Unit 3: Time Series-I

3.1 Introduction to Time Series Data

A time series is a set of statistical observations arranged in chronological (time) order.

It's a sequence of data points, measured typically at successive, uniform time intervals (e.g., hourly, daily, monthly, quarterly, yearly).

Examples:

• Monthly sales of a company for the last 5 years.
• Daily closing price of a stock.
• Quarterly GDP of a country.
• Annual rainfall in a city.

The goal of Time Series Analysis is to study this past data to identify patterns (like trends or seasonality) and use them to make forecasts or predictions about the future.

3.2 Application of Time Series in Various Fields

Time series analysis is crucial in many fields:

• Economics & Finance: Forecasting stock prices, GDP, inflation rates, and interest rates.
• Business: Sales and demand forecasting (to manage inventory), financial planning, and resource allocation.
• Weather & Environment: Forecasting temperature, rainfall, and air pollution levels.
• Engineering: Analyzing signals (signal processing), and monitoring industrial processes for quality control.
• Health: Monitoring the spread of a disease (epidemiology), tracking patient heart rates (ECG).

3.3 Components of a Time Series

The "classical" approach to time series assumes that the data's value (Y) at any time (t) is a combination of four distinct components:

1. Secular Trend (T):
   • The long-term, smooth movement of the series in one general direction (up, down, or constant).
   • It represents the underlying growth or decline over many years.
   • Example: The general increase in a country's population over several decades.
2. Seasonal Variation (S):
   • Regular, periodic fluctuations that occur within one year and repeat each year.
   • Caused by weather (e.g., ice cream sales peak in summer) or social customs (e.g., retail sales peak before festivals).
   • The time period is less than one year (e.g., monthly or quarterly).
3. Cyclical Variation (C):
   • Long-term, "wave-like" oscillations or swings around the trend line.
   • The time period is more than one year (often 3-10 years).
   • This component is often related to the "business cycle" (boom, recession, depression, recovery).
   • It is less regular and harder to predict than seasonality.
4. Irregular/Random Variation (I):
   • The unpredictable, "noise" component. It's what's left over after the other three components are accounted for.
   • Caused by random, unforeseen events like strikes, natural disasters, or wars.
   • It is non-repeating and short-term.

3.4 Decomposition of Time Series

Decomposition is the process of breaking down an observed time series (Y) into its constituent components (T, S, C, and I).

The goal is to isolate and study each component. For example, to make a good forecast, we might first find the trend (T) and then add back the seasonal effect (S).

There are two main models for decomposition:

1. Additive Model

This model is used when the components are assumed to be independent of each other. The magnitude of the seasonal variation is constant, regardless of the trend level.

• Used when: The seasonal fluctuations are roughly the same size, no matter how high or low the trend is.
• Example: Monthly unemployment figures.

2. Multiplicative Model

This model is used when the components are inter-dependent. The magnitude of the seasonal variation is proportional to the trend.

• Used when: The seasonal fluctuations grow larger as the trend line increases. This is very common in business/economic data.
• Example: Company sales. A 10% holiday boost in a small company (low trend) is a small amount, but a 10% boost in a large company (high trend) is a very large amount.