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In-Depth Information
Principal Axis Methods and Classiication:
aUniiedView
4.5
he actual knowledge base in numerical analysis and powerful modern PCs allow
us to successfully make use of the computational aspects in multidimensional data
analysis (MDA). However, there are many analysis strategies that, without loss of
e
ciency, offer good solutions.
Intheprevioussectionswehaveshownthecentralityofthedistanceinfacto-
rial and clustering methods. his common element has been largely used to perform
two-step analysis, namely, using both factorial andclusteranalysis. Automatic classi-
fication techniques are used to group objects described by a set of variables; they do
not make any claim to optimality. Nevertheless, they give relatively fast, economical,
and easily interpretable results. PCA and other factorial methods rarely provide an
exhaustive analysis of a set of data. herefore, it is useful to perform a clustering of
the observations because this helps to reduce the FAcomplexity. Additionally, it is of
value touseclassification analysis tosummarize the configuration of points obtained
from a principal axis analysis. In other words, a further reduction in the dimension-
ality of the data is valuable and leads to results that are easier to analyze. So-called
“tandem analysis” represents a unified approach in which FA and clustering criteria,
bothbased onthe samenotion ofdistance, aresimultaneously satisfied inaniterative
model (Vichi and Kiers,
).
Allmethodsofmultivariate descriptive statistical analysis areusedinthesamesit-
uation wheretheuserisfacedwitharectangular matrix.hismatrixmaybeacontin-
gency table, a binary matrix (with values of
or
according to whether an object has
a certain attribute), or a matrix of numerical values. he use of automatic classifica-
tion techniques implies some basic underlying concepts with respect to the purpose
oftheanalysis. Eitheritisassumedthatcertaingroupsmustexistamongtheobserva-
tionsor,onthecontrary,theanalysisrequiresagroupingoftheobservations.Inother
words, a
-D continuous visualization of the statistical relationships is not enough.
here is also an interest in uncovering groups of individuals or of characteristics.
A given set of results might be reached through different steps and might lead to
different interpretations. For example, the problem may be to discover a partition
that really exists and that was hypothesized before carrying out the statistical analy-
sis. Conversely, it may be useful to employ partitions as tools or as surrogates in the
computations that make it easier to explore the data. In any case, using principal axis
methods in conjunction with classification makes it possible to identify groups and
to determine their relative positions.
Oten partitions or tree structures are used to amplify the results of preliminary
principal axis analysis during the exploratory phases of data analysis. here are sev-
eralfamiliesofclassificationalgorithms:agglomerativealgorithms,inwhichtheclus-
tersare builtbysuccessive pairwise agglomeration ofobjects andwhichprovideahi-
erarchy of partitions of the objects; divisive algorithms, which proceed by successive
dichotomizations ofentire setsofobjects andwhichalsoprovideahierarchyofparti-
