6.
Process or Product Monitoring and Control
6.5.
Tutorials
6.5.4.

Elements of Multivariate Analysis


Multivariate analysis

Multivariate analysis is a branch of statistics concerned with the
analysis of multiple measurements, made on one or several samples of
individuals. For example, we may wish to measure length, width, and
weight of a product.

Multiple measurement, or observation, as row or column vector

A multiple measurement or observation may be expressed as
$$
{\bf x} = \left[ \begin{array}{ccc}
4 & 2 & 0.6
\end{array} \right] \, , $$
referring to the physical properties of length, width, and weight,
respectively. It is customary to denote multivariate quantities with
bold letters. The collection of measurements on \({\bf x}\)
is called a vector. In this case it is a row vector. We could
have written \({\bf x}\)
as a column vector.
$$
{\bf x} = \left[ \begin{array}{c}
4 \\ 2 \\ 0.6
\end{array} \right] $$

Matrix to represent more than one multiple measurement

If we take several such measurements, we record them in a rectangular
array of numbers. For example, the \({\bf X}\)
matrix below represents five observations, on each of three variables.
$$ {\bf X} = \left[ \begin{array}{ccc}
4.0 & 2.0 & 0.60 \\
4.2 & 2.1 & 0.59 \\
3.9 & 2.0 & 0.58 \\
4.3 & 2.1 & 0.62 \\
4.1 & 2.2 & 0.63
\end{array} \right] $$

By convention, rows typically represent observations and columns
represent variables

In this case the number of rows, (\(n = 5\)),
is the number of observations, and the number of columns, (\(p=3\)),
is the number of variables that are measured. The rectangular array
is an assembly of \(n\)
row vectors of length \(p\).
This array is called a matrix, or, more specifically, a \(n\) by \(p\)
matrix. Its name is \({\bf X}\).
The names of matrices are usually written in bold,
uppercase letters, as in Section 6.5.3.
We could just as well have written \({\bf X}\) as a \(p\)
(variables) by \(n\)
(measurements) matrix as follows:
$$ {\bf X} = \left[ \begin{array}{ccccc}
4.0 & 4.2 & 3.9 & 4.3 & 4.1 \\
2.0 & 2.1 & 2.0 & 2.1 & 2.2 \\
0.60 & 0.59 & 0.58 & 0.62 & 0.63
\end{array} \right] $$

Definition of Transpose

A matrix with rows and columns exchanged in this manner is called the
transpose of the original matrix.
