Time Series Analysis Fundamentals

Master the fundamental concepts of time series decomposition: deterministic components, stochastic modeling, and mixed formulations with rigorous mathematical foundations

4-6 hoursIntermediate to Advanced3 lessons

Learning Objectives

Understand the mathematical foundations of time series decomposition
Master deterministic component analysis including trend and seasonality
Apply stochastic modeling techniques for residual components
Distinguish between additive and multiplicative model formulations
Implement practical forecasting workflows with mixed models
Analyze real-world time series data using decomposition methods

1. Time Series Decomposition Definition & Structure

Core Definition

A time series $X_t$ can be decomposed into distinct components that capture different aspects of the underlying data generating process. This decomposition forms the foundation for understanding, modeling, and forecasting time series data.

General Decomposition:

Any observed time series can be expressed as a combination of:

X_t = f(T_t, S_t, \varepsilon_t)

where $T_t$ is the trend component, $S_t$ is the seasonal component, and $\varepsilon_t$ is the stochastic (random) component.

Deterministic Components

Trend $T_t$ : Long-term directional movement.
Seasonality $S_t$ : Regular, predictable patterns that repeat over fixed periods.

Examples:

• Economic growth trend
• Holiday sales spikes

Stochastic Component

Random term $\varepsilon_t$ : Unpredictable fluctuations that remain after removing deterministic patterns. Often modeled using ARIMA or other stochastic processes.

Characteristics:

May exhibit serial correlation, heteroskedasticity, or other dependencies

2. Decomposition Model Formulations

Three Primary Model Types

Different ways to combine deterministic and stochastic components based on their interaction patterns

Additive Model

X_t = T_t + S_t + \varepsilon_t

Used when seasonal fluctuations remain roughly constant over time, independent of the trend level. Each component contributes separately to the observed value.

Example: Temperature data where seasonal amplitude stays constant regardless of long-term warming trend

Multiplicative Model

X_t = T_t \times S_t \times \varepsilon_t

Appropriate when seasonal effects scale with the overall level of the series. As the trend increases, seasonal fluctuations become proportionally larger.

Example: Retail sales where holiday peaks grow larger as the business expands

Mixed (Hybrid) Models

X_t = T_t \times S_t + \varepsilon_t

Flexible formulations allowing different types of interactions between components. Other forms include $X_t = T_t + S_t \times \varepsilon_t$ or $X_t = T_t \times (S_t + \varepsilon_t)$ .

Example: Economic indicators where trend and seasonality interact multiplicatively but noise is additive

3. Practical Implementation Workflow

Step-by-Step Decomposition Process

Systematic approach for decomposing and modeling time series data

Step 1: Data Exploration & Visualization

• Plot the raw time series to identify obvious patterns
• Check for missing values, outliers, and structural breaks
• Examine ACF/PACF plots for initial insights

Step 2: Choose Decomposition Method

• STL decomposition: Robust, handles varying seasonal patterns
• Classical decomposition: Simple moving averages, assumes constant seasonality
• X-13-ARIMA-SEATS: Advanced method for official statistics

Step 3: Model Selection Criteria

• Additive: Use when seasonal amplitude is roughly constant
• Multiplicative: Apply when seasonal effects scale with trend level
• Log transformation: Convert multiplicative to additive structure

Step 4: Residual Analysis & Forecasting

• Check residuals for whiteness (no remaining patterns)
• Apply ARIMA modeling if residuals show structure
• Combine forecasted components for final predictions

Next Steps & Applications

Practice Problems

Reinforce understanding with decomposition exercises, model selection problems, and forecasting challenges.

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Formula Reference

Quick access to key formulas for time series decomposition and modeling techniques.

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Advanced Topics

Continue with advanced forecasting methods, ARIMA modeling, and specialized time series techniques.

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