5 - Temporal Scaling and Evolutionary Mode

Summary

Temporal scaling is a method for analysis of: (1) the quantitative dependence of differences between samples in a time series on the amount of time separating the samples; and (2) the quantitative dependence of rates of change in a time series on the amount of time separating the samples (the denominator of the rate). Robust linear modeling provides efficient recovery of temporal scaling slopes and intercepts. Temporal scaling of differences and temporal scaling of rates are complementary and interchangeable in the sense that they recover complementary slopes and lead to the same inference of time-series ‘mode.’ The two are interchangeable too in the sense that they recover the same step-rate intercept reflecting time-series ‘tempo.’ Stationary, random, and directional time series have differences that scale with slopes of or near 0.000, 0.500, or 1.000 on a log difference versus log interval or LDI plot. The corresponding rates scale with slopes of or near −1.000, −0.500, 0.000 on a log rate versus log interval or LRI plot. Rates must have the same numerator or the same denominator if they are to be compared, and that numerator or denominator is integral to the comparison.