When the blind curve is finite: dimension estimation and model inference based on empirical waveforms

A PLEA FOR STRONG INFERENCE The recent research topic “Fractal Analyses: Statistical and Methodological Innovations and Best Practices” reveals there is no consensus among experts about the best procedure to estimate self-affine structure in trial and time series data. One of the recurring issues pertains to the validity of inferences based on analysis results about the physical change processes that generated the empirical waveforms. In this paper I argue that none of these approaches can be used to validate such inferences outside of the context of theory evaluation by strong inference (e.g., Platt, 1964). Two arguments warrant this claim: (1) All procedures make an assumption about the physics of the system under scrutiny. This is arguably most prominent in ARFIMA modeling, but associating an estimated scaling exponent to a fractal dimension is also based on assumptions (e.g., fGn vs. fBm; Mandelbrot and Van Ness, 1968); (2) given infinitesimal measurement resolution and infinite observation time, properties like dimension and self-affinity are not unique descriptors of a process, pattern or object (cf. Vicsek, 2001). Multiple mathematical models of physical processes can be constructed to generate a waveform with exactly the dynamical and invariant properties as observed in the finite sample (e.g., Mandelbrot, 2001; Kantz and Schreiber, 2003; Thornton and Gilden, 2005; Morrison, 2008). The second issue pertains to a general problem of model-based inference: a good fit to a finite sample of measurement outcomes can never be conclusive in the evaluation of predictions by theories (cf. Roberts and Pashler, 2000, 2002; Fiedler et al., 2012). Using results of (fractal) analyses to answer questions about the physics of the observed system is an attempt to evaluate the ontology of a theory, ex post facto; let’s leave ontology evaluation to the metaphysicians (cf. Poincaré, 1905, p. 211). The scientific method is not a competition for mathematical models constructed to produce the best fit to measurement outcomes; instead, theoretical predictions about the observed system compete for highest empirical precision and accuracy in order to gain scientific credibility. In what follows I evaluate to what extent fractal analyses are used in the context of strong inference given the current empirical record of human physiology and performance. Subsequently I will explore what may be gained when implicit ontology falsification is removed from fractal analyses by introducing the concepts of intuitive dimension and informed dimension estimates.