sigmaxy

Sigmaxy, commonly written as σxy, is a statistical quantity representing the cross-covariance between two random variables X and Y, or between two stochastic processes. It measures the degree to which X and Y vary together.

Definition and relation to other measures: For variables with means μx and μy and standard deviations σx and σy, the cross-covariance is defined as σxy = E[(X − μx)(Y − μy)]. The correlation coefficient is ρxy = σxy / (σx σy), which normalizes the covariance to a dimensionless measure of linear association.

Computation: In a sample of n paired observations (xi, yi), the sample cross-covariance is (1/(n−1)) ∑ (xi − x̄)(yi − ȳ). In matrix form, σxy is an element of the covariance matrix, with the (i, j) entry often denoted σij.

Applications: σxy appears across statistics, time series analysis, signal processing, and multivariate methods. It informs questions about how two variables co-vary, supports estimation of relationships in Kalman filters, and contributes to the structure of covariance matrices used in principal component analysis and other multivariate techniques.

Notes and distinctions: σxy is distinct from, though related to, cross-spectral density Sxy(ω), which is a frequency-domain analogue of covariance. The notation may vary by context, sometimes using subscripts or different conventions for σx and σy. Because covariance depends on the units of X and Y, it does not by itself indicate the strength of association without normalization.

Limitations: Covariance can be influenced by scale and outliers, and a nonzero covariance does not necessarily imply a strong or linear relationship. Context and supplementary statistics (such as correlation) are often required.