Methods for Time Series Decomposition

Time series decomposition is a fundamental technique for breaking down a time series $Y_t$ into its underlying, unobservable components. Think of it like taking apart a machine to understand how each gear and lever contributes to the overall function. The goal is to isolate patterns like trend, seasonality, and the remaining irregular fluctuations (residuals or noise). This process provides a valuable understanding of the data's structure and helps prepare it for modeling, particularly for identifying and removing aspects that cause non-stationarity.

There are two primary structural models used for decomposition, based on how the components combine:

Additive Decomposition

The additive model assumes that the components sum together to form the observed time series. It's represented as:

$Y_t = T_t + S_t + R_t$

Where:

• $Y_t$ is the observed value at time $t$.
• $T_t$ is the trend component at time $t$.
• $S_t$ is the seasonal component at time $t$.
• $R_t$ is the residual (or irregular/random) component at time $t$.

An additive model is most appropriate when the magnitude of the seasonal fluctuations or the variance of the residuals around the trend remains relatively constant over time. If you plot the data and the seasonal swings don't seem to get wider as the overall level of the series increases, an additive approach might be suitable. Imagine monthly sales data where the holiday season adds roughly the same amount (say, $10,000) to sales each December, regardless of whether the baseline sales are low or high.

Multiplicative Decomposition

The multiplicative model assumes that the components multiply together:

$Y_t = T_t \times S_t \times R_t$

This model is often more fitting when the seasonal variation or the residual fluctuations appear to be proportional to the level of the time series. As the trend increases, the amplitude of the seasonal swings or the random noise also tends to increase. For instance, if holiday sales represent a percentage increase (say, 20%) over the baseline monthly sales, the absolute size of the December spike will grow as the overall sales trend upwards. In such cases, a multiplicative model provides a better description.

It's also common practice to transform a multiplicative relationship into an additive one by taking the logarithm of the time series:

$\log(Y_t) = \log(T_t) + \log(S_t) + \log(R_t)$

This allows additive decomposition methods to be applied to the log-transformed data, which can simplify the analysis.

Common Decomposition Techniques

Several algorithms exist to perform decomposition. Here are two widely used ones:

1. Classical Decomposition: This is a relatively simple approach often based on moving averages.

   • Estimate Trend ($T_t$): Calculate a moving average with a window size matching the seasonal period (e.g., 12 for monthly data) to smooth out seasonality and noise. Centered moving averages are typically used.
   • Detrend: Remove the trend estimate from the original series. For an additive model, calculate $Y_t - T_t$. For a multiplicative model, calculate $Y_t / T_t$.
   • Estimate Seasonality ($S_t$): Average the detrended values for each season across all years. For example, average all January values, all February values, etc., from the detrended series. Adjust these seasonal averages so they sum to 0 (additive) or average to 1 (multiplicative) over a full cycle.
   • Calculate Residuals ($R_t$): Subtract (additive) or divide (multiplicative) the estimated trend and seasonal components from the original series: $R_t = Y_t - T_t - S_t$ (additive) or $R_t = Y_t / (T_t \times S_t)$ (multiplicative).

   While easy to understand, classical decomposition has drawbacks. It struggles with the beginning and end of the series (where centered moving averages can't be computed without assumptions), assumes seasonality repeats identically each cycle, and can be sensitive to unusual values.

2. STL Decomposition (Seasonal and Trend decomposition using Loess): This is a more sophisticated and versatile method developed by Cleveland et al. (1990). STL uses Loess (Locally Estimated Scatterplot Smoothing), a non-parametric regression technique, to estimate the trend and seasonal components iteratively.

   • It performs an inner loop that estimates and removes seasonality, then estimates and removes the trend from the seasonally adjusted data.
   • An outer loop refines the estimates and calculates robustness weights to reduce the influence of outliers.
   • Main advantages of STL include its ability to handle any type of seasonality (not just fixed periods like 12 months), user control over the smoothness of the trend and seasonal components, and robustness to outliers. It's generally preferred over classical decomposition for most practical applications.

Visualizing Decomposition Results

Decomposition is most effective when visualized. Typically, you'll plot the original time series along with its estimated trend, seasonal, and residual components on separate panels. This allows for easy inspection of the underlying patterns.

Example output of an additive time series decomposition, showing the original observed data, the estimated trend, the repeating seasonal pattern, and the remaining residual component.

Understanding these components is essential. The trend and seasonality often represent the non-stationary parts of the series. By identifying them through decomposition, we can take steps, such as differencing (which we'll discuss next), to remove them and achieve the stationarity required by many forecasting models like ARIMA. The residual component ideally represents stationary noise; examining its properties helps validate the decomposition and model assumptions.