To truly understand a time series, you must take it apart. Decomposition is the mathematical procedure that extracts the underlying components from raw data.
1The Additive Approach
An Additive Model assumes that the components of the time series are independent and simply add up to the total value: $Y(t) = Trend + Seasonality + Noise$. This model is the best fit when the seasonal fluctuations (the 'peaks' and 'valleys') stay roughly the same size regardless of whether the trend is high or low. It is common in datasets with stable, linear growth.
2The Multiplicative Approach
A Multiplicative Model assumes that the components interact with each other: $Y(t) = Trend imes Seasonality imes Noise$. Use this when the seasonal variation increases or decreases in proportion to the trend. For example, in airline passenger data, the seasonal 'holiday peaks' get much larger as the total number of travelers (the trend) grows over the years.
3The Residual Test
The Residuals (or Noise) are what is left over after the trend and seasonality are removed. Analyzing the residuals is the primary way to evaluate your decomposition. If the residuals contain a trend or a repeating cycle, it means your model is 'underfitting'—missing a piece of the signal. A perfect decomposition results in 'White Noise' residuals that contain no information whatsoever.
4Step-by-Step Breakdown
Decomposition is the process of physically separating a time series into its Trend, Seasonality, and Residual components. It's like taking apart a car to see how the engine works.
There are two main types: Additive and Multiplicative. In an Additive model, the components are added together. It's best when the seasonal variations are constant over time.
In a Multiplicative model, the components are multiplied. Use this when the seasonal peaks grow or shrink as the trend increases.
Checkpoint: If the seasonal peaks of a stock's volume get larger as the stock price (trend) increases, which model should you use?
- →Additive
- →Multiplicative
Once decomposed, you can look at the 'Residuals' plot. If you see a clear pattern there, your decomposition is incomplete. The residuals should look like random noise.
Decomposition is the 'X-ray' of time series analysis. It allows you to see the true trend without being distracted by seasonal noise.
Checkpoint: In an Additive model, the equation is Y(t) = Trend + Seasonality + ???
- →Error
- →Noise (Residuals)
Decomposition mastered! You've learned to peel back the layers of time. Ready to smooth out the noise with Moving Averages?
Decompose a Real Time Series. Finish computing the residual left over once trend and seasonal components are removed from the actual value.
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