Eliminating a Confounding Factor in Power Law Parameter Interpretation

Abstract:

Power law models of the form y = -ax^b are used to represent a wide range of phenomena in the physical, biological and social sciences. Power laws are well known to display a “scale-free property”, meaning that the value of the exponent b is independent of the unit of measurement (the “scale”) of the state variable x. While this makes estimation of the exponent robust, it raises significant estimation challenges for the linear multiplier a. Specifically, if both the multiplier a and exponent b are allowed to vary when undertaking an empirical fitting procedure, then physical units used to measure x induce a formal (i.e., entirely nonphysical) dependence between a and b, because the fitted value of a will contain a (potentially large) multiplicative factor that varies depending on the scale of x. This is problematic for two reasons: (i) if a is to be empirically estimated, but admits a physical interpretation, then the scale-dependent factor can confound the physically meaningful value (ii) the formal relationship between a and b due to the scaling of the relationship obscures any true relationship between the parameters and could motivate a spurious interpretation of their co-variation.

To correct this issue, we present a technique to remove the formal correlation between a and b, and demonstrate an application of the technique in the context of streamflow recession analysis, where the falling limb of the hydrograph is modeled using a simple power law relationship, dq/dt = -aq^b. Following the decorrelation of (a, b) recession parameter pairs, physically intuitive seasonal patterns, greatly obscured in the original data, are clearly found.