Data matrix

In the statistics which is the data matrix , also experimental design matrix , the design matrix (from English design research : German experimental design ) model matrix , observation matrix or regressor called a matrix , the data on a plurality of features of several persons or objects ( statistical units ) contains. It is the basis of the classical model of linear multiple regression .

The term test plan or design matrix (denoted by ) comes from the sub-area of statistical test planning, which deals with the statistically optimal design of experiments (see optimal test planning ). When the values ​​of the are planned (determined by the researcher), the matrix essentially contains the design and is therefore sometimes referred to as the design matrix. ${\displaystyle \mathbf {X} }$ ${\displaystyle x_{ij}}$ ${\displaystyle \mathbf {X} }$

definition

Assuming that there are units of investigation on which variables were observed, then the value observed on the -th unit of investigation is the -th variable . The data matrix is defined as the matrix ${\displaystyle n}$ ${\displaystyle p=k+1}$ ${\displaystyle i}$ ${\displaystyle j}$ ${\displaystyle x_{ij}}$ ${\displaystyle n\times p}$

${\displaystyle \mathbf {X} =(x_{ij})_{n\times p}={\begin{pmatrix}1&x_{11}&x_{12}&\cdots &x_{1k}\\1&x_{21}&x_{22}&\cdots &x_{2k}\\\vdots &\vdots &\vdots &\ddots &\vdots \\1&x_{n1}&x_{n2}&\cdots &x_{nk}\end{pmatrix}}}$.

The th row of the data matrix is the - with the most educated - observed th object variable values row vector , . The -th object can be represented geometrically as a point by interpreting the elements as the coordinates of a point in a -dimensional feature space that is spanned by feature axes arranged at right angles. If all line vectors of are represented as points in this way , a distribution of points in the feature space that represents the objects (units of investigation) results. ${\displaystyle i}$ ${\displaystyle \mathbf {X} }$ ${\displaystyle i}$ ${\displaystyle \mathbf {x} _{i\mathbf {.} }^{\top }=(x_{i0},x_{i1},\dotsc ,x_{ik})}$ ${\displaystyle i=1,\dotsc ,n}$ ${\displaystyle i}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle \mathbf {X} }$

Likewise, you can see the data matrix as a summary of column vectors , interpret. Each column vector is assigned to a variable and contains the values ​​of this variable observed on the units of investigation. With these values, the variables can be represented as points in a right-angled coordinate system in which the axes represent the units of investigation. The relationships between the variables can be illustrated in the object space spanned by the axes . ${\displaystyle \mathbf {x} _{\mathbf {.} j}=(x_{1j},x_{2j},\dotsc ,x_{nj})^{\top }}$ ${\displaystyle j=0,\dotsc ,k}$ ${\displaystyle X_{j}}$ ${\displaystyle n}$ ${\displaystyle n}$

Alternative representations

The data matrix can be expressed as a partitioned matrix with respect to its columns as ${\displaystyle \mathbf {X} }$ ${\displaystyle p=k+1}$

${\displaystyle \mathbf {X} =(\mathbf {1} ,\mathbf {x} _{(1)},\mathbf {x} _{(2)},\dotsc ,\mathbf {x} _{(k)})}$.

The columns of the data matrix including the one- vector are all -dimensional vectors and therefore points in the data space. Since it is usually assumed to be of rank , the vectors are linearly independent . The set of all possible linear combinations of the columns of form a subset of the data space. ${\displaystyle \mathbf {X} }$ ${\displaystyle \mathbf {1} }$ ${\displaystyle n}$ ${\displaystyle \mathbf {X} }$ ${\displaystyle k+1}$ ${\displaystyle \mathbf {X} }$

Individual evidence

1. ↑ ^{a b} design matrix. Glossary of statistical terms. In: International Statistical Institute . June 1, 2011, accessed May 19, 2020 . 
2. ^ Rencher, Alvin C., and G. Bruce Schaalje: Linear models in statistics. , John Wiley & Sons, 2008., p. 139
3. ↑ Werner Timischl : Applied Statistics. An introduction for biologists and medical professionals. 3. Edition. 2013, p. 420.
4. ↑ Werner Timischl: Applied Statistics. An introduction for biologists and medical professionals. 3. Edition. 2013, p. 420.
5. ^ Rencher, Alvin C., and G. Bruce Schaalje: Linear models in statistics. , John Wiley & Sons, 2008., p. 153.