A Tutorial on Multifractality, Cascades, and Interactivity for Empirical Time Series in Ecological Science

Abstract

Interactivity is a central theme of ecological psychology. According to Gibsonian views, behavior is the emergent property of interactions between organism and environment. Hence, an important challenge for ecological psychology has been to identify physical principles that provide an empirical window into interactivity. We suspect that multifractality, a concept from statistical physics, may be helpful in this regard, and we offer this article as a tutorial on multifractality with 2 main goals. First, we aim to describe multifractality with a series of simple, concrete, but progressively more elaborate examples that will incrementally elucidate the relationship between multifractality and interactivity. Second, we aim to describe a direct estimation method for computing the multifractal spectrum (e.g., Chhabra & Jensen, 1989), presenting it as an alternative that avoids the pitfalls of more popular methods and that may address more appropriately the measurements traditionally taken by ecological psychologists. In sum, this tutorial aims to unpack the theoretical background for an analytical method allowing rigorous test of interactivity in a variety of empirical settings.

Notes

^{1A speech gesture is a constriction formation or release event in the vocal tract that is produced by coordination among a set of synergistic speech articulators, for example, the bilabial lip-closing gesture for /b/ produced by the upper lip, lower lip, and jaw.}

²A foot is comprised of a group of one-to-several syllables, one of which typically receives primary lexical stress; for example, “parrot” and “regard” are two-syllable feet with stress on the first and second syllables, respectively.

³Proportions in Example 1 scale in the same way as Example 2, and so the plots for Example 2 would be equivalent to those for Example 1.

⁴Discrete rather than continuous spectra may be found in cases where singular fluctuations (e.g., scale-invariant fluctuations consistent with fractality) are interrupted by nonsingular fluctuations (e.g., oscillations with characteristic scales). Such cases yield “bifractal” spectra in which there are two distinct scaling relationships, often over two separate ranges of scales (Halsey et al., 1986). Because such a discrete case does not necessarily entail a continuous range of singular relations but may rather reflect the superposition of disjoint processes (i.e., one singular and the other not), we do not focus on potential links between bifractality and interactivity, and we instead limit our considerations to the relationship between continuous multifractal spectra and interactivity across scales.

⁵Whereas readers may recognize Examples 1 and 2 as an adaptation of a standard (i.e., monofractal) Cantor set, we intend our example of the binomial multiplicative cascade as an adaptation of Halsey et al.'s (1986) two-scale Cantor set. See Halsey et al.'s Section II.C.4.

⁶It is important to keep in mind that the tosses of coins, biased or not, bear no intrinsic probability distribution independent of the constraints we put upon them. We discuss only the constraint of a mathematical example's construction, but the reader may find interesting a discussion by Diaconis, Holmes, and Montgomery (2007) of how the outcome of a coin toss may depend wholly on its initial conditions. Diaconis et al. constructed a mechanical flipper arm that, short of mechanical failure, could replicate completely the dynamics of a toss, and they found that the same coin tossed in a mechanically identical manner will provide the exact same outcome 100% of the time. That is to say, what randomness that statisticians might want to convince us belongs to the coin is more properly attributed to the randomness of the coin's placement in the hand and the dynamics of the coin-tossing action. Whatever entailments this may have for perceptions of randomness, the moral for present considerations is that a biased coin in and of itself bears absolutely no necessary entailment on the distributional structure of the outcome of its being tossed.

⁷Note that H is elsewhere designated ζ (e.g., Ihlen & Vereijken, 2010).

⁸Hence, for q = 2, H(q) in the multifractal version of the analysis is equivalent to the monofractal case.

⁹Note that α(q) and f(α(q)) are elsewhere designated h and D(h), respectively (e.g., Ihlen & Vereijken, 2010).

¹⁰WTMM analysis involves a statistical adjustment called the “supremum condition” that essentially forces the fluctuation function to increase monotonically (e.g., Oswiecimka et al., 2006). Although this may smooth the route to estimating H(q) and does not require the judgment calls involved in selecting frequencies to filter, it may lead to the error of conflating the sinusoidal trend with the fractal fluctuations.

¹¹An anonymous reviewer brought to our attention a resemblance between the schematic of the “ugly” curve in the top-right panel of Figure 9 and the depiction of the binomial multiplicative cascade in Figure 8. First, it is important to note that the binomial multiplicative cascade is a theoretical and not measured process that might be instantiated in many different ways, sometimes as a continuous series of a ratio-scaled variable as in this work and elsewhere (Schumann & Kantelhardt, 2011), sometimes as discontinuous series (or more plainly, a histogram) of populated bins interspersed with gaps where bins might simply be empty (e.g., Halsey et al., 1986), and sometimes in terms of cumulative proportions over successive bins giving rise to a much smoother profile (Troutman & Over, 2001). We do not explicitly intend any particular comparison between Figure 8 and the top-right panel of Figure 9, and we do not present analysis of the series in Figure 8 using variance-based fractal methods. That said, if the “ugly case” in Figure 9 resembles a subset of the series in Figure 8, this resemblance may exemplify another informative aspect about the constraints of measurement conditions on the outcome of multifractal analysis: the “ugly” case schematized here may be due not only to systems driven by many underlying sinusoidal functions but also to too-short observations of systems that, under longer periods of observation (i.e., for which there are longer series), might show themselves to be less “ugly.” Put another way, observed quasiperiodicity may reflect (a) true system properties or (b) artifacts of experimental measurement that inadequately represent whatever the real system properties may be. We do not know how the measurement schematized in top-right panel of Figure 9 might continue if observation were to continue, and so we do not know whether more prolonged observation might resolve the “ugly” case into a “bad” or a “good” one.

¹²Because sampling intervals were constant, this measure may be understood as a sort of normalized velocity related to tangential velocity discussed in the motor-control literature (Schaal & Sternad, 2001; Viviani & Terzuolo, 1980). Note that, by definition, all such measurement values are ≥ 0.

¹³Bootstrap t tests comparing original spectrum width to surrogate spectrum widths may be less biased than ordinary t tests. Bootstrapping involves constructing very many (e.g., 1,000) new “artificial” samples by randomly selecting from the actual sample with replacement. However, for the time series reported in this article as well as various simulations not reported here, bootstrap t tests agreed with ordinary t tests in suggesting a significant difference between original and surrogate spectrum widths.