Resampled Time Series

 

Multiple time series can be selected and for each selected time series a number of resampled time series are generated. Resampled time series are time series with samples formed randomly, but possibly under given conditions, on the basis of the original time series. They can be formed by bootstrapping or by randomization (or permutation) of the original time series, the latter referred to as surrogate time series in the literature of nonlinear analysis of time series.

A number of algorithms generating resampled time series can be chosen as explained below. Each selected resampling algorithm generates the same number of resampled time series for each time series.

 

Select one or more time series ...

Multiple time series names can be selected from this list. This can be done by standard use of the mouse: a click on the first name and Shift^click on the last name selects all names between the first and the last name; a click on the first name and subsequent Ctrl^clicks on other names select these names including the first name; Ctrl^a selects the whole list.

 

Random Permutation (RP)

Simple random permutation or so-called shuffling, preserving the marginal distribution of the original time series but destroying any possible correlations in the time series. The RP surrogate time series can be used for the test of independence of a time series.

 

Fourier Transform (FT)

Surrogate data are generated that preserve the linear correlation structure of the original time series but are otherwise random. To achieve this, the original time series if Fourier transformed, the phases are randomized, and the resulting Fourier series is transformed back to the time domain. The generated time series possess the original power spectrum (up to some inaccuracy due to the mismatch of data end-points, which is typically insignificant). Note that the marginal distribution of the FT surrogate time series is always Gaussian and may differ in general from the original marginal distribution. The FT surrogate time series can be used to test the null hypothesis that the time series is generated by a Gaussian process.

The original paper for this surrogate type is

Theiler J., Eubank S., Longtin A. and Galdrikian, B. (1992), Testing for Nonlinearity in Time Series: the Method of Surrogate Data, Physica D, 58, 77-94.

 

Amplitude Adjusted Fourier Transform (AAFT)

Surrogate data are generated that are supposed to preserve the linear correlation structure of the original time series and the marginal distribution, and be otherwise random. To achieve this, the original time series is first rank ordered matching the ranks of a Gaussian white noise time series, the FT surrogate of this time series is taken and the derived time series is rank ordered matching the ranks of the original time series. The generated time series possess the original marginal distribution exactly and the original power spectrum approximately. Note that the linear structure of the AAFT surrogate time series may differ significantly from the original linear structure at cases. The AAFT surrogate time series can be used to test the null hypothesis that the time series is generated by a Gaussian process undergoing a monotonic static transform.

The algorithm is according to

Theiler J., Eubank S., Longtin A. and Galdrikian B. (1992), Testing for Nonlinearity in Time Series: the Method of Surrogate Data, Physica D, 58, 77-94.

Refer also to the Appendix of 

Theiler J. et al (1992), Using Surrogate Data to Detect Nonlinearity  in Time Series", in Nonlinear Modeling and Forecasting, edited by  Casdagli M. and Eubank S., Addison-Wesley, Reading, MA, 163-188. 

 

Iterated Amplitude Adjusted Fourier Transform (IAAFT)

Surrogate data are generated that are supposed to preserve the linear correlation structure of the original time series and the marginal distribution, and be otherwise random. The IAAFT constitutes an improvement of the AAFT. A two stage process is applied iteratively starting with white noise, where at the first stage the Fourier amplitudes are replaced by the original Fourier amplitudes and at the second stage the derived time series is reordered to the original marginal distribution until convergence of  both power spectrum and marginal distribution is reached. The algorithm terminates if complete convergence (same reordering in two consecutive steps) is succeeded or if a maximum number of  iterations is reached. An IAAFT surrogate time series matches the original power spectrum, preserves the exact original marginal distribution and is otherwise random. The linear structure of the IAAFT surrogate time series exhibits little variance and may differ only slightly from the original linear structure but at cases this difference may be significant favoring rejection of the test. The IAAFT surrogate time series can be used to test the null hypothesis that the time series is generated by a Gaussian process undergoing a static transform (not only monotonic).

The IAAFT algorithm is proposed in 

Schreiber T. and Schmitz A. (1996), Improved Surrogate Data for Nonlinearity Tests, Physical Review Letters, Vol 77, 635-638.

 

Statically Transformed Autoregressive Process (STAP) 

Surrogate data are generated that are supposed to preserve the linear correlation structure of the original time series and the marginal distribution, and be otherwise random. STAP generates stochastic surrogates for a given time series as statically transformed realizations of a Gaussian (autoregressive)  process. The generated surrogate time series have exactly the same marginal distribution and approximately the same autocorrelation as those of the original time series. The linear structure (power spectrum or autocorrelation) of the STAP surrogate time series exhibits larger variance than IAAFT, but no significant difference from the original autocorrelation. The STAP surrogate time series can be used to test the null hypothesis that the time series is generated by a Gaussian process undergoing a static transform (not only monotonic).

In the STAP algorithm the following parameters are used:

- pol : the degree of the polynomial to approximate the sample transform from the Gaussian marginal distribution to the sample marginal distribution. The default value is 5 and it should work well for all practical purposes.

- arm : the order of the AR model to generate the Gaussian time series (which is then reordered to obtain the STAP surrogates). The range of delays for which the original autocorrelation has to be matched determines the 'arm' parameter, so if the match of autocorrelation is wanted for a long range of delays a large 'arm' should be given.

The STAP algorithm is proposed in 

Kugiumtzis D (2002), Surrogate Data Test for Nonlinearity using Statically Transformed Autoregressive Process, Physical Review E, Vol 66, 025201.

 

Autoregressive Model Residual Bootstrap (AMRB)

Bootstrap data are generated that are supposed to preserve the linear correlation structure of the original time series and the marginal distribution, and be otherwise random. AMRB is a model-based (or residual-based) bootstrap approach that uses a fitted AR model and resamples with replacement from the model residuals to form the bootstrap time series. The generated bootstrap time series have approximately the same marginal distribution and  autocorrelation as those of the original time series, but deviations in both can be at cases significant. The AMRB time series can be used to test the null hypothesis that the time series is generated by a linear autoregressive process.

In the AMRB algorithm the following parameter is used:

- arm : the order of the AR model to fit to the original time series and form the residuals.

The AMRB algorithm is described in 

Politis D.N. (2003), The Impact of Bootstrap Methods on Time Series Analysis, Statistical Science, Vol 18, No 2, pp 219-230

Hjellvik V. and Tjøstheim D. (1995), Nonparametric Tests of Linearity for Time Series, Biometrika, Vol 82, No 2, pp 351-368

 

Number of resampled time series

The number of resampled time series to generate for each selected type and for each selected time series.

 

Running Messages

The progress in the generation of the resampled time series for the selected time series is displayed.

 

OK

By pressing this button, the generation of the selected types of resampled time series for the selected time series starts and when it is completed the window is closed.
 

Cancel

The window is closed without generating any resampled time series.
 

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