Benutzer:JonskiC/Zentrierende Matrix

Dieser Artikel ist im Entstehen und noch nicht Bestandteil der freien Enzyklopädie Wikipedia.

Solltest du über eine Suchmaschine darauf gestoVorlage:SSen sein, bedenke, dass der Text noch unvollständig sein und Fehler oder ungeprüfte Aussagen enthalten kann. Wenn du Fragen zum Thema hast, nimm Kontakt mit dem Autor JonskiC auf.

In der Statistik ist die zentrierende Matrix eine Projektionsmatrix, d. h. eine symmetrische und idempotente Matrix.

Definition

Die zentrierende Matrix der Dimension n ist definiert als die folgende $n\times n$-Matrix

$C_{n}=I_{n}-{\tfrac {1}{n}}\mathbb {O}$

where $I_{n}\,$ is the identity matrix of size n and $\mathbb {O}$ is an n-by-n matrix of all 1's. This can also be written as:

$C_{n}=I_{n}-{\tfrac {1}{n}}\mathbf {1} \mathbf {1} ^{\top }$

where $\mathbf {1}$ is the column-vector of n ones and where $\top$ denotes matrix transpose.

For example

$C_{1}={\begin{bmatrix}0\end{bmatrix}}$,

$C_{2}=\left[{\begin{array}{rrr}1&0\\0&1\end{array}}\right]-{\frac {1}{2}}\left[{\begin{array}{rrr}1&1\\1&1\end{array}}\right]=\left[{\begin{array}{rrr}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{array}}\right]$ ,

$C_{3}=\left[{\begin{array}{rrr}1&0&0\\0&1&0\\0&0&1\end{array}}\right]-{\frac {1}{3}}\left[{\begin{array}{rrr}1&1&1\\1&1&1\\1&1&1\end{array}}\right]=\left[{\begin{array}{rrr}{\frac {2}{3}}&-{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {2}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&-{\frac {1}{3}}&{\frac {2}{3}}\end{array}}\right]$

Properties

Given a column-vector, $\mathbf {v} \,$ of size n, the centering property of $C_{n}\,$ can be expressed as

$C_{n}\,\mathbf {v} =\mathbf {v} -({\tfrac {1}{n}}\mathbf {1} '\mathbf {v} )\mathbf {1}$

where ${\tfrac {1}{n}}\mathbf {1} '\mathbf {v}$ is the mean of the components of $\mathbf {v} \,$.

$C_{n}\,$ is symmetric positive semi-definite.

$C_{n}\,$ is idempotent, so that $C_{n}^{k}=C_{n}$, for $k=1,2,\ldots$. Once the mean has been removed, it is zero and removing it again has no effect.

$C_{n}\,$ is singular. The effects of applying the transformation $C_{n}\,\mathbf {v}$ cannot be reversed.

$C_{n}\,$ has the eigenvalue 1 of multiplicity n − 1 and eigenvalue 0 of multiplicity 1.

$C_{n}\,$ has a nullspace of dimension 1, along the vector $\mathbf {1}$.

$C_{n}\,$ is a projection matrix. That is, $C_{n}\mathbf {v}$ is a projection of $\mathbf {v} \,$ onto the (n − 1)-dimensional subspace that is orthogonal to the nullspace $\mathbf {1}$. (This is the subspace of all n-vectors whose components sum to zero.)

Application

Although multiplication by the centering matrix is not a computationally efficient way of removing the mean from a vector, it forms an analytical tool that conveniently and succinctly expresses mean removal. It can be used not only to remove the mean of a single vector, but also of multiple vectors stored in the rows or columns of a matrix. For an m-by-n matrix $X\,$, the multiplication $C_{m}\,X$ removes the means from each of the n columns, while $X\,C_{n}$ removes the means from each of the m rows.

The centering matrix provides in particular a succinct way to express the scatter matrix, $S=(X-\mu \mathbf {1} ')(X-\mu \mathbf {1} ')'$ of a data sample $X\,$, where $\mu ={\tfrac {1}{n}}X\mathbf {1}$ is the sample mean. The centering matrix allows us to express the scatter matrix more compactly as

$S=X\,C_{n}(X\,C_{n})'=X\,C_{n}\,C_{n}\,X\,'=X\,C_{n}\,X\,'.$

$C_{n}$ is the covariance matrix of the multinomial distribution, in the special case where the parameters of that distribution are $k=n$, and $p_{1}=p_{2}=\cdots =p_{n}={\frac {1}{n}}$.