Multivariate recurrence plots for visualizing and quantifying the dynamics of spatially extended ecosystems

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Abstract

Few methods for quantifying the dynamics of temporal processes are readily applicable to spatially extended systems when equations governing the motion are unknown. The objective of this paper is to illustrate how the MRP-RQA (multivariate recurrence plot-recurrence quantification analysis) approach may serve to characterize ecosystems driven by both deterministic and stochastic forces. The strength of the MRP-RQA approach resides in its independence from constraining assumptions regarding outliers, noise, stationarity and transients. Its utility is demonstrated by means of two spatiotemporal series (summer and spring datasets) of light intensity variations in an old growth forest ecosystem. Results revealed qualitative differences in homogeneity, transiency, and drift typologies between the MRPs derived from each dataset. RQA estimates of determinism and Kolmogorov entropy supported the idea that mixed chaotic–stochastic dynamics may be common in mesoscale forest habitats. Advantages and inconveniences of the MRP-RQA approach are also discussed in the more general context of monitoring ecosystems.

Introduction

Among mathematical tools used to characterize the dynamics of complex natural processes for which equations governing the motion are unknown, many involve the technique of attractor reconstruction in state space (Godfray and Blythe, 1990, Hastings et al., 1993). Recurrence plots (RP) were initially introduced as a means of representing graphically the trajectories of a reconstructed dynamical system in state space (Eckmann et al., 1987). Given appropriate choices of parameter values, patterns appear in RPs which may not be detected otherwise. These patterns help the investigator to understand the dynamical nature of the source signal (e.g., fixed points, periodic, quasi-periodic, chaotic, and random). Because RPs do not require fitting any linear model to experimental data they may provide a valuable alternative for studying non-linear, non-stationary, high-dimensional ecological processes through space and time.

The expression recurrence quantification analysis (RQA) was first coined by Zbilut and Webber (1992) to encompass quantitative measures developed to describe patterns present in RPs. The majority of RQA measures are somehow related to the distribution of segment lengths (i.e., uninterrupted series of ones on a background of zeros) running parallel to the main diagonal in the RP (Webber and Zbilut, 1994, Zbilut et al., 2002, March et al., 2005). These diagonal segments represent the predictability time; i.e., the time that two trajectories are nearby before diverging out in state space. The longest diagonal segment represents the maximum timespan until the effects of a given perturbation die out in the system. The reciprocal of this segment is proportional to the dominant Lyapunov exponent (LE) (Eckmann et al., 1987), and this relationship was later related to the Kolmogorov entropy (Faure and Korn, 1998, Thiel et al., 2003). Similar measures were also derived for vertical segments (Marwan et al., 2002). The length of vertical segments represents the laminarity time; i.e., the amount of time two nearby trajectories remain parallel to each other in state space.

The RP-RQA approach forms an efficient strategy for both visualizing and quantifying temporal dynamics. This approach is advantageous compared to other methods for calculating attractor dimensions and LE spectra in systems subjected to external driving forces (Casdagli, 1997): (1) it provides robust RQA estimates in the presence of stochastic noise (Thiel et al., 2002, von Bloh et al., 2005); (2) it does not require stationarity as a prerequisite (Trulla et al., 1996); (3) it remains effective over short data series (Zbilut et al., 2000, Marwan et al., 2002); and (4) dynamical invariants like the correlation dimension, mutual information and Kolmogorov entropy are easily estimated (Faure and Korn, 1998, Thiel et al., 2003, March et al., 2005). Interestingly, the above attributes (i.e., stochasticity, non-stationarity, and short datasets) are often associated to experimental data. For this reason, the interest in RPs has grown steadily in various physical, medical, geographical, and biological applications, but the method is still relatively unknown.

To our knowledge only a few attempts have been made to adapt the RP-RQA approach to spatially extended processes. Two recent studies have expanded the RP-RQA approach to spatial snapshots obtained at fixed times (Marwan et al., 2005, Vasconcelos et al., 2006). However, it is still unclear how temporal and spatial RPs could be unified for characterizing spatiotemporal dynamics as a whole. Following another route, Romano et al. (2004) have developed a method for quantifying bivariate systems which builds on the concept of joint entropy in the so-called multivariate recurrence plot (MRP). Although the MRP method is straightforward and seems well suited for most experimental situations, its applicability to spatiotemporal systems has never been previously assessed.

We suggest that the multivariate dataset composed of the dominant spatial modes in a dynamical system can be used directly in the MRP-RQA approach. Our principal objective is to describe and evaluate this novel technique on experimental data driven by both deterministic and stochastic processes captured by a spatiotemporal series of light intensity changes in an old-growth forest ecosystem. More specifically, we aim to demonstrate how MRPs can be constructed from raster data and to introduce useful measures for characterizing their spatiotemporal dynamics. We believe this methodological framework could eventually form the basis of a surveillance network for monitoring ecosystem dynamics in space and time.

Section snippets

Multivariate recurrence plots

Let us first describe the construction of the RP from a single time series. Takens's theorem provides a way to reconstruct system dynamics in state space using the method of time delay coordinates (Takens, 1981). Given a one-dimensional time series of length n: μ = {μ1, μ2, μ3, …, μn} the attractor is defined from a series of vectors such that:vi=μi,μi+τ,,μi+(m1)τi=1n(m1)τ,where τ is the reconstruction delay and m is the embedding dimension. The RP is a N × N binary matrix denoted rεμ(i,j),

Characteristic length scale

In both the spring and summer datasets, the characteristic length scale was observed for subsystem sizes of 101–102 pixels (Fig. 2). Reported in true distance units, the characteristic length scale for these habitats is around 2–4 m wide; the size of the sampling grid used in vegetation surveys. At this scale there is little more gain to make in terms of spatial compression and we observe a leveling-off in the relationship DSVD versus subsystem size (Fig. 2). The characteristic length scale of

Concluding remarks

Although intricate chaotic–stochastic dynamics may turn out to be more the rule than the exception in ecosystems (Solé and Bascompte, 2006), convincing empirical evidence of spatiotemporal chaos in nature is still extremely hard to find. In that view, the two prototypical examples used in this study represent the sort of challenge faced by field scientists since they embody both observational and stochastic noise in addition to endogenous deterministic behaviours. While a strict deterministic

Acknowledgements

This work was financed by a grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada to L. Parrott. R. Proulx was partly supported by a graduate scholarship from the Fond Québécois de la Recherche sur la Nature et les Technologies (FQRNT). Special thanks to Véronique Tremblay and Gabriel Valois for field assistance and to everyone at the Mount St-Hilaire Reserve. Clément Chion and Pedro Peres-Neto also provided constructive comments on a previous version.

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    Present address: Geomatics and Landscape Ecology Lab, Carleton University, Nesbitt Building, 1125 Colonel By Drive, Ottawa, Ontario K1S 5B6, Canada.

    2

    Present address: Group for Research in Decision Analysis, HEC, Université de Montréal, C.P. 6128 succursale Centre-ville, Montréal, QC H3C 3J7, Canada.

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