Time Series Analysis: Classical Methods

📈 Decomposing Time, Forecasting the Future

A comprehensive tour of classical time series methods applied to benchmark datasets

Project Overview

Time series data — sequences of observations indexed by time — appear throughout science, economics, and industry. This project provides a rigorous yet accessible walkthrough of the classical toolkit: from the humble exponential smoother through to full ARIMA-family models and a brief introduction to state-space representations. Each method is motivated theoretically, implemented in Python, and applied to canonical benchmark datasets so that behaviour is immediately interpretable.

Datasets used include:

• Air Passengers (Box & Jenkins, 1976): monthly international airline passenger counts 1949–1960 — the drosophila of time series, exhibiting clear trend, multiplicative seasonality, and increasing variance

• Monthly Milk Production (Cryer, 1986): monthly pounds of milk per cow 1962–1975 — additive seasonality with linear trend

• Sunspot Numbers (WDC-SILSO): annual sunspot counts from 1700 — stationary-ish with quasi-periodic ~11-year solar cycle; classic AR/ARMA showcase

• Mauna Loa CO₂ (Keeling, NOAA): monthly atmospheric CO₂ 1958–present — strong trend plus annual seasonality; good for Fourier decomposition

• UK Retail Sales (ONS): monthly index values — real-world irregularity, trading-day effects, and the challenges of official statistics

Project Structure

This analysis is organised into eight interconnected notebooks:

📐 1. Introduction & Data Loading

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Loading datasets, visualising raw series, ACF/PACF, classical and STL decomposition

📉 2. Simple Exponential Smoothing

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Level smoothing, α optimisation, ARIMA(0,1,1) equivalence

🌡️ 3. Holt-Winters Smoothing

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Trend and seasonal extensions, ETS framework, AIC model selection

🔊 4. Fourier Analysis

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DFT, periodograms, spectral density, Fourier regressors

🔁 5. ARMA & ARIMA

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Stationarity testing, Box-Jenkins methodology, AR/MA identification

❄️ 6. SARIMA

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Seasonal differencing, airline model, auto-ARIMA, production seasonal adjustment

🔮 7. State-Space Models

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Kalman filter, structural time series, Harvey's framework

🏆 8. Model Comparison

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Cross-validation, forecast metrics, Diebold-Mariano test

Key Questions

• 🔍 How does the choice of smoothing parameter α affect forecast responsiveness vs stability?

• 🔍 What does the autocorrelation structure of a residual series tell us about model adequacy?

• 🔍 When does seasonal differencing outperform explicit Fourier seasonality?

• 🔍 How do ARIMA and structural state-space models relate mathematically, and do they diverge in practice?

• 🔍 Which classical method produces the best out-of-sample forecasts on benchmark data?

Datasets

Dataset	Frequency	Length	Key Features
Air Passengers	Monthly	144 obs (1949–1960)	Multiplicative trend+season, growing variance
Milk Production	Monthly	168 obs (1962–1975)	Additive trend+season
Sunspots	Annual	300+ obs (1700–present)	Quasi-periodic, near-stationarity
Mauna Loa CO₂	Monthly	700+ obs (1958–present)	Strong nonlinear trend, annual seasonality
UK Retail Sales	Monthly	~300 obs	Real-world noise, calendar effects

Technical Approach

📊 Exploratory Decomposition
Classical additive/multiplicative decomposition, STL

📐 Stationarity Testing
ADF, KPSS, variance-ratio tests

🔧 Model Fitting
statsmodels, pmdarima auto-ARIMA

🔊 Spectral Analysis
scipy.signal, manual DFT, Fourier regressors

✅ Validation
Time-series cross-validation, rolling windows

Expected Outcomes

✅ Understood the theoretical basis of each classical method

✅ Identified which method suits which data generating process

✅ Produced interpretable, well-diagnosed models

✅ Benchmarked all methods head-to-head on common datasets

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Ready to explore?

Start with Introduction & Data Loading →