Time Series Analysis

Overview

A time series decomposes into four components: trend (long-term direction), seasonality (fixed-period patterns like daily or yearly cycles), cyclical (irregular long-term fluctuations), and noise (random variation). Most time series methods require stationarity — constant mean and variance over time. The ADF test checks for a unit root (non-stationarity). The ARIMA framework applies differencing to achieve stationarity, autoregressive terms to capture dependence on past values, and moving-average terms to capture dependence on past errors. ACF/PACF plots guide model order selection. Once fitted, the model extrapolates future values, but confidence intervals widen with forecast horizon.

Key Concepts

Time Series Components

Component	Description	Example
Trend	Long-term direction	Increasing user base
Seasonality	Fixed-period pattern	Daily traffic peaks at noon
Cyclical	Irregular long-term swings	Business cycles (years)
Noise	Random variation	Weather fluctuations

ACF/PACF Pattern Guide

Model	ACF Pattern	PACF Pattern
AR(p)	Tails off (exponential/sinusoidal)	Cuts off after lag p
MA(q)	Cuts off after lag q	Tails off
ARMA(p,q)	Tails off	Tails off

Quick Example

Key Takeaways

Deep Dive

For detailed explanations, worked examples, and Python implementations, explore the dedicated statistics lessons:

Stationarity

Time Series Stationarity — ADF test, KPSS test, differencing strategies, and achieving stationarity

Model Identification

ACF and PACF — Autocorrelation and partial autocorrelation functions for identifying ARIMA order with examples

ARIMA Models

ARIMA Models — Fitting, diagnostics, forecasting, order selection, and Python implementation

Seasonal Models

Seasonal Decomposition — STL decomposition, seasonal patterns, trend estimation, and decomposition methods

Smoothing Methods

Exponential Smoothing — Simple, double, and triple (Holt-Winters) exponential smoothing for forecasting

Causality

Granger Causality — Testing whether one time series helps predict another

Time Series Analysis