Graphics Reference
In-Depth Information
Taking into account Eq. . and premultypling the let and the right members of
the equation by Y , we obtain the following expression:
YY
YU
=
YU Λ .
( . )
Let us denote V
=
YU ;( . )thenbecomes
YY
V
=
V Λ .
( . )
he above expression identifies in R n an orthogonal space in which to represent
the p variables.
We call principal components the coordinates of n points in the orthogonal basis
U q . heyare defined as Ψ
=
YU q .Equivalently, variable coordinates are obtained by
Φ
=
Y
V q .Matrices U q and V q are obtained by the first q
p columns of U and V
,
respectively.
FA output interpretation is a rather ticklish question. Very frequently we stumble
upon analyses limiting interpretation to the graphical representation of points over
thefirstorfirsttwofactorialplans.Evenifthisisthemostevidentaspectoftheanal-
ysis, there is nothing worse than ignoring the rest of the output. Distances projected
over factorial plans represent approximations of real distances. A correct interpreta-
tion, hence, should combine graphical and analytical results.
Unbiased output interpretation assumes knowledge of the following two basic
principles: (i) Principal components are orthogonal by construction, so any orthog-
onal space defined by two or more axes represents an additive model. he additivity
of the modelallows us todetermine the explained inertia associated to each factorial
subspaceasthesumoftherespective eigenvalues. (ii)Original variables arecentered,
so the axes' origin corresponds to the average statistical unit. hen the distance from
the origin reveals the deviation with respect to the mean vector.
hese are very important and remarkable properties.
Taking into account these properties, let us consider the output of the OECD
countries. Figures . and . show respectively the configuration of the variables
and statistical units with respect to the first two factors of the OECD data table.
Standardization implies all variables are plotted inside a hypersphere having a ra-
dius equal to one; angles between variables and factors express the correlation. he
first factor has a strength positive association with variables GDP and UR and nega-
tivewiththeNNSindicator.IndicatorsIRandTBhaverespectivelydirectandinverse
relation with factor and are almost uncorrelated with factor . We remark that the
LI indicator does not fit the first factorial plan. Taking into account the variable po-
sitioning with respect to the factors, we interpret the characteristics of the statistical
units according to their positioning. For example, the position of Spain in the lower
rightarea impliesthat thiscountry hasaURindicatorvalue significantly greaterthan
the variable mean. Looking at the data table, we notice that Spain has the greatest
value ( . ).
Figure . shows the OECD countries with respect to the first three factorial axes.
Notice the position of Portugal with respect to factor ; it contributes to the orienta-
tion of the third factor and has a coordinate equal to
. .
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