Time Series for Power-Law Decay

Power-law decay time series are characterized by autocorrelation functions that decay as

-α

k

, where

k

is the lag and

α≥0

is the decay parameter. When

α∈(0,1)

, the time series exhibits strong persistence, with smaller values of

α

indicating stronger persistence. When

α>1

, the time series exhibits high-frequency (or alternating) behavior and is said to be anti-persistent. Such time series are also characterized by a spectral density function that is proportional to

α-1

λ

, where

λ

is the radial frequency.

This Demonstration investigates four basic types of power-law decay time series—HD (hyperbolic decay model), FD (fractionally differenced white noise), FGN (fractional Gaussian noise), and PPL (pure power-law process)—by simulating a series of length

n

and displaying one of five selected graphs: the time series plot (data), the autocorrelation function (ACF), the partial autocorrelation (PACF), the periodogram, and the spectral density function (SDF). In each case, blue represents the data or sample estimate and magenta represents the expected value.