The detrending approach#
In the detrending, approach we separate a time series \(X_t\) into deterministic and stochastic components. In other words, the time series \(X_t\) can be expressed as the sum of a deterministic component \( D(t) \), which captures the systematic effects such as trends or seasonality, and a stochastic component \( Y_t \):
By estimating \(D(t)\) and removing it from \(X_t\), the stochastic component \(Y_t\) can be analyzed using standard statistical models, which require the data to be stationary.
For the specific case of a linear trend in the data, which could be tested for significance using the MK trend test, \(X_t\) can be written as: $\( X_t = \beta_0 + \beta_1 t + Y_t\)$
where \(\beta_0\) and \(\beta_1\) are the intercept and slope of the deterministic component. In this framework, one can estimate \(\beta_0\) and \(\beta_1\) (e.g., using least squares), subtract the trend from the observations, and then develop statistical models exclusively for \(Y_t\). This approach is particularly useful when the focus is on understanding variability or extremes in the stochastic part without being influenced by the underlying trend.
Following a similar approach to the example of linear trend, data with a strong seasonal signal can be investigated, e.g., monthly temperature. In this case, \(D(t)\) can be modeled as a sinusoidal function.