Moment matrix

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 optimization, since positive semidefinite moment matrices correspond to polynomials which are sums of squares, and econometrics.^[1]

Application in regression

A multiple linear regression model can be written as

y=\beta _{0}+\beta _{1}x_{1}+\beta _{2}x_{2}+\dots \beta _{k}x_{k}+u

where $y$ is the explained variable, $x_{1},x_{2}\dots x_{k}$ are the explanatory variables, $u$ is the error, and $\beta _{0},\beta _{1}\dots \beta _{k}$ are unknown coefficients to be estimated. Given observations $\left\{y_{i},x_{1i},x_{2i},\dots x_{ki}\right\}_{i=1}^{n}$ , we have a system of $n$ linear equations that can be expressed in matrix notation.^[2]

{\begin{bmatrix}y_{1}\\y_{2}\\\vdots \\y_{n}\end{bmatrix}}={\begin{bmatrix}1&x_{11}&x_{12}&\dots &x_{1k}\\1&x_{21}&x_{22}&\dots &x_{2k}\\\vdots &\vdots &\vdots &\ddots &\vdots \\1&x_{n1}&x_{n2}&\dots &x_{nk}\\\end{bmatrix}}{\begin{bmatrix}\beta _{0}\\\beta _{1}\\\vdots \\\beta _{k}\end{bmatrix}}+{\begin{bmatrix}u_{1}\\u_{2}\\\vdots \\u_{n}\end{bmatrix}}

or

\mathbf {y} =\mathbf {X} {\boldsymbol {\beta }}+\mathbf {u}

where $\mathbf {y}$ and $\mathbf {u}$ are each a vector of dimension $n\times 1$ , $\mathbf {X}$ is the design matrix of order $N\times (k+1)$ , and ${\boldsymbol {\beta }}$ is a vector of dimension $(k+1)\times 1$ . Under the Gauss–Markov assumptions, the best linear unbiased estimator of ${\boldsymbol {\beta }}$ is the linear least squares estimator $\mathbf {b} =\left(\mathbf {X} ^{\mathsf {T}}\mathbf {X} \right)^{-1}\mathbf {X} ^{\mathsf {T}}\mathbf {y}$ , involving the two moment matrices $\mathbf {X} ^{\mathsf {T}}\mathbf {X}$ and $\mathbf {X} ^{\mathsf {T}}\mathbf {y}$ defined as

\mathbf {X} ^{\mathsf {T}}\mathbf {X} ={\begin{bmatrix}n&\sum x_{i1}&\sum x_{i2}&\dots &\sum x_{ik}\\\sum x_{i1}&\sum x_{i1}^{2}&\sum x_{i1}x_{i2}&\dots &\sum x_{i1}x_{ik}\\\sum x_{i2}&\sum x_{i1}x_{i2}&\sum x_{i2}^{2}&\dots &\sum x_{i2}x_{ik}\\\vdots &\vdots &\vdots &\ddots &\vdots \\\sum x_{ik}&\sum x_{i1}x_{ik}&\sum x_{i2}x_{ik}&\dots &\sum x_{ik}^{2}\end{bmatrix}}

and

\mathbf {X} ^{\mathsf {T}}\mathbf {y} ={\begin{bmatrix}\sum y_{i}\\\sum x_{i1}y_{i}\\\vdots \\\sum x_{ik}y_{i}\end{bmatrix}}

where $\mathbf {X} ^{\mathsf {T}}\mathbf {X}$ is a square matrix of dimension $(k+1)\times (k+1)$ , and $\mathbf {X} ^{\mathsf {T}}\mathbf {y}$ is a vector of dimension $(k+1)\times 1$ .

References

↑ 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

Hazewinkel, Michiel, ed. (2001) [1994], "Moment matrix", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

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[1] Goldberger, Arthur S. (1964). "Classical Linear Regression". Econometric Theory. New York: John Wiley & Sons. pp. 156–212. ISBN 0-471-31101-4.

[2] Huang, David S. (1970). Regression and Econometric Methods. New York: John Wiley & Sons. pp. 52–65. ISBN 0-471-41754-8.

Matrix classes
Explicitly constrained entries	(0,1) Alternant Anti-diagonal Anti-Hermitian Anti-symmetric Arrowhead Band Bidiagonal Binary Bisymmetric Block-diagonal Block Block tridiagonal Boolean Cauchy Centrosymmetric Conference Complex Hadamard Copositive Diagonally dominant Diagonal Discrete Fourier Transform Elementary Equivalent Frobenius Generalized permutation Hadamard Hankel Hermitian Hessenberg Hollow Integer Logical Markov Metzler Monomial Moore Nonnegative Partitioned Parisi Pentadiagonal Permutation Persymmetric Polynomial Positive Quaternionic Sign Signature Skew-Hermitian Skew-symmetric Skyline Sparse Sylvester Symmetric Toeplitz Triangular Tridiagonal Unitary Vandermonde Walsh Z
Constant	Exchange Hilbert Identity Lehmer Of ones Pascal Pauli Redheffer Shift Zero
Conditions on eigenvalues or eigenvectors	Companion Convergent Defective Diagonalizable Hurwitz Positive-definite Stability Stieltjes
Satisfying conditions on products or inverses	Congruent Idempotent or Projection Invertible Involutory Nilpotent Normal Orthogonal Orthonormal Singular Unimodular Unipotent Totally unimodular Weighing
With specific applications	Adjugate Alternating sign Augmented Bézout Carleman Cartan Circulant Cofactor Commutation Coxeter Derogatory Distance Duplication Elimination Euclidean distance Fundamental (linear differential equation) Generator Gramian Hessian Householder Jacobian Moment Payoff Pick Random Rotation Seifert Shear Similarity Symplectic Totally positive Transformation Wedderburn X–Y–Z
Used in statistics	Bernoulli Centering Correlation Covariance Design Dispersion Doubly stochastic Fisher information Hat Precision Stochastic Transition
Used in graph theory	Adjacency Biadjacency Degree Edmonds Incidence Laplacian Seidel adjacency Skew-adjacency Tutte
Used in science and engineering	Cabibbo–Kobayashi–Maskawa Density Fundamental (computer vision) Fuzzy associative Gamma Gell-Mann Hamiltonian Irregular Overlap S State transition Substitution Z (chemistry)
Related terms	Jordan canonical form Linear independence Matrix exponential Matrix representation of conic sections Perfect matrix Pseudoinverse Quaternionic matrix Row echelon form Wronskian
List of matrices Category:Matrices

Moment matrix

Application in regression

See also

References

External links