Unit 1 - Introduction to Time Series

Forecasting

Forecasting means estimating future values or future events from available information. In TSF, the information is usually historical observations ordered by time. Forecasting is used because organizations must plan before the future is known.

Why forecast?

Planning: production, staffing, budgets, and inventory.
Decision making: choose an action before the actual demand or risk is seen.
Risk reduction: prepare for inflation, competition, policy changes, recession, or sudden market shifts.
Resource allocation: decide how much money, material, time, or manpower to reserve.

Forecasting examples

Marketing: estimate future sales before a campaign.
Production: decide raw material and stock levels.
Finance: plan cash flow and risk exposure.
Aviation: estimate passenger volume for route and crew planning.

Qualitative and Quantitative Methods

Forecasting methods are usually divided into judgment-based methods and data-model-based methods. This comparison is a common theory question.

Basis	Qualitative forecasting	Quantitative forecasting
Input	Judgment, experience, opinion, expert knowledge.	Historical numerical data and mathematical models.
Nature	Subjective.	Objective and repeatable.
Useful when	Data is absent, product is new, or future depends on expert judgment.	Past data is available and patterns are expected to continue.
Examples	Market survey, expert opinion, Delphi method.	Regression model, time series model, smoothing model.

Exam line: qualitative methods depend on human judgment; quantitative methods convert historical data into a model and use that model for future prediction.

What Is a Time Series?

A time series is a sequence of observations on the same variable, recorded in time order, usually at equally spaced intervals. The order matters. If the order is changed, the meaning of the data changes.

Time series = {y1, y2, y3, ..., yt}, where t is the time index.

Time series examples

Daily stock closing price.
Monthly sales or passenger count.
Quarterly revenue.
Annual GDP or per-capita GDP.
Hourly air quality index.

Not a time series

Observations taken at irregular event times without regular spacing.
Mixing annual and quarterly observations inside the same single series.
Values of many different stocks on one single date; that is cross-sectional data.

Type	Meaning	Example
Univariate	One variable observed through time.	Room temperature every minute.
Multivariate	More than one variable observed over the same time period.	GDP, unemployment, and inflation measured yearly.

Special Features of Time Series Data

Time series analysis is different from ordinary cross-sectional analysis because observations are usually dependent on nearby observations.

Order matters. Rearranging the rows destroys the structure because 2024 after 2023 is meaningful.
Dependence exists. Today's demand may depend on yesterday's demand, last week's demand, or last year's same season.
Equal spacing is expected. Daily, weekly, monthly, quarterly, and yearly series are easier to model because the time step is consistent.
History horizon matters. Too little history gives weak forecasts; very old history may be irrelevant after structural changes.

The first question in EDA is not "which model?". It is "what pattern is visible when I plot it?"

EDA and Missing Values

Exploratory data analysis starts with a time plot. The plot reveals trend, seasonality, unusual observations, sudden level shifts, and changes in variation.

EDA checklist

Plot the series against time.
Mark missing observations and outliers.
Check if the level is increasing or decreasing.
Check if a pattern repeats every fixed period.
Decide whether recent history is more relevant than old history.

Missing values

Time series should be contiguous. If an observation is missing, do not simply delete the time point because the spacing changes. Impute using domain knowledge, interpolation, nearby averages, or a model-based method.

Components of a Time Series

A time series can be understood as a combination of interpretable systematic parts and random leftover variation.

Component	Meaning	How to recognize
Trend (T)	Long-term upward or downward movement.	The general direction remains visible after short-term ups and downs.
Seasonality (S)	Fixed-period pattern that repeats at known intervals.	Same month, quarter, week, or hour behaves similarly across cycles.
Cyclical (C)	Long wave-like movement without fixed period.	Business cycles or economic expansions and recessions.
Irregular (I)	Random error or white-noise-like variation not explained by systematic components.	Unpredictable spikes, drops, and residual fluctuations.

Do not mix up seasonality and cycle. Seasonality has a fixed known period, such as 12 months or 4 quarters. A cycle has no fixed calendar length.

Time Series Decomposition

Decomposition separates a series into components so that the long-run movement and seasonal effect can be understood separately. It helps compare series fairly, remove seasonality, and build better forecasts.

Additive model: Yt = Tt + St + Ct + It

Multiplicative model: Yt = Tt x St x Ct x It

Model	Use when	Seasonal index property
Additive	Seasonal fluctuation size is roughly constant as the level changes.	Seasonal effects usually sum to 0 over one full season.
Multiplicative	Seasonal fluctuation grows or shrinks with the level of the series.	Seasonal indices usually average to 1, or sum to season length.

Simple decomposition workflow

Plot the data and decide whether the seasonal amplitude looks constant or proportional to the level.
Estimate the trend, often with moving averages or a fitted trend curve.
Remove the trend: use Yt - Tt for additive, or Yt / Tt for multiplicative.
Average the de-trended values by season to estimate seasonal indices.
Adjust indices so additive seasonal values sum to 0, or multiplicative indices average to 1.
Forecast trend forward and reapply seasonality.

Trend Fitting and Regression Link

Trend can be estimated by graphical inspection, moving averages, semi-average methods, or least squares. The question bank includes straight-line and parabola least-squares trend problems.

Straight-line trend: y = a + bx

Parabolic trend: y = a + bx + cx^2

Straight-line shortcut with coded x

If the coded time values are chosen so that sum x = 0, then:

a = sum y / n, b = sum xy / sum x^2

Regression note

R^2 measures how much variation in y is explained by the predictors. Adjusted R^2 is preferred when comparing models with different numbers of predictors because it can penalize unnecessary variables.

Exam Answer Frames

Write briefly about components of time series

Start with: "A time series can be decomposed into components that explain different patterns in the observed data." Then define trend, seasonal, cyclical, and irregular components with one example each. End by saying trend and seasonality are systematic; irregular variation is random error.

Describe decomposition process

Define decomposition, state additive and multiplicative models, explain when each is used, list steps to estimate trend and seasonal indices, then mention de-seasonalization and forecasting.

Qualitative vs quantitative forecasting

Make a two-column comparison: basis, nature, data requirement, examples, advantages, and limitations. Use market survey/expert opinion/Delphi for qualitative and regression/time-series/smoothing for quantitative.