Where does long-range dependence collide?

In hydrology, the Hurst effect is the tendency of river flows, lake levels, and other geophysical records to stay above or below their long-run average for stretches far longer than short-memory models predict, so that wet and dry spells cluster over decades. Harold Hurst reported it in his 1951 Long-term storage capacity of reservoirs after studying centuries of Nile records, finding a rescaled-range scaling exponent near 0.7 rather than the 0.5 expected of independent years. Mandelbrot and Wallis later modelled it with fractional Gaussian noise in their 1968 Noah, Joseph, and operational hydrology. The referent is a slowly decaying autocorrelation that keeps distant observations linked, the same persistence that physics calls 1/f noise and econometrics calls long memory.[1][2]

Unit

The rescaled-range (Hurst) exponent H of a long geophysical record

Recovery means

Not applicable

Measured by

Estimated by rescaled-range (R/S) analysis and by the low-frequency slope of the power spectrum

In physics, 1/f noise, also called flicker or pink noise, is a fluctuation whose power spectral density rises toward low frequencies roughly as the inverse of frequency, so that slow, large excursions never fully average out and the signal carries structure across a huge range of timescales. Voss and Clarke's 1975 1/f noise in music and speech showed that even loudness and pitch fluctuations follow this law, one instance of a spectrum seen in resistors, semiconductors, heartbeats, and traffic. The referent is a scale-free persistence in a fluctuating signal, and a 1/f spectrum near zero frequency is mathematically the same long-range dependence that hydrology measures as the Hurst effect and econometrics as long memory.[3]

Unit

The low-frequency exponent β in a power spectral density proportional to 1/f^β

Recovery means

Not applicable

Measured by

Characterised by fitting the power spectral density at low frequencies and estimating its slope on log–log axes

In econometrics, long memory (or fractional integration) describes a time series whose autocorrelations decay so slowly, at a hyperbolic power-law rate rather than the geometric rate of standard ARMA models, that shocks influence the series far into the future without making it permanently non-stationary. Granger's 1980 Long memory relationships and the aggregation of dynamic models showed how aggregating many short-memory components can generate exactly this behaviour, motivating fractionally differenced models indexed by a parameter d between 0 and one half. The referent is the same slowly decaying dependence found in geophysical and physical signals, carried here by inflation, volatility, and interest-rate series and renamed for the language of time-series analysis.[4]

Unit

The fractional differencing parameter d of a long-memory time series

Recovery means

Not applicable

Measured by

Estimated from the fractional differencing parameter d, the spectral density near zero frequency, and rescaled-range or log-periodogram regressions