L6 Time Series Decomposition

Time Series Decomposition

Any time series can be decomposed into the following four aspects/groups:

1. Secular Trend (\(T_{t}\))
   • Long term movement (doesn't happen in a shorter time-span)
2. Seasonal Component (\(S_{t}\)), repeats after a certain time-point
   • Regular periodic variation (period of cycle \(\leq 1\) year)
   • E.g. Summer and Winter (monthly temperature within an year)
3. Cyclical Component (\(C_{t}\))
   • Repetition cycle is longer than 1 year (and NOT at regular intervals)
   • E.g. Business Cycle
   • Gradual and relatively long term up and down movements of a series
4. Random/irregular Component (\(I_{t}\))
   • Purely random and irregular

(1), (2) and (3) are due to permanent causes and are predictable → Systematic Components. (4) is due to "noise" → Unsystematic components.

Example: "Nottem" Dataset

Contains 20 years of monthly temperature in Nottingham Castle (UK)

1. No trend
2. Seasonality is dominant

• Remainder is what remains after all others have been isolated, here we see that the remainder is indeed \(I_{t}\) is completely random.

Decomposition: Trend

• Only upward or downward (not upward in summer and downward in winter) → Without calendar related influences
• Without random movement

The average/mean level changes

Decomposition: Seasonality

• Seasonal Effect
  • systematic, calendar related effect
  • higher sales during festive season
• Complex seasonal effect
  • weekly sales of last week
  • number of trading days in a month effect (irregular repetitions)

Causes of seasonality

• Natural conditions
  • water consumption during summer
  • milk production during winter
• Business Admin process
  • Higher traveling during summer
• Social/ cultural behavior
  • Dhanteras gold purchase

Decomposition: Cyclical component

• Long term variations
• Wave shaped curve (Expansion and contractions)

• Repetitions are irregular (peaks are not equal, and spaced irregularly)

Decomposition: Irregular component

• Residual or random component.
• What remains after removing seasonal and trend component.
• Difference between peaks and troughs are completely random

Types of Decomposition

• Additive Decomposition
  • \(Y_{t} = T_{t} + S_{t}+C_{t}+I_{t}\)
  • Represents an absolute amount. (What is the absolute change?)
  • e.g. Manufacturer produces 10,000 more machine parts
• Multiplicative decomposition
  • \(Y_{t} = T_{t} \times S_{t} \times C_{t}\times I_{t}\)
  • Represents a relative amount. (What is the relative change?)
  • e.g. Manufacturer produces 20% more machine parts