timean

Timean, also referred to as the time mean, is the time average of a quantity over a specified interval. It is used to summarize a signal or function by its central tendency in time and is a common tool in signal processing, physics, and time series analysis.

For a continuous-time function f(t) defined on an interval [a, b], the timean is

timean(f) = (1/(b - a)) ∫_a^b f(t) dt.

For a discrete-time sequence x_n, n = 1,...,N, the timean is

timean(x) = (1/N) ∑_{n=1}^N x_n.

In the case of a sliding window, the term timean can refer to the average of the values within the current window, sometimes called a moving average when the window slides.

Key properties include linearity: the timean of af + bg equals a timean(f) + b timean(g). For periodic functions with period aligning with the window, the timean equals the average over a full period (often zero for symmetric waves like sin t over [0, 2π]). In stochastic contexts, if a process is stationary and ergodic, the timean of a single sufficiently long realization converges to the ensemble mean.

Applications include characterizing signals, estimating long-term trends in climate data, and providing smooth representations of time series. Limitations arise when the underlying signal is non-stationary or when the chosen time window is inappropriate, since the timean can vary with window length and location.

See also: time series analysis, moving average, ensemble average, ergodic theorem, temporal mean.