Abstract
An arbitrary nonmonotonic trend is composed by a succession of monotonic segments limited by its local extrema. A superposed noise breaks up the trend monotonic variations into many small fluctuations, but the global shape of the trend is recognizable because the trend local extrema have a larger time scale than those induced by noise. By rigorously defining the time scale of a local extremum we design an automatic algorithm to estimate the trend local extrema from a noisy time series. The estimation accuracy is improved if the noisy time series is first smoothed such that the noise fluctuations are damped. Using the ACD algorithm for monotonic trend estimation presented in the previous chapter we evaluate the significance of the estimated local extrema. As an example we analyze a biophysical time series for which we estimate the large scale monotonic segments of the trend.
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Notes
- 1.
In this chapter we use the notation \(x(n)\) instead of \(x_n\).
- 2.
The automatic algorithm to build a partition of time scale \(\varDelta n\) is implemented by the MATLAB program local_extrema freely accessible on web.
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Vamos, C., Craciun, M. (2012). Estimation of Monotonic Trend Segments from a Noisy Time Series. In: Automatic trend estimation. SpringerBriefs in Physics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4825-5_6
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DOI: https://doi.org/10.1007/978-94-007-4825-5_6
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