394
M. Forina . S. Lanteri . R. Todeschini
component El and with negative loading to component E2. The location of object 20
can be explained with very large values of X and Z. The location of object 5 is due to
small values of all the original variables.
Moreover, three objects, namely 18, 19, 20, are clearly separated from the cluster of
the other objects. A new direction, VI, explains the difference between the three objects
and the others. This direction, and the second direction V2, is obtained by further rotation with a technique of the family of varimax rotations, and for this reason they are
called varivectors. As PCs, varivectors have their loadings of the original variables, so
that the three objects are characterized by very large values of X, and slightly large values of Y and Z. Fig. 21-5' shows the projection of the objects on the two varivectors.
Factor Analysis (FA) works by means of further rotations (orthogonal or not) in the
inner space, with the aim of helping in the interpretation and in the further use of the structured information. Varimax rotations are only one example of rotation techniques of FA.
21.2.4
Class-Modelling Techniques
This is a family of techniques used in classification problems (e.g. in quality control
the classes are acceptable/rejected; in toxicology they are often toxic/non-toxic).
Class-modelling techniques compute a mathematical model of the studied class.
In Fig. 21.6 the mathematical model of a class is a point, the centroid of the class. The
mathematical model of the other class is a suitable range of its first principal component. An object falling exactly on the mathematical model can be considered an ideal
object. The effect of some factors produces a more or less large distance from the
mathematical model for the other objects. A statistical procedure computes the maximum permitted distance, the boundary of the class space. This maximum distance
corresponds to a critical value of the used statistics. In the case of the centroid model
(UNEQ method) the distance is a Mahalanobis distance, which takes into account the
different variances of the variables and their correlation, and t statistics is used to
compute the critical value. In the case of PC model (SIMCA method) the class boundary is obtained by the variance in the outer space, and by F statistics.
The class model accepts an object which falls within the class space, with a significance level decreasing with the distance from the class centroid. The class model does
not accept an object which falls outside the class space. It can be considered as a nontypical object, or as an object very different from the other objects of the class according to its significance level.
In Fig. 21.6, object A fits both class models. This double possibility depends on the
not perfectly separated two-class space. Object A can be considered more similar to
class modelled by SIMCA. At first, it was considered an object of the class modelled
by UNEQ, but the assignment can be modified, and the models computed again (refinement of class model).
21.2.5
Regression Techniques, Responses and Predictors
Regression techniques are used to model the relationship between one (or more) response variable and some predictor variables.
M. Forina . S. Lanteri . R. Todeschini
component El and with negative loading to component E2. The location of object 20
can be explained with very large values of X and Z. The location of object 5 is due to
small values of all the original variables.
Moreover, three objects, namely 18, 19, 20, are clearly separated from the cluster of
the other objects. A new direction, VI, explains the difference between the three objects
and the others. This direction, and the second direction V2, is obtained by further rotation with a technique of the family of varimax rotations, and for this reason they are
called varivectors. As PCs, varivectors have their loadings of the original variables, so
that the three objects are characterized by very large values of X, and slightly large values of Y and Z. Fig. 21-5' shows the projection of the objects on the two varivectors.
Factor Analysis (FA) works by means of further rotations (orthogonal or not) in the
inner space, with the aim of helping in the interpretation and in the further use of the structured information. Varimax rotations are only one example of rotation techniques of FA.
21.2.4
Class-Modelling Techniques
This is a family of techniques used in classification problems (e.g. in quality control
the classes are acceptable/rejected; in toxicology they are often toxic/non-toxic).
Class-modelling techniques compute a mathematical model of the studied class.
In Fig. 21.6 the mathematical model of a class is a point, the centroid of the class. The
mathematical model of the other class is a suitable range of its first principal component. An object falling exactly on the mathematical model can be considered an ideal
object. The effect of some factors produces a more or less large distance from the
mathematical model for the other objects. A statistical procedure computes the maximum permitted distance, the boundary of the class space. This maximum distance
corresponds to a critical value of the used statistics. In the case of the centroid model
(UNEQ method) the distance is a Mahalanobis distance, which takes into account the
different variances of the variables and their correlation, and t statistics is used to
compute the critical value. In the case of PC model (SIMCA method) the class boundary is obtained by the variance in the outer space, and by F statistics.
The class model accepts an object which falls within the class space, with a significance level decreasing with the distance from the class centroid. The class model does
not accept an object which falls outside the class space. It can be considered as a nontypical object, or as an object very different from the other objects of the class according to its significance level.
In Fig. 21.6, object A fits both class models. This double possibility depends on the
not perfectly separated two-class space. Object A can be considered more similar to
class modelled by SIMCA. At first, it was considered an object of the class modelled
by UNEQ, but the assignment can be modified, and the models computed again (refinement of class model).
21.2.5
Regression Techniques, Responses and Predictors
Regression techniques are used to model the relationship between one (or more) response variable and some predictor variables.