forecast_res = results.get_forecast(steps=len(test))
forecast_df = forecast_res.summary_frame(alpha=0.05)
forecast_df.index = test.index
forecast_df["actual"] = test.values
print(forecast_df[["mean", "mean_ci_lower", "mean_ci_upper", "actual"]].head())Temp mean mean_ci_lower mean_ci_upper actual Date 1990-01-01 12.887473 8.135428 17.639518 14.8 1990-01-02 13.212491 7.685577 18.739405 13.3 1990-01-03 13.341927 7.671745 19.012109 15.6 1990-01-04 13.366339 7.645590 19.087088 14.5 1990-01-05 13.363559 7.600908 19.126209 14.3
This example trains on historical data, forecasts the holdout period, and returns both point forecasts and confidence intervals. Using a holdout period gives a more honest estimate of real-world forecast behavior than evaluating on training data. ## Visualizing forecast vs actual
import pandas as pd
import matplotlib.pyplot as plt
from statsmodels.tsa.arima.model import ARIMA
# Refit on train, forecast test, then compare visually
df = pd.read_csv("daily-min-temperatures.csv")
df["Date"] = pd.to_datetime(df["Date"])
df = df.set_index("Date")
series = df["Temp"].asfreq("D")
train = series.iloc[:-365]
test = series.iloc[-365:]
model = ARIMA(train, order=(2, 1, 2))
results = model.fit()
forecast_res = results.get_forecast(steps=len(test))
forecast_df = forecast_res.summary_frame(alpha=0.05)
forecast_df.index = test.index
plt.figure(figsize=(11, 5))
plt.plot(train.index[-500:], train.values[-500:], label="Train (recent)", alpha=0.7)
plt.plot(test.index, test.values, label="Actual", color="black")
plt.plot(forecast_df.index, forecast_df["mean"], label="Forecast", color="tab:blue")
plt.fill_between(
forecast_df.index,
forecast_df["mean_ci_lower"],
forecast_df["mean_ci_upper"],
color="tab:blue",
alpha=0.2,
label="95% CI",
)
plt.title("ARIMA Forecast vs Actual")
plt.xlabel("Date")
plt.ylabel("Temperature")
plt.legend()
plt.tight_layout()
plt.show()
Overlaying forecast and actual values helps you evaluate practical forecast quality and whether uncertainty intervals are realistic. The chart also makes bias and under/over-shoot patterns easier to spot than tables alone.