$T=43115258$. The computation of ${\bm f}_{eq} $ is compared with shorter sequences of the same time series of length $T^* = 10^6 ,\, 5\times 10^6,\, \& \,  20\times 10^6$. (b) The set of 257 eigenvalues of the $\hat W$-matrix for the same conditions of (b). Naturally, $\lambda =1$ is present by the definition of the probability transfer matrix $\hat W$. ](Lamb_N_16x16_E_292_T.eps "fig:"){width="7cm"}

        $T*$                 $Q_1$             $Q_2$
  ----------------- ----------------------- -----------
       $10^6$        $3.95\times 10^{-5} $   $0.0038 $
   $5\times 10^6$    $3.91\times 10^{-5}$     0.0020
   $20\times 10^6$   $3.84\times 10^{-5} $   $0.0002$

  : \[TableTest\] Error estimation of the equilibrium distribution and the spectral decomposition of the $\hat W$ matrix for different lengths of the time series.

We have noticed that the coarse-graining approach applied to the full cycle with period $T$ presents an important characteristic. Namely, the eigenvalues of the $\hat W$ matrix are real numbers. However, as we apply the same approach to a partial sequence of the same cycle of a length less than $T$, some eigenvalues become complex values (typically located near the origin in the complex plane). An important consequences of the present work is the following conjecture: For complete cycles the probability transfer matrix possesses real eigenvalues, but for incomplete sequences the 