Selfmoments

Selfmoments are a class of statistical descriptors that summarize the shape and temporal structure of a dataset by computing moments using only information derived from the data itself. They are designed to reflect central tendency, dispersion, asymmetry, and tail behavior without relying on external reference distributions.

In practice, a selfmoment of order k for a finite sample X = {X_i} within a window W is defined as M_k(W) = (1/|W|) sum_{i in W} (X_i - m_W)^k, where m_W is a data-derived center statistic such as the mean, median, or a robust estimator computed from W. When computed in a moving window across a time series, these selfmoments form a time-varying portrait of local distributional shape, highlighting changes in variance, skewness, or kurtosis over time. Selfmoments can also be computed with self-normalized scales to improve comparability across heterogeneous segments.

Selfmoments are distinct from traditional moments anchored to a fixed population or an external reference, because both the center and often the scale are determined directly from the data. This makes them particularly useful for nonstationary signals, non-Gaussian distributions, or datasets where standardization against a global reference is inappropriate.

Applications include time-series analysis, anomaly detection, financial volatility studies, and texture or signal analysis in images. Limitations include sensitivity to window size and to outliers, and the lack of universal conventions for choosing the center statistic or the order of moments.

See also: Moments, Self-normalization, Time-series analysis, Local descriptors.