Momentswithin

Momentswithin is a term used in statistics and related fields to denote the computation of statistical moments of a data distribution restricted to a specified subset or neighborhood of the domain. Conceptually, it generalizes the global moments of a measure to local regions, allowing the description of spatially varying properties of data.

Formally, for a measure μ on a space X and a region R ⊆ X with μ(R) > 0, the k-th moment within R is m_k(R) = ∫_R φ_k(x) dμ(x), where φ_k(x) is x^k in one dimension or a multivariate monomial x_1^{k1} x_2^{k2} ... in higher dimensions. Centered moments within R are defined using the local mean μ_R = m_1(R)/m_0(R).

In practice, momentswithin can be estimated from samples by summing over points x_i ∈ R or by kernel-weighted integrals m_k(R) = ∑ w_R(x_i) φ_k(x_i), where w_R(x) is a weight function supported on R. This leads to local descriptors such as local mean, variance, skewness, and kurtosis, and to higher-order moments for texture and shape analysis. When applied to images, R may be a window or neighborhood around a pixel, enabling local texture analysis and feature extraction.

Applications include computer vision, image and texture analysis, geostatistics, and anomaly detection in sensory data. The method requires careful choice of region and weighting to balance bias and variance, and can be computationally intensive for high-dimensional data. See also moments, local statistics, kernel methods, and windowed analysis.