Power spectra [of random processes]

Many random processes in nature show power spectra Abs[Fourier[data]]² with fairly simple forms. Most common are white noise uniform in frequency and 1/f² noise associated with random walks. Other pure power laws 1/f^α are also sometimes seen; the pictures below show some examples. (Note that the correlations in such data in some sense go like t^{α - 1}.) Particularly over the past few decades all sorts of examples of "1/f noise" have been identified with α ≃ 1, including flicker noise in resistors, semiconductor devices and vacuum tubes, as well as thunderstorms, earthquake and sunspot activity, heartbeat intervals, road traffic density and some DNA sequences. A pure 1/f^α spectrum presumably reflects some form of underlying nesting or self-similarity, although exactly what has usually been difficult to determine. Mechanisms that generally seem able to give α ≃ 1 include random walks with exponential waiting times, power-law distributions of step sizes (Lévy flights), or white noise variations of parameters, as well as random processes with exponentially distributed relaxation times (as from Boltzmann factors for uniformly distributed barrier heights), fractional integration of white noise, intermittency at transitions to chaos, and random substitution systems. (There was confusion in the late 1980s when theoretical studies of self-organized criticality failed correctly to take squares in computing power spectra.) Note that the Weierstrass function of page 918 yields a 1/f spectrum, and presumably suitable averages of spectra from any substitution system should also have 1/f^α forms (compare page 586).