13 Statistical Techniques for the Interpretation of Analytical Data 693
of these techniques can have the following objectives: (1)to reduce the dimensions
of the datawithout losing important information, (2)to look for clustering of obser-
vations or variablesbased on some similar measure, (3)to define the rules to decide
on the assignation of an observationto a given group, (4)to study measures of
dependencebetween variables, (5)to predict the valuesof the variables from others
by applying a mathematical model, and (6)to develop and compare hypotheses
about some population parameters. As working tools these methods use algebraic
geometry, matricial calculus and numerical calculus.
13.3.1.1 Data Matrix
To apply these multivariate techniques, we require a data matrix with the informa-
tion corresponding tonobservations ofpquantitative variables (X 1 ,X 2 ,...,Xp).
We could, also, have some qualitative variables, coded numerically, to classify the
observations into groups. From a geometric perspective, thenobservations of the
data matrix would correspond tonpoints of the Euclidean space of thepvariables,
and the Euclidean distance between observations would correspond to a measure of
proximity (similarity).
13.3.1.2 Graphical Representation of the Data
The bidimensional methods of representation most used by multivariate techniques
are:direct methods, such as matricial dispersion diagrams, and icon plots based on
histograms, profiles or stars;projection approach techniques, that represent obser-
vations in the new variables obtained, and which fulfil a specific objective (principal
components, canonical variables, etc.) anddendrogramsthat inform about the sim-
ilarity of observations or variables (Krzanowski 1988).
13.3.1.3 Classification of Methods
Taking into account the groups of variables studied and the origin of the observa-
tions of the data matrix, we can havenon-supervised techniquesfor matrices with
nobservations derived from one population and a single group of variables,super-
vised methodsforkmatrices withnkobservations derived fromkpopulations and a
single group of variables, and methods for thedependence studyfor matrices with
nobservations from a single population and two groups of variables.
13.3.2 Multivariate Statistical Non Supervised Techniques
To apply these techniques, we have a data matrix withnobservations inpvariables
(X 1 ,X 2 , ...,Xp):