Unit 6: Time-Based Data

Time Series, Components & Trend Analysis

1. Introduction to Time Series

Definition A set of data collected, recorded, or observed over successive intervals of time (chronological data).
Components The forces affecting the values of a time series are classified into four components:

  • Secular Trend (T): The long-term tendency of data to increase or decrease (e.g., population growth).
  • Seasonal Variation (S): Short-term, periodic fluctuations occurring within a year (e.g., sales of woolens in winter).
  • Cyclical Variation (C): Oscillatory movements extending over more than a year (e.g., business cycles of boom and depression).
  • Irregular Variation (I): Random, unpredictable fluctuations (e.g., floods, strikes, wars).
Decomposition Models Additive Model: $$Y = T + S + C + I$$
Multiplicative Model: $$Y = T \times S \times C \times I$$

2. Moving Average Method (Detailed)

This method is used to smooth out fluctuations and find the trend by averaging values over a specific period ($$m$$).

Case A: When Period ($$m$$) is ODD (e.g., 3-Year, 5-Year)
Steps 1. Calculate the Moving Total of $$m$$ years.
2. Place the total against the middle year of the group.
3. Divide the total by $$m$$ to get the Moving Average (Trend Value).
Example (3-Year) For years $$Y_1, Y_2, Y_3$$:
$$MA_1 = \frac{Y_1 + Y_2 + Y_3}{3}$$(Placed against$$Y_2$$)
$$MA_2 = \frac{Y_2 + Y_3 + Y_4}{3}$$(Placed against$$Y_3$$)
Case B: When Period ($$m$$) is EVEN (e.g., 4-Year) – “Centering” Required
The Problem The middle of an even number of years falls between two time points (e.g., between 2010 and 2011). We need to “Center” it back to a specific time point.
Steps 1. Calculate $$m$$-Year Moving Totals (place between middle years).
2. Calculate 2-Period Moving Totals of the results from Step 1 (Sum of pairs).
3. Divide this final sum by $$2 \times m$$(e.g., for 4-year, divide by$$2 \times 4 = 8$$).
Formula (4-Year) $$T_t = \frac{(Y_{t-1} + Y_t + Y_{t+1} + Y_{t+2}) + (Y_t + Y_{t+1} + Y_{t+2} + Y_{t+3})}{8}$$

3. Method of Least Squares

Used for fitting a straight-line trend and estimating future values.

Straight Line Equation $$Y_c = a + bX$$
Normal Equations To find constants $$a$$and$$b$$:
1. $$\Sigma Y = n a + b \Sigma X$$
2. $$\Sigma XY = a \Sigma X + b \Sigma X^2$$
Short-Cut Method If the origin (middle year) is chosen such that $$\Sigma X = 0$$:
$$a = \frac{\Sigma Y}{n}$$
$$b = \frac{\Sigma XY}{\Sigma X^2}$$
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