Multivariate Analysis for Metabolomics and Proteomics Data - Proteomic and Metabolomic Approaches to Biomarker Discovery - page 293

TABLE 1 Example Data Set (cont'd)

TABLE 1 Example Data Set

Length Mass

Length Mass

0.824 124.100

0.801 121.410

0.788 120.800

0.824 127.700

0.782 107.400

0.841 129.200

0.795 120.700

0.816 131.800

0.805 121.910

0.840 135.100

0.836 122.310

0.842 131.500

0.788 110.600

0.820 126.700

0.772 103.510

0.802 115.100

0.776 110.710

0.828 130.800

0.758 113.800

0.819 124.600

0.826 118.310

0.802 114.200

are related to each other. Principal component
plots, also called loading plots, provide informa-
tion about how different variables are related to
each other. Because we are working with scaled
variables, the principal components (PCs) and
scores are dimensionless variables.
The mathematics of PCA can be clearly
described using linear algebra. 4 An excellent
discussion of linear algebra can be found in the
topic by Strang. 5 By convention, the data matrix,
X , has p columns and n rows, and each column
represents another variable and new rows for
each ob ser vation or sample. The average data
matrix, X , is the average of each individual
column (i.e., variable) in the data set. Mean
centering is written as
X X

0.810 120.300

0.802 115.700

0.832 117.510

0.796 109.810

0.759 109.100

0.770 115.100

0759 118.310

0.772 112.600

0.806 116.200

0.803 118.000

0.845 131.000

0.822 125.700

(2)

0.971 126.100

0.816 125.800

The covariance matrix is written as

0.836 125.500

T

(3)
where an uppercase T represents a matrix trans-
pose. The covariance matrix is a square,
symmetric, p

C ¼ðX XÞ ðX XÞ

0.815 127.800

0.822 130.500

0.822 127.900

p matrix. The covariance matrix
provides information about the relationship
between different variables. For example, the

0.843 123.900

( Continued )