A Strategy for the Use of the Cross Recurrence Quantification Analysis

Teresa Aparicio; Eduardo F. Pozo; Dulce Saura

doi:10.1515/snde-2018-0103

Abstract

In this work, our goal is to analyze the use of the Cross Recurrence Plot (CRP) and its quantification (CRQA) as tools to detect the possible existence of a relationship between two systems. To do that, we define three tests that are a bivariate extension of those proposed by Aparicio et al. (Aparicio, T., E. Pozo, and D. Saura. 2008. “Detecting Determinism Using Recurrence Quantification Analysis: Three Test Procedures.” Journal of Economic Behavior & Organization 65: 768–787, Aparicio, T., E. F. Pozo, and D. Saura. 2011. “Detecting Determinism Using Recurrence Quantification Analysis: A Solution to the Problem of Embedding.” Studies in Nonlinear Dynamics and Econometrics 15: 1–10) within the context of the Recurrence Quantification Analysis. These tests, based on the diagonal lines of the CRP, are applied to a large number of simulated pairs of series. The results obtained are not always satisfactory, with problems being detected specifically when the series have a high degree of laminarity. We study the identified problems and we implement a strategy that we consider adequate for the use of these tools. Finally, as an example, we apply this strategy to several economic series.

Keywords: cross recurrence quantification analysis; Diks and Panchenko test; independence test; laminarity

Appendix: A brief description of the dynamic systems employed

This Appendix provides details of the equations that generate the dynamic systems used in our study.

Lorenz System:

ẋ = 10·(y – x)

ẏ = 28·(x – y – x·z)

ż = x·y – 8 3 x·z

Initial conditions:

x = 0.1, y = 0.1, z = 0.1

Time step (fixed) = 0.015.

Rossler System:

ẋ = – y – z

ẏ = x + 0.2 · y

ż = 0.4 + z · (x – 5.7)

Initial conditions:

x = 8.6274, y = −1.8956,

z = 0.9850.

Time step (fixed) = 0.105.

Henon System:

x_t+1 = 1 – 1.4·x_t ² + y_t

y_t+1 = 0.3·x_t

Initial conditions:

x = 1, y = 0.

Martin System:

x_t ₊₁ = y_t – sin(t)

y_t ₊₁ = 3.14 – x_t

Initial conditions:

x = 0, y = 0.

Ikeda System:

t = 0.4 − 6 1 + x t 2 + y t 2

x_t ₊₁ = 1 + 0.9·[x_t ·cos(t) – y_t ·sin(t)]

y_t ₊₁ = 0.9·[x_t ·sin(t) + y_t ·cos(t)]

Initial conditions:

x = 1, y = 0.

Lu Chen System

ẋ = 36·(y – x)

ẏ = x – x·z + 20·y

ż = x·y – 3·z

Initial conditions:

x = 0.1, y = 0.3, z = − 0.6

Time step (fixed) = 0.005.

Pickover System:

x_t ₊₁ = sin(2.87·y_t ) + 0.76·sin(2.87·x_t )

y_t ₊₁ = sin(−0.96·x_t ) + 0.74·sin(−0.96·y_t )

Initial conditions:

x = 1, y = 1.

Duffing System:

ẋ = y

ẏ = x – x ³ – 0.25·y + 0.3·sin(t)

Initial conditions:

x = 1, y = 1, t = 0

Time step (fixed) = 0.05.

Peter de Young System:

x_t ₊₁ = sin(1.76·y_t ) – sin(1.67·x_t )

y_t ₊₁ = sin(−0.85·x_t ) – cos(2.1·y_t )

Initial conditions:

x = 0, y = 0.5.

Hopalong System:

x_t ₊₁ = y_t – [abs(0.5·x_t – 9.25)]·sign(x_t – 1)

y_t ₊₁ = −0.85 – x_t

Initial conditions:

x = 0, y = 0.

Van der Pol – Duffing System:

ẋ = y

ẏ = −0.1·y – [x ³ + 12·cos(2·π·t)]

Initial conditions:

x = 2, y = 0.1, t = 0

Time step (fixed) = 0.02.

Ueda System:

ẋ = y

ẏ = −0.05·y – x ³ + 0.75·sin(t)

Initial conditions:

x = 2.5, y = 0, t = 0

Time step (fixed) = 0.02.

Rayleigh System:

ẋ = y

ẏ = – x + y – y ³

Initial conditions:

x = 1, y = 1

Time step (fixed) = 0.5.

Tinkerbell System:

x _t+1 = x_t ² – y_t ² + 0.9·x_t – 0.6013·y_t

y _t+1 = 2·x_t ·y_t + 2·x_t + 0.5·y_t

Initial conditions:

x = −0.72, y = −0.64.

Chua modified System

f = − 0.11 ⋅ sin ⁡ [ π ⋅ x 2.6 ]

ẋ = 10.814·(y – f)

ẏ = x – y + z

ż = −14.286·y

Initial conditions:

x = 1, y = 1, z = 0

Time step (fixed) = 0.1.

Marwan et al. (2007) System:

x_t : variable x of the Lorenz System

y _t+1 = 0.86·y_t + 0.2·x_t ² + 0.5·z_t

z_t : Gaussian N(0,1) signal.

Initial conditions:

x = 0.1, y = 0.

Bilinear model:

x _t+1 = e _t+1 + 0.3·x_t ·e_t

e_t : Gaussian N(0,1) signal.

Initial condition: x = 0.1.

ARCH1: ARCH(1) model with constant = 0.05 and ARCH coefficient = 0.2.

ARCH2: ARCH(1) model with constant = 0.05 and ARCH coefficient = 0.7.

GARCH1: GARCH(1,1) model with constant = 0.05, ARCH coefficient = 0.2 and GARCH coefficient = 0.7.

GARCH2: GARCH(1,1) model with constant = 0.05, ARCH coefficient = 0.7 and GARCH coefficient = 0.2.

EGARCH1: EGARCH(1,1) model with constant = 0.05, ARCH coefficient = 0.2, GARCH coefficient = 0.7 and leverage coefficient = −0.03.

EGARCH2: EGARCH(1,1) model with constant = 0.05, ARCH coefficient = 0.7, GARCH coefficient = 0.2 and leverage coefficient = −0.03.

Bivariate VAR(1): Stationary bivariate VAR(1) model, with Constant = [0.05; 0], AR coefficients = [0.5, 0; 0.1, 0.3]} and Covariance matrix = I(2).

BEKK-GARCH(1,1,1): Bivariate scalar BEKK-GARCH(1,1,1), with CCp = [1, 0.5; 0.5, 4], A² = 0.05, G² = 0.1 and B² = 0.88.

CCC-GARCH(1,0,1): Multivariate 3 × 3 CCC-GARCH(1,0,1) model: GARCH_coefficients = [0.1, 0.1, 0.8] and constant conditional correlation = [1, 0.2, 0.5; 0.2, 1, 0.3; 0.5, 0.3, 1].

Vect-GARCH(1,1,1),a: Bivariate vector-GARCH(1,1,1) model: GARCH parameters = (0.2, 0.4, 0.4) and Intercept = [1, 0.8; 0.8, 1].

Vect-GARCH(1,1,1),b: Bivariate vector-GARCH(1,1,1) model: GARCH parameters = (0.1, 0.1, 0.1) and Intercept = [1, 0.8; 0.8, 1].

References

Aparicio, T., E. Pozo, and D. Saura. 2008. “Detecting Determinism Using Recurrence Quantification Analysis: Three Test Procedures.” Journal of Economic Behavior & Organization 65: 768–787.10.1016/j.jebo.2006.03.005Search in Google Scholar

Aparicio, T., E. F. Pozo, and D. Saura. 2011. “Detecting Determinism Using Recurrence Quantification Analysis: A Solution to the Problem of Embedding.” Studies in Nonlinear Dynamics and Econometrics 15: 1–10.10.2202/1558-3708.1719Search in Google Scholar

Athreya, D. N., M. A. Riley, and T. J. Davis. 2014. “Visual Influences on Postural and Manual Interpersonal Coordination During a Joint Precision Task.” Experimental Brain Research 232: 2741–2751.10.1007/s00221-014-3957-2Search in Google Scholar PubMed

Bai, A., S. Hira, and S. Deshpande-Parag. 2017. “Recurrence Based Similarity Identification of Climate Data.” Discrete Dynamics in Nature and Society 2017: 7836720.10.1155/2017/7836720Search in Google Scholar

Bruni, R., A. Costantino, E. Tritarelli, C. Marcantonio, M. Ciccozzi, and M. Rapicetta. 2009. “A Computational Approach Identifies Two Regions of Hepatitis C Virus E1 Protein as Interacting Domains Involved in Viral Fusion Process.” BMC Structural Biol 9: art 48.10.1186/1472-6807-9-48Search in Google Scholar PubMed PubMed Central

Caraiani, P., and E. Haven. 2013. “The Role of Recurrence Plots in Characterizing the Output-Unemployment Relationship: An Analysis.” Plos One 8 (2): e56767.10.1371/journal.pone.0056767Search in Google Scholar PubMed PubMed Central

Crowley, P. 2008. “Analyzing Convergence and Synchronicity of Business and Growth Cycles in the Euro Area Using Cross Recurrence Plots.” The European Physical Journal Special Topics 164: 67–84.10.1140/epjst/e2008-00835-3Search in Google Scholar

Crowley, P. M., and A. Schultz. 2011. “Measuring the Intermittent Synchronicity of Macroeconomic Growth in Europe.” International Journal of Bifurcation and Chaos 21: 1215–1231.10.1142/S0218127411028957Search in Google Scholar

Dale, R., A. S. Warlaumont, and D. C. Richardson. 2011. “Nominal Cross Recurrence as a Generalized Lag Sequential Analysis for Behavioral Streams.” International Journal of Bifurcation and Chaos 21: 1153–1161.10.1142/S0218127411028970Search in Google Scholar

Diks, C., and V. Panchenko. 2005. “A note on the Hiemstra-Jones test for Granger Non-causality.” Studies in Nonlinear Dynamics and Econometrics 9 (2): 1–9.10.2202/1558-3708.1234Search in Google Scholar

Diks, C., and V. Panchenko. 2006. “A new Statistic and Practical Guidelines for Nonparametric Granger Causality Testing.” Journal of Economics Dynamics and Control 30 (9): 1647–1669.10.1016/j.jedc.2005.08.008Search in Google Scholar

Eckmann, J. P., S. Kamphorst, and D. Ruelle. 1987. “Recurrence Plots of Dynamical Systems.” Europhysics Letters 4: 973–977.10.1142/9789812833709_0030Search in Google Scholar

Elias, J., and V. N. Narayanan. 2014. “Cross-Recurrence Plot Quantification Analysis of Input and Output Signals for the Detection of Chatter in Turning.” Nonlinear Dynamics 76: 255–261.10.1007/s11071-013-1124-0Search in Google Scholar

Gao, J., and H. Cai. 2000. “On the Structures and Quantification of Recurrence Plots.” Physics Letters A 270: 75–87.10.1016/S0375-9601(00)00304-2Search in Google Scholar

Grzybowski, H., and R. Mosdorf. 2016. “Application of Cross Recurrence Plot for Identification of Temperature Fluctuations Synchronization in Parallel Minichannels.” Journal of Physics: Conference Series 745: 032126.10.1088/1742-6596/745/3/032126Search in Google Scholar

Hiemstra, C., and J. D. Jones. 1994. “Testing for Linear and Nonlinear Granger Causality in the Stock Price-Volume Relation.” Journal of Finance 49 (5): 1639–1664.10.1111/j.1540-6261.1994.tb04776.xSearch in Google Scholar

Li, K., and Z. Li. 2013. “Cross Recurrence Quantification Analysis of Precision Grip Following Peripheral Median Nerve Bloc.” Journal of NeuroEngineering and Rehabilitation 10: art. 28.10.1186/1743-0003-10-28Search in Google Scholar PubMed PubMed Central

Marwan, N., N. Wessel, U. Meyerfeldt, A. Schirdewan, and J. Kurths. 2002. “Recurrence Plot Based Measures of Complexity and its Application to Heart Rate Variability Data.” Physical Review E 66: 026702.10.1103/PhysRevE.66.026702Search in Google Scholar PubMed

Marwan, N., M. C. Romano, M. Thiel, and J. Kurts. 2007. “Recurrence Plots for the Analysis of Complex Systems.” Physics Reports 438: 237–329.10.1016/j.physrep.2006.11.001Search in Google Scholar

Marwan, N. 2011. “How to avoid potential pitfalls in recurrence plot based data Analysis.” International Journal of Bifurcation and Chaos 21 (4): 1003–1017.10.1142/S0218127411029008Search in Google Scholar

Namboodiri, S., A. Giuliani, A. S. Nair, and P. K. Dhar. 2012. “Looking for a Sequence Based Allostery Definition: A Statistical Journey at Different Resolution Scales.” Journal of Theoretical Biology 304: 211–218.10.1016/j.jtbi.2012.03.005Search in Google Scholar PubMed

Packard, N., J. P. Crutchfield, J. D. Farmer, and R. S. Shaw. 1980. “Geometry from a Time Series.” Physical Review Letters 45: 712–716.10.1103/PhysRevLett.45.712Search in Google Scholar

Pen, Y., and Z. Sun. 2011. “Characterization of QT and RR Interval Series During Acute Myocardial Ischemia by Means of Recurrence Quantification Analysis.” Medical and Biological Engineering and Computing 49: 25–31.10.1007/s11517-010-0671-5Search in Google Scholar

Romano, M. C. 2004. “Synchronization Analysis by Means of Recurrences in Phase Space.” PhD. Dissertation. University of Potsdam, Germany.10.1063/1.1846494Search in Google Scholar

Schinkel, S., O. Dimigen, and N. Marwan. 2008. “Selection of Recurrence Threshold for Signal Detection.” The European Physical Journal Special Topics 164: 45–53.10.1140/epjst/e2008-00833-5Search in Google Scholar

Shockley, K. 2005. “Cross Recurrence Quantification of Interpersonal Postural Activity.” In Tutorials in Contemporary Nonlinear Methods for the Behavioral Sciences, edited by M. A. Riley and C. G. Van Orden, 142–177. Retrieved March 1, 2005, from http://www.nsf.gov/sbe/bcs/pac/nmbs/nmbs.jsp.Search in Google Scholar

Shockley, K., M. Butwill, J. P. Zbilut, and C. L. Webber Jr. 2002. “Cross Recurrence Quantification of Coupled Oscillators.” Physics Letters A 305: 59–69.10.1016/S0375-9601(02)01411-1Search in Google Scholar

Spiridonov, A. 2017. “Recurrence and Cross Recurrence Plots Reveal the Onset of the Mulde Event (Silurian) in the Abundance Data for Baltic Conodonts.” The Journal of Geology 125: 381–398.10.1086/691184Search in Google Scholar

Takens, F. 1981 Detecting Strange Attractors in Turbulence. In Dynamical Systems and Turbulence, edited by D. Rand and L. Young, 366–381. Berlin: Springer-Verlag.10.1007/BFb0091924Search in Google Scholar

Thiel, M., M. C. Romano, and J. Kurths. 2006. “Spurious structures in recurrence plots induced by embedding.” Nonlinear Dynamics, 44: 299–305.10.1007/s11071-006-2010-9Search in Google Scholar

Webber Jr., C. L., and J. P. Zbilut. 1985. “Dynamical Assessment of Physiological Systems and States Using Recurrence Plot Strategies.” Journal of Applied Physiology 76: 965–973.10.1152/jappl.1994.76.2.965Search in Google Scholar

Zbilut, J. P., and C. L. Webber Jr. 1992. “Embeddings and Delays as Derived Quantification of Recurrence Plots.” Physics Letters A 171: 199–203.10.1016/0375-9601(92)90426-MSearch in Google Scholar

Zbilut, J. P., A. Giuliani, and C. L. Webber Jr. 1998. “Detecting Deterministic Signals in Exceptionally Noisy Environments Using Cross-Recurrence Quantification.” Physics Letters A 246: 122–128.10.1016/S0375-9601(98)00457-5Search in Google Scholar

Supplementary Material

The online version of this article offers supplementary material (DOI: https://doi.org/10.1515/snde-2018-0103).

Published Online: 2020-04-23

You are currently not able to access this content.

Supplementary Material Details

A Strategy for the Use of the Cross Recurrence Quantification Analysis

Abstract

Appendix: A brief description of the dynamic systems employed

References

Supplementary Material

Articles in the same Issue

Articles in the same Issue