Seasonal constraint comparison

Shift a Period d

The observed series X_t = m_t + s_t + Y_t (line is the mean m_t + s_t). This is the same under both decompositions.

With the constraint

trend m_t (line), seasonal s_t (dots)

Σ s_t over one period =

Without it (shifted by a)

trend m_t − a (line), seasonal s_t + a (dots)

Σ s_t over one period =

Comparison of two decompositions of the same series into trend and seasonal parts: one where the seasonal component sums to zero over a period, and one shifted by a constant a so that it does not.