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Finally we analyze the influence of the time series resolution and the number of the monotonic parts on the estimated polynomial trend. The minimum η min plotted in Fig. 5 is determined on statistical ensembles of 100 numerically generated time series with φ = 0 and different values for N , P, and r . For P = 5 monotonic segments, from Fig. 5a it follows that by increasing the number N of the time series values, a better resemblance with the real trend is obtained, at least for time series dominated by noise (r ≤ 1).

1 0 10 20 i 30 40 0 10 20 i 30 40 0 10 20 i 30 40 Fig. 6 The average index η for trends estimated by the RCMA. The average of the minima of the index η for each time series is marked with a star. The semi-lengths of the RCMA are K = 10, 20, 30 corresponding to the increasing values of the serial correlation parameter φ time series ηmin (marked with a star in the figure) showing that the variability of individual time series due to the noise is not significant.

The simplest one is the arithmetic mean of the values within a finite averaging window with all the coefficients equal to a constant wk = T −1 . 2) k=−K identical with Eq. 14). Another commonly used MA is the exponential weighted MA defined by the recursive formula ϑx (n) = αxn + (1 − α)ϑx (n − 1), where 0 < α < 1 is a real parameter. The weighting coefficients in this case are wk = α(1 − α)−k for k ≤ K + = 0 and K − = −∞, hence only the past values of the time series are averaged. In fact all the time series encountered in practice are finite and when the averaging window contains a time series boundary we cannot apply formulas like Eq.

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