AveragingFunktionen

AveragingFunktionen, a term used in mathematical contexts, designate functions that map a finite set or sequence of numbers to a single value that represents central tendency. They appear in statistics, data processing, and signal analysis to summarize data and smooth signals.

Common types include the arithmetic mean mu = (1/n) sum xi, the weighted mean mu_w = sum w_i xi / sum w_i, the geometric mean (prod xi)^{1/n} for positive xi, and the harmonic mean n / sum (1/xi). Time-series variants include the simple moving average S_t = (1/k) sum_{i=0}^{k-1} x_{t−i} and the exponential moving average E_t = alpha x_t + (1−alpha) E_{t−1}.

Properties: an averaging function maps a data collection to a value between its minimum and maximum; many such functions are monotone with respect to inputs; linear averaging is homogeneous with respect to scaling; some variants (trimmed or robust means) reduce sensitivity to outliers.

Applications include data smoothing in statistics and time series, noise reduction in signals, indicators in finance, preprocessing in experimental science, and image processing.

Computation: averaging functions are typically inexpensive to evaluate and can be updated incrementally for streaming data or windowed analyses.