2 - Basic Mathematical and Statistical Concepts

Summary

SOME DEFINITIONS

The exposition of factor analysis may be approached in several ways. We shall treat it in a manner combining linear algebra, multidimensional geometry, and statistics. Many of the ideas involved will certainly be unfamiliar to some of you, and it is the purpose of this chapter to define and illustrate some of the fundamental concepts underlying the technique of factor analysis. We make no pretext of writing a treatise on linear algebra and so, if you wish to delve deeply into the algebra of vectors and matrices, you should consult a formal text on the subject. A very readable account is that of Davis (1965); another is Smith (1978).

We use a simple, geologically relevant example as a frame of reference in illustrating the examples of the various operations considered here.

The chart of numbers listed in Table 2.1 is typical of data collected by geologists, apart from its smallness. It contains information on the mineralogical composition of several rock specimens and it will be used to illustrate many of the basic concepts necessary for understanding factor analysis.

Data matrix

The table of numbers of Table 2.1 is an example of a data matrix. There are eight rows, one for each rock specimen, and four columns, one for each mineral. A data matrix is then an array of p characters measured on N specimens.

Matrix

A matrix is defined as any array of numbers with one or more rows and one or more columns. Table 2.1 is a matrix with eight rows and four columns.