⏱️ Time Series
Modelling and forecasting data that arrives in order
0.5.dev0+git.20260808.d6e9440 - August 08, 2026 18:54 UTC

Time Series

A time series is a sequence of observations indexed by time, where order and dependence carry information. This hub walks the classical Box–Jenkins path the source corpus follows — from stationarity and autocorrelation, through the AR / MA / ARMA / ARIMA / SARIMA model family, to estimation, diagnostics and forecasting — as an ordered, self-contained course of 18 lessons.

Read it at any depth:

• newcomers — what makes time-series data special, and stationarity;

• practitioners — reading the ACF / PACF and fitting ARIMA in statsmodels;

• researchers — estimation (Yule–Walker, Gaussian MLE), order selection and residual diagnostics.

Warning

Time series breaks the i.i.d. assumption behind ordinary cross-validation. Never shuffle: validate forward in time (walk-forward) so the future never leaks into the past.

Note

Follow the lessons in order with Next ▶, or jump in by stage below. Snippets use real statsmodels / pandas / numpy calls. This course pairs with the Terminology reference (Signal Processing & Time Series).

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🧭 Stage 1 — Orientation · 2 lessons

What time-series data is, why order carries information, and setting up the tools. · beginner

• 01 · What Are Time Series, and How Are They Used? — Sequences of time-ordered observations, their trend / seasonal / residual parts, and where forecasting applies.

• 02 · Getting Started with R — Setting up a working environment — the source’s R tooling and the Python (statsmodels) path used here.

📐 Stage 2 — Stationarity · 2 lessons

The property that makes a series learnable — how to recognise it and how to achieve it. · beginner

• 03 · A Gentle Introduction to Stationarity — Why a stable mean, variance and autocovariance make a series learnable — and how differencing helps.

• 04 · Weak and Strong Stationarity — The precise definitions: strict distributional invariance versus the weaker second-order (covariance) form.

🔗 Stage 3 — Linear & ARMA Processes · 4 lessons

The building blocks: linear processes, the AR / MA / ARMA family, and their autocorrelation. · intermediate

• 05 · Linear Processes — Series written as a linear filter of white noise — the general form underlying AR, MA and ARMA.

• 06 · Understanding ARMA Processes — Combining autoregressive and moving-average terms; causality, invertibility and what each parameter does.

• 07 · Computing ACFs of Causal AR(2) Processes Using Difference Equations — Solving the autocorrelation of an AR(2) by treating its recursion as a linear difference equation.

• 08 · Understanding ACFs via Difference Equations for AR(p) and ARMA(p, q) — Generalising the difference-equation method to the autocorrelation of any AR(p) or ARMA(p, q).

🎯 Stage 4 — Prediction & the Sample ACF / PACF · 2 lessons

Optimal linear forecasting and the empirical correlation tools used to identify model order. · intermediate

• 09 · Best Linear Predictor of a Stationary Process — The optimal linear forecast, the projection principle, and how the PACF emerges from it.

• 10 · Sample ACF and Sample PACF — Estimating autocorrelation from data, their sampling behaviour, and reading them to pick model order.

🧮 Stage 5 — Estimation · 2 lessons

Fitting parameters: Yule–Walker for AR models, Gaussian maximum likelihood for ARMA. · advanced

• 11 · Preliminary Estimation for AR Models and the Yule–Walker Equations — Method-of-moments AR fitting by solving the Yule–Walker system from sample autocovariances.

• 12 · Maximum Likelihood Estimation for ARMA Models (Gaussian MLE) — Fitting ARMA by maximising the Gaussian likelihood — the standard, efficient estimator.

🏗️ Stage 6 — Building & Forecasting Models · 6 lessons

Diagnostics, order selection, ARIMA / SARIMA, multi-step forecasting and exponential smoothing. · advanced

• 13 · Diagnostics After Fitting a Time Series Model — Checking standardized residuals for leftover structure: normality, autocorrelation and the Ljung–Box test.

• 14 · Order Selection for Time Series Models — Choosing p, d, q with information criteria (AIC / BIC) balanced against parsimony and diagnostics.

• 15 · ARIMA Models: How Nonstationary Models Are Built from Stationary Ones — Differencing to remove trend, turning a nonstationary series into an ARMA-modellable one.

• 16 · SARIMA Models: Seasonal ARIMA — Extending ARIMA with seasonal AR, differencing and MA terms for periodic patterns.

• 17 · Beyond One-Step Ahead Predictions — Multi-step forecasting, how uncertainty compounds with horizon, and forecast intervals.

• 18 · Exponential Smoothing Models — Weighted-average forecasting (simple, Holt, Holt–Winters) as a practical complement to ARIMA.

🔤 Every lesson, A–Z index

🔠 Open the full alphabetical index

A Gentle Introduction to Stationarity ARIMA Models: How Nonstationary Models Are Built from Stationary Ones Best Linear Predictor of a Stationary Process Beyond One-Step Ahead Predictions Computing ACFs of Causal AR(2) Processes Using Difference Equations Diagnostics After Fitting a Time Series Model Exponential Smoothing Models Getting Started with R Linear Processes

Maximum Likelihood Estimation for ARMA Models (Gaussian MLE) Order Selection for Time Series Models Preliminary Estimation for AR Models and the Yule–Walker Equations Sample ACF and Sample PACF SARIMA Models: Seasonal ARIMA Understanding ACFs via Difference Equations for AR(p) and ARMA(p, q) Understanding ARMA Processes Weak and Strong Stationarity What Are Time Series, and How Are They Used?

Tags: purpose: reference topic: time series