timecorrelated

Timecorrelated describes processes, signals, or data in which observations taken at different times are statistically related. In time-series analysis and stochastic modeling, timecorrelated behavior means that values at one time carry information about values at other times, so the joint distribution cannot be decomposed into independent components. By contrast, white noise is an idealization with zero correlation between distinct times.

The degree and structure of timecorrelation are typically summarized by the autocorrelation function, R(τ), or the autocovariance function, γ(τ), which describe how the correlation between values decays as the time lag τ increases. If R(τ) depends only on τ, the process may be stationary in the weak sense; many models of timecorrelated data assume this property. Timecorrelated processes can exhibit short-range dependence, where correlation decays quickly, or long-range dependence, where correlation persists over long periods, often characterized by a slow, power-law decay and a Hurst exponent greater than 0.5.

Common models that generate timecorrelated behavior include autoregressive and moving-average processes (AR, MA, ARMA), which combine lagged values and noise to produce memory effects. Fractional Gaussian noise and fractional Brownian motion are examples with strong, long-range time correlations. Colored noise, such as pink (1/f) or brownian (1/f^2) noise, explicitly incorporates time correlation into its spectral properties.

Applications span engineering, physics, finance, climate science, and neuroscience, where understanding timecorrelation informs prediction, filtering, anomaly detection, and the modeling of memory effects. Estimation often relies on the sample autocorrelation function, spectral analysis, or fitting autoregressive-type models to observed data.