In mathematics, a moment matrix is a special symmetric square matrix whose rows and columns are indexed by monomials. The entries of the matrix depend on the product of the indexing monomials only (cf. Hankel matrices.)
Moment matrices play an important role in polynomial fitting, polynomial optimization (since positive semidefinite moment matrices correspond to polynomials which are sums of squares)[1] and econometrics.[2]
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ECE6340 Lecture 20-5: Filling the Moment Matrix
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ECE6340 Lecture 20-1: Introduction to the Method of Moments
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ECE6340 Lecture 20-2: Unknowns with Method of Moments
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ECE6340 Lecture 20-3: Basis functions with the Method of Moments
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ECE6340 Lecture 20-4: Weight functions with the Method of Moments
Transcription
Application in regression
A multiple linear regression model can be written as
where is the explained variable, are the explanatory variables, is the error, and are unknown coefficients to be estimated. Given observations , we have a system of linear equations that can be expressed in matrix notation.[3]
or
where and are each a vector of dimension , is the design matrix of order , and is a vector of dimension . Under the Gauss–Markov assumptions, the best linear unbiased estimator of is the linear least squares estimator , involving the two moment matrices and defined as
and
where is a square normal matrix of dimension , and is a vector of dimension .
See also
References
- ^ Lasserre, Jean-Bernard, 1953- (2010). Moments, positive polynomials and their applications. World Scientific (Firm). London: Imperial College Press. ISBN 978-1-84816-446-8. OCLC 624365972.
{{cite book}}
: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link) - ^ Goldberger, Arthur S. (1964). "Classical Linear Regression". Econometric Theory. New York: John Wiley & Sons. pp. 156–212. ISBN 0-471-31101-4.
- ^ Huang, David S. (1970). Regression and Econometric Methods. New York: John Wiley & Sons. pp. 52–65. ISBN 0-471-41754-8.
External links
- "Moment matrix", Encyclopedia of Mathematics, EMS Press, 2001 [1994]