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6.
Process or Product Monitoring and Control
6.5. Tutorials
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| 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
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 x is called a vector. In this case it is a row vector. We could have written x as a column vector. ![X=[4 2 0.6]](eqns/mvar1.gif)
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| 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 X matrix below represents
5 observations, on each of three variables.
![X = [4.0 2.0 .60; 4.2 2.1 .59; 3.9 2.0 .58; 4.3 2.1 .62;
4.1 2.2 .63]](eqns/mvar2.gif)
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| 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 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 X as a p (variables)
by n (measurements) matrix as follows:
![X = [4.0 4.2 3.9 4.3 4.1; 2.0 2.1 2.0 2.1 2.2;
.60 .59 .58 .62 .63]](eqns/mvar3.gif)
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| Definition of Transpose | A matrix with rows and columns exchanged in this manner is called the transpose of the original matrix. | ||
