By Calin Vamos, Maria Craciun
Our booklet introduces a mode to guage the accuracy of pattern estimation algorithms less than stipulations just like these encountered in actual time sequence processing. this system relies on Monte Carlo experiments with synthetic time sequence numerically generated through an unique set of rules. the second one a part of the booklet comprises a number of automated algorithms for pattern estimation and time sequence partitioning. The resource codes of the pc courses imposing those unique automated algorithms are given within the appendix and should be freely on hand on the net. The e-book includes transparent assertion of the stipulations and the approximations lower than which the algorithms paintings, in addition to the correct interpretation in their effects. We illustrate the functioning of the analyzed algorithms by way of processing time sequence from astrophysics, finance, biophysics, and paleoclimatology. The numerical test strategy widely utilized in our publication is already in universal use in computational and statistical physics.
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Extra info for Automatic Trend Estimation
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.