6. Process or Product Monitoring and Control
6.5. Tutorials

## 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.