“1/fα noise” is equivalent to an eigenstructure power relation

The discovery that the power spectrum of a time series of observations has a 1/fα character has been thought to imply that the generating process has some hidden, remarkable, nature, such as self-organized criticality or interaction across multiple scales. We show that 1/fα noise is equivalent to a Markovian eigenstructure power relation for natural systems. Fluctuations of a stationary reversible Markov process are characterized in terms of the eigenvalues, λ, and eigenfunctions, Pλ, of its generator. The power relation states that the product of the density of the eigenvalues and the squared first moment of the eigenfunctions is approximately a power function, λ−α, if and only if the power spectral density is approximately 1/fα. This characterization of 1/fα noise goes some distance toward explaining its ubiquity in natural systems.