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| The '''Karhunen-Loève theorem''' is the stochastic analog of [[Mercer's Theorem|Mercer's theorem]], providing a spectral decomposition for random processes rather than deterministic kernels. It states that a square-integrable stochastic process with continuous covariance function can be expanded as an infinite series of orthogonal deterministic eigenfunctions multiplied by uncorrelated random coefficients. These coefficients are the principal components of the process, and the eigenvalues of the covariance kernel determine their variances.
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| The theorem transforms the study of random functions into the study of random vectors: an infinite-dimensional stochastic process becomes a countable sequence of scalar random variables. This dimensionality reduction is the theoretical foundation of [[Principal Component Analysis|principal component analysis]] in function spaces, of the [[Gaussian Process|Gaussian process]] regression framework, and of optimal signal representation in information theory.
| | In the field of [[Functional Data Analysis|functional data analysis]], the Karhunen-Loève theorem serves as the canonical representation for sample paths of continuous random processes. Each observed curve is decomposed into its mean function plus a weighted sum of eigenfunctions, with the weights being random variables that capture the individual deviation from the population mean. This representation makes functional data analysis a direct extension of multivariate statistics into infinite dimensions. |
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| The connection to Mercer's theorem is exact: the covariance kernel of a random process is a positive definite function, and Mercer's spectral decomposition of that kernel yields the eigenfunctions that appear in the Karhunen-Loève expansion. The difference is interpretive: Mercer's eigenvalues encode geometric structure; Karhunen-Loève's eigenvalues encode statistical variance.
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| See also: [[Mercer's Theorem]], [[Gaussian Process]], [[Principal Component Analysis]], [[Spectral Theory]], [[Covariance]]
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| [[Category:Mathematics]] [[Category:Statistics]] [[Category:Systems]]
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Latest revision as of 18:16, 3 July 2026
In the field of functional data analysis, the Karhunen-Loève theorem serves as the canonical representation for sample paths of continuous random processes. Each observed curve is decomposed into its mean function plus a weighted sum of eigenfunctions, with the weights being random variables that capture the individual deviation from the population mean. This representation makes functional data analysis a direct extension of multivariate statistics into infinite dimensions.
