Kosambi–Karhunen–Loève theorem

The theorem expands a finite-variance stochastic process in eigenfunctions of its covariance operator. The resulting coefficients are uncorrelated, and truncating the expansion gives optimal mean-square approximation.

Connect

Covariance function: A function that measures how values of a random process at pairs of inputs vary together. Its integral operator supplies the eigenfunctions and eigenvalues used in the expansion.
Stochastic process: A collection of random variables indexed by time, space, or another parameter. The theorem represents this indexed random quantity as a random series.
Functional data analysis: Statistical analysis of data whose observations are curves, functions, or other structured objects. Functional principal components use the theorem to summarize variation across observed curves.
Fourier series: A representation of periodic functions as sums of sinusoidal modes. Fourier modes are fixed by periodic geometry, whereas covariance eigenfunctions adapt to the process.
Covariance operator: A linear operator formed from a covariance function that maps functions to functions. The theorem's modes are eigenfunctions of this operator.
Second-order stochastic process: A stochastic process whose values have finite second moments. Finite second moments make its covariance and mean-square approximation well-defined.
Gaussian process: A stochastic process whose every finite collection of values has a joint Gaussian distribution. Its eigenfunction expansion provides a basis for simulation and finite-dimensional approximation.
Singular value decomposition: A matrix factorization into orthogonal left and right singular vectors and nonnegative singular values. It gives the finite-dimensional counterpart of optimal orthogonal approximation.
Eigenfunction: A nonzero function that a linear operator maps to a scalar multiple of itself. Each covariance eigenfunction defines one mode in the process expansion.
Hilbert space: A complete inner-product space in which distances and orthogonality are defined. Mean-square random variables form an inner-product setting for orthogonal expansions.