Surrogate data method applied to nonlinear time series

Michael Small; Tomomichi Nakamura; Xiaodong Luo

REVIEW

Surrogate data method applied to nonlinear time series

Xiaodong Luo, Tomomichi Nakamura, Michael Small

Chaotic Dynamics · Submitted 1 Mar 2006 · 46 references

Pith's verdict: Useful review of surrogate methods held back by an overclaimed and unproven assertion about temporal-shift surrogates.

T0 · AI review

How this review was made: An AI referee (deepseek-v4-flash) read the paper on August 28, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 3 major objections · 4 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv IDnlin/0603004 (version 1)

Pith ID23WQPPPM, Pith's permanent record of this paper and its review

CitationMichael Small, Tomomichi Nakamura, Xiaodong Luo. "Surrogate data method applied to nonlinear time series." arXiv:nlin/0603004 (2006). https://pith.science/paper/23WQPPPM

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

Quick take: this is a review, not a new result, and the best part is the conceptual organization. It lays out the hierarchical surrogate nulls, the distinction between constrained realizations and pivotal statistics, and a menu of algorithms for experimental time series. That is genuinely useful. The weak spot is Section 2.2.1: the temporal-shift algorithm is claimed to generate constrained realizations under the null, and that claim is not proven and, as stated, probably false.

What the paper does well: it explains why standard Fourier-based surrogates only test a narrow set of nulls, and it covers cycle-shuffled, ATS/PPS, and small-shuffle methods with enough detail to be a practical starting point. It also has the right caveat in Section 3.2, admitting that for pseudo-periodic data the temporal shift algorithm is not constrained and recommending pivotal statistics. That suggests the authors know the issue, but they don't carry the same caution into Section 2.2.1.

The main problem: the superposition argument for two independent time series does not transfer to two subsets of one realization. Even if τ is large enough to make d_i and d_{i+τ} nearly independent, the construction s_i = αd_i + βd_{i+τ} has an autocovariance with extra cross terms; it is not a realization of the same ARMA process. So the sentence 'the algorithm ensures that the surrogates are generated from the same process under the null hypothesis' is doing heavy lifting with no proof. A pivotality argument might save it—if the chosen statistic has the same distribution for the original and the shifted-sum process—but none is supplied. That should be fixed.

Everything else is minor: the simulated annealing section defers details, there are typos and unresolved references, and the footnote on extending temporal shift to NH2 is too sketchy to be actionable. The self-citation pattern is fine; the authors are summarizing their own methods and citing the original papers.

Bottom line: this is a serviceable review for experimentalists who want to know what surrogate tests are available and why. It does not deserve a desk reject, but it does deserve a referee who will push back on the temporal-shift claim. If I were handling it, I would send it to review and ask for either a proof of constrainedness/pivotality or a corrected, weaker statement.

Referee report

Significance

If the methods reviewed are valid, the paper serves as a useful survey of modern surrogate techniques that address hypotheses relevant to experimental time series, such as aperiodicity and correlations with trends, rather than only testing for nonlinearity. The explicit discussion of pivotality and constrained realizations is helpful for practitioners, and the inclusion of explicit algorithmic steps for the attractor-trajectory and small-shuffle methods is a practical strength. The manuscript is also honest in places: Section 3.2 states that the temporal-shift algorithm is not constrained for pseudo-periodic data, and Section 3.3 acknowledges that attractor-trajectory surrogates may not preserve gross-scale dynamics. However, the mathematical support is uneven: several key claims about constrainedness are asserted without proof, and at least one of these assertions appears to be false. Because the central selling point of the review is that the reviewed methods test hypotheses of interest, a load-bearing technical error in one of the core algorithms calls for substantial revision.

Major objections (3)

[Section 2.2.1] The claim that the temporal-shift algorithm produces constrained-realization surrogates is unsupported and, as stated, incorrect. For a linear process such as an AR(1) process x_t = φ x_{t−1} + e_t, the surrogate s_t = α x_t + β x_{t+τ} contains innovations e_{t+1},...,e_{t+τ} because x_{t+τ} = φ^τ x_t + Σ_{j=0}^{τ−1} φ^j e_{t+τ−j}. Thus the surrogate series is not generated from the same ARMA process as the original data unless τ is so large that the overlap is negligible and a pivotal statistic is used; the two subsets d1 and d2 are overlapping, not independent, so the motivating superposition argument for independent series does not transfer. The sentence "the algorithm ensures that the surrogates are generated from the same process under the null hypothesis" is therefore not reliable. Section 3.2 later concedes that for pseudo-periodic data the temporal-shift algorithm is not constrained, but no such caveat is given here. The authors should either provide a proof of pivotality for the intended statistics or explicitly restrict the claim to pivotal test statistics and revise the assertion accordingly.
[Section 2.2.2] The simulated-annealing method is described only at a sketch level: the initialization of the control parameter, the acceptance criterion, the cooling schedule, and the stopping criterion are explicitly omitted ("To save time, we skip the introduction..."). The paper nonetheless concludes that "thus generated surrogates are also constrained realizations." This is a nontrivial claim that depends on the details of the annealing configuration and on convergence to an appropriate optimum; it is stated without proof or a precise reference to the exact algorithm configuration. Since this method is one of the reviewed contributions, the review would benefit either from a complete algorithmic description or from a clearly labeled statement that the constrainedness claim is heuristic and depends on the configuration described in [29].
[Section 2.1.3 and general] Several theoretical statements throughout the review are asserted without proof or supporting argument. The most central is the definition in Section 1.3 that for constrained-realization surrogates "any adopted test statistic T will appear as if it were pivotal for the processes {F_d, F_s1, F_s2, ...}"; this statement conflates the finite-sample empirical distribution of a statistic over surrogates with the theoretical pivotality property. The review would be strengthened by a precise statement of when surrogate-based tests control false rejection rates, particularly for non-pivotal statistics, and by explicitly identifying which of the reviewed algorithms satisfy constrainedness and which do not. As it stands, the paper moves between asserting constrainedness and conceding non-constrainedness (Section 3.2, Section 3.3) without a unifying theorem or even a precise definition of what is meant by "appear as if pivotal."

Minor objections (4)

[Section 2.1.3] The text contains reference placeholders "[?,?]" in "The interested readers are referred to [6,31, ?,?]." This is an incomplete citation and should be fixed before publication.
[Throughout] There are numerous typos and misspellings, including "depende nt" in the abstract, "Ho ng Kong" in the author affiliations, "reply" instead of "answer" in the Introduction, "Kuiumtzis" instead of "Kugiumtzis" in the footnote in Section 2.1.3, and "Recontructed" in the caption of Fig. 3. The manuscript should be carefully proofread.
[Section 2.1.1 and Fig. 1] The paper refers to Fig. 1(a) as "NMR laser data" but later describes it as an irregular fluctuation; the caption could be clearer about whether this is an example of a chaotic or stochastic signal. Also, the related discussion in Section 1.1 says "one needs to reply some elementary questions" and should be rewritten for clarity.
[Section 4] In the description of the small-shuffle surrogate algorithm, step (2) says "Sort r in the ascendant order to produce a new data set t" and then "Re-ordering the index set h will lead to the disturbed index set k" — the notation is a bit compressed. A small example or a clarified definition of k would help readability.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

Surrogate data tests can be built around the hypotheses experimental scientists actually care about, not just generic nonlinearity.

The reading

Surrogate data testing is a Monte Carlo procedure for asking whether a measured time series is consistent with a specified null hypothesis, such as 'produced by a linear stochastic process.' The review argues that the standard toolbox mostly tests null hypotheses about generic nonlinearity, which is often not the question an experimentalist wants answered. It presents four recent surrogate-generation algorithms—temporal shift, simulated annealing, pseudo-periodic (attractor trajectory) surrogates, and small-shuffle surrogates—each built so that the hypothesis under test is directly meaningful for real experimental data. The shared thread is the constrained-realization idea: surrogates should be drawn from the same process as the data under the null, which keeps the test interpretable even when the test statistic is not pivotal. If the claims hold, experimentalists can use surrogate tests to detect aperiodicity in pseudo-periodic signals and correlations in trended data, not merely to establish nonlinearity.

What carries the argument

The carrying mechanism is the constrained-realization surrogate ensemble. Given a null hypothesis, a surrogate is constrained when the process that generated it matches the process that generated the data (in practice, when the processes estimated from the surrogate and from the data coincide); the chosen test statistic then appears pivotal over the ensemble and the original series. The four algorithms are constructions of such ensembles without relying on Fourier phase randomization: temporal-shift additions exploit the superposition principle for linear processes; simulated annealing preserves the amplitude distribution while matching linear autocorrelations; the pseudo-periodic, attractor-trajectory method contaminates a trajectory on the reconstructed attractor with dynamical noise; and small-shuffle surrogates reorder the index set by tiny Gaussian perturbations to keep long-term structure while breaking local dependence. The review also stresses that when surrogates are not constrained, the test statistic itself must be pivotal, and it points to correlation dimension estimated by a Gaussian kernel algorithm as a practical choice.

Extended reading notes

Core claim

The paper's central claim is that the surrogate data method has been, and can be, extended beyond the classic hierarchical tests for nonlinearity. Under a composite null hypothesis, a surrogate algorithm should produce constrained realizations—series generated from the same process (or an estimate of it) as the original data—so that any test statistic behaves as if it were pivotal. The review describes algorithms that meet this requirement or that are paired with pivotal statistics: the temporal-shift algorithm adds overlapping subsets of the original series, exploiting the superposition principle that holds for linear but not nonlinear processes; simulated annealing exchanges data values to match the linear autocorrelation while preserving the amplitude distribution; the pseudo-periodic, attractor-trajectory algorithm walks along a reconstructed attractor to destroy fine-scale dynamics while keeping gross-scale structure; and the small-shuffle algorithm perturbs the index set by small Gaussian amounts to preserve a trend while disturbing irregular fluctuations. In each case, the hypothesis being tested—aperiodicity, noise-contaminated periodicity, or independence of irregular fluctuations—is one an experimental scientist would care about.

Load-bearing premise

The temporal-shift algorithm rests on the premise that adding two overlapping subsets of a series produced by the same linear stochastic process yields a new series from that same process; if the overlap or the chosen coefficients $\alpha$, $\beta$, and $\tau$ break that, the surrogate test's false-rejection rate is not controlled.

Editorial extensions

If this is right

An experimentalist with a pseudo-periodic recording can test for aperiodicity (chaos) rather than merely for nonlinearity, using temporal-shift or attractor-trajectory surrogates with a pivotal statistic.
For data with long-term trends, small-shuffle surrogates permit a test of whether the irregular fluctuations are correlated, a question standard Fourier-based surrogates cannot address because they assume stationarity.
The temporal-shift algorithm avoids the Fourier transform and therefore the wraparound artifacts of phase randomization, but it no longer preserves the amplitude distribution exactly.
Because non-constrained surrogates require pivotal statistics, algorithm choice and test-statistic choice are coupled; the review makes this coupling explicit.
Simulated annealing can enforce the 'static nonlinear transform of a linear process' null without inverting the nonlinear filter, at the cost of increased computation time.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same-process assumption behind temporal-shift surrogates suggests a calibration check: scan over shift $\tau$ and coefficient ratio $\alpha/\beta$ on known linear processes and verify the false-rejection rate stays at its nominal level.
Small-shuffle surrogates could be combined with rank-based or other non-pivotal statistics if the null distribution is estimated conditionally on the shuffled index sets; the review's framework indicates the resulting error rate would need explicit calibration.
The constrained-realization principle generalizes beyond the four algorithms: any resampling scheme that can draw fresh realizations from a fitted process under the null—parametric or nonparametric—fits the same testing framework.
For multivariate or multichannel recordings, the same design goals (preserve global structure, destroy local dependence) would suggest extensions of the small-shuffle and attractor-trajectory ideas, though the review does not develop them.

Original abstract

The surrogate data method is widely applied as a data dependent technique to test observed time series against a barrage of hypotheses. However, often the hypotheses one is able to address are not those of greatest interest, particularly for system known to be nonlinear. In the review we focus on techniques which overcome this shortcoming. We summarize a number of recently developed surrogate data methods. While our review of surrogate methods is not exhaustive, we do focus on methods which may be applied to experimental, and potentially nonlinear, data. In each case, the hypothesis being tested is one of the interests to the experimental scientist.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,732 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.92 · human participants 0.05 · controlled comparison 0.11 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.11 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The surrogate data method is widely applied as a data dependent technique to test observed time series against a barrage of hypotheses. However, often the hypotheses one is able to address are not those of greatest interest, particularly for system known to be nonlinear. In the review we focus on techniques which overcome this shortcoming. We summarize a number of recently developed surrogate data methods. While our review of surrogate methods is not exhaustive, we do focus on methods which may be applied to experimental, and potentially nonlinear, data. In each case, the hypothesis being tested is one of the interests to the experimental scientist.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The surrogate data method is widely applied as a data dependent technique to test observed time series against a barrage of hypotheses.	background 1.00	0.04	0.04	asserted 1.00
s1	However, often the hypotheses one is able to address are not those of greatest interest, particularly for system known to be nonlinear.	gap 0.62	0.05	0.07	asserted 0.83
s2	In the review we focus on techniques which overcome this shortcoming.	contribution 0.92	0.33	0.11	asserted 0.79
s3	We summarize a number of recently developed surrogate data methods.	contribution 0.99	0.73	0.07	asserted 0.99
s4	While our review of surrogate methods is not exhaustive, we do focus on methods which may be applied to experimental, and potentially nonlinear, data.	limitation 0.91	0.73	0.07	hedged possibility 0.87
s5	In each case, the hypothesis being tested is one of the interests to the experimental scientist.	contribution 0.91	0.42	0.30	asserted 1.00

What kind of paper the classifier reports this is: review 1.00 · evidence from literature 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** (a) Nuclear magnetic resonance (NMR) laser data, (b)

**Figure 2.** Schematic representation of the hierarchical hypot

**Figure 3.** An illustration of that the addition of two subsets of

Figures reproduced from arXiv:nlin/0603004 by the authors.

References

Works cited (46)

[29]

T. Schreiber, Phys. Rev. Lett. 80, 2105 (1998)

View on Pith 1998
[1]

E. Aarts and J. Korst, Simulated Annealing and Boltzmann Machines : A Stochastic Approach to Combinatorial Optimization and Neu ral Computing (Wiley, 1989). 24

View on Pith 1989
[2]

G.A. Barnard, J. Roy. Statis Soc. B 25, 294 (1963)

View on Pith 1963
[3]

G.E.P. Box, G.M. Jenkins and G.C. Reinsel, Time Series An alysis: Forecasting and Control 3rd Ed. (Prentice-Hall, 1994)

View on Pith 1994
[4]

C. Diks, Phys. Rev. E 53, 4263 (1996)

View on Pith 1996
[5]

K. Dolan, A. Witt, M.L. Spano, A. Neiman and F. Moss, Phys. Rev. E 59, 5235 (1999)

View on Pith 1999
[6]

A. Galka, Topics in Nonlinear Time Series Analysis with I mplications for EEG Analysis (World Scientiﬁc, 2000)

View on Pith 2000
[7]

P. Good, Permutation, Parametric, and Bootstrap Tests o f Hypotheses 3rd Ed. (Springer, 2005)

View on Pith 2005
[8]

P. Grassberger and I. Procaccia, Phys. Rev. Lett. 50, 346 (1983)

View on Pith 1983
[9]

P. Grassberger and I. Procaccia, Physica D 9, 189 (1983)

View on Pith 1983
[10]

A.C.A. Hope, J. Roy. Statis Soc. B 30, 582 (1968)

View on Pith 1968
[11]

K. Judd, Building optimal models of time series, in Chao s and its Reconstruction (ed. G. Gousebet, S. Meunier-Guttin-Cluze l and O. Menard) (Nova Science Pub Inc, 2003)

View on Pith 2003
[12]

X. Luo, T. Nakamura and M. Small, Phys. Rev. E 71: 026230 ( 2005)

View on Pith 2005
[13]

X. Luo, J. Zhang and M. Small, submitted

View on Pith
[14]

D.T. Kaplan and L. Glass, Phys. Rev. Lett. 68, 427 (1992)

View on Pith 1992
[15]

D.T. Kaplan, Physica D 73, 38 (1994)

View on Pith 1994
[16]

S. Kirkpatrick, C.D. Gelatt and M.P. Vecchi, Science 22 0, 671 (1983)

View on Pith 1983
[17]

D. Kugiumtzis, Phys. Rev. E 62, 25 (2000)

View on Pith 2000
[18]

H.R. K¨ unsch, Annals of Statistics 17, 1271 (1989). 25

View on Pith 1989
[19]

R. Maˆ n´ e, On the dimension of the compact invariant sets of certain non-linear maps, in Lecture Notes in Math. Vol. 898 (Springer, New York, 1981)

View on Pith 1981
[20]

T. Nakamura, X. Luo and M. Small, Phys. Rev. E 72: 055201 ( 2005)

View on Pith 2005
[21]

T. Nakamura and M. Small, Phys. Rev. E 72: 056216 (2005)

View on Pith 2005
[22]

A.I. Mees, Nonlinear Dynamics and Statistics (Birkh¨ a user, 2000)

View on Pith 2000
[23]

E.W. Noreen, Computer-Intensive Methods for Testing H ypotheses: An Introduction (Wiley, 1989)

View on Pith 1989
[24]

A.R. Osborne and A. Provenzale, Physica D 35, 357 (1989)

View on Pith 1989
[25]

M. Paluˇ s V. Albrecht and I. Dvoˇ r´ ak, Phys. Lett. A 175,203 (1993)

View on Pith 1993
[26]

T. Sauer, J.A. Yorke and M.C. Casdagli, J. Stat. Phys. 65 , 579 (1991)

View on Pith 1991
[27]

T. Schreiber and A. Schmitz, Phys. Rev. E 55, 5443 (1997)

View on Pith 1997
[28]

T. Schreiber and A. Schmitz, Phys. Rev. Lett. 77, 635 (19 96)

View on Pith
[30]

T. Schreiber and A. Schmitz, Physica D 142, 346 (2000)

View on Pith 2000
[31]

M. Small and K. Judd, Physica D 120, 386 (1998)

View on Pith 1998
[32]

M. Small, D.J. Yu and R. G. Harrison, Phys. Rev. Lett. 87: 188101 (2001)

View on Pith 2001
[33]

M. Small, Applied Nonlinear Time Series Analysis: Appl ications in Physics, Physiology and Finance (World Scientiﬁc, 2005)

View on Pith 2005
[34]

F. Takens, Detecting Strange Attractors in Turbulence , in Lecture Notes in Math. Vol. 898 (Springer, New York, 1981)

View on Pith 1981
[35]

F. Takens, International Journal of Bifurcation and Ch aos 3, 241 (1993)

View on Pith 1993
[36]

J. Theiler, S. Eubank, A. Longtin, B. Gaidrikian and J.D . Farmer, Physica D 58, 77 (1992). 26

View on Pith 1992
[37]

J. Theiler, Phys. Lett. A 196, 335 (1995)

View on Pith 1995
[38]

J. Theiler and D. Prichard, Physica D 94, 221 (1996)

View on Pith 1996
[39]

J. Theiler and P.E.Rapp, Electroenc. Clin. Neurophys. 98, 213 (1996)

View on Pith 1996
[40]

J. Theiler and D. Prichard, Fields Inst. Commun. 11, 99 ( 1997)

View on Pith 1997
[41]

C.P. Unsworth., M.R. Cowper, S. McLaughlin and B. Mulgr ew, Physica D 155, 51 (2001)

View on Pith 2001
[42]

P. van Laarhoven and E. Aarts, Simulated Annealing: The ory and Applications (Kluwer Academic Publishers, 1988)

View on Pith 1988
[43]

R. Wayland, D. Bromley, D. Pickett and A. Passamante, Ph ys. Rev. Lett. 70, 580 (1993)

View on Pith 1993
[44]

E.W. Weisstein, Hypothesis testing, from MathWorld (http://mathworld.wolfram.com/HypothesisTesting.html)

View on Pith
[45]

D.J. Yu, M. Small, R.G. Harrison and C. Diks, Phys. Rev. E 61, 3750 (2000)

View on Pith 2000
[46]

J. Zhang, X. Luo and M. Small, Phys. Rev. E. 73, 016216 (20 06). 27

View on Pith

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [29]

T. Schreiber, Phys. Rev. Lett. 80, 2105 (1998)

View on Pith 1998

[2] [1]

E. Aarts and J. Korst, Simulated Annealing and Boltzmann Machines : A Stochastic Approach to Combinatorial Optimization and Neu ral Computing (Wiley, 1989). 24

View on Pith 1989

[3] [2]

G.A. Barnard, J. Roy. Statis Soc. B 25, 294 (1963)

View on Pith 1963

[4] [3]

G.E.P. Box, G.M. Jenkins and G.C. Reinsel, Time Series An alysis: Forecasting and Control 3rd Ed. (Prentice-Hall, 1994)

View on Pith 1994

[5] [4]

C. Diks, Phys. Rev. E 53, 4263 (1996)

View on Pith 1996

[6] [5]

K. Dolan, A. Witt, M.L. Spano, A. Neiman and F. Moss, Phys. Rev. E 59, 5235 (1999)

View on Pith 1999

[7] [6]

A. Galka, Topics in Nonlinear Time Series Analysis with I mplications for EEG Analysis (World Scientiﬁc, 2000)

View on Pith 2000

[8] [7]

P. Good, Permutation, Parametric, and Bootstrap Tests o f Hypotheses 3rd Ed. (Springer, 2005)

View on Pith 2005

[9] [8]

P. Grassberger and I. Procaccia, Phys. Rev. Lett. 50, 346 (1983)

View on Pith 1983

[10] [9]

P. Grassberger and I. Procaccia, Physica D 9, 189 (1983)

View on Pith 1983

[11] [10]

A.C.A. Hope, J. Roy. Statis Soc. B 30, 582 (1968)

View on Pith 1968

[12] [11]

K. Judd, Building optimal models of time series, in Chao s and its Reconstruction (ed. G. Gousebet, S. Meunier-Guttin-Cluze l and O. Menard) (Nova Science Pub Inc, 2003)

View on Pith 2003

[13] [12]

X. Luo, T. Nakamura and M. Small, Phys. Rev. E 71: 026230 ( 2005)

View on Pith 2005

[14] [13]

X. Luo, J. Zhang and M. Small, submitted

View on Pith

[15] [14]

D.T. Kaplan and L. Glass, Phys. Rev. Lett. 68, 427 (1992)

View on Pith 1992

[16] [15]

D.T. Kaplan, Physica D 73, 38 (1994)

View on Pith 1994

[17] [16]

S. Kirkpatrick, C.D. Gelatt and M.P. Vecchi, Science 22 0, 671 (1983)

View on Pith 1983

[18] [17]

D. Kugiumtzis, Phys. Rev. E 62, 25 (2000)

View on Pith 2000

[19] [18]

H.R. K¨ unsch, Annals of Statistics 17, 1271 (1989). 25

View on Pith 1989

[20] [19]

R. Maˆ n´ e, On the dimension of the compact invariant sets of certain non-linear maps, in Lecture Notes in Math. Vol. 898 (Springer, New York, 1981)

View on Pith 1981

[21] [20]

T. Nakamura, X. Luo and M. Small, Phys. Rev. E 72: 055201 ( 2005)

View on Pith 2005

[22] [21]

T. Nakamura and M. Small, Phys. Rev. E 72: 056216 (2005)

View on Pith 2005

[23] [22]

A.I. Mees, Nonlinear Dynamics and Statistics (Birkh¨ a user, 2000)

View on Pith 2000

[24] [23]

E.W. Noreen, Computer-Intensive Methods for Testing H ypotheses: An Introduction (Wiley, 1989)

View on Pith 1989

[25] [24]

A.R. Osborne and A. Provenzale, Physica D 35, 357 (1989)

View on Pith 1989

[26] [25]

M. Paluˇ s V. Albrecht and I. Dvoˇ r´ ak, Phys. Lett. A 175,203 (1993)

View on Pith 1993

[27] [26]

T. Sauer, J.A. Yorke and M.C. Casdagli, J. Stat. Phys. 65 , 579 (1991)

View on Pith 1991

[28] [27]

T. Schreiber and A. Schmitz, Phys. Rev. E 55, 5443 (1997)

View on Pith 1997

[29] [28]

T. Schreiber and A. Schmitz, Phys. Rev. Lett. 77, 635 (19 96)

View on Pith

[30] [30]

T. Schreiber and A. Schmitz, Physica D 142, 346 (2000)

View on Pith 2000

[31] [31]

M. Small and K. Judd, Physica D 120, 386 (1998)

View on Pith 1998

[32] [32]

M. Small, D.J. Yu and R. G. Harrison, Phys. Rev. Lett. 87: 188101 (2001)

View on Pith 2001

[33] [33]

M. Small, Applied Nonlinear Time Series Analysis: Appl ications in Physics, Physiology and Finance (World Scientiﬁc, 2005)

View on Pith 2005

[34] [34]

F. Takens, Detecting Strange Attractors in Turbulence , in Lecture Notes in Math. Vol. 898 (Springer, New York, 1981)

View on Pith 1981

[35] [35]

F. Takens, International Journal of Bifurcation and Ch aos 3, 241 (1993)

View on Pith 1993

[36] [36]

J. Theiler, S. Eubank, A. Longtin, B. Gaidrikian and J.D . Farmer, Physica D 58, 77 (1992). 26

View on Pith 1992

[37] [37]

J. Theiler, Phys. Lett. A 196, 335 (1995)

View on Pith 1995

[38] [38]

J. Theiler and D. Prichard, Physica D 94, 221 (1996)

View on Pith 1996

[39] [39]

J. Theiler and P.E.Rapp, Electroenc. Clin. Neurophys. 98, 213 (1996)

View on Pith 1996

[40] [40]

J. Theiler and D. Prichard, Fields Inst. Commun. 11, 99 ( 1997)

View on Pith 1997

[41] [41]

C.P. Unsworth., M.R. Cowper, S. McLaughlin and B. Mulgr ew, Physica D 155, 51 (2001)

View on Pith 2001

[42] [42]

P. van Laarhoven and E. Aarts, Simulated Annealing: The ory and Applications (Kluwer Academic Publishers, 1988)

View on Pith 1988

[43] [43]

R. Wayland, D. Bromley, D. Pickett and A. Passamante, Ph ys. Rev. Lett. 70, 580 (1993)

View on Pith 1993

[44] [44]

E.W. Weisstein, Hypothesis testing, from MathWorld (http://mathworld.wolfram.com/HypothesisTesting.html)

View on Pith

[45] [45]

D.J. Yu, M. Small, R.G. Harrison and C. Diks, Phys. Rev. E 61, 3750 (2000)

View on Pith 2000

[46] [46]

J. Zhang, X. Luo and M. Small, Phys. Rev. E. 73, 016216 (20 06). 27

View on Pith