Automatic trend estimation by C˘alin Vamos¸, Maria Cr˘aciun

By C˘alin Vamos¸, Maria Cr˘aciun

Our e-book introduces a style to guage the accuracy of pattern estimation algorithms less than stipulations just like these encountered in genuine time sequence processing. this system relies on Monte Carlo experiments with man made time sequence numerically generated by way of an unique set of rules. the second one a part of the publication includes a number of computerized algorithms for development estimation and time sequence partitioning. The resource codes of the pc courses enforcing those unique automated algorithms are given within the appendix and should be freely to be had 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 right kind interpretation in their effects. We illustrate the functioning of the analyzed algorithms via processing time sequence from astrophysics, finance, biophysics, and paleoclimatology. The numerical test procedure greatly utilized in our publication is already in universal use in computational and statistical physics.

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The trend estimation reduces to the computation of the unknown parameters βk . One can determine the values of βk in Eq. 1) by means of the ordinary least square (OLS) method [3]. The OLS estimates of βk (denoted bk ) are obtained by minimizing the sum of squared residuals N xn − S= 2 K n=1 βk akn . k=1 By differentiating with respect to βl and equating the derivatives with zero we obtain the following system of equations for bk K N bk k=1 N aln akn n=1 xn aln , l = 1, 2, . . , K . 3) n=1 The analysis of this system of equations is presented in Appendix A.

3 for strongly correlated noise. 9 when the stochastic trend is much larger than the deterministic one. Hence the errors of the estimated trend become very large when the series is dominated by noise and become intolerable if the noise is also strongly correlated. If we compare these results with those obtained for polynomial fitting (see Fig. 4) we observe that many of the properties presented above are the same. The accuracy of the trend estimated by the RCMA mainly depends on the ratio r . Its accuracy is high when the time series is dominated by trend (r > 1) and it is only slightly influenced by the noise serial correlation.

Near the boundaries (i) the coefficients cn,m become asymmetric (Fig. 2). The asymmetry always occurs at m = K + 1 and the maximum values of the coefficients always occur inside the boundaries of the time series. Therefore the values near the boundaries are influenced with a greater weight by the values located inside the time series. An important feature of any MA method is that it introduces a spurious serial correlation in the smoothed time series. d. noise averaged by a MA and of the noise estimated from a time series containing a trend.

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