## Automatic trend estimationOur book introduces a method to evaluate the accuracy of trend estimation algorithms under conditions similar to those encountered in real time series processing. This method is based on Monte Carlo experiments with artificial time series numerically generated by an original algorithm. The second part of the book contains several automatic algorithms for trend estimation and time series partitioning. The source codes of the computer programs implementing these original automatic algorithms are given in the appendix and will be freely available on the web. The book contains clear statement of the conditions and the approximations under which the algorithms work, as well as the proper interpretation of their results. We illustrate the functioning of the analyzed algorithms by processing time series from astrophysics, finance, biophysics, and paleoclimatology. The numerical experiment method extensively used in our book is already in common use in computational and statistical physics. |

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### Contents

1 Introduction | 1 |

2 Monte Carlo Experiments | 14 |

3 Polynomial Fitting | 31 |

4 Noise Smoothing | 43 |

5 Automatic Estimation of Monotonic Trends | 61 |

6 Estimation of Monotonic Trend Segments from a Noisy Time Series | 81 |

7 Automatic Estimation of Arbitrary Trends | 98 |

Appendix A Statistical Properties of the Linear Regression | 111 |

Appendix B Spurious Serial Correlation Induced by MA | 113 |

Appendix C Continuous Analogue of the ACD Algorithm | 117 |

Appendix D Standard Deviation of a Noise Superposed over a Monotonic Trend | 120 |

Appendix E Construction of a Partition of Scale Delta n | 127 |

Appendix F Estimation of the Ratio Between the Trend and Noise Magnitudes | 129 |

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### Common terms and phrases

accuracy ACD algorithm amplitude analyze approximation artificial time series autocovariance function automatic algorithm Automatic Trend Estimation AutRCMA trend averaged time series averaging window averagings number coefficients compute confidence intervals denote distribution dominated by noise dominated by trend estimated local extrema estimated noise estimated polynomial trend estimated trend extremum finite AR(1 Gaussian increases interval linear local maxima logreturns maxima maxima and minima maximum minimum monotonic component monotonic segments monotonic trend Monte Carlo experiments moving average noise fluctuations noisy time series nonmonotonic trend null hypothesis obtained partitioning error periodogram polynomial fitting random variable ratio RCMA trend real local extrema real time series real trend sample autocorrelation function Sect serial correlation series is dominated series length series values series xn smaller smoothing SpringerBriefs in Physics standard deviation stationary stationary process statistical ensemble stochastic process successive local extrema superposed trend fn trend local extrema trend variations white noise