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We will then attempt to reject this null hypothesis by comparing the value of a nonlinear parameter estimated on the data with its probability distribution for the hypothesis. Since only exceptional cases allow for the exact or asymptotic derivation of this distribution unless strong additional assumptions are made, we have to estimate it by a Monte Carlo resampling technique. This procedure is known in the nonlinear time series literature as the method of surrogate data, see Refs. [104,180,181].

One of the most well-known objectives is the prediction of future values of some quantity, for instance the price of an entity at the stock market. 3. It is almost identical to the problem of estimating the dynamics underlying a time series. An intermediate step in most time series studies is to filter the data in order to enhance the relevant information. Prediction and filtering, or noise reduction, have many things in common, but there are notable differences. In particular, for noise reduction it is not enough to have a description of the dynamics.

Unless quite narrow assumptions are made, the probability distribution of these quantities is not known exactly. Therefore, we cannot usually assess the significance of changes in these quantities in a rigorous way. Also, a signal might be considered stationary for some purpose, but not for another. A typical case is dimension estimation which requires stationarity in the probability of close recurrences. The authors of Ref. [161] use a -test for the difference of histograms on sections of the data.

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Physics Reports vol.308

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