Time Series Analysis: Trend, Seasonality, Cyclical & Irregular Components

Core Concepts

1. What is a Time Series?

A time series is a collection of observations recorded at successive points in time or over equal intervals (e.g., daily tourist arrivals, monthly hotel bookings, quarterly GDP growth). It helps analyze patterns over time to make predictions.

Example in Tourism:

• Nepal Tourism Board tracks monthly tourist arrivals to predict peak seasons (e.g., October–November for Dashain/Tihar festivals).
• Ncell analyzes call data trends to forecast network demand during Dasain and Tihar.

2. Components of a Time Series

Every time series can be broken into four components:

1. Trend (Long-term movement) – Gradual increase or decrease over time (e.g., rising hotel occupancy due to infrastructure growth).
2. Seasonality (Short-term, repeating cycles) – Patterns that repeat yearly (e.g., higher bookings in December for Christmas).
3. Cyclical (Medium-term fluctuations) – Economic or business cycles (e.g., global recessions affecting tourism).
4. Irregular (Random shocks) – Unexpected events (e.g., COVID-19 lockdowns, political unrest).

3. Methods to Extract Trend

(A) Semi-Average Method (Simple Moving Average)

How it works:

1. Split data into two equal halves (odd years: first half +1).
2. Calculate average of each half.
3. Fit a straight-line trend using the two averages.

Worked Example 1: Semi-Average Method (Exam Style) Data: Yearly sales (in Rs. 000) from 2061–2066:

Steps:

1. Split into two halves (first 3 years vs. last 3 years):
   • First half (2061–2063): 22, 26, 24 → Average = (22+26+24)/3 = 24
   • Second half (2064–2066): 27, 26, 28 → Average = (27+26+28)/3 = 27
2. Assign mid-years (since data is yearly):
   • First half midpoint = 2062 (between 2061 and 2063)
   • Second half midpoint = 2065 (between 2064 and 2066)
3. Calculate slope (b): $$b=\frac{Y_2-Y_1}{X_2-X_1}=\frac{27-24}{2065-2062}=\frac{3}{3}=1$$
4. Find intercept (a) using $Y=a+bX$: $$24=a+1\times 2062⟹a=24-2062=-2038$$ Trend equation: $\hat{Y}=-2038+1\times X$
5. Predict for 2067 (X = 2067): $$\hat{Y}=-2038+2067=29\text{ (Rs. 000)}$$

(B) Moving Averages (Smoothing)

How it works:

• Use a 3-year or 5-year moving average to smooth out seasonality and reveal the trend.
• Centering: Adjust averages to align with the middle year.

Worked Example 2: 3-Year Moving Average Data: Quarterly profits (in Rs. 000) for a travel agency:

Steps:

1. Calculate 3-year moving average (not possible here; use 3-quarter moving average instead):
   • Q2 average = (Q1 + Q2 + Q3)/3 = (50 + 60 + 45)/3 = 51.67
   • Q3 average = (Q2 + Q3 + Q4)/3 = (60 + 45 + 70)/3 = 58.33
   • Q4 average = (Q3 + Q4 + Q1)/3 = (45 + 70 + 55)/3 = 56.67
2. Center the averages (shift back by 1 quarter):
   • Adjusted Q2 = 51.67 → aligns with 2020-Q2
   • Adjusted Q3 = 58.33 → aligns with 2020-Q3
3. Plot moving averages to see trend.

4. Seasonal Variations

Definition: Repeating patterns within a year (e.g., higher bookings in December for Christmas, lower in January).

Methods to Measure Seasonality:

1. Seasonal Index (SI): $$SI=\frac{\text{Actual Value}}{\text{Trend Value}}\times 100$$
2. Ratio-to-Moving-Average (RMA): $$RMA=\frac{\text{Actual Value}}{\text{Moving Average}}$$

Worked Example 3: Seasonal Index for Hotel Bookings Data: Quarterly bookings (in units) and trend values:

Steps:

1. Calculate Seasonal Index (SI) for each quarter:
   • Q1: $SI=(120/130)\times 100=92.31$
   • Q2: $SI=(150/140)\times 100=107.14$
   • Q3: $SI=(180/150)\times 100=120$
   • Q4: $SI=(200/160)\times 100=125$
2. Interpretation:
   • Q1 has 92.31% of trend (low season).
   • Q4 has 125% of trend (peak season).

5. Cyclical and Irregular Components

Example: Cyclical Effect on Airline Tickets (Nepal Airlines)

• 2015–2019: Rising ticket sales due to economic growth.
• 2020: Sharp drop due to COVID-19 (irregular).
• 2021–2022: Recovery but not back to pre-2020 levels (cyclical effect).

6. Decomposition of Time Series

Additive Model: $Y=T+S+C+I$ Multiplicative Model: $Y=T\times S\times C\times I$

Worked Example 4: Decomposing Tourist Arrivals Data: Monthly tourist arrivals (in 000s) for 2023:

Steps:

1. Extract Trend (using moving averages).
2. Remove Trend to get Seasonal + Irregular.
3. Calculate Seasonal Index (as in Example 3).
4. Residuals = Actual – (Trend + Seasonal) → Shows irregularity.

flowchart TD
    A["Time Series Data"] --> B["Extract Trend\n(Moving Averages)"]
    B --> C["Subtract Trend\n→ Seasonal + Irregular"]
    C --> D["Calculate Seasonal Index\n(Q1=92, Q4=125)"]
    D --> E["Residuals = Actual - (Trend + Seasonal)\n→ Irregular Component"]
    E --> F["Final Decomposition:\nY = Trend + Seasonal + Cyclical + Irregular"]

## In the Real World

1. eSewa & Khalti (Digital Payments in Nepal)

   • Trend: Rising transaction volumes due to smartphone adoption.
   • Seasonality: Peak in Dashain/Tihar (Oct–Nov) and New Year (April).
   • Irregular: Sudden drops during load-shedding or app crashes.

2. Daraz (E-Commerce)

   • Trend: Increasing orders as internet grows.
   • Seasonality: Sales spikes in Dashain (Oct) and Christmas (Dec).
   • Cyclical: Economic slowdowns reduce discretionary spending.

3. Nepal Tourism Board (NTB) Forecasting

   • Uses time series to predict:
     • Trend: More foreign tourists yearly.
     • Seasonality: Peak in Oct–Nov (festivals) vs. low in monsoon (Jun–Sep).
     • Irregular: Earthquake (2015) caused sudden drop in arrivals.

4. Ncell & NTC (Telecom Data)

   • Trend: More data usage as 4G expands.
   • Seasonality: Highest in evenings (7–10 PM).
   • Irregular: Network congestion during Dashain calls.

## Exam Tip

What Examiners Look For:

✅ Correct method selection (semi-average vs. moving averages). ✅ Accurate calculations (show all steps for trend line, seasonal index). ✅ Interpretation (explain what trend/seasonality means in context). ✅ Prediction (always estimate future values if asked).

Common Mistakes to Avoid:

❌ Ignoring mid-years in semi-average (leads to wrong slope). ❌ Forgetting to center moving averages. ❌ Mixing additive/multiplicative models without justification. ❌ Not labeling axes properly in graphs.

Quick Revision Checklist:

1. Can you split data into two halves for semi-average?
2. Can you calculate a 3-year moving average and center it?
3. Can you compute seasonal indices for each quarter?
4. Can you decompose a time series into T, S, C, I?
5. Can you predict future values using the trend equation?

## Practice Questions (Exam Style)

1. Given yearly sales data (2018–2023), use the semi-average method to find the trend line and predict sales for 2024.
2. For quarterly hotel occupancy (2022), calculate the 4-quarter moving average and identify seasonal patterns.
3. Decompose monthly tourist arrivals into trend, seasonality, and irregular components.
4. A travel agency’s profits follow $\hat{Y}=50+3X$ (origin year 2010). Estimate profit for 2025.

## Key Formulas to Memorize

## Final Visual Summary

mindmap
  root((Time Series Analysis))
    Components
      Trend
        Definition: Long-term growth/decline
        Methods: Semi-average, Moving averages
      Seasonality
        Definition: Repeating yearly patterns
        Example: Dashain peak in Oct–Nov
      Cyclical
        Definition: Medium-term fluctuations
        Example: Economic recessions
      Irregular
        Definition: Random shocks
        Example: COVID-19 dip
    Methods
      Semi-Average
        Steps: Split → Average → Trend line
      Moving Averages
        Steps: Smooth data → Center → Trend
      Decomposition
        Models: Additive vs. Multiplicative
    Applications
      Tourism: Predict tourist arrivals
      Business: Forecast sales/profits
      Finance: Analyze stock trends