About: Kosambi–Karhunen–Loève theorem

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In the theory of stochastic processes, the Karhunen–Loève theorem (named after Kari Karhunen and Michel Loève), also known as the Kosambi–Karhunen–Loève theorem is a representation of a stochastic process as an infinite linear combination of orthogonal functions, analogous to a Fourier series representation of a function on a bounded interval. The transformation is also known as Hotelling transform and eigenvector transform, and is closely related to principal component analysis (PCA) technique widely used in image processing and in data analysis in many fields.

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rdfs:label	Kosambi–Karhunen–Loève theorem (en)
rdfs:comment	In the theory of stochastic processes, the Karhunen–Loève theorem (named after Kari Karhunen and Michel Loève), also known as the Kosambi–Karhunen–Loève theorem is a representation of a stochastic process as an infinite linear combination of orthogonal functions, analogous to a Fourier series representation of a function on a bounded interval. The transformation is also known as Hotelling transform and eigenvector transform, and is closely related to principal component analysis (PCA) technique widely used in image processing and in data analysis in many fields. (en)
dbp:wikiPageUsesTemplate	dbt:= dbt:Cite_book dbt:Cite_journal dbt:Definition_needed dbt:Expand_section dbt:Main_article dbt:Math dbt:Mvar dbt:Reflist dbt:Short_description dbt:Typo_help_inline dbt:Math_proof
Subject	Probability theorems Theorems in statistics Signal estimation
Link from a Wikipa... related subject.	dbpedia-fr:Transformée_de_Karhunen-Loève
prov:wasDerivedFrom	http://en.wikipedia.org/wiki/Kosambi–Karhunen–Loève_theorem?oldid=1073640876&ns=0
Wikipage page ID	767253 (xsd:integer)
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Wikipage revision ID	1073640876 (xsd:integer)
Link from a Wikipage to another Wikipage	Fourier series Probability space Proper orthogonal decomposition Uncorrelatedness (probability theory) Adaptive optics Additive white Gaussian noise Statistic Wiener process Wiener–Hopf method Brownian bridge Brownian motion Probability theorems Geophysics Meteorology An Introduction to the Study of Indian History Linear map Principal component analysis Square-integrable function Trigonometric functions Convergence of random variables Eigenvalues and eigenvectors Michel Loève Neyman–Pearson lemma Orthogonal functions Orthogonality principle Orthonormal basis Covariance function Empirical orthogonal functions Entropy (information theory) Functional derivative Gaussian process Independence (probability theory) Kari Karhunen Harold Hotelling Harvey Mudd College Integral equation Variance Theorems in statistics Stochastic process Mercer's theorem Maximum likelihood estimation Mean Mean squared error Polynomial chaos Random variable Bergman kernel Sine and cosine Singular value decomposition Signal estimation
has abstract	In the theory of stochastic processes, the Karhunen–Loève theorem (named after Kari Karhunen and Michel Loève), also known as the Kosambi–Karhunen–Loève theorem is a representation of a stochastic process as an infinite linear combination of orthogonal functions, analogous to a Fourier series representation of a function on a bounded interval. The transformation is also known as Hotelling transform and eigenvector transform, and is closely related to principal component analysis (PCA) technique widely used in image processing and in data analysis in many fields. Stochastic processes given by infinite series of this form were first considered by Damodar Dharmananda Kosambi. There exist many such expansions of a stochastic process: if the process is indexed over [a, b], any orthonormal basis of L2([a, b]) yields an expansion thereof in that form. The importance of the Karhunen–Loève theorem is that it yields the best such basis in the sense that it minimizes the total mean squared error. In contrast to a Fourier series where the coefficients are fixed numbers and the expansion basis consists of sinusoidal functions (that is, sine and cosine functions), the coefficients in the Karhunen–Loève theorem are random variables and the expansion basis depends on the process. In fact, the orthogonal basis functions used in this representation are determined by the covariance function of the process. One can think that the Karhunen–Loève transform adapts to the process in order to produce the best possible basis for its expansion. In the case of a centered stochastic process {Xt}t ∈ [a, b] (centered means E[Xt] = 0 for all t ∈ [a, b]) satisfying a technical continuity condition, X admits a decomposition where Zk are pairwise uncorrelated random variables and the functions ek are continuous real-valued functions on [a, b] that are pairwise orthogonal in L2([a, b]). It is therefore sometimes said that the expansion is bi-orthogonal since the random coefficients Zk are orthogonal in the probability space while the deterministic functions ek are orthogonal in the time domain. The general case of a process Xt that is not centered can be brought back to the case of a centered process by considering Xt − E[Xt] which is a centered process. Moreover, if the process is Gaussian, then the random variables Zk are Gaussian and stochastically independent. This result generalizes the Karhunen–Loève transform. An important example of a centered real stochastic process on [0, 1] is the Wiener process; the Karhunen–Loève theorem can be used to provide a canonical orthogonal representation for it. In this case the expansion consists of sinusoidal functions. The above expansion into uncorrelated random variables is also known as the Karhunen–Loève expansion or Karhunen–Loève decomposition. The empirical version (i.e., with the coefficients computed from a sample) is known as the Karhunen–Loève transform (KLT), principal component analysis, proper orthogonal decomposition (POD), empirical orthogonal functions (a term used in meteorology and geophysics), or the Hotelling transform. (en)
foaf:isPrimaryTopicOf	http://en.wikipedia.org/wiki/Kosambi–Karhunen–Loève_theorem
is Wikipage redirect of	Karhunen-Loeve decomposition Karhunen-Loève theorem Karhunen-Loève transform Karhunen-loeve expansion Karhunen–Loeve theorem Karhunen–Loève theorem Loève–Karhunen theorem Kosambi-Karhunen-Loève theorem Loève-Karhunen theoem Loève-Karhunen theorem Loève–Karhunen theoem Karhunen-Loeve expansion Karhunen-Loeve theorem Karhunen-Loeve transform Karhunen-Loève decomposition Karhunen-Loève expansion Karhunen-loève expansion Karhunen–Loeve expansion Karhunen–Loeve transform Karhunen–Loève expansion Karhunen–Loève transform
is Link from a Wikipage to another Wikipage of	List of statistics articles List of stochastic processes topics List of theorems List of transforms Non-negative matrix factorization Nonlinear dimensionality reduction

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